#  The geometric constraints on Filippov algebroids

Yanhui Bi  Center for Mathematical Sciences, College of Mathematics and
Information Science, Nanchang Hangkong University<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Zhixiong Chen  College of Mathematics and Information
Science, Nanchang Hangkong University<EMAIL_ADDRESS>,  Zhuo
Chen  Department of Mathematical Sciences, Tsinghua University
<EMAIL_ADDRESS>and  Maosong Xiang  School of Mathematics and
Statistics, Huazhong University of Science and Technology<EMAIL_ADDRESS>
###### Abstract.

Filippov $n$-algebroids are introduced by Grabowski and Marmo as a natural
generalization of Lie algebroids. In this note, we characterize Filippov
$n$-algebroid structures by considering certain multi-input connections, which
we call Filippov connections, on the underlying vector bundle. Through this
approach, we are able to express the $n$-ary bracket of any Filippov
$n$-algebroid using a torsion-free type formula. Additionally, we transform
the generalized Jacobi identity of the Filippov $n$-algebroid into the
Bianchi-Filippov identity. Furthermore, in the case of rank $n$ vector
bundles, we provide a characterization of linear Nambu-Poisson structures
using Filippov connections.

Supported by the Natural Science Foundation of China (NSFC) grants
11961049(Bi), 11901221(Xiang), and 12071241(Zhuo Chen), and by the Key Project
of Jiangxi Natural Science Foundation grant 20232ACB201004(Bi, Xiang and
Zhixiong Chen).

Keywords: Filippov algebroids, Filippov connections, Bianchi-Filippov
identity, Nambu-Poisson structures.

###### Contents

  1. 1 Preliminaries: Anchored bundles and Filippov algebroids
  2. 2 The geometric constraints of Filippov algebroids
    1. 2.1 The main theorem
    2. 2.2 Proof of Theorem 2.3
    3. 2.3 Construction of Filippov connections
  3. 3 A construction of linear Nambu-Poisson structures
    1. 3.1 From Filippov algebroids to linear Nambu-Poisson structures
    2. 3.2 Proof of Theorem 3.3

## Introduction

Filippov introduced a generalized Jacobi identity for $n$-ary skew-symmetric
operation, which acts as a replacement for the classical Jacobi identity in
the context of Lie algebras [ref10]. He also proposed the concept of $n$-Lie
algebra, also known as Filippov $n$-algebra, with the corresponding
generalized Jacobi identity referred to as the Filippov identity. Nambu and
Takhtajan extended the concept of Poisson manifold to an $n$-ary
generalization called Nambu-Poisson structure in order to study Hamiltonian
mechanics more comprehensively [ref16, Tak]. It is worth noting that both the
Nambu-Poisson structure and the $n$-Lie algebra share the same generalized
Jacobi identity. Grabowski and Marmo introduced the concept of Filippov
$n$-algebroids, an $n$-ary generalization of Lie algebroids, in order to
determine the relationship between linear Nambu-Poisson structures and
Filippov algebras [ref12]. Consequently, it is reasonable to anticipate that
many tools used to study Lie algebroids could be enhanced or upgraded to the
realm of Filippov algebroids. Therefore, this paper aims to address the
absence of the concepts of connections and curvatures of Filippov algebroids
in the literature and provide a primitive analysis of these topics from a
geometric point of view.

Recall that a Lie algebroid is a (real) vector bundle $A\to M$ together with a
bundle map $\rho\colon A\to TM$, called anchor, and a Lie bracket
$[\cdotp,\cdotp]$ on the section space $\Gamma(A)$ of $A$, satisfying
$\rho\colon\Gamma(A)\to\Gamma(TM)$ is a morphism of Lie algebras and the
Leibniz rule

$[X,fY]=f[X,Y]+(\rho(X)f)Y,\qquad\forall X,Y\in\Gamma(A)\mbox{ and }f\in
C^{\infty}(M).$

By an easy smooth analysis, the bracket $[\cdotp,\cdotp]$ can always be
reformulated in the form

$[X,Y]=\nabla_{X}Y-\nabla_{Y}X,$ (1)

where $\nabla\colon\Gamma(A)\times\Gamma(A)\to\Gamma(A)$ satisfies the
properties

$\nabla_{fX}Y=f\nabla_{X}Y\mbox{ and
}\nabla_{X}(fY)=f\nabla_{X}Y+(\rho(X)f)Y.$

One calls $\nabla$ a connection on the anchored bundle $(A,\rho)$. (This is
indeed a straightforward generalization of connections on vector bundles.)

When the bracket $[~{},~{}]$ of a Lie algebroid $A$ is expressed in the form
(1), one says that the connection $\nabla$ is torsion free. The curvature form
$R^{\nabla}\in\Gamma(\wedge^{2}A^{*}\otimes\operatorname{End}(A))$ of such a
connection $\nabla$ is defined in the standard manner:

$R^{\nabla}(X,Y)(Z)=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,$

for all $X,Y,Z\in\Gamma(A)$. Now, $\rho$ being a morphism of Lie algebras is
equivalent to the condition that $R^{\nabla}$ is a tensor in its third
argument. Moreover, the Jacobi identity for $[~{},~{}]$ is transformed into
the following Lie-Bianchi identity

$R^{\nabla}(X,Y)(Z)+R^{\nabla}(Y,Z)(X)+R^{\nabla}(Z,X)(Y)=0.$

Therefore, Lie algebroids can be realized as anchored bundles equipped with
special connections [PP]. We wish to find an analogous characterization of the
$n$-ary bracket of any Filippov algebroid. A significant difference between
Lie algebroids and Filippov $n$-algebroids (for $n\geqslant 3$) is that the
bracket and anchor of the latter are of more arguments (see Definition 1.2).
So there is not an obvious way to extend Equation (1). We come up with a
solution in Section 2. Below is a quick summary:

  * •

First, we define (multi-input) connections compatible with a given (multi-
input) $n$-anchor (see Definition 2.1). This is a quite straightforward
extension of usual connections of Lie algebroids (when $n=2$).

  * •

Second, we introduce the curvature form $R^{\nabla}$ stemming from a
connection $\nabla$ (see Equation (7)). We believe that this is a highly
nontrivial invention of this note.

  * •

Third, we prove in Theorem 2.3 that certain good connections, which we call
Filippov connections, fully determines Filippov algebroid structures. This
includes two points: (1) The $n$-ary bracket of any Filippov algebroid can be
realized in a torsion free manner (see Equation (6)); (2) The generalized
Jacobi identity is transformed to a constraint, called the Bianchi-Filippov
identity (see Equation (8)) about the associated curvature $R^{\nabla}$.

We illustrate a simple method to construct Filippov connections in Section
2.3.

Finally, we show that there exists a one-to-one correspondence between
Filippov $n$-algebroid structures on a vector bundle $A$ of rank $n\geqslant
3$ and linear Nambu-Poisson structures on its dual bundle $A^{*}$ (see Theorem
3.3). As an interesting application of our result, one is able to construct
linear Nambu-Poisson structures from Filippov connections (Corollary 3.4).

In short summary, torsion-free connections subject to the Bianchi-Filippov
identity are implicit geometric constraints for Filippov algebroids. We know
that torsion free connections for Lie algebroids play a crucial role in
various mathematical constructions. They are particularly important in the
construction of Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg manifolds
for Lie algebroid pairs [LSX]. Additionally, Bianchi identities are not only
significant in Riemannian geometry, but also in Poisson geometry [BDPR].
Therefore, we believe that our approach to Filippov algebroids will be
beneficial in this context.

##  1\. Preliminaries: Anchored bundles and Filippov algebroids

In this section, we recall the definition of Filippov algebroids from [ref12].
There is an alternative characterization of Filippov algebroids in terms of
certain $1$-derivations [ref14]. It is important to note that $n\geqslant 2$
is an integer, although the only interesting situation is when $n\geqslant 3$.
Let us start with a notion of $n$-anchored bundles.

######  Definition 1.1.

An $n$-anchored vector bundle over a smooth manifold $M$ is pair $(A,\rho)$,
where $A$ is a vector bundle over $M$ and $\rho\colon\wedge^{n-1}A\to TM$ is a
vector bundle morphism, called $n$-anchor of $A$.

######  Definition 1.2.

A Filippov $n$-algebroid over a smooth manifold $M$ is an $n$-anchored bundle
$(A,\rho)$ over $M$ together with an $\mathbb{R}$-multilinear and skew-
symmetric $n$-bracket on the section space $\Gamma(A)$ of $A$:

$[\cdotp,\cdots,\cdotp]:~{}\underbrace{\Gamma(A)\times\cdotp\cdotp\cdotp\times\Gamma(A)}_{n-\mbox{copies}}\to\Gamma(A)$

satisfying the following compatibility conditions:

  1. (1)

The $n$-anchor $\rho$ intertwines the $n$-bracket and the standard Lie bracket
$[\cdotp,\cdotp]_{TM}$ on $\Gamma(TM)$:

$\displaystyle[\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]_{TM}$ (2)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge[X_{1},\cdots,X_{n-1},Y_{i}]\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1});$

  2. (2)

The $n$-bracket is a derivation with respect to
$C^{\infty}(M)$-multiplications:

$[X_{1},\cdots,X_{n-1},fY]=f[X_{1},\cdots,X_{n-1},Y]+\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})(f)Y;$ (3)

  3. (3)

The following equation holds, to be called the (generalized) Jacobi identity
(or Filippov identity):

$\displaystyle[X_{1},\cdots,X_{n-1},[Y_{1},\cdots,Y_{n}]]$ (4)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}[Y_{1},\cdots,Y_{i-1},[X_{1},\cdots,X_{n-1},Y_{i}],Y_{i+1},\cdots,Y_{n}],$

for all $X_{\cdot},Y_{\cdot}\in\Gamma(A)$ and $f\in C^{\infty}(M)$.

Note that any Lie algebroid is a Filippov 2-algebroid. A Filippov $n$-algebra
is a Filippov $n$-algebroid over the one-point base manifold. The following
examples (due to [ref12]) illustrate two Filippov $n$-algebroid structures on
the trivial tangent bundle $T\mathbb{R}^{m}$ of $\mathbb{R}^{m}$ for
$m\geqslant n\geqslant 2$.

######  Example 1.3.

Consider the trivial n-anchored bundle $(T\mathbb{R}^{m},\rho=0)$. For each
Filippov $n$-algebra structure on $\mathbb{R}^{m}$ with structure constants
$\\{c^{j}_{i_{1},\cdots,i_{n}}\\}$ and each a smooth function $g\in
C^{\infty}(\mathbb{R}^{m})$, we have a Filippov $n$-algebroid
$(T\mathbb{R}^{m},0)$ whose bracket is defined by

$\left[f_{1}\frac{\partial}{\partial
x_{i_{1}}},\cdots,f_{n}\frac{\partial}{\partial x_{i_{n}}}\right]=gf_{1}\cdots
f_{n}\sum_{j=1}^{m}c^{j}_{i_{1},\cdots,i_{n}}\frac{\partial}{\partial x_{j}}.$

######  Example 1.4.

Equip $T\mathbb{R}^{m}$ with the $n$-anchor map $\rho$ defined by the tensor
field

$dx_{1}\wedge\cdotp\cdotp\cdotp\wedge dx_{n-1}\otimes\frac{\partial}{\partial
x_{1}},$

where $x_{1},\cdots,x_{n-1},x_{n},\cdots,x_{m}$ are coordinates of
$\mathbb{R}^{m}$. Then the $n$-anchored bundle $(T\mathbb{R}^{m},\rho)$
together with the trivial $n$-bracket on generators $\frac{\partial}{\partial
x_{i}}$ produces a (nontrivial) Filippov $n$-algebroid over $\mathbb{R}^{m}$.

We emphasize a crucial but often overlooked point in the literature: the
presence of a Filippov $n$-bracket on an $n$-anchored bundle $(A,\rho)$
imposes a constraint on the rank of $\rho$ for every integer $n\geqslant 3$.

######  Proposition 1.5.

Let $(A,[\cdotp,\cdotp\cdotp\cdotp,\cdotp],\rho)$ be a Filippov $n$-algebroid
for $n\geqslant 3$. Then the rank of the image of $\rho$ as a distribution on
$M$ can not exceed $1$, i.e.,
$\operatorname{rank}(\rho(\wedge^{n-1}A))\leqslant 1$.

###### Proof.

Suppose that the image of $\rho$ at $p\in M$ is not trivial. So we can find an
open neighborhood $U$ of $p$ and some $Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}\in\Gamma(\wedge^{n-1}A)|_{U}$ such that
$\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$ is nowhere vanishing on
$U$. The desired statement amounts to show that, if
$\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})$ is also nowhere vanishing
on $U$, then there exists some $c\in C^{\infty}(U)$ such that

$\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})=c\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1}).$

In fact, by the definition of Filippov $n$-algebroids, we obtain

$\displaystyle[\rho(fX_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]$
$\displaystyle\xlongequal[]{\mbox{\eqref{eq1}}}$
$\displaystyle\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{i-1}\wedge[fX_{1},X_{2},\cdots,X_{n-1},Y_{i}]\wedge
Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$
$\displaystyle\xlongequal[]{\mbox{\eqref{eq2}}}$ $\displaystyle
f\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{i-1}\wedge[X_{1},X_{2},\cdots,X_{n-1},Y_{i}]\wedge
Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$ $\displaystyle\
+\sum_{i=1}^{n-1}(-1)^{n-1}\rho(X_{2}\wedge\cdots\wedge X_{n-1}\wedge
Y_{i})(f)\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{i-1}\wedge X_{1}\wedge
Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$
$\displaystyle\xlongequal[]{\mbox{\eqref{eq1}}}$ $\displaystyle
f[\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]$ $\displaystyle\
+\sum_{i=1}^{n-1}(-1)^{n-1}\rho(X_{2}\wedge\cdots\wedge X_{n-1}\wedge
Y_{i})(f)\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{i-1}\wedge X_{1}\wedge
Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1}).$

Meanwhile, since $\rho$ is a morphism of vector bundles, we have

$\displaystyle[\rho(fX_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]$ $\displaystyle=$
$\displaystyle[f\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]$ $\displaystyle=$
$\displaystyle f[\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})]-\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(f)\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}).$

Setting $Y_{1}=X_{1}$ in the above two Equations, we obtain

$\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})(f)\rho(X_{1}\wedge
Y_{2}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})=-\rho(X_{1}\wedge
Y_{2}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(f)\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}).$ (5)

Using Equation (5), we have

$\displaystyle\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})$
$\displaystyle=g_{1}\rho(X_{1}\wedge Y_{2}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})=-g_{1}\rho(Y_{2}\wedge X_{1}\wedge\cdots\wedge Y_{n-1})$
$\displaystyle=-g_{1}g_{2}\rho(Y_{2}\wedge Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})=g_{1}g_{2}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1}),$

for some $g_{1},g_{2}\in C^{\infty}(U)$.

  1. (1)

If $\rho(X_{1}\wedge Y_{2}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$ is nowhere
vanishing on $U$, then the vector fields
$\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})$ and
$\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$ must be
$C^{\infty}(U)$-linearly dependent.

  2. (2)

If it happens that $\rho(X_{1}\wedge Y_{2}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})|_{p}=0$, then we let $\tilde{X}_{1}=X_{1}+Y_{1}$ and consider
$\rho(\tilde{X}_{1}\wedge Y_{2}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$, which
is nowhere vanishing on $U$. By arguments in (1) as above,
$\rho(\tilde{X}_{1}\wedge X_{2}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})$ and
$\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$ are
$C^{\infty}(U)$-linearly dependent, and we again get the desired statement.

∎

##  2\. The geometric constraints of Filippov algebroids

###  2.1. The main theorem

In this section, we characterize Filippov algebroids via connections on the
underlying anchored bundles.

######  Definition 2.1.

A connection on an $n$-anchored bundle $(A,\rho)$ is a bilinear map
$\nabla:\Gamma(\wedge^{n-1}A)\times\Gamma(A)\to\Gamma(A)$ satisfying two
conditions:

$\displaystyle\nabla_{fX_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}}X_{n}$
$\displaystyle=$ $\displaystyle f\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n},$ $\displaystyle\mbox{ and
}~{}\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}}(fX_{n})$
$\displaystyle=$ $\displaystyle f\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}+\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})(f)X_{n}$

for all $X_{1},\cdots,X_{n-1}\in\Gamma(A)$ and $f\in C^{\infty}(M)$.

To see the existence of such a connection, one takes a $TM$-connection on $A$,
say $\nabla^{TM}$, and then define $\nabla$ on the $n$-anchored bundle
$(A,\rho)$ as the pullback of $\nabla^{TM}$:

$\nabla_{X_{1}\wedge\cdots\wedge
X_{n-1}}X_{n}:=\nabla^{TM}_{\rho(X_{1}\wedge\cdots\wedge X_{n-1})}X_{n}.$

The key point of this note is that any connection $\nabla$ on $(A,\rho)$
induces a skew-symmetric $n$-bracket on $\Gamma(A)$ defined by

$\displaystyle[X_{1},\cdots,X_{n}]^{\nabla}$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{n}(-1)^{n+i}\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\widehat{X_{i}}\cdotp\cdotp\cdotp\wedge
X_{n}}X_{i}$ (6) $\displaystyle=$
$\displaystyle\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}+(-1)^{n-1}\nabla_{X_{2}\wedge\cdotp\cdotp\cdotp\wedge
X_{n}}X_{1}+\cdots$ $\displaystyle\ +\nabla_{X_{n-1}\wedge X_{n}\wedge
X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-3}}X_{n-2}+(-1)^{n-1}\nabla_{X_{n}\wedge
X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-2}}X_{n-1}.$

For computational convenience, we denote the covariant derivative on
$\Gamma(A)$ along $X_{1},\cdots,X_{n-1}\in\Gamma(A)$ by

$X^{\nabla}_{1\cdots
n-1}:=[X_{1},\cdots,X_{n-1},-]^{\nabla}\colon\Gamma(A)\to\Gamma(A),$

It extends to all sections in $\wedge^{\bullet}A$ by

${X_{1\cdots n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{m}):=\sum_{i=1}^{m}Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{i-1}\wedge
X^{\nabla}_{1\cdots n-1}(Y_{i})\wedge Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{m}.$

We then introduce the curvature form of $\nabla$, an operation

$R^{\nabla}(-\cdots-,-)(-):~{}\underbrace{\Gamma(A)\times\cdots\Gamma(A)}_{(n-1)-\mbox{copies}}\times\Gamma(\wedge^{n-1}A)\times\Gamma(A)\to\Gamma(A)$

defined by

$\displaystyle
R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(Z)$ (7) $\displaystyle:=$ $\displaystyle[{X_{1\cdots
n-1}^{\nabla}},\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}](Z)-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})}Z$
$\displaystyle:=$ $\displaystyle X^{\nabla}_{1\cdots
n-1}\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}Z-\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}X^{\nabla}_{1\cdots n-1}Z-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})}Z,$

for all $X_{1},\cdots,X_{n-1},Y_{1},\cdots,Y_{n-1},Z\in\Gamma(A)$ and
$n\geqslant 3$.

When $n=3$, it reads

$\displaystyle R^{\nabla}(X_{1},X_{2},Y_{1}\wedge Y_{2})(Z)$ $\displaystyle=$
$\displaystyle X^{\nabla}_{12}\nabla_{Y_{1}\wedge Y_{2}}Z-\nabla_{Y_{1}\wedge
Y_{2}}X^{\nabla}_{12}Z-\nabla_{{X_{12}^{\nabla}}(Y_{1}\wedge Y_{2})}Z$
$\displaystyle=$ $\displaystyle[X_{1},X_{2},\nabla_{Y_{1}\wedge
Y_{2}}Z]^{\nabla}-\nabla_{Y_{1}\wedge Y_{2}}[X_{1},X_{2},Z]^{\nabla}$
$\displaystyle\ -\nabla_{[X_{1},X_{2},Y_{1}]^{\nabla}\wedge
Y_{2}+Y_{1}\wedge[X_{1},X_{2},Y_{2}]^{\nabla}}Z.$

When $n=4$, the expression of $R^{\nabla}$ consists of twenty terms. As $n$
gets larger, more terms are involved.

It is easy to verify from the defining Equation $\eqref{eq10}$ that
$R^{\nabla}$ is $C^{\infty}(M)$-linear with respect to the argument
$Y_{1}\wedge\cdots\wedge Y_{n-1}$.

######  Definition 2.2.

A connection $\nabla$ on an $n$-anchored bundle $(A,\rho)$ is called a
Filippov connection if the following two conditions are true:

  * (1)

The curvature $R^{\nabla}$ is $C^{\infty}(M)$-linear with respect to its last
argument, i.e., for all $f\in C^{\infty}(M)$ we have

$R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(fZ)=fR^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(Z);$

  * (2)

The following equality holds, to be called the Bianchi-Filippov identity:

$\displaystyle\sum_{i=1}^{n}(-1)^{n+i}R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\widehat{Y_{i}}\cdotp\cdotp\cdotp\wedge
Y_{n})Y_{i}$ (8) $\displaystyle=$ $\displaystyle
R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})Y_{n}+(-1)^{n-1}R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{2}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n})Y_{1}$ $\displaystyle\
+\cdots+R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{n-1}\wedge Y_{n}\wedge
Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-3})Y_{n-2}$ $\displaystyle\ \
+(-1)^{n-1}R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{n}\wedge
Y_{1}\wedge\cdot\cdot\cdot\wedge Y_{n-2})Y_{n-1}$ $\displaystyle=$
$\displaystyle 0.$

We are ready to state our main theorem, which characterizes Filippov
algebroids fully by Filippov connections.

######  Theorem 2.3.

Let $(A,\rho)$ be an $n$-anchored bundle. If $\nabla$ is a Filippov connection
on $(A,\rho)$, then $(A,\rho,[\cdotp,\cdots,\cdotp]^{\nabla})$ is a Filippov
$n$-algebroid, where $[\cdotp,\cdots,\cdotp]^{\nabla}$ is the $n$-bracket
given by Equation (6). Moreover, any Filippov $n$-algebroid structure on
$(A,\rho)$ arises from a Filippov connection in this way.

###  2.2. Proof of Theorem 2.3

The proof of Theorem 2.3 is divided, and will follow immediately from the
three lemmas below.

######  Lemma 2.4.

Let $\nabla$ be a connection on an $n$-anchored bundle $(A,\rho)$. The
curvature $R^{\nabla}$ satisfies the first condition of Definition 2.2 if and
only if the anchor $\rho$ intertwines the induced $n$-bracket
$[\cdotp,\cdots,\cdotp]^{\nabla}$ and the Lie bracket $[\cdotp,\cdotp]_{TM}$
on $\Gamma(TM)$, i.e.,

$\displaystyle[\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}),\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})]_{TM}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge[X_{1},\cdots,X_{n-1},Y_{i}]^{\nabla}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}),$

for all $X_{1},\cdots,X_{n-1},Y_{1},\cdots,Y_{n-1}\in\Gamma(A)$.

###### Proof.

By the definition of curvature, we have

$\displaystyle
R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(fZ)$ $\displaystyle\xlongequal[]{\mbox{\eqref{eq10}}}$
$\displaystyle[{X_{1\cdots
n-1}^{\nabla}},\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}](fZ)-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})}(fZ)$
$\displaystyle=$ $\displaystyle X^{\nabla}_{1\cdots
n-1}\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}(fZ)-\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}X^{\nabla}_{1\cdots n-1}(fZ)-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})}(fZ)$
$\displaystyle=$ $\displaystyle
fR^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})(Z)+\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})(f)Z$ $\displaystyle\
-\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})(f)Z-\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
X^{\nabla}_{1\cdots n-1}(Y_{i})\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})(f)Z.$

Hence, the curvature $R^{\nabla}$ is $C^{\infty}(M)$-linear with respect to
its last argument if and only if

$\displaystyle\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})-\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1})\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
X^{\nabla}_{1\cdots n-1}(Y_{i})\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\rho(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge[X_{1},\cdots,X_{n-1},Y_{i}]^{\nabla}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}).$

∎

######  Lemma 2.5.

Let $\nabla$ be a connection on an $n$-anchored bundle $(A,\rho)$. The
curvature $R^{\nabla}$ satisfies the second condition of Definition 2.2, i.e.
the Bianchi-Filippov identity (8), if and only if the induced $n$-bracket
$[\cdotp,\cdots,\cdotp]^{\nabla}$ satisfies the (generalized) Jacobi identity
(4).

###### Proof.

The statement follows directly from the following lines of computation:

$\displaystyle\sum_{i=1}^{n}(-1)^{n+i}R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{1}\wedge\cdotp\cdotp\cdotp\widehat{Y_{i}}\cdotp\cdotp\cdotp\wedge
Y_{n})Y_{i}$ $\displaystyle\xlongequal[]{\mbox{\eqref{eq12}}}$
$\displaystyle\sum_{i=1}^{n-1}(-1)^{(n-1)i}R^{\nabla}(X_{1},\cdots,X_{n-1},Y_{i+1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n}\wedge Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{i-1})Y_{i}$
$\displaystyle\xlongequal[]{\mbox{\eqref{eq10}}}$
$\displaystyle\sum_{i=1}^{n}(-1)^{(n-1)i}([X^{\nabla}_{1\cdots
n-1},\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge\widehat{Y_{i}}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n}}](Y_{i})-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge\widehat{Y_{i}}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n})}Y_{i})$ $\displaystyle=$
$\displaystyle[X_{1},\cdots,X_{n-1},\nabla_{Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-1}}]^{\nabla}(Y_{n})-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-1})}Y_{n}$
$\displaystyle+(-1)^{n-1}([X_{1},\cdots,X_{n-1},\nabla_{Y_{2}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n}}]^{\nabla}(Y_{1})-\nabla_{{X_{1\cdots
n-1}^{\nabla}}(Y_{2}\wedge\cdotp\cdotp\cdotp\wedge Y_{n})}Y_{1})+\cdots$
$\displaystyle+[X_{1},\cdots,X_{n-1},\nabla_{Y_{n-1}\wedge Y_{n}\wedge
Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-3}}]^{\nabla}(Y_{n-2})-\nabla_{{X_{1\cdots n-1}^{\nabla}}(Y_{n-1}\wedge
Y_{n}\wedge Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-3})}Y_{n-2}$
$\displaystyle+(-1)^{n-1}([X_{1},\cdots,X_{n-1},\nabla_{Y_{n}\wedge
Y_{1}\wedge\cdotp\cdotp\cdotp\wedge
Y_{n-2}}]^{\nabla}(Y_{n-1})-\nabla_{{X_{1\cdots n-1}^{\nabla}}(Y_{n}\wedge
Y_{1}\wedge\cdotp\cdotp\cdotp\wedge Y_{n-2})}Y_{n-1})$
$\displaystyle\xlongequal[]{\mbox{\eqref{eq9}}}$
$\displaystyle[X_{1},\cdots,X_{n-1},[Y_{1},\cdots,Y_{n}]^{\nabla}]^{\nabla}-\sum_{i=1}^{n}[Y_{1},\cdotp,[X_{1},\cdots,X_{n-1},Y_{i}]^{\nabla},\cdots,Y_{n}]^{\nabla}.$

∎

The next lemma shows that any Filippov algebroid can be realized by a Filippov
connection.

######  Lemma 2.6.

Let $(A,[\cdotp,\cdotp\cdotp\cdotp,\cdotp],\rho)$ be a Filippov $n$-algebroid.
Then there exists a Filippov connection $\nabla$ on the underlying
$n$-anchored bundle $(A,\rho)$ such that
$[\cdotp,\cdots,\cdotp]=[\cdotp,\cdots,\cdotp]^{\nabla}$ (the torsion-free
property).

###### Proof.

Given a connection $\nabla^{\circ}$ on $(A,\rho)$, we are able to obtain an
$\mathbb{R}$-multilinear operation $K(\cdotp,\cdotp\cdotp\cdotp,\cdotp)$ on
$\Gamma(A)$ by

$\displaystyle K(X_{1},\cdots,X_{n})$ $\displaystyle:=$
$\displaystyle[X_{1},\cdots,X_{n}]-[X_{1},\cdots,X_{n}]^{\nabla^{\circ}}.$

Using axioms of Filippov algebroids, it is easy to see that
$K(\cdotp,\cdots,\cdotp)$ is indeed $C^{\infty}(M)$-multilinear. Then we
define a new connection $\nabla$ on $(A,\rho)$ by

$\displaystyle\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}}X_{n}$
$\displaystyle:=$
$\displaystyle\frac{1}{n}K(X_{1},\cdots,X_{n})+\nabla^{\circ}_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}.$

It remains to check the desired equality:

$\displaystyle[X_{1},\cdots,X_{n}]^{\nabla}$ $\displaystyle=$
$\displaystyle\nabla_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}+\sum_{i=1}^{n-1}(-1)^{(n-1)i}\nabla_{X_{i+1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n}\wedge X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{i-1}}X_{i}$
$\displaystyle=$ $\displaystyle
K(X_{1},\cdots,X_{n})+\nabla^{\circ}_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}+\sum_{i=1}^{n-1}(-1)^{(n-1)i}\nabla^{\circ}_{X_{i+1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n}\wedge X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{i-1}}X_{i}$
$\displaystyle=$ $\displaystyle[X_{1},\cdots,X_{n}].$

∎

The following examples illustrate three Filippov connections on the trivial
tangent bundle $T\mathbb{R}^{m}$ of $\mathbb{R}^{m}$ for $m\geqslant
n\geqslant 2$.

######  Example 2.7.

Consider the trivial $n$-anchored vector bundle $(T\mathbb{R}^{m},\rho=0)$.
Suppose that the vector space $\mathbb{R}^{m}$ is endowed with a Filippov
$n$-algebra structure whose structure constants are
$\\{c^{j}_{i_{1},\cdots,i_{n}}\\}$ with respect to the standard basis of
$\mathbb{R}^{m}$. Given a smooth function $g\in C^{\infty}(\mathbb{R}^{m})$
and a set of constants $\\{a^{j}_{i_{1}\cdots i_{n-1}}\\}$ satisfying the
equality:

$a^{j}_{i_{1}\cdots i_{n-1}}+\sum_{k=1}^{n-1}(-1)^{(n-1)k}a^{j}_{i_{k+1}\cdots
i_{n}\cdots i_{1}i_{k-1}}=c^{j}_{i_{1}\cdots i_{n}},$ (9)

we are able to obtain a connection on $(T\mathbb{R}^{m},\rho=0)$ generated by
the only one nontrivial relation:

$\nabla_{\frac{\partial}{\partial
x_{i_{1}}}\wedge\cdots\wedge\frac{\partial}{\partial
x_{i_{n-1}}}}\frac{\partial}{\partial
x_{i_{n}}}:=g\sum_{j=1}^{m}a^{j}_{i_{1}\cdots i_{n-1}}\frac{\partial}{\partial
x_{j}}.$

It follows from the recipe in Equation (6) and Equation (9) that

$\left[\frac{\partial}{\partial x_{i_{1}}},\cdots,\frac{\partial}{\partial
x_{i_{n}}}\right]^{\nabla}=g\sum_{i=1}^{m}c^{j}_{i_{1}\cdots
i_{n}}\frac{\partial}{\partial x_{j}}.$

So, what we recover is the Filippov structure on $T\mathbb{R}^{m}$ as in
Example 1.3. Hence, $\nabla$ is indeed a Filippov connection.

######  Example 2.8.

Consider the $n$-anchor map $\rho$ on the tangent bundle $T\mathbb{R}^{m}$
defined by the tensor field $dx_{1}\wedge\cdotp\cdotp\cdotp\wedge
dx_{n-1}\otimes\frac{\partial}{\partial x_{1}}$, where
$x_{1},\cdots,x_{n-1},x_{n},\cdots,x_{m}$ are coordinate functions of
$\mathbb{R}^{m}$. It is obvious that the (nontrivial) connection on
$(T\mathbb{R}^{m},\rho)$ generated by the trivial relation:

$\nabla_{\frac{\partial}{\partial
x_{i_{1}}}\wedge\cdots\wedge\frac{\partial}{\partial
x_{i_{n-1}}}}\frac{\partial}{\partial x_{i_{n}}}:=0.$

produces a nontrivial $n$-bracket which is compatible with $\rho$. Indeed,
what we recover is the Filippov structure on $T\mathbb{R}^{m}$ as in Example
1.4, and the said connection $\nabla$ is a Filippov connection.

######  Example 2.9.

Continue to work with the anchored bundle $(A,\rho)$ as in the previous
example. We consider a different connection with the only nontrivial
generating relations:

$\nabla_{\frac{\partial}{\partial
x_{i_{1}}}\wedge\cdots\wedge\frac{\partial}{\partial
x_{i_{n-1}}}}\frac{\partial}{\partial x_{k}}=\left\\{\begin{aligned}
&\frac{\partial}{\partial x_{k}},\mbox{ if }\quad\frac{\partial}{\partial
x_{i_{1}}}\wedge\cdots\wedge\frac{\partial}{\partial
x_{i_{n-1}}}=\frac{\partial}{\partial
x_{1}}\wedge\cdots\wedge\frac{\partial}{\partial x_{n-1}};\\\ &0,\mbox{
otherwise,}\end{aligned}\right.$

for all $x_{k}\in\\{x_{1},\cdots,x_{m}\\}$. Then, the associated $n$-bracket
can be derived:

$\left[\frac{\partial}{\partial x_{i_{1}}},\cdots,\frac{\partial}{\partial
x_{i_{n}}}\right]^{\nabla}=\left\\{\begin{aligned} &\frac{\partial}{\partial
x_{i_{n}}},\quad\text{if}\quad i_{j}=j,~{}\forall 1\leqslant j\leqslant
n-1;\\\ &0,\text{otherwise}.\end{aligned}\right.$

for all $i_{1}<i_{2}<\cdots<i_{n}$. By subtle analysis, one can find that the
associated curvature $R^{\nabla}$ is just zero. Hence $\nabla$ is truly a
Filippov connection and the above bracket defines a Filippov algebroid
structure on $(A,\rho)$.

###  2.3. Construction of Filippov connections

Let $A\to M$ be a vector bundle. Consider the bundle $\operatorname{CDO}(A)$
of covariant differential operators (cf. [Lie]*III, [pseudoalgebras], see also
[Differential], where the notation $\mathcal{D}(A)$ is used instead of
$\operatorname{CDO}(A)$). An element $D$ of $\Gamma(\operatorname{CDO}(A))$,
called a covariant differential operator, is an $\mathbb{R}$-linear operator
$\Gamma(A)\rightarrow\Gamma(A)$ together with a vector field
$\hat{D}\in\Gamma(TM)$, called the symbol of $D$, satisfying

$\displaystyle D(fX)=fD(X)+\hat{D}(f)\cdot X,\qquad\forall X\in\Gamma(A),f\in
C^{\infty}(M).$

The operator $D$ can be extended naturally to an operator
$D\colon\Gamma(\wedge^{n-1}A^{*})\rightarrow\Gamma(\wedge^{n-1}A^{*})$,
defined by

$\displaystyle\langle X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|D(\bar{\eta})\rangle$ $\displaystyle=$ $\displaystyle\hat{D}\langle
X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|\bar{\eta}\rangle-\sum_{i=1}^{n-1}\langle
X_{1}\wedge\cdotp\cdotp\cdotp\wedge D(X_{i})\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|\bar{\eta}\rangle,$

for all $X_{1},\cdots,X_{n-1}\in\Gamma(A)$ and
$\bar{\eta}\in\Gamma(\wedge^{n-1}A^{*})$.

Given a pair $(D,\bar{\xi})$, where $D\in\Gamma(\operatorname{CDO}(A))$ and
$\bar{\xi}\in\Gamma(\wedge^{n-1}A^{*})$, one is able to construct a map

$\displaystyle\rho^{(D,\bar{\xi})}\colon\Gamma(\wedge^{n-1}A)$
$\displaystyle\to$ $\displaystyle\Gamma(TM),$ $\displaystyle
X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-1}$ $\displaystyle\mapsto$
$\displaystyle\langle X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|\bar{\xi}\rangle\hat{D}.$

It is clear that $\rho^{(D,\bar{\xi})}$ makes $A$ an $n$-anchored bundle. And
the rank of the image of $\rho^{(D,\bar{\xi})}$ does not exceed $1$.

Define a connection on the $n$-anchored bundle $(A,\rho^{(D,\bar{\xi})})$ by

$\nabla^{(D,\bar{\xi})}_{X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}}X_{n}:=\langle X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|\bar{\xi}\rangle D(X_{n}).$ (10)

######  Proposition 2.10.

If the pair $(D,\bar{\xi})$ is subject to the relation

$D(\bar{\xi})=g\bar{\xi},\quad\mbox{ for some }g\in C^{\infty}(M),$ (11)

then $\nabla^{(D,\bar{\xi})}$ defined as in (10) is a Filippov connection on
the $n$-anchored bundle $(A,\rho^{(D,\bar{\xi})})$.

We omit the proof of this proposition as it follows from a straightforward but
lengthy verification. As a consequence of Theorem 2.3, a pair $(D,\bar{\xi})$
subject to Condition (11) produces a Filippov $n$-algebroid structure on $A$.
Its $n$-bracket reads:

$\displaystyle[X_{1},\cdots,X_{n}]^{\nabla^{(D,\bar{\xi})}}$ $\displaystyle=$
$\displaystyle\langle X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1}|\bar{\xi}\rangle D(X_{n})+(-1)^{n-1}\langle
X_{2}\wedge\cdotp\cdotp\cdotp\wedge X_{n}|\bar{\xi}\rangle D(X_{1})+\cdots$
$\displaystyle\ +\langle X_{n-1}\wedge X_{n}\wedge
X_{1}\wedge\cdotp\cdotp\cdotp\wedge X_{n-3}|\bar{\xi}\rangle
D(X_{n-2})+(-1)^{n-1}\langle X_{n}\wedge X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-2}|\bar{\xi}\rangle D(X_{n-1}).$

##  3\. A construction of linear Nambu-Poisson structures

In this section, we unravel under certain conditions a relation between linear
Nambu-Poisson structures and Filippov connections.

###  3.1. From Filippov algebroids to linear Nambu-Poisson structures

######  Definition 3.1.

[Tak, ref15] A Nambu-Poisson structure of order $n$ on a smooth manifold $P$
is an $\mathbb{R}$-multilinear and skew-symmetric $n$-bracket on the smooth
function space $C^{\infty}(P)$:

$\\{\cdotp,\cdotp\cdotp\cdotp,\cdotp\\}\colon\underbrace{C^{\infty}(P)\times\cdotp\cdotp\cdotp\times
C^{\infty}(P)}_{n-\mbox{copies}}\to C^{\infty}(P)$

satisfying the following two conditions:

  1. (1)

The $n$-bracket is a derivation with respect to
$C^{\infty}(P)$-multiplications:

$\\{f_{1},\cdots,f_{n-1},g_{1}g_{2}\\}=g_{1}\\{f_{1},\cdots,f_{n-1},g_{2}\\}+\\{f_{1},\cdots,f_{n-1},g_{1}\\}g_{2};$

  2. (2)

The (generalized) Jacobi identity (also known as the fundamental identity)

$\\{f_{1},\cdots,f_{n-1},\\{g_{1},\cdots,g_{n}\\}\\}=\sum_{i=1}^{n}\\{g_{1},\cdots,\\{f_{1},\cdots,f_{n-1},g_{i}\\},\cdots,g_{n}\\},$

holds for all $f_{i}$ and $g_{j}\in C^{\infty}(P)$.

The pair $(P,\\{\cdotp,\cdotp\cdotp\cdotp,\cdotp\\})$ is called a Nambu-
Poisson manifold.

Alternatively, one could express the said bracket via an $n$-vector field
$\pi$ on $P$ such that

$\\{f_{1},\cdots,f_{n}\\}=\pi(df_{1},\cdots,df_{n}),\quad\forall
f_{1},\cdots,f_{n}\in C^{\infty}(P).$ (12)

Given a smooth vector bundle $p\colon A\to M$, the section space $\Gamma(A)$
are identified as the space $C^{\infty}_{\mathrm{lin}}(A^{*})$ of fiberwise
linear functions on $A^{*}$, the dual vector bundle of $A$; while elements in
$p^{\ast}(C^{\infty}(M))$ are called basic functions on $A^{*}$. To fix the
notations, for any section $X\in\Gamma(A)$, let $\phi_{X}\in
C^{\infty}_{\mathrm{lin}}(A^{*})$ be the corresponding linear function on
$A^{\ast}$.

######  Definition 3.2.

[BBDM] A Nambu-Poisson structure of order $n$ on the vector bundle
$A^{\ast}\to M$ is said to be linear, if it satisfies the following three
conditions:

  1. (1)

The bracket of $n$ linear functions is again a linear function;

  2. (2)

The bracket of $(n-1)$ linear functions and a basic function is a basic
function;

  3. (3)

The bracket of $n$ functions is zero if there are more than one basic
functions among the arguments.

Note that any Poisson manifold is a Nambu-Poisson manifold of order $2$. A
well-known fact is the following: A Lie algebroid $A$ over $M$ gives rise to a
linear Poisson manifold $A^{*}$, and vice versa. It is pointed out in
[BBDM]*Theorem 4.4 that a linear Nambu-Poisson structure of order $n$ on
$A^{*}$ corresponds to a Filippov $n$-algebroid structure on $A$ (see also
[BL]). However, the reverse process is generally not valid for the cases of
$n\geqslant 3$, mainly because the condition of a Nambu-Poisson structure is
very strong (cf. [ref13]). Nevertheless, in this paper, we will require that
$A$ be a rank $n$ vector bundle and establish the one-to-one correspondence
between Filippov $n$-algebroid structures on $A$ and linear Nambu-Poisson
structures on $A^{*}$. In specific, under the said condition, our main theorem
below serves as a complement to [BBDM]*Theorem 4.4.

######  Theorem 3.3.

Let $(A,\rho,[\cdotp,\cdots,\cdotp])$ be a Filippov $n$-algebroid over a
smooth manifold $M$, where $A\to M$ is a vector bundle of rank $n\geqslant 3$.
Then there exists a unique linear Nambu-Poisson structure on the dual bundle
$A^{*}\to M$ such that for all sections $X_{1},\cdots,X_{n}\in\Gamma(A)$,

$\\{\phi_{X_{1}},\cdots,\phi_{X_{n}}\\}=\phi_{[X_{1},\cdots,X_{n}]}.$ (13)

Note that, if Equation (13) holds, then it is easy to deduce that the linear
Nambu-Poisson structure on $A^{*}$ and the anchor map $\rho$ are also related:

$\\{\phi_{X_{1}},\cdots,\phi_{X_{n-1}},p^{\ast}f\\}=p^{\ast}(\rho(X_{1}\wedge\cdotp\cdotp\cdotp\wedge
X_{n-1})(f)),$ (14)

for all $f\in C^{\infty}(M)$ and $X_{1},\cdots,X_{n-1}\in\Gamma(A)$.

As a direct application of Theorems 2.3 and 3.3, one can construct linear
Nambu-Poisson structures out of Filippov connections:

######  Corollary 3.4.

If $\nabla$ is a Filippov connection on an $n$-anchored bundle $(A,\rho)$,
where $A\to M$ is a vector bundle of rank $n\geqslant 3$, then the dual bundle
$A^{\ast}$ admits a unique linear Nambu-Poisson structure of order $n$ defined
by

$\\{\phi_{X_{1}},\cdots,\phi_{X_{n}}\\}=\phi_{[X_{1},\cdots,X_{n}]^{\nabla}},$

for all $X_{1},\cdots,X_{n}\in\Gamma(A)$.

###  3.2. Proof of Theorem 3.3

The proof is divided into three steps.

Step 1 — Since functions of type $\phi_{X}$ (for $X\in\Gamma(A)$) and $p^{*}f$
(for $f\in C^{\infty}(M)$) generate $C^{\infty}(A^{*})$, there exists a unique
$\mathbb{R}$-multilinear $n$-bracket $\\{\cdotp,\cdots,\cdotp\\}$ on
$C^{\infty}(A^{*})$ satisfying Equations (13) and (14). We wish to write the
corresponding $n$-vector field $\pi$ on $A^{*}$ explicitly.

To this end, we work locally and consider the trivialization $A|_{U}\cong
U\times\mathbb{R}^{n}$ over an open subset $U\subset M$ with coordinates
$x^{1},\cdots,x^{m}$; let $\\{X_{1},\cdots,X_{n}\\}$ be a local basis of
$\Gamma(A|_{U})$. Then

$\\{y_{1}=\phi_{X_{1}},\cdots,y_{n}=\phi_{X_{n}},p^{*}x^{1},\cdots,p^{*}x^{m}\\}$

forms a chart on $A^{*}|_{U}$. For convenience, $p^{*}x^{i}$ is still denoted
by $x^{i}$.

Suppose further that the Filippov algebroid $A|_{U}$ is described by the
structure functions $c^{k}$ and $f_{1\cdots\widehat{l}\cdots n}\in
C^{\infty}(U)$ such that

$[X_{1},\cdots,X_{n}]=\sum_{k=1}^{n}c^{k}X_{k}$ $\mbox{and
}\quad\rho(X_{1}\wedge\cdots\widehat{X_{l}}\cdots\wedge
X_{n})=f_{1\cdots\widehat{l}\cdots n}\frac{\partial}{\partial x^{1}}.$

Here we have utilized Proposition 1.5. Then one is able to find the expression
of the $n$-vector field $\pi$ on $A^{*}|_{U}$:

$\pi=\sum_{k=1}^{n}c^{k}y_{k}\frac{\partial}{\partial
y_{1}}\wedge\cdots\wedge\frac{\partial}{\partial
y_{n}}+\sum_{l=1}^{n}f_{1\cdots\widehat{l}\cdots n}\frac{\partial}{\partial
y_{1}}\wedge\cdots\wedge\frac{\partial}{\partial\widehat{y_{l}}}\wedge\cdots\wedge\frac{\partial}{\partial
y_{n}}\wedge\frac{\partial}{\partial x^{1}}$ (15)

which corresponds to the $n$-bracket $\\{\cdots\\}$ on $A^{*}|_{U}$.

Step 2 — We need to set up a preparatory lemma.

######  Lemma 3.5.

There exists a local basis $\\{X_{1},\cdots,X_{n}\\}$ of $\Gamma(A|_{U})$ such
that the corresponding structure functions $c^{k}$ and
$f_{1\cdots\widehat{l}\cdots n}$ satisfy the following relations: for all
$i\neq j$ (in $\\{1,\cdots,n\\}$),

$\displaystyle f_{1\cdots\widehat{i}\cdots n}\frac{\partial
f_{1\cdots\widehat{j}\cdots n}}{\partial x^{1}}$ $\displaystyle=$
$\displaystyle f_{1\cdots\widehat{j}\cdots n}\frac{\partial
f_{1\cdots\widehat{i}\cdots n}}{\partial x^{1}};$ (16)
$\displaystyle(-1)^{i}f_{1\cdots\widehat{i}\cdots n}c^{j}$ $\displaystyle=$
$\displaystyle(-1)^{j}f_{1\cdots\widehat{j}\cdots n}c^{i}.$ (17)

###### Proof.

Consider the map $\rho:\wedge^{n-1}A|_{U}\to TU$. By Proposition 1.5, for any
point $p\in U$, we have $\operatorname{rank}(\rho(\wedge^{n-1}A)_{p})\leqslant
1$, and hence $\dim(\ker(\rho_{p}))\geqslant(n-1)$. So, we are able to find a
local basis $\\{Z_{1},\cdots,Z_{n}\\}$ of $\Gamma(\wedge^{n-1}A|_{U})$ such
that $\rho(Z_{2})$, $\cdots$, $\rho(Z_{n})$ are trivial.

Take an arbitrary $\Omega\in\Gamma(\wedge^{n}A|_{U})$ which is nowhere
vanishing on $U$. Consider

$\Omega^{\sharp}:A^{\ast}|_{U}\to\wedge^{n-1}A|_{U},\ \ \ \
\Omega^{\sharp}(\alpha):=i_{\alpha}\Omega,$

which is an isomorphism of vector bundles. Then we obtain a basis
$\\{\alpha_{1}$, $\cdots$, $\alpha_{n}\\}$ of $\Gamma(A^{\ast}|_{U})$ by
setting $\alpha_{i}:=(\Omega^{\sharp})^{-1}(Z_{i})$. Let
$\\{X_{1},\cdots,X_{n}\\}$ be the dual basis of $\Gamma(A|_{U})$ corresponding
to $\\{\alpha_{1},\cdots,\alpha_{n}\\}$. There exists a nowhere vanishing
smooth function $g\in C^{\infty}(U)$ such that
$\Omega=gX_{1}\wedge\cdots\wedge X_{n}$, and hence
$Z_{i}=i_{\alpha_{i}}\Omega=gX_{1}\wedge\cdots\widehat{X_{i}}\wedge\cdots\wedge
X_{n}$.

Since we have $\rho(Z_{i})=0$ ($\forall i\in\\{2,\cdots,n\\}$), we also have

$\rho(X_{1}\wedge\cdots\widehat{X_{i}}\wedge\cdots\wedge X_{n})=0,\quad\forall
i\in\\{2,\cdots,n\\}.$ (18)

Using the axiom of a Filippov algebroid, we have a relation

$\displaystyle f_{1\cdots\widehat{i}\cdots n}\frac{\partial
f_{1\cdots\widehat{j}\cdots n}}{\partial x^{1}}\frac{\partial}{\partial
x^{1}}-f_{1\cdots\widehat{j}\cdots n}\frac{\partial
f_{1\cdots\widehat{i}\cdots n}}{\partial x^{1}}\frac{\partial}{\partial
x^{1}}$ $\displaystyle=$ $\displaystyle[f_{1\cdots\widehat{i}\cdots
n}\frac{\partial}{\partial x^{1}},f_{1\cdots\widehat{j}\cdots
n}\frac{\partial}{\partial x^{1}}]_{TM}$ $\displaystyle=$
$\displaystyle[\rho(X_{1}\wedge\cdots\wedge\widehat{X_{i}}\wedge\cdots\wedge
X_{n}),\rho(X_{1}\wedge\cdots\wedge\widehat{X_{j}}\wedge\cdots\wedge
X_{n})]_{TM}$ $\displaystyle\xlongequal[]{\mbox{\eqref{eq1}}}$
$\displaystyle\sum_{k=1,k<j}^{n}\rho(X_{1}\wedge\cdots\wedge[X_{1},\cdots,\widehat{X_{i}},\cdots,X_{n},X_{k}]\wedge\cdots\wedge\widehat{X_{j}}\wedge\cdots
X_{n})$ $\displaystyle\
+\sum_{k=1,k>j}^{n}\rho(X_{1}\wedge\cdots\wedge\widehat{X_{j}}\wedge\cdots\wedge[X_{1},\cdots,\widehat{X_{i}},\cdots,X_{n},X_{k}]\wedge\cdots
X_{n})$ $\displaystyle=$
$\displaystyle(-1)^{(n-i)}c^{i}f_{1\cdots\widehat{j}\cdots
n}\frac{\partial}{\partial x^{1}}-(-1)^{(n-j)}c^{j}f_{1\cdots\widehat{i}\cdots
n}\frac{\partial}{\partial x^{1}}.$

According to the previous fact (18), all the lines above must be trivial, and
thus the desired two equalities (16) and (17) are proved. ∎

Step 3 — We wish to show that the $n$-bracket $\\{\cdotp,\cdots,\cdotp\\}$
given in Step 1, or the $n$-vector field $\pi$ locally given in Equation (15),
is a linear Nambu-Poisson structure on $A^{*}$.

We need the following proposition due to Dufour and Zung [DZ]*Proposition 2.1.

######  Proposition 3.6.

[DZ] Let $\Omega$ be a volume form on an $l$-dimensional manifold $P$, and
$\pi$ an $n$-vector filed on $P$, where $l>n\geqslant 3$. Consider the
$(l-n)$-form $\omega:=i_{\pi}\Omega$ on $P$. Then $\pi$ defines a Nambu-
Poisson structure (via Equation (12)) if and only if $\omega$ satisfies the
following two conditions:

$\displaystyle(i_{K}\omega)\wedge\omega=0,$ (19)
$\displaystyle(i_{K}\omega)\wedge d\omega=0$ (20)

for any $(l-n-1)$-vector field $K$ on $P$.

Consider the volume form $\Omega=dy_{1}\wedge\cdots dy_{n}\wedge
dx^{1}\wedge\cdots\wedge dx^{m}$ on $A^{\ast}|_{U}$, where $U$, $y_{i}$, and
$x^{j}$ are as earlier, and we suppose that such a coordinate system stems
from $\\{X_{1},\cdots,X_{n}\\}$ fulfills Lemma 3.5. According to Proposition
3.6, we need to examine the $m$-form defined by:

$\omega:=i_{\pi}\Omega=\sum_{k=1}^{n}c^{k}y_{k}dx^{1}\wedge\cdots\wedge
dx^{m}+\sum_{j=1}^{n}(-1)^{n-j+1}f_{1\cdots\widehat{j}\cdots n}dy_{j}\wedge
dx^{2}\wedge\cdots\wedge dx^{m}.$

We can easily check that $\omega$ satisfies Equation (19) and hence it remains
to check Equation (20). One first finds that

$\displaystyle d\omega$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}y_{k}dc^{k}\wedge dx^{1}\wedge\cdots\wedge
dx^{m}+\sum_{k=1}^{n}c^{k}dy_{k}\wedge dx^{1}\wedge\cdots\wedge dx^{m}$
$\displaystyle\ +\sum_{j=1}^{n}(-1)^{n-j+1}df_{1\cdots\widehat{j}\cdots
n}\wedge dy_{j}\wedge dx^{2}\wedge\cdots\wedge dx^{m}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}c^{k}dy_{k}\wedge dx^{1}\wedge\cdots\wedge
dx^{m}+\sum_{j=1}^{n}(-1)^{n-j+1}\frac{\partial f_{1\cdots\widehat{j}\cdots
n}}{\partial x^{1}}dx^{1}\wedge dy_{j}\wedge dx^{2}\wedge\cdots\wedge dx^{m}.$

Consider the following special type of $(m-1)$-vector field on
$A^{\ast}|_{U}$: $K=\frac{\partial}{\partial
x^{2}}\wedge\cdots\wedge\frac{\partial}{\partial x^{m}}$. Then one computes:

$\displaystyle(i_{K}\omega)\wedge d\omega$ $\displaystyle=$
$\displaystyle(-1)^{m-1}\left(\sum_{k=1}^{n}c^{k}y_{k}dx^{1}+\sum_{j=1}^{n}(-1)^{n-j+1}f_{1\cdots\widehat{j}\cdots
n}dy_{j}\right)\wedge d\omega$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}(-1)^{m+n-j}f_{1\cdots\widehat{j}\cdots
n}dy_{j}\wedge\sum_{i=1}^{n}c^{i}dy_{i}\wedge dx^{1}\wedge\cdots\wedge dx^{m}$
$\displaystyle\ +\sum_{j=1}^{n}(-1)^{m+n-j}f_{1\cdots\widehat{j}\cdots
n}dy_{j}\wedge\sum_{i=1}^{n}(-1)^{n-i+1}\frac{\partial
f_{1\cdots\widehat{i}\cdots n}}{\partial x^{1}}dx^{1}\wedge dy_{i}\wedge
dx^{2}\wedge\cdots\wedge dx^{m}$ $\displaystyle=$
$\displaystyle(-1)^{m}\sum_{j=1}^{n}\sum_{i=1,i\neq
j}^{n}((-1)^{(n-i)}c^{i}f_{1\cdots\widehat{j}\cdots
n}-(-1)^{(n-j)}c^{j}f_{1\cdots\widehat{i}\cdots n})dy_{j}\wedge dy_{i}$
$\displaystyle\ +\sum_{j=1}^{n}\sum_{i=1,i\neq
j}^{n}(-1)^{m+i+j}(f_{1\cdots\widehat{j}\cdots n}\frac{\partial
f_{1\cdots\widehat{i}\cdots n}}{\partial x^{1}}-f_{1\cdots\widehat{i}\cdots
n}\frac{\partial f_{1\cdots\widehat{j}\cdots n}}{\partial x^{1}})dy_{j}\wedge
dy_{i}$ $\displaystyle\xlongequal[]{\mbox{\eqref{volume-1},\eqref{volume-2}}}$
$\displaystyle 0.$

This justifies Equation (20) for this particular $K$. For other types of $K$,
it is easy to verify Equation (20) as well. This completes the proof of
Theorem 3.3.

## References


# Extraction of the Coulomb Sum Rule, Transverse Enhancement, and Longitudinal
Quenching from an Analysis of all Available e-12C and e-16O Cross Section
Data.

A. Bodek  Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627, USA  M. E. Christy  Thomas Jefferson National Accelerator
Facility, Newport News, VA 23606, USA

###### Abstract

We report on a phenomenological analysis of all available electron scattering
data on ${\rm{}^{12}C}$ (about 6600 differential cross section measurements)
and on ${\rm{}^{16}O}$ (about 250 measurements) within the framework of the
quasielastic (QE) superscaling model (including Pauli blocking). All QE and
inelastic cross section measurements are included down to the lowest momentum
transfer $\bf q$ (including photo-production data). We find that there is
enhancement of the transverse QE response function ($R_{T}^{QE}$) and
quenching of the QE longitudinal response function ($R_{L}^{QE}$) at low $\bf
q$ (in addition to Pauli blocking). We extract parameterizations of a
$multiplicative$ low $\bf q$ ”Longitudinal Quenching Factor” and an $additive$
”Transverse Enhancement” contribution. Additionally, we find that the
excitation of nuclear states contribute significantly (up to 30%) to the
Coulomb Sum Rule $SL({\bf q})$. We extract the most accurate determination of
$SL({\bf q})$ to date and find it to be in disagreement with random phase
approximation (RPA) based calculations but in reasonable agreement with recent
theoretical calculations such as ”First Principle Green’s Function Monte
Carlo”.

###### pacs:

13.38.Dg,13.85.Fb,14.60.Cd,14.70.Hp

We report on a fit to all available electron scattering data on $\rm{}^{12}C$
(about 6600 differential cross section measurements) and $\rm{}^{16}O$ (about
250 measurements) within the framework of the quasielastic (QE) superscaling
model (including Pauli blocking). The cross sections measurements include the
available data on QE (down to the lowest momentum transfer $\bf q$
($\equiv|\vec{q}|$), inelastic production, and photoproduction. The fit
includes inelastic structure functions and empirical parameters to model both
an enhancement of the transverse (T) QE response function $R_{T}^{QE}$ and
quenching of the longitudinal (L) QE response function $R_{L}^{QE}$ at low
$\bf q$. As the fit provides an accurate description of the data, it can be
used as a proxy to validate modeling of cross sections in Monte Carlo event
generators for electron and neutrino ($\nu_{e,\mu}$) scattering. Careful
consideration of nuclear excitations is critical for an accurate extraction of
the normalized Coulomb Sum RuleCSR  $SL({\bf q})$ at low $\bf q$ as these
states can contribute up to 30%. After accounting for the dominant
excitations, we extract the most accurate determination of $SL({\bf q})$) as
function of $\bf q$ for $\rm{}^{12}C$ and $\rm{}^{16}O$ based on the global
fit and compare to theoretical models. In addition, the ”Transverse
Enhancement” (TE) of $R_{T}^{QE}$ and the ”Quenching Factor” of $R_{L}^{QE}$
are also of great interest to $\nu_{e,\mu}$ scattering experimentsminiboone ;
minerva ; microboone1 ; microboone2 .

The electron scattering differential cross section can be writtenGEp  in terms
of $R_{L}({\bf q},\nu)$ and $R_{T}({\bf q},\nu)$ as:

$\displaystyle\frac{d^{2}\sigma}{d\nu d\Omega}$ $\displaystyle=$
$\displaystyle\sigma_{M}[AR_{L}({\bf q},\nu)+BR_{T}({\bf q},\nu)],$ (1)
$\displaystyle\sigma_{M}$ $\displaystyle=$
$\displaystyle\alpha^{2}\cos^{2}(\theta/2)/[4E_{0}^{2}\sin^{4}(\theta/2)].$

Here, $E_{0}$ is the incident electron energy, $E^{\prime}$ and $\theta$ are
energy and angle of the final state electron, $\nu=E_{0}-E^{\prime}$, $Q^{2}$
is the square of the 4-momentum transfer (defined to be positive),
$\bf{q}^{2}$=$Q^{2}+\nu^{2}$, $A=(Q^{2}/\bf{q}^{2})^{2}$ and
$B=\tan^{2}(\theta/2)+Q^{2}/2\bf{q}^{2}$. In the analysis we also use the
invariant hadronic mass $W^{2}=M_{p}^{2}+2M_{p}\nu-Q^{2}$.

The inelastic Coulomb Sum Rule is the integral of $R_{L}({\bf q},\nu)d\nu$,
excluding the elastic peak and pion production processes. It has contributions
from QE scattering and from electro-excitations of nuclear states:

$\displaystyle{\rm CSR}({\bf q})=\int R_{L}({\bf q},\nu)d\nu$ $\displaystyle=$
$\displaystyle\int R_{L}^{QE}({\bf q},\nu)d\nu+G^{\prime 2}_{E}(Q^{2})\times
Z^{2}\sum_{all}^{L}F^{2}_{i}({\bf q})$ $\displaystyle=$ $\displaystyle
G^{\prime 2}_{E}(Q^{2})\times\big{[}Z\int V^{QE}_{L}({\bf
q},\nu)d\nu+Z^{2}\sum_{all}^{L}F^{2}_{i}({\bf q})\big{]}.$

We define $V^{QE}({\bf q},\nu)$ as the reduced longitudinal QE response, which
integrates to unity in the absence of any suppression (e.g. Pauli blocking).
The charge form factors for the electro-excitation of nuclear states
$F^{2}_{iC}({\bf q})$ is $G^{2}_{Ep}(Q^{2})\times F^{2}_{i}({\bf q})$. In
order to account for the small contribution of the neutron and relativistic
effects $G^{\prime 2}_{E}(Q^{2})$ is given byGEp :

$G^{\prime
2}_{E}(Q^{2})=[G^{2}_{Ep}(Q^{2})+\frac{N}{Z}G^{2}_{En}(Q^{2})]\frac{1+\tau}{1+2\tau},$
(3)

where, $G_{Ep}$ and $G_{En}$ are the electric form factors BBBA  of the proton
and neutron respectively and $\tau=Q^{2}/4M_{p}^{2}$. By dividing Eq.
Extraction of the Coulomb Sum Rule, Transverse Enhancement, and Longitudinal
Quenching from an Analysis of all Available e-12C and e-16O Cross Section
Data. by $ZG^{\prime 2}_{E}{\bf q})$ we obtain the normalized inelastic
Coulomb Sum Rule $SL({\bf q})$ :

$\displaystyle SL({\bf q})=\int V^{QE}_{L}({\bf
q},\nu)d\nu+Z\sum_{all}^{L}F^{2}_{i}({\bf q}).$ (4)

At high $\bf q$ it is expected that $S_{L}\rightarrow 1$ because both nuclear
excitation form factors and Pauli suppression are small. At small $\bf q$ it
is expected that $S_{L}\rightarrow 0$ because the all form factors for
inelastic processes (QE and nuclear excitations) must be zero at $\bf q$=0.

We begin by parameterizing the measurements of the L and T form factors for
the electro-excitation of all nuclear states in $\rm{}^{12}C$ with excitation
energies ($E_{x}$) less than 16.0 MeV (the approximate proton removal energy
from $\rm{}^{12}C$). For these states the measurements are straightforward
since the QE cross section is zero for $E_{x}<$ 16 MeV.

For $E_{x}>$ 16 MeV the extractions of form factors require corrections for
the QE contribution. We perform a reanalysis of all published cross sections
with in $E_{x}<$ 55 MeV and use our fitted QE model (described below) to
extract L and T form factors. For $E_{x}>$ 20 MeV (region of the Giant Dipole
resonances) we group the strength from multiple excitations into a few states
with a large width $E_{x}$ and extract effective form factors accounting for
all states in this region. The top two panels of Fig. 1 show comparisons of
our fit (red) to $R_{L}$ measurements by Yamaguchi (blue) 1996Yamaguchi  with
$E_{x}>$ 14 MeV. An estimated resolution smearing of 600 keV has been applied
to the excitations in the fit to match the data. While individual states are
well reproduced at low excitation energy, Above 20 MeV the effect of grouping
several excitations together into broad effective states in the fit can be
seen. While the fit does not capture the structure from individual states, the
total strength is seen to be well reproduced. A similar analysis has been done
for $\rm{}^{16}O$. The fits to the form factors for $\rm{}^{12}C$) and
$\rm{}^{16}O$ are included in a longer paper (in preparation).

The contribution of nuclear excitation to $SL({\bf q})$ (factor
$Z\sum_{all}^{L}F^{2}_{i}({\bf q})$ in eq. 4) is calculated using the
parametrizations of the form factors. The bottom two panels of Fig. 1 show the
contributions of nuclear excitations to $SL({\bf q})$ for $\rm{}^{12}C$ and
$\rm{}^{16}O$. The contribution of all excitations is largest ($\approx$ 0.29)
at $\bf q$=0.22 GeV. Although the contributions of different $E_{x}$ regions
to $SL({\bf q})$ is different for $\rm{}^{12}C$ and $\rm{}^{16}O$, the total
contribution turns out to be similar for the two nuclei. The total
contribution of excitations to $S_{L}({\bf q})$ in $\rm{}^{12}C$ can be
parameterized as:

$\displaystyle Z$ $\displaystyle\sum_{all}^{L}F^{2}_{i}({\bf
q})=N_{1}e^{-(x-C_{1})^{2}/D_{1}^{2}}$ $\displaystyle+$ $\displaystyle
N_{2}e^{-(x-C_{2})^{2}/D_{2}^{2}}+N_{3}e^{-(x-C_{3})^{2}/D_{3}^{2}}$

where x= ${\bf q}/K_{F}$ ($K_{F}$= 0.228 GeV), $N_{1}$= 0.260, $C_{1}$=1.11,
$D_{1}$=0.50, $N_{2}$= 0.075, $C_{2}$=0.730, $D_{2}$=0.30, and $N_{3}$= 0.01,
$C_{3}$=2.0, $D_{3}$=0.30. The uncertainty in the total contribution of
excited states was estimated to 15% plus a systematic error to account for the
choice of parametrization at very low $\bf q$ ($\pm$0.01) added in quadrature.

Figure 1: Top two panels: Comparison of $R_{L}({\bf q},\nu)$ extracted from
our $\rm{}^{12}C$ fit (red) to a sample of experimental data (blue)Yamaguchi .
For $E_{x}$ less than 12 MeV the values are multiplied by 1/6. Bottom two
panels: The contributions of longitudinal nuclear excitations (between 2 and
55 MeV) to the Coulomb sum rule ($Z\sum_{all}^{L}F^{2}_{i}({\bf q})$) in
equation 4 for $\rm{}^{12}C$ and $\rm{}^{16}O$. Figure 2: Comparison of the
fit to electron scattering $\frac{d^{2}\sigma}{d\Omega d\nu}$
measurementsarchive ; Barreau ; Baran  at $\bf q$ values close to 0.30, 0.38
and 0.57 GeV (and different scattering angles). Shown are total
$\frac{d^{2}\sigma}{d\Omega d\nu}$ (solid-purple line), total minus the
contribution of the nuclear excitations (solid-blue), the QE cross section
without TE (dashed-blue), the TE contribution (solid-red) and inelastic pion
production (dot-dashed black line). Additional comparisons are included in
supplemental materialsSupplemental .

The universal fit to the $\rm{}^{12}C$ data is an update of the 2012 fit by
Bosted and Mamyan Bosted . The QE contribution is modeled by the superscaling
approachDonnelly ; Amaro1 ; Amaro2 ; Megias  with Pauli blocking calculated
using the RosenfelderMegias ; Rosenfelder ; Megias2  method. The superscaling
function extracted from the fit is similar to the superscaling functions of
Amaro 2005Amaro1  and Amaro 2020Amaro2  and yields similar Pauli suppression.

In modeling the QE response we use the same scaling function for both
$R_{L}^{QE}({\bf q},\nu)$ and $R_{T}^{QE}({\bf q},\nu)$ and fit for empirical
corrections to the response functions. For $R_{T}^{QE}$ we extract an additive
”Transverse Enhancement/MEC” TE(${\bf q},\nu$) contribution (which includes
both single nucleon and two nucleon final states). As shown in ref. transverse
. TE(${\bf q},\nu$) increases $R_{T}^{QE}$ with the largest fractional
contribution around $Q^{2}$=0.3 GeV2. For $R_{L}^{QE}$ we extract a
multiplicative $\bf q$ dependent ”Longitudinal Quenching Factor”,
$F_{quench}({\bf q})$, which decreases $R_{L}^{QE}$ at low $\bf q$.

Since $\frac{d^{2}\sigma}{d\Omega d\nu}$ measurements span a large range of
$\theta$ and $\bf q$, parametrizations of both TE(${\bf q},\nu$) and
$F_{quench}^{L}({\bf q})$ can be extracted. The analysis includes all data for
a large range of nuclei. However, in this paper we only include data on
$\rm{}^{12}C$ and $\rm{}^{16}O$. Briefly, the updated fit includes:

Figure 3: Top left panel: QE ”Longitudinal Quenching Factor” (dotted-black
line with yellow error band). Top right panel: The various contributions to
$SL({\bf q})$ for $\rm{}^{12}C$ (dotted blue with yellow error band) including
QE with Pauli suppression only (dotted-purple), QE suppressed by both ”Pauli”
and ”Longitudinal Quenching” (solid-green), and the contribution of nuclear
excitations (red-dashed with green error band). Bottom left panel: $SL({\bf
q})$ for $\rm{}^{12}C$ (dotted-blue with yellow error band) compared to
theoretical calculations including Lovato 2016 Lovato2016  (solid-purple),
(Mihaila 2000microscopic  (dashed-green), and RPA Cloet 2016Cloet  (solid-
red). Bottom right panel: $SL({\bf q})$ for $\rm{}^{16}O$ (dotted-black with
green error band) compared to theoretical calculations of Sobczyk 2020Coupled
(red-dashed) and Mihaila 2000 (dotted-dashed).

  1. 1.

All electron scattering data on $\rm{}^{1}H$, $\rm{}^{2}H$, $\rm{}^{12}C$ and
$\rm{}^{16}O$ in addition to the data in the QEarchive  and resonanceresonance
data archives.

  2. 2.

Coulomb correctionsCoulomb  using the Effective Momentum Approximation (EMA)
in modeling scattering from nuclear targets.

  3. 3.

Updated nuclear elastic+excitations form factors.

  4. 4.

Superscaling $FN(\psi^{\prime})$ parameters are re-extracted including the
Fermi broadening parameter $K_{F}$.

  5. 5.

Parameterizations of the free nucleon form factorsFreeN  are re-derived from
all $\rm{}^{1}H$ and $\rm{}^{2}H$ data.

  6. 6.

Rosenfelder Pauli suppressionMegias ; Rosenfelder ; Megias2  which reduces and
changes the QE distribution at low $\bf{q}$ and $\nu$.

  7. 7.

Updates of fitsFreeN  to inelastic electron scattering data (in the nucleon
resonance region and inelastic continuum) for $\rm{}^{1}H$ and $\rm{}^{2}H$.

  8. 8.

A $\bf q$ dependent $E_{\mathrm{shift}}^{QE}(\bf q)$ parameter for the QE
process to account for the optical potentialremoval  of final state nucleons.

  9. 9.

Photo-production data in the nucleon resonance region and inelastic
continuumChristyD .

  10. 10.

Gaussian Fermi smeared nucleon resonance and inelastic continuumChristyD . The
$K_{F}$ parameters for pion production and QE can be different.

  11. 11.

Parametrizations of the medium modifications of both the L and T structure
functions responsible for the EMC effect (nuclear dependence of inelastic
structure functions). These are applied to the free nucleon cross sections
prior to application of the Fermi smearing.

  12. 12.

Parametizations of TE(${\bf q},\nu$) and $F_{quench}^{L}({\bf q})$ as
described below.

  13. 13.

QE data at all values of $Q^{2}$ down to $Q^{2}$=0.01 GeV2 ($\bf q$=0.1 GeV)
(which were not included in the Bosted-Mamyan fit).

The average (over $\nu$) Pauli suppression factor for $x<2.5$ ( $x={\bf
q}/K_{F}$, $K_{F}$=0.228 GeV) is described by:

$\langle F_{Pauli}^{This-analysis}({\bf
q})\rangle=\sum_{j=0}^{j=3}k_{j}(x)^{j}.$ (6)

Using the Rosenfeld method with supercaling function used in this analysis, we
find $k_{o}$=0.3054, $k_{1}$=0.7647, $k_{2}$=${\bf-}$0.2768 and
$k_{3}$=0.0328. The Pauli suppression factor for $x>2.5$ is 1.0.

Comparisons of the fit to electron scattering $\frac{d^{2}\sigma}{d\Omega
d\nu}$ measurementsarchive ; Barreau ; Baran  at different values of $\theta$
for $\bf q$ values close to 0.30, 0.38 and 0.57 GeV (corresponding to
extractions of $R_{L}$ and $R_{T}$ by JourdanJourdan ) are shown in Fig. 2.
Shown are the total $\frac{d^{2}\sigma}{d\Omega d\nu}$ cross section (solid-
purple line), the total minus the contribution of nuclear excitations (solid-
blue), the QE cross section without TE (dashed-blue), the TE contribution
(solid-red), and inelastic pion production (dot-dashed black). An estimated
resolution smearing of 3.5 MeV has been applied to the excitations to better
match the data.

The fit is in good agreement with all electron scattering data for both small
and large $\theta$.

The extracted QE ”Longitudinal Quenching Factor” $F^{L}_{quench}({\bf q})$ is
unity for x$>$3.75, and is zero for x$<$0.35. For $0.35<x<4.0$ it is
parameterized by:

$\displaystyle F^{L}_{quench}({\bf
q})=\frac{(x-0.2)^{2}}{(x-0.18)^{2}}\big{[}1.0+A_{1}(3.75-x)^{1.5}$
$\displaystyle+A_{2}(3.75-x)^{2.5}+A_{3}(3.75-x)^{3.5}\big{]}\vspace{-10pt}$
(7)

with $A_{1}$=${\bf-}$0.13152 $A_{2}$=0.11693, and $A_{3}$=$\bf-$0.03675. The
top-left panel of Fig. 3 shows the extracted $F^{L}_{quench}({\bf q})$.
(black-dotted line). The yellow band includes the statistical,
parameterization and a normalization error of 2% (all added in quadrature).

If another formalism is used to model QE scattering (e.g. RFG or spectral
functions) then the quenching factor for the model $F^{L-model}_{quench}({\bf
q})$ is given by:

$\displaystyle F^{L-model}_{quench}({\bf q})=\frac{\langle F_{Pauli}^{This-
analysis}({\bf q})\rangle}{\langle F_{Pauli}^{model}({\bf
q})\rangle}F^{L}_{quench}({\bf q})$ (8)

The top right panel of Fig. 3 shows the various contributions to the measured
$SL({\bf q})$ for $\rm{}^{12}C$ (dotted blue line with yellow error band).
Shown are the QE contribution with only Pauli suppression (dotted-purple), QE
suppressed by both ”Pauli Suppression” and $F^{L}_{quench}({\bf q})$ labeled
as QE total suppression (solid-green), and the contribution of nuclear
excitations (red-dashed line). The green error band is 15% plus 0.01 added in
quadrature.

Figure 4:  Comparisons between our extraction of $R_{L}(\bf q,\nu)$ and
$R_{T}(\bf q,\nu)$ and the extraction (for only three values of $\bf q$) by
JourdanJourdan  (the Jourdan analysis includes data from only two
experiments). Also shown are comparisons to 1b+2b GFMCLovato2016  and ED-
RMFED-RMF  theoretical predictions. In these two models the curves labeled 2b
are the only contribution of 2-body currents to $TE({\bf q},\nu)$. The
transverse enhancement in both 1b and 2b currents is included in the total.

The left panel on the bottom of Figure 3 shows a comparison of the extracted
$SL({\bf q})$ for $\rm{}^{12}C$ (dotted-blue curve with yellow error band) to
theoretical calculations. These include the Lovato 2016Lovato2016  ”First
Principle Green’s Function Monte Carlo” (GFMC) calculation (solid-purple
line), Mihailamicroscopic  2000 Coupled-Clusters based calculation (AV18+UIX
potential, dashed-green), and Cloet 2016Cloet  RPA calculation (RPA solid-
red). Our measurement for $\rm{}^{12}C$ are in disagreement with Cloet 2016
RPA, and in reasonable agreement with Lovato 2016 and Mihaila 2000 except near
$\bf{q}\approx$ 0.30 GeV where the contribution from nuclear excitations is
significant.

There is not enough QE data for $\rm{}^{16}O$ to perform a complete analysis.
We find that the QE fit parameters for $\rm{}^{12}C$ also describe all
available data on $\rm{}^{16}O$. A difference in $SL({\bf q})$ between
$\rm{}^{12}C$ and $\rm{}^{16}O$ could be the contribution of nuclear
excitations. However as shown in Fig. 1 the contributions of nuclear
excitations to the $SL({\bf q})$ for $\rm{}^{12}C$ and $\rm{}^{16}O$ are
consistent with being equal.

The bottom right panel of Fig. 3 shows $SL({\bf q})$ for $\rm{}^{16}O$
(dotted-blue with green error band) compared to theoretical calculations.
These include the Sobczyk 2020Coupled  ”Coupled-Cluster with Singles-and
Doubles (CCSD) NNLOsat” (red-dashed line), and Mihaila 2000microscopic
Coupled-Cluster calculation with (AV18+UIX potential, dashed green line). The
data are in reasonable agreement with Sobczyk 2020 and Mihaila 2000
calculations for $\rm{}^{16}O$ except near $\bf{q}\approx$ 0.30 GeV where the
contribution from nuclear excitations is significant.

The $TE({\bf q},\nu)$ contribution to the QE transverse structure function
$F_{1}({\bf q},\nu)$ for $\rm{}^{12}C$ is parameterized as a distorted
Gaussian centered around $W\approx 0.88$ GeV and a Gaussian at $W\approx 1.2$
GeVMEC2p2h  with $Q^{2}$ dependent width and amplitude. $F_{1}^{MEC}$=0 for
$\nu<\nu_{min}$ ($\nu_{min}$=16.5 MeV). For $\nu>\nu_{min}$ it is given by:

$\displaystyle F_{1}^{MEC}$ $\displaystyle=$ $\displaystyle
max((f_{1}^{A}+f_{1}^{B}),0.0)$ (9) $\displaystyle f_{1}^{A}$ $\displaystyle=$
$\displaystyle a_{1}Y\cdot[(W^{2}-W^{2}_{min})^{1.5}\cdot
e^{-(W^{2}-b_{1})^{2}/2c_{1}^{2}}]$ $\displaystyle f_{1}^{B}$ $\displaystyle=$
$\displaystyle
a_{2}Y\cdot(Q^{2}+q_{o}^{2})^{1.5}\cdot[e^{-(W^{2}-b_{2})^{2}/2c_{2}^{2}}]$
$\displaystyle Y$ $\displaystyle=$ $\displaystyle
Ae^{-Q^{4}/12.715}\frac{(Q^{2}+q_{0}^{2})^{2})}{(0.13380+Q^{2})^{6.90679}}$
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
0.091648,~{}a_{2}=0.10223$ $\displaystyle W^{2}_{min}$ $\displaystyle=$
$\displaystyle M_{p}^{2}+2M_{p}\nu_{min}-Q^{2}$

where Q2 is in units of GeV2, Mp is the proton mass, A is the atomic weight,
$q_{0}^{2}=1.0\times 10^{-4}$, $b_{1}$ = 0.77023,
$c_{1}=0.077051+0.26795Q^{2}$, $b_{2}$= 1.275, and $c_{2}$= 0.375.

The parameters of the empirical model of $TE({\bf q},\nu)$ in electron
scattering can be used to predict the $TE({\bf q},\nu)$ contribution in
neutrino scatteringMosel .

A comparison between our extraction of $R_{L}(\bf q,\nu)$ and $R_{T}(\bf
q,\nu)$ and the extraction (for only three values of $\bf q$) by
JourdanJourdan  are shown in Fig. 4. At the lowest $\bf q$ our $R_{L}(\bf
q,\nu)$ is a somewhat lower and our $R_{T}(\bf q,\nu)$ is a somewhat higher
(the Jourdan analysis includes data from only two experiments). Also shown are
two 1-body+2-body current (1b+2b) calculations: GFMCLovato2016  and ”Energy
Dependent-Relativistic Mean Field” (ED-RMF)ED-RMF . In our fit, we show each
nuclear excitation with excitation energy $E_{x}$ at $\nu=E_{x}+{\bf
q}^{2}/2M_{C12}$ where $M_{C12}$ is the mass of the carbon nucleus¿ In
contrast, the ED-RMF calculations group all excitations in two fixed $\nu$
peaks as shown in Fig.4, and the GFMC calculations do not show any nuclear
excitations. Both calculations are in reasonable agreement with our analysis
in the QE region. In both models there is enhancement of $R_{T}^{QE}(\bf
q,\nu)$ if only 1b currents are included and additional enhancement if both 1b
and 2b currents are included. The curves labeled 2b in Fig. 4 show the
enhancement in $R_{T}^{QE}(\bf q,\nu)$ from 2b currents only [i.e. (1b+2b)
minus (1b only)], while our empirical extraction of $TE({\bf q},\nu)$ shown in
Fig. 4 models the enhancement in $R_{T}(\bf q,\nu)$ from all sources.

In summary, using all available electron scattering data we extract
parameterizations of the quenching of the $R_{L}^{QE}(\bf q,\nu)$ and the
enhancement of $R_{T}^{QE}(\bf q,\nu)$ over a large range of $\bf q$ and
$\nu$, and obtain the best measurement of the Coulomb Sum Rule $SL({\bf q})$
to date. The measured $SL({\bf q})$ for $\rm{}^{12}C$ are inconsistent with
Cloet 2016 RPA, but in reasonable agreement with Lovato 2016 and Mihaila 2000
calculations. The sum rule $SL({\bf q})$ for $\rm{}^{16}O$ is in reasonable
agreement with Sobczyk 2020 and Mihaila 2000 calculations,

The contribution of nuclear excitations to $SL({\bf q})$ is significant (up to
29%). Theoretical studies show that at low $\bf q$ nuclear excitations are
also significant in $\nu_{e,\mu}$ scatteringPandey1 ; Pandey2 . Therefore,
nuclear excitations should be included in both electron and $\nu_{e,\mu}$ MC
generators. Decays of excitations with $E_{x}$ above proton removal threshold
can have a proton in the final state (which in $\nu_{e,\mu}$ experiments
cannot be distinguished from QE events).

Additional comparisons of our fit to experimental data are included in the
supplemental materialsSupplemental . New precision ${}^{12}C$ QE data from
both Hall C HallC  (at low $\bf q$ and forward angles) and Hall A
(specifically taken to examine the saturation of the CSR) are expected to be
finalized soon and to further improve the separation of longitudinal and
transverse cross sections via future fitting efforts. This Research is
supported by the U.S. Department of Energy, Office of Science, under
University of Rochester grant number DE-SC0008475, and the Office of Science,
Office of Nuclear Physics under contract DE-AC05-06OR23177.

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##  I Supplementary materials

Additional comparisons of the universal fit to some of the measurements of
$\frac{d^{2}\sigma}{d\Omega d\nu}$ on $\rm{}^{12}C$ and $\rm{}^{16}O$.

Figure 5: Radiatively corrected inelastic electron scattering cross sections
on $\rm{}^{12}C$ for excitation energies less than 50 MeV. The cross sections
for excitation energies less than 12 MeV have been multiplied by (1/6). The
pink solid line is the predicted $\frac{d^{2}\sigma}{d\nu d\Omega}$ from our
fit which include nuclear states excitation form factor, quasilelastic
scattering (dashed blue line), ”Transverse Enhancement/MEC” (solid red line)
and inelastic pion production (at higher excitation energies). Most of the
data is from Yamaguchi 1971 (Phys. Rev. D3 (1971) 1750). The data for 54 MeV
and 1800 are from Goldemberg 1964 (Phys. Rev. 134 (1964) B963), and the data
for 65 MeV and 1800 are from deForest 1965 (Phys. Letters 16 (1965), 311).

Figure 6:  Comparisons our fit to a subset of electron scattering differential
cross section data with $Q^{2}<0.12$ GeV2. The total $\frac{d^{2}\sigma}{d\nu
d\Omega}$ is shown as the solid purple line. The dashed blue line is the QE
differential cross section. The TE contribution to the QE differential cross
section is shown as the solid red line. Inelastic pion production processes
are shown as the dot-dashed black line. The fit is in good agreement with the
cross section data for both small and large angles. The values of $Q^{2}$
increase from top bottom from $Q^{2}$=0.009 to $Q^{2}$=0.121 GeV2. Data are
from Barreau 1983 (Nucl. Phys. 402A (1983) 515) and Baran 1988 (Phys. Rev.
Lett. 61 (1988) 400).

ı

Figure 7: Comparison of our fit (using $\rm{}^{12}C$ parameters) to all
available $\frac{d^{2}\sigma}{d\nu d\Omega}$ measurements on $\rm{}^{16}O$.
Data are from O’Connell 1987 (Phys. Rev. C35 (1987) 1063), Anghinolfi 1995 (J.
Phy. G: Nucl. Part. Phys. 21 L9, 1995) and Anghinolfi 1996 (Nucl. Phys. A602
(1996) 405.