Compatible $E$-differential forms on Lie algebroids over (pre-)multisymplectic
manifolds
Noriaki Ikeda ***E-mail<EMAIL_ADDRESS>

Department of Mathematical Sciences, Ritsumeikan University
Kusatsu, Shiga 525-8577, Japan

We consider higher generalizations of both a (twisted) Poisson structure and
the equivariant condition of a momentum map on a symplectic manifold. On a Lie
algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we
introduce a Lie algebroid differential form called a compatible $E$-$n$-form.
This differential form satisfies a compatibility condition, which is
consistent with both the Lie algebroid structure and the
(pre-)(multi)symplectic structure. There are many interesting examples such as
a Poisson structure, a twisted Poisson structure and a twisted R-Poisson
structure for a pre-$n$-plectic manifold. Moreover, momentum maps and momentum
sections on symplectic manifolds, homotopy momentum maps and homotopy momentum
sections on multisymplectic manifolds have this structure.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
    1. 2.1 Lie algebroids
    2. 2.2 Lie algebroid differentials
  3. 3 Compatible $E$-$n$-forms
    1. 3.1 Definition and consistency
    2. 3.2 Gauge transformations
  4. 4 Examples
  5. 5 Higher Dirac structures on $E\oplus\wedge^{n-1}E^{*}$
  6. 6 Q-manifold description
  7. 7 Realization by QP-manifold and twist
    1. 7.1 Twisting of homological functions and twisted QP-manifolds
    2. 7.2 QP-manifold description of compatible differential forms
    3. 7.3 Lie $n$-algebroid induced from QP-manifold
    4. 7.4 Lagrangian Q-submanifold and higher Dirac structure

##  1 Introduction

A Lie algebroid is a generalization of both a Lie algebra and a space of
vector fields on a tangent bundle over a smooth manifold. It is not only
important in differential geometry, especially in Poisson geometry, but also
has appeared in physics like a gauge theory, string theory and a sigma model.

In this paper, we consider a compatible geometric structure between a section
of the exterior algebra of $E^{*}$ with a Lie algebroid and a (pre-)symplectic
and a (pre-)multisymplectic structure on the base manifold $M$. It is called
the compatible condition. Here $E^{*}$ is a dual space of a Lie algebroid $E$
over and a (pre-)symplectic and a (pre-)multisymplectic smooth manifold $M$. A
section of the exterior algebra $E^{*}$ is called an $E$-differential form.

A Poisson structure is important geometric structure inspired by the
Hamiltonian mechanics. A deformation of the Poisson structure by adding a
closed $3$-form $H$ is called the twisted Poisson structure [29, 38]. We
consider a bivector field $\pi\in\Gamma(\wedge^{2}TM)$ and a closed $3$-form
$H\in\Omega^{3}(M)$ on a smooth manifold $M$. If $(\pi,H)$ satisfy

$\displaystyle\frac{1}{2}[\pi,\pi]_{S}=\langle\otimes^{3}\pi,\,H\rangle,$ (1)

$(\pi,H)$ is called a twisted Poisson structure, where $[-,-]_{S}$ is the
Schouten bracket on the space of multivector fields and $\langle-,\,-\rangle$
is the pairing of $TM$ and $T^{*}M$. The condition of the twisted Poisson
structure is regarded as a compatibility condition between a Lie algebroid and
a pre-$2$-plectic form. To consider a generalization to general
pre-$n$-plectic form is natural. However a generalization is not
straightforward since a simple generalization of $\pi$ and $H$ in Equation (1)
to a multivector field and a pre-$n$-plectic form does not give a good
condition.

A twisted Poisson structure is discovered in the analysis of a sigma model
called a twisted Poisson (or a WZ Poisson) sigma model [29], we rely on
analysis of sigma models. Recently, Chatzistavrakidis proposed a new higher
dimensional topological sigma model with WZ term whose target space was a
Poisson manifold [12, 13]. If and only if a multivector field on the target
space is satisfied a geometric condition, the topological sigma model is
consistent. A generalization to general Lie algebroid setting is formualted in
the paper [25]. Consistency conditions of these sigma models imposes a
geometric condition of an $E$-differential form with a Lie algebroid structure
and a (pre-)(multi)symplectic form.. In this paper, we investigate this
geometric structure mathematically.

Another motivation is a generalization of a momentum map on a symplectic
manifold. A momentum map is defined on a symplectic manifold $M$ with a Lie
group $G$ action. Let $\mathfrak{g}$ be a Lie algebra of the Lie group $G$.
Recently, a generalization of a momentum map theory to a Lie algebroid setting
has been proposed [5] inspired by analysis in [6, 31]. A momentum map is
$\mu:M\rightarrow\mathfrak{g}^{*}$ is generalized to a section of the dual of
a vector bundle $\mu\in\Gamma(E^{*})$ called a momentum section, where $E$ is
a Lie algebroid. If a vector bundle is the direct product
$E=M\times\mathfrak{g}$, it reduces to a momentum map.

Conditions of the momentum map are generalized. In this paper, we concentrate
on one compatibility condition called the bracket-compatibility condition,

$\displaystyle{}^{E}\mathrm{d}\mu=-\iota_{\rho}^{2}{\omega},$ (2)

where $\omega$ is a symplectic form, $\rho$ is the anchor map of a Lie
algebroid, and ${}^{E}\mathrm{d}$ is the Lie algebroid differential. This
condition looks to be curious at first, however it has natural good
properties. Moreover we claim that the condition is unified with the twisted
Poisson structure, and generalized to higher algebroids and multisymplectic
settings. A generalization of the momentum section to a (pre)-multisymplectic
manifold called a homotopy momentum section has be proposed in [21]. Our
condition for an $E$-$n$-form in this paper is consistent with the definition
of the homotopy momentum section.

A twisted Poisson structure is known as an example of the Dirac structure
[38]. A momentum map on a symplectic manifold and a homotopy momentum map [16]
on a multisymplectic manifold have reinterpretation in terms of higher
algebroids [33]. We show that a compatible $E$-$n$-form is an example of a
higher Dirac structure. Moreover, a compatible $E$-$n$-form can be formulated
as a Q-manifold and a QP-manifold. These descriptions are connected to
constructions of topological sigma models. One of constructions is so called
AKSZ sigma models [2, 36, 22].

We claim that the above structure is interesting and worth to analyze, since
it has many examples already listed above. Generalizations of momentum maps
such as momentum sections on a Lie algebroid, homotopy momentum maps [16] and
homotopy momentum sections [21] on multisymplectic manifolds are examples of
our geometric structure.

This paper is organized as follows. In Section 2, a Lie algebroid and related
notion are prepared and notation is fixed. In Section 3, a compatible
condition is defined and basic properties are analyzed. In Section 4, Examples
are listed. In Section 5, the interpretation in terms of the higher Dirac
structure is discussed. In Section 6, the compatible condition is formulated
as a Q-manifold. In Section 7, we reformulate the compatible condition as a
(twisted) QP-manifold. In the Appendix, some formulas are summarized.

##  2 Preliminary

In this section, we explain background geometry of this paper. A Lie algebroid
and Lie algebroid differentials are introduced.

###  2.1 Lie algebroids

###### Definition 2.1

Let $E$ be a vector bundle over a smooth manifold $M$. A Lie algebroid
$(E,\rho,[-,-])$ is a vector bundle $E$ with a bundle map $\rho:E\rightarrow
TM$ called the anchor map, and a Lie bracket
$[-,-]:\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)$ satisfying the Leibniz
rule,

$\displaystyle[e_{1},fe_{2}]$ $\displaystyle=$ $\displaystyle
f[e_{1},e_{2}]+\rho(e_{1})f\cdot e_{2},$ (3)

where $e_{i}\in\Gamma(E)$ and $f\in C^{\infty}(M)$.

A Lie algebroid is a generalization of a Lie algebra and the space of vector
fields on a smooth manifold.

######  Example 2.1 (Lie algebras)

Let a manifold $M$ be one point $M=\\{pt\\}$. Then a Lie algebroid is a Lie
algebra $\mathfrak{g}$.

######  Example 2.2 (Tangent Lie algebroids)

If a vector bundle $E$ is a tangent bundle $TM$ and $\rho=\mathrm{id}$, then a
bracket $[-,-]$ is a normal Lie bracket on the space of vector fields
$\mathfrak{X}(M)$ and $(TM,\mathrm{id},[-,-])$ is a Lie algebroid. It is
called a tangent Lie algebroid.

######  Example 2.3 (Action Lie algebroids)

Suppose a smooth action of a Lie group $G$ to a smooth manifold $M$, $M\times
G\rightarrow M$. The differential of the map induces an infinitesimal action
of the Lie algebra $\mathfrak{g}$ of $G$ on a manifold $M$. Since
$\mathfrak{g}$ acts as a differential operator on $M$, the differential of the
map determines a bundle map $\rho:M\times\mathfrak{g}\rightarrow TM$.
Consistency of a Lie bracket requires that $\rho$ is a Lie algebra morphism
such that

$\displaystyle~{}[\rho(e_{1}),\rho(e_{2})]$ $\displaystyle=$
$\displaystyle\rho([e_{1},e_{2}]),$ (4)

where the bracket in left hand side of (4) is a Lie bracket of vector fields.
These data gives a Lie algebroid $(E=M\times\mathfrak{g},\rho,[-,-])$. This
Lie algebroid is called an action Lie algebroid.

######  Example 2.4 (Poisson Lie algebroids)

A bivector field $\pi\in\Gamma(\wedge^{2}TM)$ is called a Poisson bivector
field if $[\pi,\pi]_{S}=0$, where $[-,-]_{S}$ is a Schouten bracket on the
space of multivector fields, $\Gamma(\wedge^{\bullet}TM)$. If a smooth
manifold $M$ has a Poisson bivector field $\pi$, $(M,\pi)$ is called a Poisson
manifold. For a Poisson bivector field $\pi$, a bundle map is defined as
$\rho=-\pi^{\sharp}:T^{*}M\rightarrow TM$ by
$\langle\pi^{\sharp}(\alpha),\,\beta\rangle=\pi(\alpha,\beta)$ for all
$\beta\in\Omega^{1}(M)$.

Let $(M,\pi)$ be a Poisson manifold. Then, a Lie algebroid structure is
induced on $T^{*}M$. $\pi^{\sharp}:T^{*}M\rightarrow TM$ is the anchor map,
and a Lie bracket on $\Omega^{1}(M)$ is defined by the so called Koszul
bracket,

$\displaystyle[\alpha,\beta]_{\pi}={\mathcal{L}}_{\pi^{\sharp}(\alpha)}\beta-{\mathcal{L}}_{\pi^{\sharp}(\beta)}\alpha-\mathrm{d}(\pi(\alpha,\beta)),$
(5)

where $\alpha,\beta\in\Omega^{1}(M)$. $(T^{*}M,-\pi^{\sharp},[-,-]_{\pi})$ is
a Lie algebroid.

######  Example 2.5 (Lie algebroids induced from twisted Poisson structures)

If a bivector field $\pi\in\Gamma(\wedge^{2}TM)$ and $H\in\Omega^{3}(M)$
satisfy

$\displaystyle\frac{1}{2}[\pi,\pi]_{S}=\langle\otimes^{3}\pi,\,H\rangle,$ (6)
$\displaystyle\mathrm{d}H=0,$ (7)

$(M,\pi,H)$ is called a twisted Poisson manifold [29, 34, 38, 26]. Here
$\langle\otimes^{3}\pi,\,H\rangle$ is defined by
$\langle\otimes^{3}\pi,\,H\rangle(\alpha_{1},\alpha_{2},\alpha_{3}):=H(\pi^{\sharp}(\alpha_{1}),\pi^{\sharp}(\alpha_{2}),\pi^{\sharp}(\alpha_{3}))$
for $\alpha_{i}\in\Omega^{1}(M)$.

If we take the same bundle map, $-\pi^{\sharp}:T^{*}M\rightarrow TM$ as in
Example 2.4, and a Lie bracket deformed by $H$,

$\displaystyle[\alpha,\beta]_{\pi,H}={\mathcal{L}}_{\pi^{\sharp}(\alpha)}\beta-{\mathcal{L}}_{\pi^{\sharp}(\beta)}\alpha-\mathrm{d}(\pi(\alpha,\beta))+\iota_{\pi^{\sharp}(\alpha)}\iota_{\pi^{\sharp}(\beta)}H,$
(8)

for $\alpha,\beta\in\Omega^{1}(M)$. Then,
$(T^{*}M,-\pi^{\sharp},[-,-]_{\pi,H})$ is a Lie algebroid.

One can refer to proper documents for basic properties of Lie algebroids, for
instance, in [32].

###  2.2 Lie algebroid differentials

For a Lie algebroid $E$, sections of the exterior algebra of $E^{*}$ are
called $E$-differential forms. Consider the spaces of $E$-differential forms
$\Gamma(\wedge^{\bullet}E^{*})$. A differential ${}^{E}\mathrm{d}$ on
$\Gamma(\wedge^{\bullet}E^{*})$ is called a Lie algebroid differential (or
simply, a differential)
${}^{E}\mathrm{d}:\Gamma(\wedge^{m}E^{*})\rightarrow\Gamma(\wedge^{m+1}E^{*})$
and is defined as follows.

###### Definition 2.2

A Lie algebroid differential
${}^{E}\mathrm{d}:\Gamma(\wedge^{m}E^{*})\rightarrow\Gamma(\wedge^{m+1}E^{*})$
is defined by

$\displaystyle{}^{E}\mathrm{d}\alpha(e_{1},\ldots,e_{m+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m+1}(-1)^{i-1}\rho(e_{i})\alpha(e_{1},\ldots,\check{e_{i}},\ldots,e_{m+1})$
(9) $\displaystyle+\sum_{1\leq i<j\leq
m+1}(-1)^{i+j}\alpha([e_{i},e_{j}],e_{1},\ldots,\check{e_{i}},\ldots,\check{e_{j}},\ldots,e_{m+1}),$

where $\alpha\in\Gamma(\wedge^{m}E^{*})$ and $e_{i}\in\Gamma(E)$.

$({}^{E}\mathrm{d})^{2}=0$ is satisfied. The Lie algebroid differential
${}^{E}\mathrm{d}$ is a generalization of the de Rham differential on $T^{*}M$
and the Chevalley-Eilenberg differential on a Lie algebra.

##  3 Compatible $E$-$n$-forms

###  3.1 Definition and consistency

In this section, we define a geometric structure on a Lie algebroid over a
pre-multisymplectic manifold. We introduce a pre-multisymplectic manifold.

###### Definition 3.1

A closed $n+1$-form $\omega\in\Omega^{n+1}(M)$ is called a pre-$n$-plectic
form. $(M,\omega)$ is called a pre-$n$-plectic manifold.

If we suppose that $\omega$ is nondegenerate additional to the above
condition, $\omega$ is called an $n$-plectic form. A $1$-plectic form is a
symplectic form.

Let
${{}^{E}\mathrm{d}}:\Gamma(\wedge^{m}E^{*})\rightarrow\Gamma(\wedge^{m+1}E^{*})$
be a Lie algebroid differential. Notation
${\iota_{\rho}^{k}\omega}\in\Omega^{n-k}(M,\wedge^{k}E^{*})$ is defined by

$\displaystyle\iota_{\rho}^{k}{\omega}(v_{k+1},\ldots,v_{n+1})(e_{1},\ldots,e_{k})$
$\displaystyle=$
$\displaystyle\iota_{\rho(e_{1})}\ldots\iota_{\rho(e_{k})}{\omega}(v_{k+1},\ldots,v_{n+1})$
(10) $\displaystyle=$
$\displaystyle\langle\otimes^{k}\rho,\,{\omega}\rangle(e_{k},\ldots,e_{1},v_{k+1},\ldots,v_{n+1})$
$\displaystyle:=$
$\displaystyle{\omega}(\rho(e_{k}),\ldots,\rho(e_{1}),v_{k+1},\ldots,v_{n+1}),$

for $e_{1},\ldots,e_{k}\in\Gamma(E)$ and
$v_{k+1},\ldots,v_{n+1}\in\mathfrak{X}(M)$.

###### Definition 3.2

Let $(M,\omega)$ be a pre-$n$-plectic manifold and $(E,\rho,[-,-])$ be a Lie
algebroid over $M$. Then, an $E$-$n$-form $J\in\Gamma(\wedge^{n}E^{*})$ is
called compatible if $J$ satisfies

$\displaystyle{}^{E}\mathrm{d}J=-\iota_{\rho}^{n+1}\omega=-\langle\otimes^{n+1}\rho,\,\omega\rangle.$
(11)

Since ${{}^{E}\mathrm{d}}^{2}=0$,

$\displaystyle{{}^{E}\mathrm{d}}(\iota^{n+1}_{\rho}\omega)=0,$ (12)

must be satisfied for consistency. The left hand side is

$\displaystyle{{}^{E}\mathrm{d}}(\iota^{n+1}_{\rho}\omega)(e_{n+2},\ldots,e_{1})$
$\displaystyle=\mathrm{d}\omega(\rho(e_{1}),\ldots\rho(e_{n+2}))$
$\displaystyle+\sum_{1\leq i<j\leq
n+2}(-1)^{i+j-1}\omega([\rho(e_{i}),\rho(e_{j})],\rho(e_{1}),\ldots,\check{\rho(e_{i})},\ldots,\check{\rho(e_{j})},\ldots,\rho(e_{n+2}))$
$\displaystyle+\sum_{1\leq i<j\leq
n+2}(-1)^{i+j}\omega(\rho([e_{i},e_{j}]),\rho(e_{1}),\ldots,\check{\rho(e_{i})},\ldots,\check{\rho(e_{j})},\ldots,\rho(e_{n+2})),$
(13)

for $e_{i}\in\Gamma(E)$. Since
$~{}[\rho(e_{1}),\rho(e_{2})]=~{}\rho([e_{1},e_{2}])$ is satisfied in a Lie
algebroid $E$, we obtain the following proposition.

###### Proposition 3.3

If $\omega$ is closed, Eq. (12) is satisfied and Equation (11) is consistent
with $\rho$, $[-,-]$ and $\omega$.

In other words, Definition 3.2 means that if $-\iota_{\rho}^{n+1}\omega$ is
${}^{E}\mathrm{d}$-exact, and its $E$-$n$-form is $J$, $(J,\omega)$ is called
compatible.

###  3.2 Gauge transformations

Recall that $\iota^{n+1}_{\rho}\omega$ is closed with respect to the Lie
algebroid differential ${{}^{E}\mathrm{d}}$,
${{}^{E}\mathrm{d}}(\iota^{n+1}_{\rho}\omega)=0$ from consistency with
Equation (11) and (12). Obvious equivalence relation of compatible
$E$-$n$-forms is difference of ${}^{E}\mathrm{d}$-exact terms.

###### Definition 3.4

For a Lie algebroid $E$, two compatible $E$-$n$-forms $J$ and $J^{\prime}$ are
equivalent if there exists an $E$-$(n-1)$-form $K\in\Gamma(\wedge^{n-1}E^{*})$
satisfying

$\displaystyle J^{\prime}=J+{}^{E}\mathrm{d}K.$ (14)

In fact, if $J$ is a compatible $E$-$n$-form $J^{\prime}$ also satisfies the
same compatible condition.

We consider another nontrivial equivalence. For a pre-$n$-plectic form
$\omega$, we consider a deformation $\omega^{\prime}=\omega+\mathrm{d}\lambda$
for an arbitrary $n$ form $\lambda\in\Omega^{n}(M)$, which is called a gauge
transformation. Obviously, if $\omega$ is a pre-$n$-plectic form,
$\omega^{\prime}$ is also a pre-$n$-plectic form.

Using the identity $[\rho(e_{1}),\rho(e_{2})]=\rho([e_{1},e_{2}])$ of a Lie
algebroid, the following equation holds,

$\displaystyle{{}^{E}\mathrm{d}}(\iota_{\rho}^{n}\lambda)=\iota_{\rho}^{n+1}\mathrm{d}\lambda.$
(15)

for $\lambda\in\Omega^{n}(M)$. Thus, if $J$ is a compatible $E$-$n$-form for
$\omega$, and $\omega^{\prime}=\omega-\mathrm{d}\lambda$, the compatible
$E$-$n$-form $J^{\prime}$ for $\omega^{\prime}$ is given by
$J^{\prime}=J-\iota_{\rho}^{n}\lambda$. In fact,

$\displaystyle{{}^{E}\mathrm{d}}J^{\prime}$ $\displaystyle=$
$\displaystyle-\iota_{\rho}^{n+1}\omega^{\prime}=-\iota_{\rho}^{n+1}\omega-(\iota_{\rho})^{n+1}\mathrm{d}\lambda=-\iota_{\rho}^{n+1}\omega-{{}^{E}\mathrm{d}}(\iota_{\rho}^{n}\lambda)$
(16) $\displaystyle=$
$\displaystyle{{}^{E}\mathrm{d}}(J-\iota_{\rho}^{n}\lambda).$

A gauge equivalence of two compatible $E$-$n$-forms is formulated as follows.

###### Definition 3.5

Two compatible $E$-$n$-forms $J$ and $J^{\prime}$ are gauge equivalent if
there exists an $n$-form $\lambda\in\Omega^{n}(M)$ such that
$J^{\prime}=J-\iota_{\rho}^{n}\lambda$.

In Section 4, we discuss gauge equivalence of the twisted Poisson structure as
an example in Example 4.2.

##  4 Examples

The compatible $E$-$n$-form (11) is considered as a universal compatibility
condition of a Lie algebroid structure with a pre-multisymplectic form. It
appears in various geometries. In this section, we list up examples.

######  Example 4.1 (Liouville $1$-forms)

Take $n=1$ and let $(M,\omega)$ be a symplectic manifold and $E=TM$ be a
tangent bundle. $TM$ is a tangent Lie algebroid with the anchor map
$\rho=\mathrm{id}$ in Example 2.2. An $E$-$1$-form $J$ is a normal
differential forms on $M$, and a Lie algebroid differential
${}^{E}\mathrm{d}=\mathrm{d}_{\mathrm{dR}}$ is the de Rham differential. The
compatible condition (11) is

$\displaystyle\mathrm{d}_{\mathrm{dR}}J=-\omega,$ (17)

which means that $J$ is the Liouville $1$-form for the symplectic form
$\omega$.

######  Example 4.2 (Poisson and twisted Poisson structures)

Let $(\pi,H)$ be a twisted Poisson structure on $M$ in Example 2.5. Note that
it reduces to a Poisson structure if $H=0$. The cotangent bundle $T^{*}M$ has
a Lie algebroid structure as explained in Examples 2.4 and 2.5. Equation (6)
is equivalent to

$\displaystyle{}^{E}\mathrm{d}\pi=-\langle\otimes^{3}\pi,\,H\rangle,$ (18)

where ${}^{E}\mathrm{d}$ is the Lie algebroid differential of this Lie
algebroid $T^{*}M$. Thus, $J=\pi$ is a compatible $TM$-$2$-form on a
pre-$2$-plectic manifold with a pre-$2$-plectic form $H$.

Next let us analyze gauge equivalence. A gauge transformation of the twisted
Poisson structure is $H^{\prime}=H+\mathrm{d}B$ with a $2$-form $B$. Then, the
compatible $E$-$2$-form with respect to $H^{\prime}$ is
$\pi^{\prime}=\pi-\langle\otimes^{2}\pi,\,B\rangle$ from Definition 3.5.

Note that gauge transformations in Definition 3.5 are different from the
definition of the gauge transformation in the twisted Poisson structure [38,
9]. In the standard definition, the bivector field changes to
$\pi^{\prime}=\pi(1+\pi B)^{-1}$ under $H^{\prime}=H+\mathrm{d}B$.

In the standard definition, Equation (6) gives the Lie algebroid structure on
$T^{*}M$ and a compatible $E$-$n$-form simultaneously. If the gauge
transformation $\pi^{\prime}=\pi(1+\pi B)^{-1}$ consistently deforms the Lie
algebroid structure defined by $(\pi^{\prime},H)$ and a compatible
$E$-$n$-form to $J^{\prime}=\pi^{\prime}$ simultaneously.

However, our gauge transformation
$\pi^{\prime}=\pi-\langle\otimes^{2}\pi,\,B\rangle$ only deforms the
compatible $E$-$n$-form to $J^{\prime}=\pi^{\prime}$ under the fixed Lie
algebroid constructed from $(\pi,H)$. The reason is for generalizations to
higher pre-$n$-plectic cases. In a pre-$n$-plectic case, the simultaneous
deformation of the Lie algebroid and the compatible $E$-$n$-form like the
twisted Poisson structure does not work. The twisted Poisson structure is a
very special case and our gauge transformation is applicable to general
$E$-$n$-form on a higher $n$-plectic manifold.

######  Example 4.3 (Twisted $R$-Poisson structures)

A twisted $R$-Poisson structure $(\pi,H,J)$ on a smooth manifold $M$ is
defined as follows [12]. ††† Since $J$ is denoted by $R$ in the paper [12], it
is called the $R$-Poisson structure.

$M$ is a smooth manifold and $\pi\in\Gamma(\wedge^{2}TM)$ is a Poisson
bivector field. $H$ is a closed $(n+1)$-form, and $J\in\Gamma(\wedge^{n}TM)$
is an $n$-multivector field. As explained in Example 2.4, the Poisson bivector
field $\pi$ induces a Lie algebroid structure on $T^{*}M$, where
$\rho=-\pi^{\sharp}$ and the Lie bracket is given by Eq. (5). Under this Lie
algebroid structure, $(H,J)$ is the twisted $R$-Poisson structure if it
satisfies

$\displaystyle{}^{E}\mathrm{d}J=(-1)^{n}\langle\otimes^{n+1}\pi,\,H\rangle.$
(19)

$J$ is a compatible $TM$-$n$-form with the pre-$(n+1)$-plectic form
$\omega=(-1)^{n+1}H$.

######  Example 4.4 (Momentum maps)

We assume a smooth action of a Lie group $G$ on a symplectic manifold
$(M,\omega)$. Let $\mathfrak{g}$ be a Lie algebra for $G$. The Lie group
action induces the Lie algebra action $\rho:\mathfrak{g}\rightarrow TM$. A map
$\mu:M\rightarrow\mathfrak{g}^{*}$ is called a momentum map if it satisfies

$\displaystyle\mathrm{d}\mu(e)$ $\displaystyle=$
$\displaystyle-\iota_{\rho(e)}\omega,$ (20)
$\displaystyle\rho(e_{1})\langle\mu,\,e_{2}\rangle$ $\displaystyle=$
$\displaystyle\langle\mu,\,[e_{1},e_{2}]\rangle,$ (21)

for $e,e_{1},e_{2}\in\mathfrak{g}$. Here $\langle-,\,-\rangle$ is the pairing
of $\mathfrak{g}^{*}$ and $\mathfrak{g}$. As explained in Example 2.3, the Lie
group action induces an action Lie algebroid structure on a trivial bundle
$E=M\times\mathfrak{g}$. The anchor map is induced from the map of the Lie
algebra action as $\rho:M\times\mathfrak{g}\rightarrow TM$. The momentum map
is regarded as a section of the dual trivial bundle
$\mu\in\Gamma(M\times\mathfrak{g}^{*})$. Under the condition (20), Equation
(21) is changed to [5]

$\displaystyle{}^{E}\mathrm{d}\mu(e_{1},e_{2})$ $\displaystyle=$
$\displaystyle-\iota_{\rho}^{2}\omega(e_{1},e_{2}),$ (22)

i.e., ${}^{E}\mathrm{d}\mu=-\iota_{\rho}^{2}\omega$. Here ${}^{E}\mathrm{d}$
is a Lie algebroid differential with respect to the action Lie algebroid. Thus
we obtain the following result.

###### Proposition 4.1

A momentum map $\mu:M\rightarrow\mathfrak{g}^{*}$ is a compatible $E$-$1$-form
for the action Lie algebroid induced from a Lie group action on $M$.

######  Example 4.5 (Homotopy moment(um) maps)

[16] Let $(M,{\omega})$ be an $n$-plectic manifold. Assume an action of a Lie
group $G$ on $M$ as in the case of the momentum map. The action of $G$ induces
the corresponding infinitesimal Lie algebra action on $M$ as vector fields,
$\rho:\mathfrak{g}\rightarrow TM$. We consider the space,
$\Omega^{k}(M,\wedge^{n-k}\mathfrak{g}^{*})=\Omega^{k}(M)\otimes\wedge^{n-k}\mathfrak{g}^{*}$,
and two differentials $\mathrm{d}:=\mathrm{d}\otimes 1$ and
$\mathrm{d}_{\mathrm{CE}}:=1\otimes\mathrm{d}_{\mathrm{CE}}$ where
$\mathrm{d}$ is the de Rham differential on $\Omega^{k}(M)$ and
$\mathrm{d}_{CE}$ is the Chevalley-Eilenberg differential on
$\wedge^{n-k}\mathfrak{g}^{*}$. Introduce a $k$-form taking a value in
$\wedge^{n-k}\mathfrak{g}^{*}$,
${\widehat{\mu}}_{k}\in\Omega^{k}(M,\wedge^{n-k}\mathfrak{g}^{*})$, where
$k=0,\ldots,n-1$, and their formal sum,
${\widehat{\mu}}=\sum_{k=0}^{n-1}{\widehat{\mu}}_{k}$.

###### Definition 4.2

A homotopy momentum map ${\widehat{\mu}}$ is defined by ‡‡‡Sign factors of the
equation are different from [16]. Sign factors are arranged to original ones
by proper redefinitions.

$\displaystyle(\mathrm{d}+\mathrm{d}_{CE}){\widehat{\mu}}=\sum_{k=0}^{n}(-1)^{n-k+1}\iota_{\rho}^{n+1-k}{\omega}.$
(23)

A momentum map on an $n$-plectic manifold [10, 18] is a homotopy momentum map.
Expanded by the form degree, Equation (23) becomes the following $n$
equations,

$\displaystyle\mathrm{d}{\widehat{\mu}}_{n-1}=-\iota_{\rho}^{1}{\omega},$ (24)
$\displaystyle\mathrm{d}{\widehat{\mu}}_{n-2}+\mathrm{d}_{CE}{\widehat{\mu}}_{n-1}=\iota_{\rho}^{2}{\omega},$
(25) $\displaystyle\vdots$
$\displaystyle\mathrm{d}{\widehat{\mu}}_{k-1}+\mathrm{d}_{CE}{\widehat{\mu}}_{k}=(-1)^{n-k+1}\iota_{\rho}^{n+1-k}{\omega},$
(26) $\displaystyle\vdots$
$\displaystyle\mathrm{d}_{CE}{\widehat{\mu}}_{0}=(-1)^{n+1}\iota_{\rho}^{n+1}{\omega}.$
(27)

The Lie group action induces an action Lie algebroid structure on the trivial
bundle $E=M\times\mathfrak{g}$. The Lie algebroid differential on the action
Lie algebroid is ${}^{E}\mathrm{d}=\mathrm{ad}^{*}_{\rho}+\mathrm{d}_{CE}$,
where $\mathrm{ad}^{*}$ is the following action on $\alpha$ induced from the
Lie algebra action on $M$,

$\displaystyle\mathrm{ad}^{*}_{\rho}\alpha(e_{1},\ldots,e_{m+1})$
$\displaystyle:=$
$\displaystyle\sum_{i=1}^{m+1}(-1)^{i-1}\rho(e_{i})\alpha(e_{1},\ldots,\check{e_{i}},\ldots,e_{m+1}),$
(28)

for $e_{i}\in\mathfrak{g}$, We can rewrite $n$ equations (24)–(27) to
equivalent $n$ equations [21],

$\displaystyle\mathrm{d}\mu_{n-1}=-\iota_{\rho}^{1}{\omega},$ (29)
$\displaystyle\mathrm{d}\mu_{n-2}+{}^{E}\mathrm{d}\mu_{n-1}=-\iota_{\rho}^{2}{\omega},$
(30) $\displaystyle\vdots$
$\displaystyle\mathrm{d}\mu_{k-1}+{}^{E}\mathrm{d}\mu_{k}=-\iota_{\rho}^{n+1-k}{\omega},$
(31) $\displaystyle\vdots$
$\displaystyle{}^{E}\mathrm{d}\mu_{0}=-\iota_{\rho}^{n+1}{\omega},$ (32)

where ${\widehat{\mu}}_{k}=(-1)^{n-k+1}\mu_{k}$. The last equation (32) shows
that $J=\mu_{0}$ is a compatible $E$-$n$-form.

###### Proposition 4.3

${\widehat{\mu}}_{0}=(-1)^{n+1}\mu_{0}$ in a homotopy momentum map is a
compatible $E$-$n$-form.

######  Example 4.6 (Momentum sections)

A momentum map is generalized to a momentum section [5]. See also [31, 23,
24]. A momentum section is also an example of a compatible $E$-$n$-form.

Suppose that a base manifold $M$ is a pre-symplectic manifold with a pre-
symplectic form $\omega$. Moreover, suppose a Lie algebroid $(E,\rho,[-,-])$
over $M$. In order to define a momentum section, we introduce a connection on
the vector bundle $E$, $\nabla:\Gamma(E)\rightarrow\Gamma(T^{*}M\otimes E)$,
which is a ${\mathbb{R}}$-linear map satisfying the Leibniz rule,

$\displaystyle\nabla(fe)=f\nabla e+(\mathrm{d}f)\otimes e,$ (33)

for $e\in\Gamma(E)$ and $f\in C^{\infty}(M)$. A dual connection on $E^{*}$ is
defined by the equation,

$\displaystyle\mathrm{d}\langle\mu,\,e\rangle=\langle\nabla\mu,\,e\rangle+\langle\mu,\,\nabla
e\rangle,$ (34)

for all sections $\mu\in\Gamma(E^{*})$ and $e\in\Gamma(E)$.

A momentum section is defined as follows.

###### Definition 4.4

A section $\mu\in\Gamma(E^{*})$ of $E^{*}$ is called a bracket-compatible
momentum section if $\mu$ satisfies the following two conditions. §§§One more
condition $\nabla^{2}\mu=0$ is imposed in [5]. The condition is dropped in
this paper, since it is not necessary in our context.

$\displaystyle\nabla\mu=-\iota_{\rho}\omega,$ (35)
$\displaystyle{}^{E}\mathrm{d}\mu=-\iota_{\rho}^{2}{\omega}.$ (36)

The second equation (36) is nothing but the condition (11) of a compatible
$E$-$1$-form for $J=\mu$.

###### Proposition 4.5

A momentum section $\mu$ is a compatible $E$-$1$-form.

######  Example 4.7 (Homotopy momentum sections)

[21] A homotopy momentum map on a pre-multisymplectic manifold is generalized
to a homotopy momentum section.

As the preparation to introduce a homotopy momentum section, we define another
derivation called an $E$-connection on a Lie algebroid.

###### Definition 4.6

Let $E$ be a Lie algebroid over a smooth manifold $M$ and $E^{\prime}$ be a
vector bundle over the same base manifold $M$. An $E$-connection on a vector
bundle $E^{\prime}$ with respect to the Lie algebroid $E$ is a
${\mathbb{R}}$-linear map
${}^{E}\nabla:\Gamma(E^{\prime})\rightarrow\Gamma(E^{*}\otimes E^{\prime})$
satisfying

$\displaystyle{}^{E}\nabla_{e}(fe^{\prime})=f{}^{E}\nabla_{e}e^{\prime}+(\rho(e)f)e^{\prime},$
(37)

for $e\in\Gamma(E)$, $e^{\prime}\in\Gamma(E^{\prime})$ and $f\in
C^{\infty}(M)$.

The ordinary connection is regarded as an $E$-connection for $E=TM$,
$\nabla={}^{TM}\nabla$.

The standard $E$-connection on $E^{\prime}=E$,
${}^{E}\nabla:\Gamma(E)\rightarrow\Gamma(E^{*}\otimes E)$ is defined by

$\displaystyle{}^{E}\nabla_{e}e^{\prime}:=\nabla_{\rho(e)}{e^{\prime}}+[e,e^{\prime}],$
(38)

for $e,e^{\prime}\in\Gamma(E)$. A general $E$ connection ${}^{E}\nabla$ on $E$
is given by

$\displaystyle{}^{E}\nabla_{e}e^{\prime}=\nabla_{\rho(e)}{e^{\prime}}+[e,e^{\prime}]-\chi(e,e^{\prime}),$
(39)

with some tensor $\chi\in\Gamma(E\otimes E^{*}\otimes E^{*})$.

If a normal connection $\nabla$ on $E$ as a vector bundle is given, we can
define the following canonical $E$-connection for the tangent bundle
$E^{\prime}=TM$. An (canonical) $E$-connection on a tangent bundle,
${}^{E}\nabla:\Gamma(TM)\rightarrow\Gamma(E^{*}\otimes TM)$ is defined by

$\displaystyle{}^{E}\nabla_{e}v$ $\displaystyle:=$
$\displaystyle{\mathcal{L}}_{\rho(e)}v+\rho(\nabla_{v}e)=[\rho(e),v]+\rho(\nabla_{v}e),$
(40)

where $e\in\Gamma(E)$ and $v\in\mathfrak{X}(M)$. It is also called the
opposite connection. One can refer to [1, 14, 17] about the general theory of
connections on a Lie algebroid.

Given a connection $\nabla$, a covariant derivative is generalized to the
derivation $\mathrm{d}^{\nabla}$ on the space of differential forms
$\Omega^{k}(M,\wedge^{m}E^{*})$ called an exterior covariant derivative¶¶¶The
exterior covariant derivative is also denoted by $\nabla$ in [21].. Similarly,
given an $E$-connection ${}^{E}\nabla$, the Lie algebroid differential
${}^{E}\mathrm{d}$ can be generalized to the operation on the space
$\Gamma(\wedge^{k}E^{\prime}\otimes\wedge^{m}E^{*})$,
${}^{E}\mathrm{d}^{\nabla}:\Gamma(\wedge^{k}E^{\prime}\otimes\wedge^{m}E^{*})\rightarrow\Gamma(\wedge^{k}E^{\prime}\otimes\wedge^{m+1}E^{*})$.
For instance, see [1]. The concrete equation for $E^{\prime}=T^{*}M$ is as
follows.

###### Definition 4.7

For
$\Omega^{k}(M,\wedge^{m}E^{*})=\Gamma(\wedge^{k}T^{*}M\otimes\wedge^{m}E^{*})$,
the $E$-exterior covariant derivative
${}^{E}\mathrm{d}^{\nabla}:\Omega^{k}(M,\wedge^{m}E^{*})\rightarrow\Omega^{k}(M,\wedge^{m+1}E^{*})$
is defined by

$\displaystyle{}^{E}\mathrm{d}^{\nabla}\alpha(e_{1},\ldots,e_{m+1})$
$\displaystyle:=$
$\displaystyle\sum_{i=1}^{m+1}(-1)^{i-1}{}^{E}\nabla_{e_{i}}(\alpha(e_{1},\ldots,\check{e_{i}},\ldots,e_{m+1}))$
(41) $\displaystyle+\sum_{1\leq i<j\leq
m+1}(-1)^{i+j}\alpha([e_{i},e_{j}],e_{1},\ldots,\check{e_{i}},\ldots,\check{e_{j}},\ldots,e_{m+1}),$

for $\alpha\in\Omega^{k}(M,\wedge^{m}E^{*})$ and $e_{i}\in\Gamma(E)$.

Let $\mu_{k}\in\Omega^{k}(M,\wedge^{n-k}E^{*})$ be a $k$-form taking values in
$\wedge^{n-k}E^{*}$, where $k=0,\ldots,n-1$. Both the exterior covariant
derivative $\nabla$ and the $E$-exterior covariant derivative
${}^{E}\mathrm{d}^{\nabla}$ act on the space
$\Omega^{k}(M,\wedge^{n-k}E^{*})$.

###### Definition 4.8

A formal sum $\mu=\sum_{k=0}^{n-1}\mu_{k}$ is called a homotopy momentum
section if $\mu$ satisfies

$\displaystyle(\mathrm{d}^{\nabla}+{}^{E}\mathrm{d}^{\nabla})\mu=-\sum_{k=0}^{n}\iota_{\rho}^{n+1-k}{\omega}.$
(42)

Expanding the equation by form degree of both sides, Equation (42) is the
following $n$ equations,

$\displaystyle\mathrm{d}^{\nabla}\mu_{n-1}=-\iota_{\rho}^{1}{\omega},$ (43)
$\displaystyle\mathrm{d}^{\nabla}\mu_{n-2}+{}^{E}\mathrm{d}^{\nabla}\mu_{n-1}=-\iota_{\rho}^{2}{\omega},$
(44) $\displaystyle\vdots$
$\displaystyle\mathrm{d}^{\nabla}\mu_{k-1}+{}^{E}\mathrm{d}^{\nabla}\mu_{k}=-\iota_{\rho}^{n+1-k}{\omega},$
(45) $\displaystyle\vdots$
$\displaystyle{}^{E}\mathrm{d}^{\nabla}\mu_{0}={}^{E}\mathrm{d}\mu_{0}=-\iota_{\rho}^{n+1}{\omega}.$
(46)

The final equation (46) is the condition that $J=\mu_{0}$ is a compatible
$E$-$n$-form (11).

###### Proposition 4.9

$\mu_{0}$ in a homotopy momentum section is a compatible $E$-$n$-form.

##  5 Higher Dirac structures on $E\oplus\wedge^{n-1}E^{*}$

In this section, we reformulate a compatible $E$-$n$-form in terms of a higher
Dirac structure of a Lie $n$-algebroid. We consider a Lie $n$-algebroid on
$E\oplus\wedge^{n-1}E^{*}$ induced from a Lie algebroid $E$.

Let $(E,\rho,[-,-])$ be a Lie algebroid over $M$. Then, an almost Lie
algebroid structure is induced on $E\oplus\wedge^{n-1}E^{*}$. Three operations
are constructed from the Lie algebroid operations. They are a bilinear
symmetric form
${({{-},{-}})}:\Gamma(E\oplus\wedge^{n-1}E^{*})\times\Gamma(E\oplus\wedge^{n-1}E^{*})\rightarrow\Gamma(\wedge^{n-2}E^{*})$,
a bundle map ${\widehat{\rho}}:E\oplus\wedge^{n-1}E^{*}\rightarrow TM$ called
the anchor map, and a bilinear bracket
${[{{-},{-}}]_{D}}:\Gamma(E\oplus\wedge^{n-1}E^{*})\times\Gamma(E\oplus\wedge^{n-1}E^{*})\rightarrow\Gamma(E\oplus\wedge^{n-1}E^{*})$
called a (higher) Dorfman bracket. Let
$u+\alpha,v+\beta\in\Gamma(E\oplus\wedge^{n-1}E^{*})$, where $u,v\in\Gamma(E)$
and $\alpha,\beta\in\Gamma(\wedge^{n-1}E^{*})$. Then three operations are
defined by

$\displaystyle{({{u+\alpha},{v+\beta}})}$
$\displaystyle=\iota_{u}\beta+\iota_{v}\alpha,$ (47)
$\displaystyle{\widehat{\rho}}(e)f$ $\displaystyle=\rho(u)f,$ (48)
$\displaystyle{[{{u+\alpha},{v+\beta}}]_{D}}$
$\displaystyle=[u,v]+{}^{E}{\mathcal{L}}_{u}\beta-\iota_{v}{}^{E}\mathrm{d}\alpha+\iota_{u}\iota_{v}(\iota_{\rho}^{n+1}H),$
(49)

where the interior product $\iota_{v}$ is the contraction of an element
$v\in\Gamma(E)$ with $\Gamma(E^{*})$, $\rho(u)$ is the anchor map of $E$, and
${}^{E}{\mathcal{L}}_{u}:=\iota_{u}{}^{E}\mathrm{d}+{}^{E}\mathrm{d}\iota_{u}$
is the $E$-Lie derivative. The Dorfman bracket (49) is not skew-symmetric but
satisfies the Jacobi type identity,

$\displaystyle{[{{{[{{u+\alpha},{v+\beta}}]_{D}}},{w+\gamma}}]_{D}}+{[{{{[{{v+\beta},{w+\gamma}}]_{D}}},{u+\alpha}}]_{D}}+{[{{{[{{w+\gamma},{u+\alpha}}]_{D}}},{v+\beta}}]_{D}}=0.$
(50)

An algebroid satisfying Equation (50) is called a Leibniz algebroid.
Espeically, $(E\oplus\wedge^{n-1}E^{*},{({{-},{-}})},\rho,{[{{-},{-}}]_{D}})$
is called a Lie $n$-algebroid or a Vinogradov algebroid induced from a Lie
algebroid $E$ [19].

Equations (47)–(49) does not contain $J$. The compatible condition (11),
${}^{E}\mathrm{d}J=-\iota_{\rho}^{n+1}\omega$, is described as an extra
structure on the Lie $n$-algebroid, a higher Dirac structure[8, 4, 43].

###### Definition 5.1

A higher Dirac structure is the subbundle $L$ of the Lie $n$-algebroid
satisfying the conditions, ${({{e_{1}},{e_{2}}})}=0$ for all
$e_{1},e_{2}\in\Gamma(L)$, and ${[{{e_{1}},{e_{2}}}]_{D}}$ is an element of
$\Gamma(L)$, i.e., $\Gamma(L)$ is involutive with respect to the bracket,
${[{{\Gamma(L)},{\Gamma(L)}}]_{D}}\subset\Gamma(L)$.

Note that we do not impose that $L$ is the maximal rank with respect to the
inner product ${({{-},{-}})}$ but only isotropic, which is different from the
definition of a Dirac structure of a Courant algebroid.

For an $E$-$n$-form $J\in\Gamma(\wedge^{n}E^{*})$, we consider the following
space of sections of $E\oplus\wedge^{n-1}E^{*}$,

$\displaystyle\Gamma(L)$
$\displaystyle=\\{u+{({{J},{u}})}\in\Gamma(E\oplus\wedge^{n-1}E^{*})|u\in\Gamma(E)\\}.$
(51)

###### Theorem 5.2

If $J\in\Gamma(\wedge^{n}E^{*})$ and $\omega\in\Omega^{n+1}(M)$ satisfy
${}^{E}\mathrm{d}J=-\iota_{\rho}^{n+1}\omega$, then $L$ is a higher Dirac
structure of a Lie $n$-algebroid $E\oplus\wedge^{n-1}E^{*}$.

###### Proof.

The inner product of two elements of $\Gamma(L)$, $u+{({{J},{u}})}$ and
$v+{({{J},{v}})}$ for $u,v\in\Gamma(E)$ is

$\displaystyle{({{u+{({{J},{u}})}},{v+{({{J},{v}})}}})}$
$\displaystyle={({{u},{{J}{v}}})}+{({{{J}{u}},{v}})}=0,$ (52)

from completely skewsymmetricity of $J$. Moreover the Dorfman bracket is
computed as

$\displaystyle{[{{u+{({{J},{u}})}},{v+{({{J},{v}})}}}]_{D}}$
$\displaystyle=[u,v]+{({{J},{[u,v]}})},$ (53)

which is the element of $\Gamma(L)$ again if the identity
${}^{E}\mathrm{d}J=-\iota_{\rho}^{n+1}\omega$ is satisfied. ∎

###### Example 5.1

Let us consider the twisted Poisson structure in Example 2.5. We take $E=TM$
and $n=2$. Then, we obtain a Lie $2$-algebroid $TM\oplus T^{*}M$, which
operations are

$\displaystyle{({{u+\alpha},{v+\beta}})}$
$\displaystyle=\iota_{u}\beta+\iota_{v}\alpha,$ (54)
$\displaystyle{\widehat{\rho}}(e)f$ $\displaystyle=u(f),$ (55)
$\displaystyle{[{{u+\alpha},{v+\beta}}]_{D}}$
$\displaystyle=[u,v]+{\mathcal{L}}_{u}\beta-\iota_{v}\mathrm{d}\alpha+\iota_{u}\iota_{v}H,$
(56)

which give in fact the standard Courant algebroid with
$({({{-},{-}})},\rho=\mathrm{id},{[{{-},{-}}]_{D}})$, where $H=\omega$ is a
closed $3$-form. Let us introduce a bivector field $\pi$ and define a
subbundle defined by

$\displaystyle\Gamma(L)$
$\displaystyle=\\{{({{\pi},{\alpha}})}+\alpha\in\Gamma(TM\oplus
T^{*}M)|\alpha\in\Omega^{1}(M)\\}.$ (57)

Then, $(M,\pi,H)$ is a twisted Poisson manifold satisfying Equation (18) if
and only if $L$ is a Dirac structure.

##  6 Q-manifold description

In this section, a compatible $E$-$n$-form is formulated by graded geometry. A
Q-manifold description based on graded geometry provides a clear method to
formulate Lie $n$-algebroids. Concretely, the compatible $E$-$n$-form is
described by a Lagrangian Q-submanifold of a Q-manifold.

A graded manifold ${\mathcal{M}}$ is a locally ringed space
$(M,{\mathcal{O}}_{{\mathcal{M}}})$ whose structure sheaf ${\mathcal{O}}_{M}$
is a ${\mathbb{Z}}$-graded commutative algebra over an ordinary smooth
manifold $M$. The grading is compatible with the supermanifold grading, that
is, a variable of even degree is commutative and a variable of odd degree is
anticommutative. By definition, the structure sheaf of $M$ is locally
isomorphic to $C^{\infty}(U)\otimes S^{\bullet}(V)$, where $U$ is a local
chart on $M$, $V$ is a graded vector space, and $S^{\bullet}(V)$ is a free
graded commutative ring on $V$. ${\mathcal{O}}_{{\mathcal{M}}}$ is also
denoted by $C^{\infty}({\mathcal{M}})$. Grading of an element $x\in
C^{\infty}({\mathcal{M}})$ is called degree and denoted by $|x|$. For two
homogeneous elements $f,g\in C^{\infty}({\mathcal{M}})$, the product is graded
commutative, $fg=(-1)^{|f||g|}gf$. We have some recent reviews of graded
manifolds, [11, 22, 36, 42] and references therein.

We concentrate on nonnegatively graded manifolds with ${\mathbb{Z}}_{\geq 0}$
grading called an N-manifold.

###### Definition 6.1

If an N-manifold ${\mathcal{M}}$ has a vector field $Q$ of degree $+1$
satisfies $Q^{2}=0$, it is called a Q-manifold, or a differential graded (dg)
manifold.

This vector field $Q$ is called a homological vector field.

Let $E$ be a vector bundle over $M$. If and only if $E$ is a Lie algebroid,
$E[1]$ is a Q-manifold with a homological vector field [41]. Concrete
correspondence is as follows. Let $(x^{i},q^{a})$ be local coordinates on
$E[1]$ of degree $(0,1)$. Here $i$ is the index on the base manifold $M$ and
$a$ is the index of the fiber. A vector field of degree $+1$ is locally given
by

$\displaystyle Q$ $\displaystyle=\rho^{i}_{a}(x)q^{a}\frac{\partial}{\partial
x^{i}}-\frac{1}{2}C_{bc}^{a}(x)q^{b}q^{c}\frac{\partial}{\partial q^{a}},$
(58)

where $\rho^{i}_{a}(x)$ and $C^{a}_{bc}(x)$ are local functions. We define a
bundle map $\rho:E\rightarrow TM$ and the Lie bracket
$[-,-]:\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)$ as

$\displaystyle\rho(e_{a})$ $\displaystyle:=\rho^{i}_{a}(x)\partial_{i},$ (59)
$\displaystyle~{}[e_{a},e_{b}]$ $\displaystyle:=C_{ab}^{c}(x)e_{c}.$ (60)

for the basis of sections $e_{a}\in\Gamma(E)$. Then, $Q^{2}=0$ is equivalent
that two operations $(\rho,[-,-])$ satisfy the definition of a Lie algebroid
on $E$ and vice versa. All the identities of the anchor map and the Lie
bracket in the Lie algebroid $E$ are obtained from the $Q^{2}=0$ condition.

We have another description of a Lie algebroid based on a QP-manifold.

###### Definition 6.2

If an N-manifold ${\mathcal{M}}$ has a graded symplectic form
${\omega}_{grad}$ of degree $n$ and a vector field $Q$ of degree $+1$
satisfies $Q^{2}=0$, and ${\mathcal{L}}_{Q}\omega=0$,
$({\mathcal{M}},{\omega}_{grad},Q)$ is called a QP-manifold of degree $n$.

In a QP-manifold, if $n\neq 0$, there exist a Hamiltonian function $\Theta$
such that $Q={\\{{{\Theta},{-}}\\}}$. $Q^{2}=0$ is equivalent to
${\\{{{\Theta},{\Theta}}\\}}=0$.

For a Lie algebroid, we take an $n-1$-shifted cotangent bundle
${\mathcal{M}}=T^{*}[n-1]E[1]$ with a nonnegative integer $n-1$. Take local
coordinates on $T^{*}[n-1]E[1]$, $(x^{i},q^{a},z_{i},y_{a})$ of degree
$(0,1,n-1,n-2)$. Since $T^{*}[n-1]E[1]$ is a (graded) cotangent bundle, it has
a canonical graded symplectic form of degree $n-1$,

$\displaystyle{\omega}_{grad}$ $\displaystyle=\delta x^{i}\wedge\delta
z_{i}+\delta q^{a}\wedge\delta y_{a},$ (61)

where $\delta$ is the graded de Rham differential on the graded manifold. A
graded Poisson bracket for ${\omega}_{grad}$ is defined by

$\displaystyle{\\{{{f},{g}}\\}}$ $\displaystyle=$
$\displaystyle(-1)^{|f|+n}\iota_{X_{f}}\iota_{X_{g}}{\omega}_{grad}.$ (62)

The Poisson brackets ${\\{{{-},{-}}\\}}$ satisfy

$\displaystyle\\{x^{i},z_{j}\\}$
$\displaystyle=\delta^{i}_{j},\qquad\\{z_{j},x^{i}\\}=-\delta^{i}_{j},$ (63)
$\displaystyle\\{q^{a},y_{b}\\}$
$\displaystyle=\delta^{a}_{b},\qquad\\{y_{b},q^{a}\\}=-(-1)^{n-2}\delta^{a}_{b}.$
(64)

We take the following degree $n$ function on $T^{*}[n-1]E[1]$,

$\displaystyle\Theta_{Lie}$
$\displaystyle=\langle\rho(q),\,z\rangle+\frac{(-1)^{n-1}}{2}{({{[q,q]},{y}})}=(-1)^{n-1}\rho^{i}_{a}(x)z_{i}q^{a}+\frac{(-1)^{n-1}}{2}C_{bc}^{a}(x)q^{b}q^{c}y_{a}.$
(65)

Here $\langle-,\,-\rangle$ is the pairing of $TM$ and $T^{*}M$ an
${({{-},{-}})}$ is the pairing of $E$ and $E^{*}$.
$\\{\Theta_{Lie},\Theta_{Lie}\\}=0$ is equivalent to
$\rho(e_{a})=\rho^{i}_{a}(x)\partial_{i}$ and $C_{bc}^{a}(x)$ are the anchor
map and the structure function of the Lie bracket of a Lie algebroid.

We can prove that $\iota_{Q_{Lie}}{\omega}_{grad}=-\delta\Theta_{Lie}$,
equivalently, $Q_{Lie}=-\\{\Theta_{Lie},-\\}$ in (65) satisfies
$Q_{Lie}^{2}=0$ and ${\mathcal{L}}_{Q_{Lie}}\omega=0$ if and only if $E$ is a
Lie algebroid. The local coordinate expression is

$\displaystyle{Q_{\mathrm{Lie}}}$ $\displaystyle=-\\{\Theta_{Lie},-\\}$
$\displaystyle=\rho^{i}_{a}(x)q^{a}\frac{\partial}{\partial
x^{i}}-\frac{1}{2}C_{bc}^{a}(x)q^{b}q^{c}\frac{\partial}{\partial
q^{a}}+\left((-1)^{n}\rho^{i}_{a}z_{i}-C_{ab}^{c}(x)q^{b}y_{c}\right)\frac{\partial}{\partial
y_{a}}$
$\displaystyle-\left(\partial_{i}\rho^{j}_{a}z_{j}q^{a}-\frac{(-1)^{n}}{2}\partial_{i}C_{bc}^{a}(x)q^{b}q^{c}y_{a}\right)\frac{\partial}{\partial
z_{i}}.$ (66)

Here $\partial_{i}=\frac{\partial}{\partial x^{i}}$. ${Q_{\mathrm{Lie}}}$ is
also denoted by ${\mathcal{L}}_{Q}={Q_{\mathrm{Lie}}}$ and called the tangent
lift of the vector field $Q$ in (58). ${Q_{\mathrm{Lie}}}$ is the homological
vector field such that $Q={Q_{\mathrm{Lie}}}|_{E[1]}$. $T^{*}[n-1]E[1]$ has a
QP-manifold structure induced from a Lie algebroid structure on $E$.

We consider a function of degree $n$,

$\displaystyle{\underline{J}}{}$ $\displaystyle:=\frac{1}{n!}J_{a_{1}\ldots
a_{n}}(x)q^{a_{1}}\ldots q^{a_{n}},$ (67)

corresponding to the $E$-$n$-form $J\in\Gamma(\wedge^{n}E^{*})$, where

$\displaystyle\frac{1}{n!}J_{a_{1}\ldots a_{n}}(x)$
$\displaystyle:=J(e_{a_{1}},\cdots,e_{a_{n}}).$ (68)

The term ${\underline{J}}{}$ is summed to $\Theta_{Lie}$ as

$\displaystyle\Theta_{LieJ}$
$\displaystyle=\Theta_{Lie}+(-1)^{n-1}{\underline{J}}{},$ (69)

and consider the vector field of degree $+1$

$\displaystyle Q_{LieJ}$ $\displaystyle=-{\\{{{\Theta_{LieJ}},{-}}\\}}.$ (70)

$Q_{LieJ}$ is not necessarily a homological vector field since $Q_{LieJ}^{2}$
is not necessarily zero without any condition for ${\underline{J}}{}$. In
order to realize the condition (11) of a compatible $E$-$n$-form $J$, we
modify $Q_{LieJ}$ by adding a term induced from a (pre)-$n$-plectic form
$\omega=H$∥∥∥Here a pre-$n$-plectic form on a (normal) base manifold $M$ is
denoted by $H$ to distinguish with ${\omega}_{grad}$. as

$\displaystyle{Q_{\mathrm{comp}}}$
$\displaystyle={\\{{{-\Theta_{Lie}+(-1)^{n}{\underline{J}}{}},{-}}\\}}+(-1)^{n(n+1)/2}({\omega}_{grad}^{\flat})^{-1}\iota_{\rho(q)}^{n}H$
$\displaystyle=\rho^{i}_{a}(x)q^{a}\frac{\partial}{\partial
x^{i}}-\frac{1}{2}C_{bc}^{a}(x)q^{b}q^{c}\frac{\partial}{\partial q^{a}}$
$\displaystyle+\left((-1)^{n}\rho^{i}_{a}z_{i}-C_{ab}^{c}(x)q^{b}y_{c}+(-1)^{n}\frac{1}{(n-1)!}J_{ab_{2}\ldots
b_{n}}q^{b_{2}}\ldots q^{b_{n}}\right)\frac{\partial}{\partial y_{a}}$
$\displaystyle-\left[\partial_{i}\rho^{j}_{a}z_{j}q^{a}-\frac{(-1)^{n}}{2}\partial_{i}C_{bc}^{a}(x)q^{b}q^{c}y_{a}\right.$
$\displaystyle\left.+(-1)^{n}\frac{1}{n!}(\partial_{i}J_{a_{1}\ldots
a_{n}}-\rho^{j_{1}}_{a_{1}}\ldots\rho^{j_{n}}_{a_{n}}H_{ij_{1}\ldots
j_{n}})q^{a_{1}}\ldots q^{a_{n}}\right]\frac{\partial}{\partial z_{i}},$ (71)

where

$\displaystyle H=\frac{1}{(n+1)!}H_{i_{1}\ldots
i_{n+1}}(x)\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{n+1}},$ (72)

is a closed $n+1$-form. For a $1$-form $\alpha=\alpha_{i}(x)\mathrm{d}x^{i}$,
$({\omega}_{grad}^{\flat})^{-1}:T^{*}[n-1]E[1]\rightarrow TE[1]$ is the map
induced from the symplectic form ${\omega}_{grad}$.

$\displaystyle({\omega}_{grad}^{\flat})^{-1}\alpha=\alpha_{i}(x)\frac{\partial}{\partial
z_{i}}.$ (73)

###### Proposition 6.3

Let $E$ be a Lie algebroid and $H$ is a pre-$n$-plectic form. The vector field
of degree $+1$, ${Q_{\mathrm{comp}}}$ in (71) satisfies
${Q_{\mathrm{comp}}}^{2}=0$ if and only if $J$ is a compatible $E$-$n$-form.

###### Example 6.1

Let
$\pi=\frac{1}{2}\pi^{ij}(x)\partial_{i}\wedge\partial_{j}\in\Gamma(\wedge^{2}TM)$
be a bivector field and
$H=\frac{1}{3!}H_{ijk}(x)\mathrm{d}x^{i}\wedge\mathrm{d}x^{j}\wedge\mathrm{d}x^{k}\in\Omega^{3}(M)$
be a closed $3$-form. We consider an N-manifold $T^{*}[2]T[1]M$. Local
coordinates are $(x^{i},q_{i},z_{i},y^{i})$ of degree $(0,1,2,1)$.

$\displaystyle Q$ $\displaystyle=\pi^{ij}(x)q_{i}\frac{\partial}{\partial
x^{j}}-\frac{1}{2}\partial_{i}\pi^{jk}(x)q_{j}q_{k}\frac{\partial}{\partial
q_{i}}-\left(\pi^{ij}z_{j}+\partial_{k}\pi^{ij}(x)q_{j}y^{k}-\pi^{ij}q_{j}\right)\frac{\partial}{\partial
y_{i}}$
$\displaystyle+\left[\partial_{i}\pi^{jk}z_{j}q_{k}+\frac{1}{2}\partial_{i}\partial_{j}\pi^{kl}q_{k}q_{l}y_{j}-\frac{1}{2}(\partial_{i}\pi^{jk}-\pi^{jl}\pi^{km}H_{ilm})q_{j}q_{k}\right]\frac{\partial}{\partial
z_{i}},$ (74)

$Q^{2}=0$ is equivalent that $(\pi,H)$ is the twisted Poisson structure.

There does not necessarily exist a homological function $\Theta$ satisfying
${Q_{\mathrm{comp}}}={\\{{{\Theta},{-}}\\}}$ for the homological function in
(71), if $H$ is nonzero. A compatible $E$-$n$-form can not be constructed as a
QP-structure on $T^{*}[n-1]E[1]$.****** Since a homological function for a
compatible $E$-$n$-form does not exist, we cannot construct a topological
sigma model by using the so called AKSZ construction [2].

##  7 Realization by QP-manifold and twist

In the Q-manifold description in the previous section, a compatible condition
is not formulated as a QP-manifold, i.e., there exists no homological function
$\Theta$ such that $\iota_{Q}\omega=-\delta\Theta$ for the homological vector
field $Q$ in Equation (71) if $H\neq 0$. We consider another realization in
graded geometry as a Lagrangian Q-submanifold of a QP-manifold.

For it, we consider a shifted cotangent bundle of the graded bundle in Section
6. Take a shifted double cotangent bundle $T^{*}[n]{\mathcal{M}}$ of
${\mathcal{M}}=T^{*}[n-1]E[1]$ in the previous section 6. We consider a QP-
structure on $T^{*}[n]{\mathcal{M}}$, i.e., we take the canonical graded
symplectic form ${\widetilde{\omega}}_{grad}$ and ${\widetilde{Q}}$ such that
${\mathcal{L}}_{{\widetilde{Q}}}{\widetilde{\omega}}_{grad}=0$.

###  7.1 Twisting of homological functions and twisted QP-manifolds

In this section, a general theory to describe a Lagrangian submanifold and its
geometric structure of a QP-manifold in terms of graded geometry and applied
to a compatible $E$-$n$-form. We use twisting of a homological function
introduced by [35] and analyzed in [40, 30, 27].

Let ${pr}:T^{*}[n]{\mathcal{M}}\rightarrow{\mathcal{M}}$ be a natural
projection map. ${\mathcal{M}}$ is a trivial Lagrangian graded submanifold
since $\\{{pr}^{*}f,{pr}^{*}g\\}=0$ for all $f,g\in C^{\infty}({\mathcal{M}})$
and ${\mathcal{M}}$ is of half dimensions of $T^{*}[n]{\mathcal{M}}$.

If $(T^{*}[n]{\mathcal{M}},{\widetilde{\omega}}_{grad},{\widetilde{Q}})$ is a
QP-manifold of degree $n$, an induced graded Poisson structure of degree $-n$
is defined on ${\mathcal{M}}$ as follows. A bilinear graded bracket of degree
$-n+1$ is defined on ${\mathcal{M}}$ by a derived bracket,

$\displaystyle\\{f,g\\}_{{\mathcal{M}}}$
$\displaystyle:={pr}_{*}\\{\\{{pr}^{*}f,{\widetilde{\Theta}}\\},{pr}^{*}g\\},$
(75)

where $\\{-,-\\}$ is the graded Poisson bracket induced from
${\widetilde{\omega}}_{grad}$ and ${\widetilde{\Theta}}$ is a homological
function for ${\widetilde{Q}}$. $\\{-,-\\}_{{\mathcal{M}}}$ is a graded
Poisson bracket of degree $-n$ by using $\\{{pr}^{*}f,{pr}^{*}g\\}=0$. In
fact, $\\{-,-\\}_{{\mathcal{M}}}$ satisfies identities of a graded Poisson
bracket,

$\displaystyle\\{f,g\\}_{{\mathcal{M}}}$
$\displaystyle=-(-1)^{(|f|-n+1)(|g|-n+1)}\\{g,f\\}_{{\mathcal{M}}},$ (76)
$\displaystyle\\{f,gh\\}_{{\mathcal{M}}}$
$\displaystyle=\\{f,g\\}_{{\mathcal{M}}}h+(-1)^{(|f|-n+1)|g|}g\\{f,h\\}_{{\mathcal{M}}},$
(77) $\displaystyle\\{f,\\{g,h\\}_{{\mathcal{M}}}\\}_{{\mathcal{M}}}$
$\displaystyle=\\{\\{f,g\\}_{{\mathcal{M}}},h\\}_{{\mathcal{M}}}+(-1)^{(|f|-n+1)(|g|-n+1)}\\{g,\\{f,h\\}_{{\mathcal{M}}}\\}_{{\mathcal{M}}},$
(78)

for $f,g,h\in C^{\infty}({\mathcal{M}})$.

In order to construct a general geometric structure realized in a Lagrangian
submanifold ${\mathcal{L}}\subset T^{*}[n+1]{\mathcal{M}}$, we use twist of a
homological function ${\widetilde{\Theta}}$.

###### Definition 7.1

Let $\phi\in C^{\infty}(T^{*}[n]{\mathcal{M}})$. Then the twist
$e^{\mathrm{ad}\phi}$ of any function $\alpha\in C^{\infty}({\mathcal{M}})$ by
$\phi$ is defined by the following exponential of the following adjoint
action, [35]

$\displaystyle
e^{\mathrm{ad}\phi}\alpha:=\alpha+\\{\alpha,\phi\\}+\frac{1}{2}\\{\\{\alpha,\phi\\},\phi\\}+\ldots+\frac{1}{n!}\\{\ldots\\{\\{\alpha,\phi\\},\phi\\},\ldots,\phi\\}+\ldots.$
(79)

If a function $\phi\in C^{\infty}(T^{*}[n]{\mathcal{M}})$ is of degree $n$,
twist preserves degree and gives a Poisson map $\alpha\mapsto
e^{\mathrm{ad}\phi}\alpha$ satisfying

$\displaystyle\\{e^{\mathrm{ad}\phi}\alpha,e^{\mathrm{ad}\phi}\beta\\}=e^{\mathrm{ad}\phi}\\{\alpha,\beta\\},$
(80)

for all $\alpha,\beta\in C^{\infty}(T^{*}[n]{\mathcal{M}})$.

Let ${\mathcal{E}}_{\phi}:=\\{e^{\mathrm{ad}\phi}{pr}^{*}f|f\in
C^{\infty}({\mathcal{M}})\\}$ be a space of functions twisted by a degree $n$
function $\phi\in C^{\infty}(T^{*}[n]{\mathcal{M}})$. In general,
${\mathcal{E}}_{\phi}$ is identified as a space of functions on some graded
manifold ${\mathcal{L}}_{\phi}\subset T^{*}[n]{\mathcal{M}}$,
${\mathcal{E}}_{\phi}=C^{\infty}({\mathcal{L}}_{\phi})$, by applying a Serre-
Swan type theorem on a graded manifold.

###### Proposition 7.2

${\mathcal{L}}_{\phi}$ is a graded Lagrangian submanifold of
$T^{*}[n]{\mathcal{M}}$.

It is confirmed using the formula
$\\{e^{\mathrm{ad}\phi}{pr}^{*}f,e^{\mathrm{ad}\phi}{pr}^{*}g\\}=e^{\mathrm{ad}\phi}\\{{pr}^{*}f,{pr}^{*}g\\}=0$
and the dimension of ${\mathcal{L}}_{\phi}$ is the same as ${\mathcal{M}}$.

By definition, for $\alpha,\beta\in C^{\infty}({\mathcal{L}}_{-\phi})$, there
exist $f,g\in C^{\infty}({\mathcal{M}})$ such that
$\alpha=e^{-\mathrm{ad}\phi}{pr}^{*}f$ and
$\beta=e^{-\mathrm{ad}\phi}{pr}^{*}g$. The derived bracket of $\alpha$ and
$\beta$ is equal to

$\displaystyle\\{\\{e^{-\mathrm{ad}\phi}{pr}^{*}f,{\widetilde{\Theta}}\\},e^{-\mathrm{ad}\phi}{pr}^{*}g\\}$
$\displaystyle:=e^{-\mathrm{ad}\phi}\\{\\{{pr}^{*}f,e^{\mathrm{ad}\phi}{\widetilde{\Theta}}\\},{pr}^{*}g\\}.$
(81)

A bilinear bracket is defined by the following derived bracket,

$\displaystyle\\{\alpha,\beta\\}_{{\mathcal{L}}_{-\phi}}$
$\displaystyle:={pr}_{*}e^{-\mathrm{ad}\phi}\\{\\{{pr}^{*}f,e^{\mathrm{ad}\phi}{\widetilde{\Theta}}\\},{pr}^{*}g\\}.$
(82)

Since it is equivalent to
${pr}_{*}\\{\\{e^{-\mathrm{ad}\phi}{pr}^{*}f,{\widetilde{\Theta}}\\},e^{-\mathrm{ad}\phi}{pr}^{*}g\\}$,
twisting by $\phi$ gives a graded Poisson bracket on ${\mathcal{L}}_{-\phi}$
induced by ${\widetilde{\Theta}}$ equivalent to a graded Poisson bracket on
${\mathcal{M}}$ induced by $e^{\mathrm{ad}\phi}{\widetilde{\Theta}}$. We can
prove that $\\{-,-\\}_{{\mathcal{L}}_{-\phi}}$ is a graded Poisson bracket of
degree $-n+1$ similar to the proof for the bracket (75).

Thus, in the preceding discussion, we analyze the twisted homological function
$e^{\mathrm{ad}\phi}{\widetilde{\Theta}}$ on ${\mathcal{M}}$ instead of a
homological function on ${\mathcal{L}}_{-\phi}$.

The twisting function $\phi$ is an almost homological function. By (81) and
(82), the graded Poisson bracket is defined by the derived bracket
$\\{\\{-,{\widetilde{\Theta}}\\},-\\}$. On the other hand, we obtain

$\displaystyle{pr}_{*}e^{\mathrm{ad}\phi}{\widetilde{\Theta}}$
$\displaystyle={pr}_{*}\
\left({\widetilde{\Theta}}+\\{{\widetilde{\Theta}},\phi\\}+\frac{1}{2}\\{\\{{\widetilde{\Theta}},\phi\\},\phi\\}+\ldots+\frac{1}{n!}\underbrace{\\{\ldots\\{\\{{\widetilde{\Theta}},\phi\\},\phi\\},\ldots,\phi\\}}_{n\,\text{brackets}}+\ldots\right),$
(83)

The function $\phi$ is of degree $n$, and from Equation (75), the third term
of ${pr}_{*}\,e^{\mathrm{ad}\phi}\Theta$ gives

$\displaystyle\\{\phi,\phi\\}_{{\mathcal{M}}}=-{pr}_{*}\
\\{\\{{\widetilde{\Theta}},\phi\\},\phi\\}.$ (84)

If $\phi$ is a homological function on ${\mathcal{M}}$ if Equation (84) is
zero, and $({\mathcal{M}},\\{-,-\\}_{{\mathcal{M}}},\phi)$ is a QP-manifold of
degree $n-1$. However, Equation (84) is not necessarily zero.

Inspired by the above analysis and Equation (83), we define a twisted QP-
structure. We consider the following class of generalizations of a QP-manifold
which has good properties but not a QP.

###### Definition 7.3

Take a shifted cotangent bundle ${\mathcal{N}}=T^{*}[n]{\mathcal{M}}$ of a
graded manifold ${\mathcal{M}}$. Let ${\mathcal{N}}=T^{*}[n]{\mathcal{M}}$ be
a QP-manifold with a homological function $\Theta$. If a degree $n$ function
$\phi\in C^{\infty}(T^{*}[n]{\mathcal{M}})$ satisfies

$\displaystyle{pr}_{*}e^{\mathrm{ad}\phi}\Theta$ $\displaystyle=0,$ (85)

$({\mathcal{M}},\\{-,-\\}_{{\mathcal{M}}},\phi)$ is called a twisted QP-
manifold, where ${pr}:T^{*}[n]{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a
natural projection.

A function $\phi$ satisfying the condition (85) is also called a a Poisson
function or a canonical function. Obviously a QP-manifold is a twisted QP-
manifold. A twisted QP-manifold has some applications to physical theories
such as the reduction of the AKSZ sigma models and the supergeometric
construction of current algebras.[27, 28, 3]

###  7.2 QP-manifold description of compatible differential forms

Our graded manifold is $T^{*}[n]{\mathcal{M}}=T^{*}[n]T^{*}[n-1]E[1]$. In
order to compute the concrete formula, take local coordinates
$(x^{i},q^{a},z_{i},y_{a})$ of degree $(0,1,n-1,n-2)$ on $T^{*}[n-1]E[1]$ and
canonical conjugate coordinates of the fiber
$(\xi_{i},p_{a},\zeta^{i},\eta^{a})$ of degree $(n,n-1,1,2)$. The canonical
projection ${pr}:T^{*}[n]{\mathcal{M}}\rightarrow{\mathcal{M}}$ is given by
${pr}:(\xi_{i},p_{a},\zeta^{i},\eta_{a})\mapsto(0,0,0,0)$. The canonical
symplectic form of degree $n$ is

$\displaystyle{\widetilde{\omega}}_{grad}$ $\displaystyle=\delta
x^{i}\wedge\delta\xi_{i}+\delta q^{a}\wedge\delta p_{a}+\delta
z_{i}\wedge\delta\zeta^{i}+\delta y_{a}\wedge\delta\eta^{a}.$ (86)

Graded Poisson brackets induced from the above symplectic form are

$\displaystyle\\{x^{i},\xi_{j}\\}$
$\displaystyle=-\\{\xi_{j},x^{i}\\}=\delta^{i}_{j},$ (87)
$\displaystyle\\{q^{a},p_{b}\\}$
$\displaystyle=-(-1)^{n-1}\\{p_{b},q^{a}\\}=\delta^{a}_{b},$ (88)
$\displaystyle\\{z_{i},\zeta^{j}\\}$
$\displaystyle=-(-1)^{n-1}\\{\zeta^{j},z_{i}\\}=\delta_{i}^{j},$ (89)
$\displaystyle\\{y_{a},\eta^{b}\\}$
$\displaystyle=-\\{\eta^{b},y_{a}\\}=\delta_{a}^{b}.$ (90)

We take an $n+1$-form $H$ on $M$ and consider the following function of degree
$n+1$,

$\displaystyle{\widetilde{\Theta}}$
$\displaystyle=\langle\xi,\,\zeta\rangle+{({{p},{\eta}})}+(-1)^{\tfrac{n(n-1)}{2}}\tilde{H}$
$\displaystyle=\xi_{i}\zeta^{i}+p_{a}\eta^{a}+(-1)^{\tfrac{n(n-1)}{2}}\frac{1}{(n+1)!}H_{i_{1}\ldots
i_{n+1}}(x)\zeta^{i_{1}}\ldots\zeta^{i_{n+1}},$ (91)

where $\tilde{H}=\frac{1}{(n+1)!}H_{i_{1}\ldots
i_{n+1}}(x)\zeta^{i_{1}}\ldots\zeta^{i_{n+1}}$ is a degree $n+1$ function
corresponding to the $n+1$-form $H=\frac{1}{(n+1)!}H_{i_{1}\ldots
i_{n+1}}(x)\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{n+1}}\in\Omega^{n+1}(M)$.
An $n+1$-form is identified to a degree $n+1$ function on $T[1]M$.
${\widetilde{\Theta}}$ satisfies
${\\{{{{\widetilde{\Theta}}},{{\widetilde{\Theta}}}}\\}}=0$ if and only if $H$
is closed, $\mathrm{d}H=0$, i.e.,
$(T^{*}[n]{\mathcal{M}},{\widetilde{\omega}}_{grad},{\widetilde{Q}})$ is a QP-
manifold if $H$ is a pre-$n$-plectic form on $M$, where ${\widetilde{Q}}$ is a
Hamiltonian vector field for ${\widetilde{\Theta}}$.

We consider a twist using the following function of degree $n$,
$\phi={\phi_{Lie}}-{\underline{J}}{}$, where two functions are

$\displaystyle{\phi_{Lie}}$
$\displaystyle=\rho^{i}_{a}(x)z_{i}q^{a}+\frac{1}{2}C^{c}_{ab}(x)q^{a}q^{b}y_{c},$
(92) $\displaystyle{\underline{J}}{}$
$\displaystyle=\frac{1}{n!}J_{a_{1}\ldots a_{n}}(x)q^{a_{1}}\ldots q^{a_{n}}.$
(93)

and impose the condition (85),
${pr}_{*}e^{\mathrm{ad}\phi}{\widetilde{\Theta}}=0$. ${\phi_{Lie}}$ is a
similar form as $\Theta_{Lie}$ in Equation (65).

Compare to the condition in non-graded geometry by defining the following
operations. Let $\rho:=\rho^{i}_{a}(x)\tfrac{\partial}{\partial
x^{i}}:E\rightarrow TM$ be a bundle map, $[e_{a},e_{b}]:=C_{ab}^{c}e_{c}$ be a
bilinear bracket on $\Gamma(E)$ for the basis $e_{a}\in\Gamma(E)$. Moreover
let $J:=\frac{1}{n!}J_{a_{1}\ldots a_{n}}(x)e^{a_{1}}\wedge\ldots\wedge
e^{a_{n}}\in\Gamma(\wedge^{n}E^{*})$ be an $E$-$n$-form with the basis
$e^{a}\in\Gamma(E^{*})$.

Only nonvanishing terms in ${pr}_{*}e^{\mathrm{ad}\phi}{\widetilde{\Theta}}$
after the projection ${pr}$ are
$\\{\\{{\widetilde{\Theta}},{\phi_{Lie}}\\},{\phi_{Lie}}\\}$,
$\\{\\{{\widetilde{\Theta}},{\underline{J}}{}\\},{\phi_{Lie}}\\}$,
$\\{\\{{\widetilde{\Theta}},{\phi_{Lie}}\\},{\underline{J}}{}\\}$ and
$\underbrace{\\{\ldots\\{\\{\Theta,{\phi_{Lie}}\\},{\phi_{Lie}}\\},\ldots,{\phi_{Lie}}\\}}_{(n+1)\,\text{brackets}}$.
$\\{\\{{\widetilde{\Theta}},{\phi_{Lie}}\\},{\phi_{Lie}}\\}=0$ must be
satisfied independently. This equation gives conditions for $\rho^{i}_{a}$ and
$C^{a}_{bc}$ that $\rho_{a}=\rho^{i}_{a}\partial_{i}$ is an anchor map and
$C^{a}_{bc}$ is a structure function of the Lie algebroid on $E$. Another
condition

$\displaystyle{pr}_{*}\left(\frac{1}{2}\\{\\{{\widetilde{\Theta}},{\underline{J}}{}\\},{\phi_{Lie}}\\}+\frac{1}{2}\\{\\{{\widetilde{\Theta}},{\phi_{Lie}}\\},{\underline{J}}{}\\}+\frac{1}{(n+1)!}\underbrace{\\{\ldots\\{\\{\Theta,{\phi_{Lie}}\\},{\phi_{Lie}}\\},\ldots,{\phi_{Lie}}\\}}_{(n+1)\,\text{brackets}}\right)=0,$
(94)

gives the compatible condition of $J$, Equation (11) by straightforward
calculations.

We summarize the result.

###### Proposition 7.4

$(E,\rho,[-,-])$ is a Lie algebroid and $J$ is a compatible $E$-$n$-form with
a pre-$(n+1)$-form $H$ if and only if $\phi$ satisfies Equation (85)
${pr}_{*}e^{\mathrm{ad}\phi}{\widetilde{\Theta}}=0$,i.e, ${\mathcal{M}}$ is a
twisted QP-manifold with $\phi$ in Equations (92) and (93).

###  7.3 Lie $n$-algebroid induced from QP-manifold

In this section, we explain that a Lie $n$-algebroid and a higher Dirac
structure in Section 5 are constructed from the (twisted) QP-manifold in the
previous section. The general theory of construction of a Lie $n$-algebroid
and a higher Dirac structure from a Q-manifold is known [7]. We apply the
technique to a compatible $E$-$n$-form.

There are many analysis about Lie $n$-algebroids [7, 20, 39, 37, 43]. One
definition is to construct a Lie $n$-algebroid from a QP-manifold
$({\mathcal{N}},{\omega}_{grad},\Theta)$ of degree $n$. In our paper, a Lie
$n$-algebroid is defined as follows. Assume that ${\mathcal{N}}$ is an
N-manifold. Let $C_{k}^{\infty}({\mathcal{N}})$ be a set of degree $k$
functions on a graded manifold ${\mathcal{N}}$. A base manifold $M$ of a
graded manifold is a degree $0$ part of ${\mathcal{N}}$, thus,
$C^{\infty}(M)\simeq C_{0}^{\infty}({\mathcal{N}})$ and
$C^{\infty}({\mathcal{N}})=\bigoplus_{k\geq 0}C_{k}^{\infty}({\mathcal{N}})$.
We choose the space of degree $n-1$ functions,
$C^{\infty}_{n-1}({\mathcal{N}})$. If there exists an (unshifted) vector
bundle ${\widehat{E}}$ such that $\Gamma({\widehat{E}})\simeq
C^{\infty}_{n-1}({\mathcal{N}})$, an element $e\in\Gamma({\widehat{E}})$ is
mapped to an element ${\underline{e}}{}\in C^{\infty}_{n-1}({\mathcal{N}})$
shifting degree by $n-1$. The map $j:{\widehat{E}}\rightarrow{\mathcal{N}}$ is
defined by the trivial injection by shifting degree. $j$ gives the map from
the space of sections of ${\widehat{E}}$ to functions on the graded mainfold
${\mathcal{N}}$, $j_{*}:C^{\infty}(M)\oplus\Gamma({\widehat{E}})\rightarrow
C^{\infty}_{0}({\mathcal{N}})\oplus C^{\infty}_{n-1}({\mathcal{N}})$. This map
induces the so called derived bracket construction of a Lie $n$-algebroid.
Note that if ${\underline{e}}{}_{1},{\underline{e}}{}_{2}\in
C^{\infty}_{n-1}({\mathcal{N}})$, the derived bracket
${\\{{{{\\{{{{\underline{e}}{}_{1}},{\Theta}}\\}}},{{\underline{e}}{}_{2}}}\\}}$
is also of degree $n-1$. We denote $j_{*}(e)={\underline{e}}{}$.

###### Definition 7.5

Let $({\mathcal{N}},{\omega}_{grad},Q)$ be a QP-manifold of degree $n$ over a
base manifold $M$. Three operations on
$C^{\infty}(M)\oplus\Gamma({\widehat{E}})$, an inner product ${({{-},{-}})}$
on $\Gamma({\widehat{E}})\times\Gamma({\widehat{E}})$, the anchor map
${\widehat{\rho}}:{\widehat{E}}\rightarrow TM$ and the (higher) Dorfman
bracket
${[{{-},{-}}]_{D}}:\Gamma({\widehat{E}})\times\Gamma({\widehat{E}})\rightarrow\Gamma({\widehat{E}})$
are defined by graded Poisson brackets and derived brackets on
$C_{0}^{\infty}({\mathcal{N}})\oplus C_{n-1}^{\infty}({\mathcal{N}})$ as

$\displaystyle{({{e_{1}},{e_{2}}})}$
$\displaystyle:=j^{*}{\\{{{{\underline{e}}{}_{1}},{{\underline{e}}{}_{2}}}\\}},$
(95) $\displaystyle{\widehat{\rho}}(e)f$
$\displaystyle:=j^{*}{\\{{{{\\{{{{\underline{e}}{}},{\Theta}}\\}}},{f}}\\}},$
(96) $\displaystyle{[{{e_{1}},{e_{2}}}]_{D}}$
$\displaystyle:=j^{*}{\\{{{{\\{{{{\underline{e}}{}_{1}},{\Theta}}\\}}},{{\underline{e}}{}_{2}}}\\}},$
(97)

for $C_{0}^{\infty}({\mathcal{N}})=C^{\infty}(M)$,
$e,e_{1},e_{2}\in\Gamma({\widehat{E}})$ and
${\underline{e}}{},{\underline{e}}{}_{1},{\underline{e}}{}_{2}\in
C_{n-1}^{\infty}({\mathcal{N}})$.

Three operations satisfy identities induced from $\\{\Theta,\Theta\\}=0$.
Then, $({\widehat{E}},{({{-},{-}})},{\widehat{\rho}},[-,-]_{D})$ with
identities induced from $\\{\Theta,\Theta\\}=0$ is called a Lie $n$-algebroid.

Since the graded Poisson bracket is of degree $-n$, the image of the inner
product ${({{-},{-}})}$ is a function of degree $n-2$. Thus, an image of the
inner product is not an element of $C^{\infty}(M)$ except for $n=2$.

In the construction of Definition 7.5, a higher Dirac structure $L$ is
characterized by the subspace of ${\widehat{E}}$ such that $\Gamma(L)\simeq
C^{\infty}_{n-1}({\mathcal{L}})$, where ${\mathcal{L}}\subset{\mathcal{N}}$ is
a graded submanifold of ${\mathcal{N}}$ such that
${\\{{{{\underline{e}}{}_{1}},{{\underline{e}}{}_{2}}}\\}}=0$ and
${\\{{{{\\{{{{\underline{e}}{}_{1}},{\Theta}}\\}}},{{\underline{e}}{}_{2}}}\\}}\in
C^{\infty}_{n-1}({\mathcal{L}})$ for all
${\underline{e}}{}_{1},{\underline{e}}{}_{1}\in
C^{\infty}_{n-1}({\mathcal{L}})$.

Let us consider the graded manifold in our case,
${\mathcal{M}}=T^{*}[n-1]E[1]$ and its graded cotangent bundle
${\mathcal{N}}=T^{*}[n]{\mathcal{M}}=T^{*}[n]T^{*}[n-1]E[1]$. A general form
of a degree $n-1$ function in local coordinates is

$\displaystyle{\underline{f}}{}_{n-1}(x,q,z,y,\xi,p,\zeta,\eta)=X^{i}(x)z_{i}+u^{a}(x)p_{a}+\sigma^{a}_{i}(x)y_{a}\zeta^{i}+\tau^{a}_{b}(x)y_{a}q^{b}$
$\displaystyle\qquad+\sum_{2k+l+m=n-1}\frac{1}{k!l!m!}\alpha_{a_{1}\ldots
a_{k}b_{1}\ldots b_{l}i_{1}\ldots
i_{m}}(x)\eta^{a_{1}}\ldots\eta^{a_{k}}q^{b_{1}}\ldots
q^{b_{l}}\zeta^{i_{1}}\ldots\zeta^{i_{m}}.$ (98)

The function corresponds to an element of $\Gamma\Bigl{(}TM\oplus
E\oplus(E\otimes T^{*}M)\oplus(E\otimes
E^{*})\bigoplus_{2k+l+m=n-1}(S^{k}E\otimes\wedge^{l}E^{*}\otimes\wedge^{m}T^{*}M)\Bigr{)}$.
In fact, the map between all basis is given by

$\displaystyle
j:(x^{i},e^{a},\partial_{i},e_{a},\partial_{i},e_{a},\mathrm{d}x^{i},e^{a})\mapsto(x^{i},q^{a},z_{i},y_{a},\xi_{i},p_{a},\zeta^{i},\eta^{a}),$
(99)

and the section on the corresponding vector bundle is

$\displaystyle
j^{*}{\underline{f}}{}_{n-1}=X^{i}(x)\partial_{i}+u^{a}(x)e_{a}+\sigma^{a}_{i}(x)e_{a}\otimes\mathrm{d}x^{i}+\tau^{a}_{b}(x)e_{a}\otimes
e^{b}$
$\displaystyle\qquad+\sum_{2k+l+m=n-1}\frac{1}{k!l!m!}\alpha_{a_{1}\ldots
a_{k}b_{1}\ldots b_{l}i_{1}\ldots i_{m}}(x)e^{a_{1}}\odot\ldots\odot
e^{a_{k}}\otimes e^{b_{1}}\wedge\ldots\wedge
e^{b_{l}}\wedge\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{m}},$
(100)

where $\odot$ is the symmetric tensor product.

The Lie $n$-algebroid on $E\oplus\wedge^{n-1}E^{*}$ in Section 5 from the
above QP-manifold is given by the restriction to a graded submanifold.

Let us start at a QP-manifold
$(T^{*}[n]{\mathcal{M}},{\widetilde{\omega}}_{grad},{\widetilde{\Theta}})$
with the graded symplectic form (86) and the homological function in Equation
(91). We consider the graded submanifold $T^{*}[n]E[1]\subset
T^{*}[n]{\mathcal{M}}$. In local coordinates, $T^{*}[n]E[1]$ is given by
$\\{(x,q,z,y,\xi,p,\zeta,\eta)|z_{i}=\zeta^{i}=y_{a}=\eta^{a}=0\\}$. Let
${\widehat{{pr}}}:T^{*}[n]{\mathcal{M}}\rightarrow T^{*}[n]E[1]$ be the
natural projection. Then, $T^{*}[n]E[1]$ has a symplectic form
${\widehat{{pr}}}_{*}{\widetilde{\omega}}_{grad}$ induced from
${\widetilde{\omega}}_{grad}$. Local coordinates on $T^{*}[n]E[1]$ are
$(x^{i},\xi_{i},q^{a},p_{a})$ of degree $(0,n,1,n-1)$ and the graded
symplectic form

$\displaystyle{\widehat{{pr}}}_{*}{\widetilde{\omega}}_{grad}$
$\displaystyle=\delta x^{i}\wedge\delta\xi_{i}+\delta q^{a}\wedge\delta
p_{a}.$ (101)

In order to construct a nontrivial Lie $n$-algebroid, ${\widetilde{\Theta}}$
is twisted by ${\phi_{Lie}}$ in Equation (92). As discussed in Section 7.2,
$\\{e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}},e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}}\\}=0$
condition induced a Lie algebroid structure on $E$. Under this condition, we
consider a general degree $n-1$ function on $T^{*}[n]E[1]$,

$\displaystyle{\underline{e}}{}_{n-1}$
$\displaystyle=u^{a}(x)p_{a}+\frac{1}{(n-1)!}\alpha_{b_{1}\ldots
b_{n-1}}(x)q^{b_{1}}\ldots q^{b_{n-1}}$ (102)

as a restriction of Equation (98), and the corresponding section
$e\in\Gamma(E\oplus\wedge^{n-1}E^{*})$,

$\displaystyle e=j^{*}{\underline{e}}{}_{n-1}$
$\displaystyle=u^{a}(x)e_{a}+\frac{1}{(n-1)!}\alpha_{b_{1}\ldots
b_{n-1}}(x)e^{b_{1}}\wedge\ldots\wedge e^{b_{n-1}}.$ (103)

Here $e_{a}$ and $e^{a}$ are the basis of $E$ and $E^{*}$. From the general
theory, three operations of a Lie $n$-algebroid are given by

$\displaystyle{({{e},{e^{\prime}}})}$
$\displaystyle=j^{*}\\{{\underline{e}}{}_{n-1},{\underline{e}}{}_{n-1}^{\prime}\\},$
(104) $\displaystyle{\widehat{\rho}}(e)f$
$\displaystyle=j^{*}\\{\\{{\underline{e}}{}_{n-1},e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}}\\},f\\},$
(105) $\displaystyle{[{{e},{e^{\prime}}}]_{D}}$
$\displaystyle=j^{*}\\{\\{{\underline{e}}{}_{n-1},e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}}\\},{\underline{e}}{}_{n-1}^{\prime}\\},$
(106)

where $e,e^{\prime}\in\Gamma(E\oplus\wedge^{n-1}E^{*})$ and $f\in
C^{\infty}(M)$. Concrete calculations show that operations (104)–(104) are the
same as operations (47)–(49), thus they give the same Lie $n$-algebroid
structure on $E\oplus\wedge^{n-1}E^{*}$ in Section 5. We summarize our result.

###### Proposition 7.6

A QP-manifold
$(T^{*}[n]E[1],{\widehat{{pr}}}_{*}{\widetilde{\omega}}_{grad},e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}})$
induces a Lie $n$-algebroid structure on $E\oplus\wedge^{n-1}E^{*}$.
Operations are given by Equations (104)–(106).

###  7.4 Lagrangian Q-submanifold and higher Dirac structure

The compatible $E$-$n$-form condition (11) as a higher Dirac structure is
related to the formulation in terms of the twisted QP-manifold in Section 7.2.
Take a degree $n$ function,

$\displaystyle{\underline{J}}{}$ $\displaystyle=\frac{1}{n!}J_{a_{1}\ldots
a_{n}}(x)q^{a_{1}}\ldots q^{a_{n}}.$ (107)

Then, $J=j_{*}{\underline{J}}{}$ is an $E$-$n$-form. For a section
$u=u^{a}(x)e_{a}\in\Gamma(E)$, we identify a degree $n-1$ function
$j_{*}u={\underline{u}}{}=u^{a}(x)p_{a}$. We consider the following subspace
of $C^{\infty}_{n-1}(T^{*}[n]E[1])$,

$\displaystyle{\mathcal{E}}_{J}=\\{{\underline{u}}{}+\\{{\underline{J}}{},{\underline{u}}{}\\}|{\underline{u}}{}=u^{a}(x)p_{a}\\}.$
(108)

Obviously, the image of ${\mathcal{E}}_{J}$ is the space of sections
$j_{*}{\mathcal{E}}_{J}=\Gamma(L)$ in Equation (51).

The projection to $T^{*}[n]E[1]$ of the homological function
$e^{\mathrm{ad}\phi_{Lie}}{\widetilde{\Theta}}$ is

$\displaystyle{\widehat{{pr}}}_{*}e^{\mathrm{ad}{\phi_{Lie}}}{\widetilde{\Theta}}$
$\displaystyle=\rho^{i}_{a}(x)\xi_{i}q^{a}+\frac{1}{2}C_{bc}^{a}(x)p_{a}q^{b}q^{c}+\frac{1}{(m+1)!}\rho^{i_{1}}_{a_{1}}\ldots\rho^{i_{m+1}}_{a_{m+1}}H_{i_{1}\ldots
i_{m+1}}(x)q^{a_{1}}\ldots q^{a_{m+1}},$ (109)

There is no information of $J$ in Equation (109).

${\mathcal{E}}_{J}$ is isotropic from the definition of ${\underline{J}}{}$,
especially, since $\\{{\underline{a}}{},{\underline{b}}{}\\}=0$ is
automatically satisfied because $q^{a}$ is of degree $1$. We obtain another
following graded geometric formulation of the compatible $E$-$n$-form as a
condition on ${\mathcal{E}}_{J}$ as follow.

###### Proposition 7.7

If and only ${\mathcal{E}}_{J}$ satisfies
$\\{\\{{\underline{a}}{},{\widehat{{pr}}}_{*}e^{\mathrm{ad}{\phi_{Lie}}}{\widetilde{\Theta}}\\},{\underline{b}}{}\\}\in{\mathcal{E}}_{J}$
for all elements ${\underline{a}}{},{\underline{b}}{}\in{\mathcal{E}}_{J}$,
i.e., ${\mathcal{E}}_{J}$ involutive under the derived bracket,
$J=j_{*}{\underline{J}}{}\in\Gamma(\wedge E^{*})$ satisfies Equation (11),
i.e., $J$ is a compatible $E$-$n$-form.

The proposition is obtained form the fact that the derived bracket
$\\{\\{{\underline{a}}{},{\widehat{{pr}}}_{*}e^{\mathrm{ad}{\phi_{Lie}}}{\widetilde{\Theta}}\\},{\underline{b}}{}\\}\in{\mathcal{E}}_{J}$
is mapped to ${[{{u+{({{J},{u}})}},{v+{({{J},{v}})}}}]_{D}}$ and Theorem 5.2.
$\\{\\{{\underline{a}}{},{\widehat{{pr}}}_{*}e^{\mathrm{ad}{\phi_{Lie}}}{\widetilde{\Theta}}\\},{\underline{b}}{}\\}\in{\mathcal{E}}_{J}$
is equivalent to the homological vector field $Q$ induced from the homological
vector field
${\widehat{{pr}}}_{*}e^{\mathrm{ad}{\phi_{Lie}}}{\widetilde{\Theta}}$ is
tangent to the Lagrangian graded submanifold ${\mathcal{E}}_{J}$. It is called
a Lagrangian Q-submanifold [15].

### Acknowledgments

The author is grateful to the Erwin Schrödinger International Institute for
Mathematics and Physics for support within the program “Higher Structures and
Field Theory” in 2022, and National Center for Theoretical Sciences and
National Tsing Hua University, where part of this work was carried, for their
hospitality. Especially, he would like to thank to Hsuan-Yi Liao, Camille
Laurent-Gengoux and Seokbong Seol for his hospitality and useful discussion.

This work was supported by JSPS Grants-in-Aid for Scientific Research Number
22K03323.

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