# The geometric deformation of curved $L_{\infty}$ algebras and Lie algebroids

Xiaoyi Cui  School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai
519082, China<EMAIL_ADDRESS>
###### Abstract.

We investigate the deformation of an $L_{\infty}$ structure that encodes the
geometric information about a smooth vector bundle. We establish that this
specific type of deformation endows the bundle with a Lie algebroid structure.
Additionally, we compute the Atiyah class and Atiyah-Chern classes in the
geometrically deformed case. The precise relationship between the first
Atiyah-Chern class and the modular class is shown. In the case of action Lie
algebroids, the Atiyah-Chern classes at the leading order are given by the
equivariant Chern characters.

The author thanks Chenchang Zhu for helpful suggestions. XC was partially
supported by NSFC grant 11801588 and by Guangdong Natural Science Foundation
grant 2018A030313273. XC thanks Kavli IPMU and the University of Göttingen,
where part of the work was done.

###### Contents

  1. 1 Introduction
  2. 2 Curved $L_{\infty}$ structures from vector bundles and geometric deformation
  3. 3 Geometric meanings of lower brackets and nilpotency equation
    1. 3.1 Family version and homotopy equivalence
  4. 4 Atiyah class and character
    1. 4.1 Atiyah class of Lie algebroids
    2. 4.2 The Weil algebra
    3. 4.3 Atiyah-Chern class
    4. 4.4 Modular class and the first Atiyah-Chern class
    5. 4.5 Action algebroids
  5. A Solving for $d^{\nabla}$ and $d^{\nabla^{\prime}}$
  6. B Lower brackets from $d^{\rho}$ and $d^{\mu}$

##  1\. Introduction

Nonlinear sigma models are powerful field theories which take local geometric
information as input, and output interesting geometrical or topological
invariants. The Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) method is
widely used to obtain classical aspects of these models in the form of
Batalin-Vilkovisky (BV) theories, where the geometric input is given by a dg
manifold or an $L_{\infty}$ algebra [3, 11]. It is therefore natural to
investigate what happens to the corresponding geometry under algebraic
perturbations to the homological vector field or the $L_{\infty}$ algebra
structure. This question has field-theoretical importance as well. From
physical aspects, such perturbations can be generated easily by quantum
effects, or some classical defects of space-time. Then BV quantisation of such
perturbed theories shall in return bring interesting observables, which are
worth studying. In low-dimensional topological or holomorphic theories of
Chern-Simons type, it has been shown that the one-loop partition function is
expressed as a series formed by the Atiyah-Chern classes of the target
geometry and arithmetic quantities related to the world-sheet geometry [11,
19, 21]. While the arithmetic quantities depend on the type of field theory,
the Atiyah-Chern classes provide valuable insights into new potentially
emerging invariants of the target geometry even in the absence of a specific
field theory.

In this paper, we start by considering the $L_{\infty}$ algebra obtained from
a real vector bundle $V$ over a smooth manifold $M$. Throughout the paper, we
shall let $\mathfrak{h}$ denote the $L_{\infty}$ algebra
$\Omega_{M}\otimes\Gamma(T_{M}[-1])$ as in [19] and let $\mathfrak{v}$ denote
the $L_{\infty}$ module $\Omega_{M}\otimes\Gamma(V)$. Then
$\mathfrak{h}\oplus\mathfrak{v}$ has a natural minimum extended $L_{\infty}$
algebra structure from that on $\mathfrak{h}$. A deformation of the above
structure refers to perturbing the $L_{\infty}$ structure on
$\mathfrak{h}\oplus\mathfrak{v}$ in a way that extends $\mathfrak{h}$. Our
first main result shows that the geometric deformations (as defined in Section
2) correspond specifically to Lie algebroid structures on $V$, and that in
what sense a couple of $L_{\infty}$ structures on the same underlying space
are homotopically equivalent. The result is based on a recursive computation
generalised from Fedosov’s method [17, 8, 15]. By the enlightening observation
of Vaíntrob [30], there is a bijection between Lie algebroid structures on the
vector bundle $V\to M$ and the homological vector fields on the
$\mathbb{Z}$-graded manifold $V[1]$. Then our preceding findings can be viewed
as the $L_{\infty}$ version of a parallel story, suggesting that the forming
Lie algebroids can be the only feasible way of encoding further multiplicative
structures on a vector bundle. Further more, by the work of [28], the
geometric deformation necessarily induces an $L_{\infty}$ action of $V$ on
$T_{M}$.

Our second main result provides additional characterisations of Atiyah and
Atiyah-Chern classes through local covariant computations. Since the
$L_{\infty}$ structure map corresponds to specific tensors in geometric terms,
it is useful to describe some lower brackets concretely. The computation is
manifestly covariant and yields representatives of the Atiyah class and the
Atiyah-Chern classes in terms of tensors, which can serve as a primer for
studying the geometric invariants of Lie algebroids in the Chern-Weil sense.
The Atiyah-Chern classes reside within the de Rham complex of
$\mathfrak{h}\oplus\mathfrak{v}$, which has been shown to be weakly equivalent
to the Weil algebra111This was first shown by [20] using homotopical methods.
Our result follows from a local computation with an explicit homotopy
retraction. of $V$ through a homotopy retraction. Despite its seemingly
unremarkable cohomological properties, it is not a boring quantity at all, as
the equivariant character theory of Lie algebroids is still far from complete.
We also demonstrate that the first Atiyah-Chern class is the
$d^{DR}$-coboundary of the modular class [16]. As for the general Atiyah-Chern
classes originating from an action Lie algebroid, they give equivariant Chern
characters in the leading order.

There are several relevant results to our work that we’d like to highlight. It
is crucial that the $L_{\infty}$ structures explored in the current paper are
dependent on the choice of connections for both the tangent bundle and the Lie
algebroid. Similar settings have been studied and led to the theory of
secondary characters [18, 12, 14], see also [5, 26] for the discussion on the
compatible connections. Despite the fact that the secondary characters reside
in Lie algebroid cohomology rather than the Weil algebra [1, 2], the latter is
a natural location for Atiyah-Chern classes, which hopefully brings us new
information on the equivariant theory.

The Atiyah class of Lie algebroids has been studied extensively in various
settings, including smooth algebraic varieties [7], dg-manifolds [9] and Lie
pairs [27]. In both the dg-manifold and Lie pair settings, it has been shown
that the Atiyah and Atiyah-Chern classes coincide for regular Lie algebroids
[10, 31]. These results can be compared with the construction using
$L_{\infty}$ spaces [20]. Our computation serves as a complement to the above
results and is particularly applicable to non-regular Lie algebroids.

A recent paper by Calaque, Campos and Nuiten [6] establishes an equivalence of
$\infty$-categories between $L_{\infty}$ algebras that arise from graded
bundles and $L_{\infty}$ algebroids over a smooth manifold $M$. Within their
framework, we believe that the specific recursive computations for the
structure maps could be used to refine some aspects of the correspondence. For
instance, it would be intriguing to consider our perturbation approach for
more general $L_{\infty}$ algebras that arise from graded bundles in simpler
forms.

There are several exciting directions for future study. Firstly, it would be
intriguing to study the geometric deformation and the corresponding Atiyah
class of $L_{\infty}$ algebras over dg manifolds [29]. Secondly, it would be
desirable to realise all Atiyah-Chern classes in the construction of Cartan
models for equivariant cohomology theories of Lie algebroids, which is not yet
clear.

The paper is organized as follows.

In Section 2 we introduce the geometric deformation of the $L_{\infty}$
algebra $\mathfrak{h}\oplus\mathfrak{v}$ that arises from the semidirect sum
extension. We use an adjusted version of Fedosov’s method to demonstrate that
the requirement for the geometric extension can be solved recursively.

In Section 3 we compute lower brackets of the deformed $L_{\infty}$ algebra,
and show that the nilpotency of the differential results in precise conditions
for a vector bundle to become a Lie algebroid. We also investigate the
homotopy equivalences of different deformed structures.

In Section 4 we define and calculate the Atiyah and Atiyah-Chern classes of
$L_{\infty}$ algebras, which are cohomological classes that encodd both the
smooth geometry of bundles and the nonlinear products of Lie algebroids. We
discuss the connection between Atiyah-Chern classes and existing results.

In Appendix A we provide a comprehensive review of the undeformed curved
$L_{\infty}$ algebra which encodes the smooth geometry. We introduce a
modified version of Fedosov’s method, which allows for the recursive
computation of the $L_{\infty}$ structure while accommodating arbitrary gauge
choices. The homotopy equivalence of pertinent structures is discussed.

In Appendix B some lower brackets of the geometrically deformed $L_{\infty}$
algebra are computed.

##  2\. Curved $L_{\infty}$ structures from vector bundles and geometric
deformation

In this section we firstly review the construction of the curved $L_{\infty}$
algebra encoding a real vector bundle, then discuss deformations of the
geometric type.

Due to the geometric nature, we shall always consider the base field to be
$\mathbb{R}$. Let $\mathbb{V}$ be a $\mathbb{Z}$-graded vector space. For any
$n$-tuple of elements $v_{1},\cdots,v_{n}\in\mathbb{V}$ and $\tau\in S_{n}$,
the Koszul sign factor is a $\\{1,-1\\}$-valued function defined by the
equation

$v_{\tau(1)}\cdot v_{\tau(2)}\cdots
v_{\tau(n)}=\epsilon_{\tau}^{v_{1},\cdots,v_{n}}v_{1}\cdot v_{2}\cdots v_{n}$

in ${\rm Sym}^{n}\mathbb{V}$. The Koszul sign factor naturally generalises to
modules over unital commutative graded algebras. Let $Sh(p,q)$ be the set of
$(p,q)$ unshuffles inside the permutation group $S_{p+q}$ in $p+q$ letters.

######  Definition 1.

Let $(A,d_{A})$ be a non-negatively graded unital commutative differential
algebra with a nilpotent differential ideal $I$. A curved $L_{\infty}$ algebra
over $(A,d_{A})$ is a $\mathbb{Z}$-graded finitely generated projective
$A$-module $\mathbb{V}$, equipped with a collection of $(A,d_{A})$-linear maps
(i.e., the $n$-ary brackets)

$l_{n}:{\rm Sym}^{n}_{A}(\mathbb{V}[1])\to\mathbb{V}[1]$

of degree $1$ for $n\geq 0$, such that

  1. (1)

for every $k$-tuples $v_{1},\cdots,v_{k}$ of homogeneous elements in
$\mathbb{V}[1]$,

$\sum_{i=0}^{k}\sum_{\tau\in
Sh(i,k-i)}\epsilon_{\tau}^{v_{1},\cdots,v_{k}}l_{k-i+1}(l_{i}(v_{\tau(1)},\dots,v_{\tau(i)}),v_{\tau(i+1)},\dots,v_{\tau(k)})=0,$

  2. (2)

the $0$-ary bracket $l_{0}:A\to\mathbb{V}[1]$ maps $1\in A$ to an element in
the submodule $I(\mathbb{V}[1])$.

∎

The $(A,d_{A})$-linearity condition means all maps $\\{l_{n}\\}_{n=0,n\neq
1}^{\infty}$ are $A$-linear, except for $l_{1}$, which satisfies the Leibniz
rule

$l_{1}(fv)=(d_{A}f)v+(-1)^{|f|}fl_{1}(v),\forall f\in A,v\in\mathbb{V}.$

######  Remark 2.

A curved $L_{\infty}$ algebra structure defined above on $\mathbb{V}$ is the
same as a (curved) $L_{\infty}[1]$ algebra structure on $\mathbb{V}[1]$ in
literatures such as [29]. In both cases, the Koszul sign factors are
calculated according to the degree in $\mathbb{V}[1]$ as opposed to
$\mathbb{V}$. ∎

The conditions on $\\{l_{n}\\}$ operations can be encoded222By virtue of the
f.g. projective module condition, $\mathbb{V}$ becomes reflexive, and as such,
taking dual is reversible. into a coboundary condition $d^{2}=0$ on the
Chevalley-Eilenberg (CE) cochain

$C^{*}(\mathbb{V}):=\widehat{{\rm Sym}}_{A}(\mathbb{V}^{\vee}[-1])$

with the differential being a derivation (on $(A,d_{A})$-algebra) of degree
$1$ completely determined by all the (duals of) $l_{n}$ operations,

$d\omega=\sum_{n=0}^{\infty}\frac{1}{n!}l_{n}^{\vee}(\omega),\quad\forall\omega\in\mathbb{V}^{\vee}[-1].$

The last condition in Definition 2 involving the nilpotent ideal $I$ now
becomes the requirement that $d$ preserves the ideal $\widehat{{\rm
Sym}}^{\geq 1}_{A}(\mathbb{V}^{\vee}[-1])$ up to some elements in $I$.

Given this context, we define the notion of maps between two $L_{\infty}$
algebras.

######  Definition 3.

Given two $L_{\infty}$ algebras $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ over
a commutative dg algebra $(A,d_{A})$, a map from $\mathfrak{g}_{1}$ to
$\mathfrak{g}_{2}$ is a commutative dg algebra map

$C^{*}(\mathfrak{g}_{2})\to C^{*}(\mathfrak{g}_{1})$

that preserves the filtration defined by the nilpotent ideal $I\subset A$. ∎

In the current work, we will also discuss the homotopy equivalence of
different $L_{\infty}$ structures over the same underlying space.

######  Definition 4.

Two $L_{\infty}$ algebra structures on $\mathbb{V}$ are homotopic if there
exists a differentials $d^{(t)}$ on the family version of CE cochain
$\Omega_{[0,1]}\otimes_{\mathbb{R}}C^{*}(\mathbb{V})$, viewed as a curved
$L_{\infty}$ algebra over
$(\Omega_{[0,1]}\otimes_{\mathbb{R}}A,d_{t}\otimes_{\mathbb{R}}1+1\otimes_{\mathbb{R}}d_{A})$,
such that

  1. (1)

$d^{(t)}$ splits into a flat connection $d_{t}+F(t)$ on the (trivial)
$C^{*}(\mathbb{V})$-bundle over the one dimensional simplex $\Delta^{1}=[0,1]$
and a $t$-parameterised CE differential $d^{(t)}_{CE}$,

  2. (2)

$d_{t}+F(t)$ preserves the ideal
$\Omega_{[0,1]}\otimes_{\mathbb{R}}\widehat{{\rm Sym}}^{\geq
1}_{A}(\mathbb{V}^{\vee}[-1])$ up to elements in
$\Omega_{[0,1]}\otimes_{\mathbb{R}}I$,

  3. (3)

$d^{(t)}_{CE}|_{t=0}$ and $d^{(t)}_{CE}|_{t=1}$ coincide with the CE
differentials of the two given $L_{\infty}$ structures.

∎

######  Proposition 5 ([19, 21]).

Given a smooth manifold $M$ and a Levi-Civita connection $\nabla$ on the
tangent bundle $T_{M}$, there exists a curved $L_{\infty}$ algebra structure,
over the dg algebra $(\Omega_{M},d)$, on the de Rham complex333Unless
explicitly noted, the tensor product $\otimes$ is over the sheaf/ring of
smooth functions $C^{\infty}_{M}$.

$\mathfrak{h}\equiv\Omega_{M}\otimes\Gamma(T_{M})[-1]$

such that

  1. (1)

there exists an element $\mathbf{c}\in\Omega_{M}^{\geq
1}\otimes\Gamma(T_{M})\subset\mathfrak{h}[1]$ that uniquely determines the
$0$-ary operation through

$l_{0}:\Omega_{M}\equiv{\rm Sym}^{0}\mathfrak{h}[1]\to\mathfrak{h}[1]:f\mapsto
f\wedge\mathbf{c}\,\quad\forall f\in\Omega_{M},$

  2. (2)

the unary operation is determined by the $\Omega_{M}$-extended connection
$\nabla$ via

$l_{1}:\mathfrak{h}[1]\equiv{\rm
Sym}^{1}\mathfrak{h}[1]\to\mathfrak{h}[1]:fY\mapsto dfY+(-1)^{|f|}f\nabla Y$

for all $f\in\Omega_{M}$ and $Y\in\Gamma(T_{M})$,

  3. (3)

up to a gauge444The gauge choice can be understood as some further conditions
needed to uniquely specify $\\{l_{n}\\}_{n=2}^{\infty}$ once $l_{0}$ and
$l_{1}$ are chosen. Within the context of current proposition, different gauge
choices lead to homotopic $L_{\infty}$ structure, see Appendix A. choice,
further $n$-ary brackets are uniquely defined by the $L_{\infty}$ relations
and the previous two brackets.

Furthermore, different choices of connections on $T_{M}$ lead to homotopic
$L_{\infty}$ structures.

Let us choose a set of local coordinates $\\{x^{i}\\}_{i=1}^{{\rm dim}(M)}$ on
$M$, this uniquely determines the frame of local coordinate vector fields
$\\{y_{i}\equiv\frac{\partial}{\partial x^{i}}\\}_{i=1}^{{\rm dim}(M)}$ on
$T_{M}$. The curving $\mathbf{c}$ is then given by $dx^{i}\otimes
y_{i}\in\mathfrak{h}[1]$. Note that the tensor product is taken over the ring
of smooth functions. Under the coordinate change, $\\{x^{i}\\}_{i=1}^{{\rm
dim}(M)}\to\\{\tilde{x}^{i}\\}_{i=1}^{{\rm dim}(M)}$,

$d\tilde{x}^{i}=\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}dx^{j},\quad\tilde{y}_{i}=\frac{\partial
x^{j}}{\partial\tilde{x}^{i}}y_{j}$
$d\tilde{x}^{i}\otimes\tilde{y}_{i}=\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}dx^{j}\otimes\frac{\partial
x^{j^{\prime}}}{\partial\tilde{x}^{i}}y_{j^{\prime}}=\frac{\partial
x^{j^{\prime}}}{\partial\tilde{x}^{i}}\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}dx^{j}\otimes y_{j^{\prime}}$

This verifies that $\mathbf{c}$ is a well-defined object in
$\Omega^{1}_{M}\otimes\Gamma(T_{M})$.

Higher operations can be computed recursively from the CE cochain
$\widehat{{\rm Sym}}(\mathfrak{h}^{\vee}[-1])$. We demonstrate this in Lemma
61 and Proposition 62 provided in Appendix A. Additionally, we establish in
Proposition 65 that these structures are homotopy equivalent.

######  Remark 6.

A gauge choice can be determined by specifying the desired values of the dual
brackets $\\{l_{n}^{\vee}\\}_{n=2}^{\infty}$ under the homotopy contraction
map $\delta^{-1}$, which locally is represented as

$\frac{1}{p+q}y^{i}\iota_{\frac{\partial}{\partial x^{i}}}\,{\rm
on}\,\,\Omega_{M}^{p}\otimes\Gamma\big{(}{\rm
Sym}^{q}(T_{M}^{\vee})\big{)},\,\forall p+q>0.$

In the case of [17] and implicitly in the jet-bundle approach [19], a popular
choice is to require that all these images vanish. This particular choice is
referred to as the $\delta^{-1}$ gauge. ∎

######  Definition 7.

Let $\mathfrak{g}$ be a curved $L_{\infty}$ algebra over the unital
commutative dg algebra $(A,d_{A})$. An $L_{\infty}$ module of $\mathfrak{g}$
is a finitely generated $\mathbb{Z}$-graded projective $A$-module
$\mathbb{V}$, equipped with a collection of $(A,d_{A})$-linear maps (i.e., the
$n$-ary brackets)

$l^{\prime}_{n}:{\rm
Sym}^{n-1}_{A}(\mathfrak{g}[1])\otimes_{A}\mathbb{V}[1]\to\mathbb{V}[1]$

of degree $1$ for $n\geq 1$, such that for any $k$-tuple $u_{1},\cdots,u_{k}$
of homogeneous elements in $\mathfrak{g}[1]$ and for any element
$v\in\mathbb{V}[1]$

$\displaystyle\sum_{i=0}^{k}\sum_{\tau\in
Sh(i,k-i)}\epsilon_{\tau}^{u_{1},\cdots,u_{k}}l^{\prime}_{k-i+1}(l_{i}(u_{\tau(1)},\dots,u_{\tau(i)}),u_{\tau(i+1)},\dots,u_{\tau(k)},v)$
$\displaystyle+$ $\displaystyle\sum_{i=1}^{k}\sum_{\tau\in
Sh(k-i,i)}\epsilon_{\tau}^{u_{1},\cdots,u_{k}}(-1)^{(\sum_{j=1}^{k-i}|u_{\tau(j)}|)(1+\sum_{j=k-i+1}^{k}|u_{\tau(j)}|+|v|)}$
$\displaystyle\qquad\qquad\times
l^{\prime}_{k-i+1}(u_{\tau(1)},\dots,u_{\tau(k-i)},l^{\prime}_{i}(u_{\tau(k-i+1)},\dots,u_{\tau(k)},v))=0.$

∎

For any $L_{\infty}$ $\mathfrak{g}$-module $\mathbb{V}$, there exists a
natural $L_{\infty}$ algebra structure on $\mathfrak{g}\oplus\mathbb{V}$,
where the $n$-ary brackets are uniquely determined by
$\\{l_{n}^{\vee}\\}_{n=0}^{\infty}$ and
$\\{(l^{\prime}_{n})^{\vee}\\}_{n=1}^{\infty}$. Moreover, the natural maps
$\mathfrak{g}\oplus\mathbb{V}\to\mathfrak{g}$ and
$\mathfrak{g}\to\mathfrak{g}\oplus\mathbb{V}$ are morphisms of $L_{\infty}$
algebras over $(A,d_{A})$. This is referred to as a splitting Abelian
extension (, or the semidirect sum) of $L_{\infty}$ algebras $\mathfrak{g}$ by
(resp. and) $\mathbb{V}$.

######  Definition 8.

Fix a (curved) $L_{\infty}$ algebra $\mathfrak{g}$, two $\mathfrak{g}$-module
structures on $\mathbb{V}$ are homotopic, if they induces homotopic
$L_{\infty}$ algebra structures on $\mathfrak{g}\oplus\mathbb{V}$. ∎

######  Proposition 9 ([19]).

Given the $L_{\infty}$ algebra $\mathfrak{h}$ as defined in Proposition 5, and
given a vector bundle $V$ of finite rank over $M$ with a connection
$\nabla^{\prime}$, there exists a $L_{\infty}$ $\mathfrak{h}$-module
structure, over the differential dg algebra $(\Omega_{M},d)$, on the de Rham
complex

$\mathfrak{v}\equiv\Omega_{M}\otimes\Gamma(V)$

such that

  1. (1)

the unary operation is determined by the $\Omega_{M}$-extended connection
$\nabla^{\prime}$ via

$l^{\prime}_{1}:\mathfrak{v}[1]\to\mathfrak{v}[1]:f\alpha\mapsto
df\alpha+(-1)^{|f|}f\nabla^{\prime}\alpha$

for all $f\in\Omega_{M}$ and $\alpha\in\Gamma(V)$,

  2. (2)

up to a gauge choice, further $n$-ary brackets
$\\{l^{\prime}_{n}\\}_{n=2}^{\infty}$ are uniquely defined by the $L_{\infty}$
relations and the previous unary bracket.

Furthermore, different choices of connections on $V$ leads to homotopic
$L_{\infty}$ module structures.

The proof is similar to that of Proposition 5.

As a result of Proposition 9, there exists a splitting Abelian extension of
$L_{\infty}$ algebras

$\mathfrak{v}\to\mathfrak{h}\oplus\mathfrak{v}\to\mathfrak{h},$

with a section $\mathfrak{h}\to\mathfrak{h}\oplus\mathfrak{v}$. It is possible
to deform the $L_{\infty}$ structure on $\mathfrak{h}\oplus\mathfrak{v}$ while
maintaining only the $L_{\infty}$ morphism
$\mathfrak{h}\to\mathfrak{h}\oplus\mathfrak{v}$. Then the CE differential on
the deformed $L_{\infty}$ algebra must necessarily preserves the ideal
generated by $\mathfrak{v}^{\vee}[-1]$ in
$C^{*}(\mathfrak{h}\oplus\mathfrak{v})$. We also want the $L_{\infty}$ module
structure on $\mathfrak{v}$ to be easily distinguished from any new
structures. The above considerations lead to the following definition of
geometric deformation, which are useful in studying AKSZ type of quantum field
theories.

######  Definition 10.

Let $\mathfrak{h}$ be a curved $L_{\infty}$ algebra over $(A,d_{A})$ and let
$\mathfrak{v}$ be an $\mathfrak{h}$-module. Assigning weight $-1$ to
$\mathfrak{v}$, we say that an $L_{\infty}$-structure
$\\{\tilde{l}_{n}\\}_{n\geq 0}$ on $\mathfrak{h}\oplus\mathfrak{v}$
geometrically deforms the splitting Abelian extended $L_{\infty}$ algebra on
$\mathfrak{h}\oplus\mathfrak{v}$ if

  1. (1)

each of the brackets $\tilde{l}_{n}$ has non-negative weights, and

  2. (2)

the weight-$0$-components of $\tilde{l}_{n}$,

${\rm Sym}^{n}(\mathfrak{h}[1])\to\mathfrak{h}[1]$

and

$\mathfrak{v}[1]\otimes{\rm Sym}^{n}(\mathfrak{h}[1])\to\mathfrak{v}[1]$

coincide with operations in the preceding Abelian extension.

∎

It is often useful to introduce a (formal) parameter $\hbar$ that controls the
degree of deformation. In the above context, one can assign the formal power
$\hbar^{k}$ to the weight-$k$-components of the deformed
bracket$\\{\tilde{l}_{n}\\}_{n=0}^{\infty}$.

The definition above is explicitly dependent on the chosen $L_{\infty}$-module
structure on $\mathfrak{v}$. However, this dependence is not as strict as one
might think, and the $L_{\infty}$-module structure can be reconstructed from a
broader characterization provided below.

######  Proposition 11.

Let $\mathfrak{h}$ be a curved $L_{\infty}$ algebra over $(A,d_{A})$ and let
$\mathfrak{v}$ be a finitely generated $\mathbb{Z}$-graded projective
$A$-module. Suppose $\\{\tilde{l}_{n}\\}_{n=0}^{\infty}$ be any
$(A,d_{A})$-linear curved $L_{\infty}$ structure on
$\mathfrak{h}\oplus\mathfrak{v}$ such that

  1. (1)

the obvious inclusion

$\iota:\mathfrak{h}\to\mathfrak{h}\oplus\mathfrak{v}$

is an $L_{\infty}$ morphism,

  2. (2)

the curving is preserved by $\iota$.

Then there exists an $L_{\infty}$-module structure on $\mathfrak{v}$ such that
$\\{\tilde{l}_{n}\\}_{n=0}^{\infty}$ is a geometric deformation in the sense
of Definition 2.

###### Proof.

Assigning weight $-1$ to $\mathfrak{v}$ as before, the operations
$\\{\tilde{l}_{n}\\}_{n=0}^{\infty}$ decomposes accordingly. The weight
$0$-components have to satisfy the quadratic $L_{\infty}$ relation since
negatively weighted operations are not allowed, due to the property of
$\iota$. Further more, the components

${\rm Sym}^{n}(\mathfrak{h}[1])\to\mathfrak{h}[1]$

for any $n$ is entirely fixed by the $L_{\infty}$ structure on $\mathfrak{h}$.
So the remaining components of weight $0$-operations are forced to encode an
$\mathfrak{h}$-module structure on $\mathfrak{v}$.∎

######  Remark 12.

It is also interesting to look for non-splitting extensions of $\mathfrak{h}$
by $\mathfrak{v}$, which might encodes novel structures. However, such
extensions are hard to find in the context of smooth geometry. Abelian
extensions of $\mathfrak{h}$ by module $\mathfrak{v}$ are usually classified
by $H^{2}(\mathfrak{h},\mathfrak{v})$, which is uninteresting in our case due
to the acyclicity (in positive degrees) of $\mathfrak{h}$ [19]. A similar
conclusion can be drawn for nonabelian extensions, which are in general
classified by commutative dg-algebra maps of CE cochains
$C^{*}(Der(\mathfrak{v}))\to C^{*}(\mathfrak{h})$[22, 32]. The existence of
such nontrivial maps is again not promising, since the codomain is acyclic in
positive degrees. ∎

This following theorem highlights an intimate connection between the geometric
deformations of $\mathfrak{h}\oplus\mathfrak{v}$ and the Lie algebroid
structures on $V$.

######  Theorem 13.

Any $(\Omega_{M},d)$-linear $L_{\infty}$ structure geometrically deformed from
an Abelian extension $\mathfrak{h}\oplus\mathfrak{v}$ is uniquely determined
by a Lie algebroid structure on $V$.

The proof involves the following steps:

  1. (1)

Through degree analysis, we identify a particular type of nontrivial
operations allowed on top of a splitting Abelian extension.

  2. (2)

By utilising duality, these new operations can be encoded in a defining
equation and a consistency equation.

  3. (3)

The defining equation can be solved recursively and uniquely, once certain
“initial conditions” are specified.

  4. (4)

The consistency equation gives nothing but the requirement for those “initial
conditions”, which asks for $V$ to be a Lie algebroid in arbitrary gauges.

We will discuss the first three steps in Section 2 and postpone the final step
until Section 3.

Consider possible new operations

$\rho_{n+m}:{\rm Sym}^{n}(\mathfrak{v}[1])\otimes{\rm
Sym}^{m}(\mathfrak{h}[1])\to\mathfrak{h}[1],\quad n\geq 1,\,m\geq 0,$

and

$\mu_{n+m}:{\rm Sym}^{n}(\mathfrak{v}[1])\otimes{\rm
Sym}^{m}(\mathfrak{h}[1])\to\mathfrak{v}[1],\quad n\geq 2\,m\geq 0.$

For any element
$\alpha\in\mathfrak{h}[1]\oplus\mathfrak{v}[1]\cong\Omega_{M}\otimes\Gamma(T_{M}\oplus
V[1])$, we say that $\alpha$ is of form degree $k$, if
$\alpha\in\Omega^{k}_{M}\otimes\Gamma(T_{M}\oplus V[1])$, and is of inner
degree $0$ (resp. $-1$) if it takes value in $\Gamma(T_{M})$ (resp.
$\Gamma(V[1])$). The total degree is the sum of form degree and inner degree.
Similarly one determines the grading for tensors and duals of
$\mathfrak{h}[1]\oplus\mathfrak{v}[1]$.

The operation $\rho_{n+m}$ has a total degree $1$ and an inner degree $n$.
Therefore, its form degree is $1-n\geq 0$. Since $n\geq 1$, we have $n=1$.
Thus, the operation $\rho_{n+m}$ has a form degree $0$ and corresponds to a
bundle map. A similar argument applies to $\mu_{n+m}$, which has a total
degree of $1$ and an inner degree of $n-1$. So in total, only the operations
$\\{\rho_{1+m}\\}_{m=0}^{\infty}$ and $\\{\mu_{2+m}\\}_{m=0}^{\infty}$
survive, which are induced from the corresponding bundle maps555In general the
$l_{1}$ operation on an $L_{\infty}$ algebra $\mathfrak{g}$ over a commutative
dg-algebra $(A,d_{A})$ needs not to be $A$-linear, we shall have
$l_{1}(fX)=\pm fl_{1}(X)+d_{A}fX$ for $f\in A$ and $X\in\mathfrak{g}$. But the
operation $\rho_{1}$ does not have nontrivial form degree, and hence is
$\Omega_{M}$-linear. Another way to understand this is, $\rho$ is an off-
diagonal component of a connection on the graded bundle $T_{M}\oplus V[1]$..

To solve the constraints on the $L_{\infty}$ operations, we consider the CE
cochain complex

$C^{*}(\mathfrak{h}\oplus\mathfrak{v})\equiv\widehat{{\rm
Sym}}(\mathfrak{h}^{\vee}[-1]\oplus\mathfrak{v}^{\vee}[-1]).$

We shall use $d^{\nabla}$ to denote the CE differential on $\widehat{{\rm
Sym}}(\mathfrak{h}^{\vee}[-1])$, and use $d^{\nabla^{\prime}}$ to denote the
CE differential dualizing the $L_{\infty}$ module structure on $V$. Let
$\\{y_{i}\\}_{i=1}^{{\rm dim}(M)}$ and $\\{v_{a}\\}_{a=1}^{{\rm rk}(V)}$ be
local basis for bundle $T_{M}$ and $V$ respectively(, likewise
$\\{y^{i}\\}_{i=1}^{{\rm dim}(M)}$ and $\\{v^{a}\\}_{a=1}^{{\rm rk}(V)}$ for
the dual basis). Locally we have that

$d^{\nabla}=\delta+\nabla+\sum_{n>1}\frac{1}{n!}\big{(}(l_{n})^{i}_{i_{1},i_{2},\cdots,i_{n}}\,y^{i_{1}}\cdots
y^{i_{n}}\wedge-\big{)}\circ\frac{\partial}{\partial
y^{i}},\quad\delta=dx^{i}\frac{\partial}{\partial y^{i}}.$
$d^{\nabla^{\prime}}=\nabla^{\prime}+\sum_{n>0}\frac{1}{n!}\big{(}(l^{\prime}_{1+n})^{b}_{i_{1},i_{2},\cdots,i_{n},a}\,y^{i_{1}}\cdots
y^{i_{n}}v^{a}\wedge-\big{)}\circ\frac{\partial}{\partial v^{b}}.$

The coefficient $(l_{n})^{i}_{i_{1},i_{2},\cdots,i_{n}}$ is given by the
$y_{i}$-component of

$l_{n}(y_{i_{1}},y_{i_{2}},\cdots,y_{i_{n}}),$

and similarly for other coefficients. The connections $\nabla$ and
$\nabla^{\prime}$ on $T_{M}$ and $V$ induce connections on the products of
dual bundles $\widehat{{\rm Sym}}(T^{\vee}_{M})$ and $\widehat{{\rm
Sym}}(V^{\vee}[-1])$, which are again denoted by $\nabla$ and
$\nabla^{\prime}$ unless explicitly stated otherwise. The $L_{\infty}$
structure of the semi-direct product is prescribed by the condition
$(d^{\nabla}+d^{\nabla^{\prime}})^{2}=0$, and all the higher brackets can be
solved recursively as shown in Appendix A.

A geometrically extended $L_{\infty}$ structure on
$\mathfrak{h}\oplus\mathfrak{v}$ has new components of the CE differential
dualising brackets $\\{\rho_{1+m}\\}_{m=0}^{\infty}$ and
$\\{\mu_{2+m}\\}_{m=0}^{\infty}$,

$d^{\rho}:=\sum_{m}\frac{1}{m!}\big{(}(\rho_{1+m})^{i}_{i_{1},i_{2},\cdots,i_{m},a}y^{i_{1}}\cdots
y^{i_{m}}v^{a}\wedge-\big{)}\circ\frac{\partial}{\partial y^{i}},$
$d^{\mu}:=\sum_{m}\frac{1}{m!2!}\big{(}(\mu_{2+m})^{c}_{i_{1},i_{2},\cdots,i_{m},a,b}y^{i_{1}}\cdots
y^{i_{m}}v^{a}\wedge v^{b}\wedge-\big{)}\circ\frac{\partial}{\partial v^{c}}.$

To simplify the notations, we shall denote

$\frac{1}{n!}\big{(}(l_{n})^{i}_{i_{1},\cdots,i_{n}}\,y^{i_{1}}\cdots
y^{i_{n}}\wedge-\big{)}\circ\frac{\partial}{\partial y^{i}},\quad n\geq 2$

by $d^{\nabla}_{n}$, and similarly for $d^{\nabla^{\prime}}_{n+1}$ ($n\geq
1$), $d^{\rho}_{n+1}$ and $d^{\mu}_{n+2}$ ($n\geq 0$). To make notations
further compatible, we shall also use $d^{\nabla}_{1}$ and
$d^{\nabla^{\prime}}_{1}$ to denote $\nabla$ and $\nabla^{\prime}$
respectively.

We have the following weight decomposition of the quadratic relation,

(14) $[d^{\nabla}+d^{\nabla^{\prime}},d^{\rho}+d^{\mu}]=0,$ (15)
$[d^{\rho}+d^{\mu},d^{\rho}+d^{\mu}]=0.$

Equation (14) decomposes into components. For $n\geq 3$, the $n$-th component
of the equation is given by

(14.n) $-[\delta,d^{\mu}_{n}+d_{n}^{\rho}]=\sum_{1\leq k\leq
n-1}[d_{k}^{\nabla}+d^{\nabla^{\prime}}_{k},d^{\rho}_{n-k}]+\sum_{1\leq k\leq
n-2}[d_{k}^{\nabla}+d^{\nabla^{\prime}}_{k},d^{\mu}_{n-k}].$

For consistency, we have

(14.2)
$-[\delta,d^{\mu}_{2}+d_{2}^{\rho}]=[d_{1}^{\nabla}+d^{\nabla^{\prime}}_{1},d^{\rho}_{1}],$
(14.1) $-[\delta,d_{1}^{\rho}]=0.$

Fedosov’s method was initially devised to determine the “elements” of Weyl
bundles, establishing a differential using the Poisson bracket and employing
the homotopy contraction $\delta^{-1}$. However, when dealing with non-
symplectic scenarios, the task shifts to solving for derivation operators
$\\{d^{\rho}_{n}\\}_{n=1}^{\infty}$ and $\\{d^{\mu}_{n}\\}_{n=2}^{\infty}$. It
is important to note that the contraction $\delta^{-1}$ does not automatically
induce a natural contraction on the space of operators. While a precise
homotopy contraction exists for the de Rham complex of formal (poly-)vector
fields [15], extending this method directly to the deformed components
$\\{d^{\rho}_{n}\\}_{n=1}^{\infty}$ and $\\{d^{\mu}_{n}\\}_{n=2}^{\infty}$
encounters challenges due to the degree differences involved.

Here we adopt a slightly different strategy. The idea involves solving the
elements $\\{d^{\rho}_{n}(y^{i})|i=1,\cdots,{\rm dim}(M)\\}_{n=1}^{\infty}$
and $\\{d^{\mu}_{n}(v^{a})|a=1,\cdots,{\rm rk}(V)\\}_{n=2}^{\infty}$
recursively. Through this process, we can acquire the essential information
required to determine the corresponding operators. It is imperative to address
the compatibility and solvability of this set of recursive equations as a
preliminary step.

For the set of equations $\\{(\ref{eqnsol}.n)\\}_{n=1}^{\infty}$, the left-
hand sides are $\delta$-exact. We say that equation $(\ref{eqnsol}.n)$ is
compatible if its right-hand side commutes with $\delta$. Say that equation
$(\ref{eqnsol}.n)$ is solvable if its solution exists.

######  Lemma 16.

Suppose equation $(\ref{eqnsol}.k)$ is solvable for each $k\leq n$, then
equation $(\ref{eqnsol}.n+1)$ is compatible.

The proof is essentially the same as the compatibility result for equations of
$\\{d^{\nabla}_{k}\\}_{k=2}^{\infty}$ in Appendix A, therefore is omitted.

######  Proposition 17.

The extended differential $d^{\mu}+d^{\rho}$ can be solved recursively and
uniquely from equations $\\{(\ref{eqnsol}.n)\\}_{n=1}^{\infty}$ for any fixed
$d^{\rho}_{1}$ and $d^{\mu}_{2}$.

###### Proof.

Locally, as the commutative dg-algebra $C^{*}(\mathfrak{g})$ is generated by
$\\{y^{i},v^{a}\\}$’s, to solve for $d^{\rho}$ and $d^{\mu}$, one needs to
solve for $d^{\rho}(y^{i})$ and $d^{\mu}(v^{a})$, which are exactly the
coefficients in front of $\frac{\partial}{\partial y^{i}}$ and
$\frac{\partial}{\partial v^{a}}$ respectively.

Then the quadratic relation decomposes into:

(18) $-[\delta,d^{\rho}_{n+1}](y^{i})=\sum_{1\leq m\leq
n}[d^{\nabla}_{m},d^{\rho}_{n-m+1}](y^{i})+\sum_{1\leq m\leq
n}d^{\nabla^{\prime}}_{m}(d^{\rho}_{n-m+1}(y^{i})),$

where $m\leq n$ and a similar expression involving $d^{\mu}$. Since
$[\delta,d^{\rho}_{n+1}](y^{i})=\delta(d^{\rho}_{n+1}(y^{i}))$, the above
equations for

$\\{d^{\rho}_{n+1}(y^{i})\\}_{n\in\mathbb{Z}_{\geq
0}}\subset\Omega^{0}_{M}\otimes\Gamma\big{(}\widehat{{\rm
Sym}}(T^{\vee}_{M})\otimes V^{\vee}[-1]\big{)}$

can be solved by a homotopy contraction given by $\delta^{-1}$. It is easy to
check that

$\delta\circ\delta^{-1}+\delta^{-1}\circ\delta=id\,{\rm
on}\,\,\Omega^{p}\otimes\Gamma\big{(}{\rm
Sym}^{q}(T_{M}^{\vee})\otimes\widehat{{\rm
Sym}}^{\bullet}(V^{\vee}[-1])\big{)}$

for666Due to the failure in the case $p=q=0$, such equations have no
predictions on $d^{\rho}_{0+1}$ and $d^{\mu}_{0+2}$. $p+q>0$.

We claim that, suppose $\\{d^{\rho}_{k}\\}_{k=1}^{n}$ are solutions to
equations $\\{(\ref{eqnsol}.k)\\}_{k=1}^{n}$, then

(19) $-d^{\rho}_{n+1}(y^{i})=\sum_{1\leq m\leq
n}\delta^{-1}[d^{\nabla}_{m},d^{\rho}_{n-m+1}](y^{i})+\sum_{1\leq m\leq
n}\delta^{-1}d^{\nabla^{\prime}}_{m}(d^{\rho}_{n-m+1}(y^{i})),$

for all $y^{i}$ give a solution for $d^{\rho}_{n+1}$ to equation
$(\ref{eqnsol}.n+1)$.

In order to check that, observe $d_{m}^{\nabla^{\prime}}(y^{i})=0$,

$\sum_{1\leq m\leq
n}\delta\left([d_{m}^{\nabla},d^{\rho}_{n-m+1}](y^{i})\right)=\sum_{1\leq
m\leq
n}[\delta,[d_{m}^{\nabla},d^{\rho}_{n-m+1}]](y^{i})+[d_{m}^{\nabla},d^{\rho}_{n-m+1}](\delta
y^{i}),$ $\sum_{1\leq m\leq
n}\delta\left(d_{m}^{\nabla^{\prime}}(d^{\rho}_{n-m+1}(y^{i}))\right)=\sum_{1\leq
m\leq
n}[\delta,[d_{m}^{\nabla^{\prime}},d^{\rho}_{n-m+1}]](y^{i})+[d_{m}^{\nabla^{\prime}},d^{\rho}_{n-m+1}](\delta
y^{i}).$

All terms on the right-hand sides vanish by Lemma 16. So

$\displaystyle\delta\delta^{-1}\left(\sum_{1\leq m\leq
n}[d^{\nabla}_{m},d^{\rho}_{n-m+1}](y^{i})+d^{\nabla^{\prime}}_{m}(d^{\rho}_{n-m+1}(y^{i}))\right)$
$\displaystyle=\sum_{1\leq m\leq
n}[d^{\nabla}_{m},d^{\rho}_{n-m+1}](y^{i})+d^{\nabla^{\prime}}_{m}(d^{\rho}_{n-m+1}(y^{i})),$

which is
$-\delta\left(d^{\rho}_{n+1}(y^{i})\right)\equiv-[\delta,d^{\rho}_{n+1}](y^{i})$.

The induction starts with an arbitrary choice for $d^{\rho}_{1}$, which
satisfies equation $(\ref{eqnsol}.1)$ trivially.

The uniqueness of $d^{\rho}$ is established based on the uniqueness of
$d^{\rho}_{n}(y^{i})$ for each $i$, which can be attributed to the property
that $\delta^{-1}$ and $\delta$ act as inverse operations on
$d^{\rho}_{n}(y^{i})$. Hence the solution given by (19) is “isomorphically”
transformed from equation (18).

Now with solutions for $d^{\rho}$, a recursion can be applied to solve for
$d^{\mu}$, which involves choosing a set of local basis $\\{v^{a}\\}$ for
bundle $V$ and solving

$\displaystyle-\delta(d^{\mu}_{n+2}(v^{a}))=$ $\displaystyle\sum_{1\leq m\leq
n}[d^{\nabla^{\prime}}_{m},d^{\mu}_{n-m+2}](v^{a})+d^{\rho}_{m}(d^{\nabla^{\prime}}_{n-m+2}(v^{a}))$
$\displaystyle+\sum_{1\leq m<n}d^{\nabla}_{m}(d^{\mu}_{n-m+2})(v^{a})$

for $n\geq 1$. The initial condition

$-[\delta,d^{\mu}_{2}]=0$

is trivially satisfied for any choice of $d^{\mu}_{2}$. The details are
similar to the previous steps and hence are omitted. ∎

##  3\. Geometric meanings of lower brackets and nilpotency equation

We have shown that equation (14) can be solved recursively with arbitrary
choices of $d_{1}^{\rho}$ and $d_{2}^{\mu}$ as initial data. In particular,
one can easily compute some lower brackets.

######  Proposition 20.

In the $\delta^{-1}$ gauge, for arbitrarily fixed $\rho\equiv\rho_{1}$ and
$\mu\equiv\mu_{2}$, we have that

$\rho_{1+1}(X,\alpha)=\nabla_{X}\rho(\alpha)-\rho(\nabla^{\prime}_{X}\alpha),$
$\mu_{2+1}(X,\alpha,\beta)=-\big{(}{\nabla^{\prime}}_{X}^{(1)}\mu\big{)}(\alpha,\beta)+\frac{1}{2}R_{\nabla^{\prime}}(X,\rho(\alpha))(\beta)-\frac{1}{2}R_{\nabla^{\prime}}(X,\rho(\beta))(\alpha),$

for all $X\in\Gamma(T_{M})$ and $\alpha\in\Gamma(V)$, where

$\big{(}{\nabla^{\prime}}_{X}^{(1)}\mu\big{)}(\alpha,\beta)=\nabla^{\prime}_{X}\mu(\alpha,\beta)-\mu(\nabla^{\prime}_{X}\alpha,\beta)-\mu(\alpha,\nabla^{\prime}_{X}\beta).$

We now consider further constraints from equation (15) and show that they
encode precisely the data of being a Lie algebroid.

######  Proposition 21.

The equation

$(d^{\mu}+d^{\rho})^{2}=0$

is solvable by $\\{d^{\rho}_{n+1}\\}_{n=0}^{\infty}$ and
$\\{d^{\mu}_{n+2}\\}_{n=0}^{\infty}$ as given in Proposition 17 if and only if
the initial data $\rho_{1+0}$ and $\mu_{2+0}$ satisfy the first couple of
equations with the lowest hweight:

$[d^{\mu}_{2}+d^{\rho}_{2},d^{\rho}_{1}]=0,$
$\frac{1}{2}[d^{\mu}_{2},d^{\mu}_{2}]+d^{\rho}_{1}(d^{\mu}_{3})=0.$

In particular, the above two equations define the anchor and the torsion of a
Lie algebroid structure on $V$.

On $C^{1}(\mathfrak{h})$,the leading order equation gives

$\displaystyle-\mu_{2+0}(v_{b},v_{c})^{a}\rho_{1+0}(v_{a})^{i}-(\rho_{1+0}(v_{b})^{k}(\rho_{1+1}(y_{k},v_{c})^{i}$
$\displaystyle+\rho_{1+0}(v_{c})^{k}\rho_{1+1}(y_{k},v_{b})^{i}=0,\quad\forall\,b,c,i.$

Consider

$C_{bc}^{a}\equiv(\Gamma^{\prime})_{i}\,^{a}\,{}_{c}(\rho)^{i}\,_{b}-(\Gamma^{\prime})_{i}\,^{a}\,{}_{b}(\rho)^{i}\,_{c}-\mu^{a}_{bc},$

this defines the structure constants of an $\mathbb{R}$-bilinear skew-
symmetric bracket operation on $\Gamma(V)$ by $[v_{b},v_{b}]=C_{bc}^{a}v_{a}$,
and $[f\alpha,\beta]=f[\alpha,\beta]-\rho(\beta)(f)\alpha$. So the first
equation gives the compatibility between the bracket operation on
$\Gamma(T_{M})$ and that of $\Gamma(V)$,

$[\rho(v_{b}),\rho(v_{c})]=\rho(\nabla_{\rho(v_{b})}v_{c}-\nabla_{\rho(v_{c})}v_{b}-\mu(v_{b},v_{c}))=\rho([v_{b},v_{c}]).$

The next to leading order equation gives

$\displaystyle(\frac{1}{2}\mu_{2+0})^{a}_{b,c}v^{b}\wedge
v^{c}\wedge\frac{\partial}{\partial
v^{a}}\big{(}(\frac{1}{2}\mu_{2+0})^{d}_{e,f}v^{e}\wedge v^{f}\big{)}$
$\displaystyle+(\rho_{1+0})^{i}_{a}v^{a}\wedge\big{(}\frac{1}{2}(\mu_{2+1})^{d}_{i,b,c}v^{b}\wedge
v^{c}\big{)}=0.$

This is (, after taking the dual,) equivalent to the condition

$\sum_{a,b,c{\rm\,cyclic}}\mu(\mu(v_{a},v_{b}),v_{c})+\big{(}{\nabla^{\prime}}_{\rho(v_{a})}^{(1)}\mu\big{)}(v_{b},v_{c})-R_{\nabla^{\prime}}(\rho(v_{a}),\rho(v_{b}))(v_{c})=0,$

which is the Bianchi identity for Lie algberoids (see equation $(61)$ in
[18]). Note that although $\mu_{2+1}$ depends on the gauge conditions777See
Appendix B for an explicit formula in arbitrary gauge. coming from $l_{2}$ and
$l^{\prime}_{2}$, such gauge dependence does not survive under
antisymmetrisation of inputs in $\Gamma(V[1])$.

###### Proof.

We have shown that the first two components of equation (15) are equivalent to
the condition that $V$ has a Lie algebroid structure, whose anchor and torsion
are determined by $\rho_{1}(-)$ and $\mu_{2}(-,-)$.

It remains to show that components of higher weights of equation
$(d^{\mu}+d^{\rho})^{2}=0$ are satisfied automatically. By the nature of
derivation, this is true iff
$(d^{\mu}+d^{\rho})^{2}(v^{a})=(d^{\mu}+d^{\rho})^{2}(y^{i})=0$ for all
$a\in\\{1,\cdots,{\rm rk}(V)\\}$ and $i\in\\{1,\cdots,{\rm dim}(M)\\}$. For
higher brackets, one can apply a recursive argument as before, which we now
describe briefly.

Consider elements such as

$\sum_{p+q-1=n+4}\frac{1}{2}[d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}](v^{a})$

and

$\sum_{p+q-1=n+3}\frac{1}{2}[d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}](y^{i})$

where $n\geq 0$. By degree analysis, they are in the component

$\Omega^{0}\otimes\Gamma\big{(}\widehat{{\rm Sym}}^{\geq
1}(T^{\vee}_{M})\otimes\widehat{{\rm Sym}}(V^{\vee}[-1])\big{)}.$

As a linear map, $\delta$ is invertible when restricted to the above
component. So showing the vanishing of those elements is the same as showing
that

$\displaystyle\sum_{p+q-1=n+4}\frac{1}{2}\delta\left([d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}](v^{a})\right)$
$\displaystyle=$
$\displaystyle\sum_{p+q-1=n+4}\frac{1}{2}[\delta,[d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}]](v^{a})=0$

and

$\displaystyle\sum_{p+q-1=n+3}\frac{1}{2}\delta\left([d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}](y^{i})\right)$
$\displaystyle=$
$\displaystyle\sum_{p+q-1=n+3}\frac{1}{2}[\delta,[d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}]](y^{i})=0.$

By the (graded) Jacobi identity of the derivation operators and equation (14),
the commutator of type
$[\delta,[d^{\mu}_{p}+d^{\rho}_{p},d^{\mu}_{q}+d^{\rho}_{q}]]$ can be
rewritten as a sum of terms like
$-[d^{\nabla}_{s}+d^{\nabla^{\prime}}_{s},[d^{\mu}_{t}+d^{\rho}_{t},d^{\mu}_{u}+d^{\rho}_{u}]]$,
with $t+u\leq p+q-1$. Now an inductive argument can be applied here. ∎

The proof for Theorem 13 follows from Proposition 17 and Proposition 21.

######  Example 22.

A weak Lie algebra bundle is a Lie algebroid with a vanishing anchor. If
$\rho:V\to T_{M}$ vanishes, so are $\\{\rho_{1+n}\\}_{n=0}^{\infty}$ by the
recursive relation.

A Lie algebra bundle is a weak Lie algebra bundle whose Lie algebra structure
is fixed with respect to local trivialisations. As shown in [14], this is
equivalent to the condition that there exists a connection $\nabla^{\prime}$
on $V$ such that

$\displaystyle\nabla^{\prime}[-,-]-[\nabla^{\prime}(-),-]-[-,\nabla^{\prime}(-)]$
$\displaystyle=$
$\displaystyle\nabla^{\prime}\mu(-,-)-\mu(\nabla^{\prime}(-),-)-\mu(-,\nabla^{\prime}(-))=0.$

Therefore $\mu_{2+1}$, which depends on the covariant derivative of $\mu$,
vanishes. Further more, in the $\delta^{-1}$ gauge, for any $v^{a}$,

$d^{\mu}_{2+2}(v^{a})=-\delta^{-1}\left(d^{\nabla^{\prime}}_{2}(d^{\mu}_{2}(v^{a}))+d^{\mu}_{2}(d^{\nabla^{\prime}}_{2}(v^{a}))\right)=0.$

Similarly one can show that all operations in $\\{\mu_{2+n}\\}_{n=1}^{\infty}$
vanish in $\delta^{-1}$ gauge by an inductive argument. ∎

###  3.1. Family version and homotopy equivalence

By previous results, a geometrically deformed $L_{\infty}$ structure on
$\Omega_{M}\otimes\Gamma\big{(}T_{M}[-1]\oplus V\big{)}$ is determined by the
choices of connections $(\nabla,\nabla^{\prime})$ on $T_{M}\oplus V[1]$, the
gauge condition on the $\delta^{-1}$ image of CE differential, and the Lie
algebroid structure on $V$. In the case of splitting Abelian extension, the
Lie algebroid $V$ is trivial, we have shown that different $L_{\infty}$
structures are ultimately homotopic, see Appendix A.

In the presence of geometric deformation, a general argument on the homotopy
type of Fedosov differential can be used to demonstrate the following result.

######  Proposition 23.

Any two geometrically deformed $L_{\infty}$ structures on
$\Omega\otimes\Gamma\big{(}T_{M}[-1]\oplus V\big{)}$ with the same underlying
Lie algebroid structure are homotopic.

If one stays in $\delta^{-1}$ gauge and works in the contexts of infinite
jets, the result may be deduced from the formal exponential map as in [20].
The homotopy given in that context comes necessarily from a genuine
isomorphism of dg algebras. In the light of Fedosov’s method, we consider a
more computable approach with arbitrary gauge choice. By working with the
family version of $L_{\infty}$ algebra parameterised by $\Delta^{n}$ for any
$n$, we also specify the form of homotopy sits behind. The main result is
Proposition 28, from which Proposition 23 can be deduced.

We start with a family of Lie algebroid structure with chosen connections,
$(\rho_{t},[-,-]_{t},\nabla_{t},\nabla^{\prime}_{t})$ where $t\in\Delta^{n}$,
and consider a family of gauge conditions
$(\phi_{t},\psi_{t})\in\Gamma\big{(}{\rm Sym}^{\geq 3}(T_{M}^{\vee})\otimes
T_{M}\big{)}\oplus\Gamma\big{(}{\rm Sym}^{\geq 2}(T_{M}^{\vee})\otimes
V^{\vee}[-1]\otimes V[1]\big{)}$. By Lemma 64 in Appendix A, Proposition 17
and 21, there exists a $t$-parameterised CE differential

$\delta+d_{\geq 1}^{\nabla_{t}}+d_{\geq
1}^{\nabla^{\prime}_{t}}+d^{\rho_{t}}+d^{\mu_{t}},$

such that

$d_{1}^{\nabla_{t}}=\nabla_{t},\quad
d_{1}^{\nabla^{\prime}_{t}}=\nabla^{\prime}_{t},$ $\delta^{-1}(d_{\geq
2}^{\nabla_{t}}+d_{\geq 2}^{\nabla^{\prime}_{t}})=\phi_{t}+\psi_{t},$
$\delta^{-1}(d^{\rho_{t}}+d^{\mu_{t}})=0.$

Now consider an extended differential

$d_{t}+F(t)+\delta+d_{\geq 1}^{\nabla_{t}}+d_{\geq
1}^{\nabla^{\prime}_{t}}+d^{\rho_{t}}+d^{\mu_{t}}$

on

$\Omega_{\Delta^{n}}\otimes_{\mathbb{R}}\Omega_{M}\otimes\Gamma\big{(}{\rm
Sym}^{m}(T_{M}^{\vee}\oplus V^{\vee}[-1])\big{)}.$

As a derivation operator, $F(t)$ is an element of degree $1$ in

$\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Omega_{M}\otimes\Gamma\big{(}\widehat{{\rm
Sym}}(T_{M}^{\vee}\oplus V^{\vee}[-1])\otimes(T_{M}\oplus V[1])\big{)}.$

By degree analysis, one may write $F(t)=F^{0}(t)+F^{1}(t)$ with

$F^{0}(t)\in\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Gamma\big{(}\widehat{{\rm
Sym}}(T_{M}^{\vee})\otimes V^{\vee}[-1]\otimes V[1]\oplus\widehat{{\rm
Sym}}(T_{M}^{\vee})\otimes T_{M}\big{)},$
$F^{1}(t)\in\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Omega_{M}^{1}\otimes\Gamma\big{(}\widehat{{\rm
Sym}}(T_{M}^{\vee})\otimes V[1]\big{)}.$

However, the component $F^{1}(t)$ would not descend along the natural
surjective map
$\Omega_{\Delta^{n}}\otimes_{\mathbb{R}}C^{*}(\mathfrak{h}\oplus\mathfrak{v})\to\Omega_{\Delta^{n}}\otimes_{\mathbb{R}}C^{*}(\mathfrak{h})$,
and therefore should not be allowed in the extended differential of geometric
deformations. The nilpotency equation decomposes by form degrees,

(24) $[d_{t}+F^{0}(t),d_{\geq 1}^{\nabla_{t}}+d_{\geq
1}^{\nabla^{\prime}_{t}}]+[F^{0}(t),\delta]=0,$ (25)
$[d_{t}+F^{0}(t),d^{\rho_{t}}+d^{\mu_{t}}]=0,$ (26)
$d_{t}F^{0}(t)+\frac{1}{2}[F^{0}(t),F^{0}(t)]=0.$

Equation (24) contain the term $[F^{0}(t),\delta]$, which gives a recursive
way to solve for $F^{0}(t)$, given that some mild conditions are satisfied.
Equations (25) and (26) impose some consistency conditions needed to be
satisfied by $F^{0}(t)$. In homogeneous components, they contain infinite
number of components to look into. However, most are satisfied automatically
except for the first few, as we shall see in the following Lemma 27.

To facilitate future discussions, we assign hweight $k$ to the components of
$F^{0}$ in

$\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Gamma\big{(}\widehat{{\rm
Sym}}^{k}(T_{M}^{\vee})\otimes V^{\vee}[-1]\otimes V[1]\oplus\widehat{{\rm
Sym}}^{k+1}(T_{M}^{\vee})\otimes T_{M}\big{)}.$

######  Lemma 27.

Suppose $F^{0}$ is solved recursively from equation (24). Let $G(k)$ be the
hweight-$k$-component of equation (25) or (26). For any $k\geq 1$, if all
$G(m)$ with $0\leq m<k$ are satisfied, then $G(k)$ is solved automatically.

The proof is similar to that of Proposition 21 and Lemma 64 with an induction
on the homogeneous hweight of $G(m)$.

By the induction equation, it is possible to assign $F^{0}(t)$ to be an
element in

$\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Gamma\big{(}\widehat{{\rm
Sym}}^{\geq 1}(T_{M}^{\vee})\otimes V^{\vee}[-1]\otimes
V[1]\oplus\widehat{{\rm Sym}}^{\geq 2}(T_{M}^{\vee})\otimes T_{M}\big{)}.$

Indeed, locally writing

$\displaystyle F^{0}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}f^{i}_{jk}(t)y^{j}y^{k}\frac{\partial}{\partial
y^{i}}+f^{a}_{j,b}(t)y^{j}v^{b}\frac{\partial}{\partial v^{a}}+\cdots$
$\displaystyle\equiv$ $\displaystyle F^{0}_{2,0}(t)+F^{0}_{1,1}(t)+\cdots,$

the components $F^{0}_{0}$ and $F^{0}_{0,1}$ determines the subsequent
components inductively, and can be set to zero for our purpose. Consequently,
$F^{0}_{1,0}=0$.

The hweight-$0$-component of Equation (25) gives

$[d_{t},d_{1}^{\rho_{t}}]=0$

and

$[d_{t},d_{2}^{\mu_{t}}]+d_{1}^{\rho_{t}}(F^{0}_{1,1})=0.$

This is equivalent to the requirement that the $t$-dependence of the Lie
algebroid structure on $V$ is trivial, by a straight-forward calculation. The
hweight-$0$-component of Equation (26) is trivial, which immediately
guarantees that the equation holds by Lemma 27. Indeed, one checks that the
lowest hweight component of Equation (26) gives

$d_{t}F_{1,1}^{0}(t)=0$

and

$d_{t}F_{2}^{0}(t)=0,$

which can be verified explicitly. The above analysis is concluded in the
following proposition.

######  Proposition 28.

Given a family of connections $(\nabla_{t},\nabla^{\prime}_{t})$ of
$T_{M}\oplus V$ parameterised smoothly by $t\in\Delta^{n}$, and given a Lie
algebroid structure on $V$, there exists a flat family of geometric
deformations of $L_{\infty}$ algebra structures on
$\Omega_{M}\otimes\Gamma\big{(}T_{M}[-1]\oplus V\big{)}$. Conversely, any flat
family of geometrically deformed $L_{\infty}$ algebra structures on
$\Omega_{M}\otimes\Gamma\big{(}T_{M}[-1]\oplus V\big{)}$ given by a flat
connection $d_{t}+F(t)$ with $F(t)$ in

$\Omega^{1}_{\Delta^{n}}\otimes_{\mathbb{R}}\Gamma\big{(}\widehat{{\rm
Sym}}^{\geq 1}(T_{M}^{\vee})\otimes V^{\vee}[-1]\otimes
V[1]\oplus\widehat{{\rm Sym}}^{\geq 2}(T_{M}^{\vee})\otimes T_{M}\big{)}$

necessarily comes from a single Lie algebroid structure on $V$.

The fact that one can set $F^{0}_{0}$, $F^{0}_{1,0}$ and $F^{0}_{0,1}$ to zero
is not surprising — in a similar case [24], the flat connection which encodes
the homotopy equivalences was given by pulling back $d_{t}$ via an exponential
map, and there again the constant and linear terms are absent.

##  4\. Atiyah class and character

Atiyah class plays a crucial role in cotangent theories, particularly in
encoding the exterior derivatives of the trilinear and higher terms in the
action functional [11, 19]. Further more, it can be employed to construct
character classes in various contexts. Here we will firstly investigate the
relationship between Atiyah class of a geometrically deformed $L_{\infty}$
algebra and its underlying Lie algebroid structure. By utilizing this
connection, we then generate the Atiyah-Chern classes, which are relevant in
the one-loop computation of certain intriguing field theories.

Consider an $L_{\infty}$ algebra $\mathfrak{g}$, there exists a commutative
dg-algebra module $C^{*}(\mathfrak{g})\otimes_{\Omega_{M}}\mathfrak{g}[1]$
which plays the role of tangent bundle of the space $\mathfrak{g}[1]\equiv
B\mathfrak{g}$. There is a natural flat connection $\nabla_{B\mathfrak{g}}$ on
$T_{B\mathfrak{g}}$ given by a map 888Here the construction is slightly
different from [19]. We are realising the notion of connection in a purely
algebraic setting, where the splitting is naturally given. In the reference
the authors considered the connection over infinite jet bundle, whose local
definition depends on a choice of local trivialization of $T_{M}$.

$C^{*}(\mathfrak{g})\otimes\mathfrak{g}[1]\to\Omega^{1}_{B\mathfrak{g}}\otimes\mathfrak{g}[1]$

satisfying that $\nabla_{B\mathfrak{g}}(f\otimes s)=d^{DR}f\otimes s$ for all
$f\in C^{*}(\mathfrak{g})$ and $s\in\mathfrak{g}[1]$, where
$\Omega^{1}_{B\mathfrak{g}}$ is the module of Kähler differentials of
$C^{*}(\mathfrak{g})$. The Atiyah class of $\nabla_{B\mathfrak{g}}$ is defined
as the obstruction to the compatibility between $\nabla_{B\mathfrak{g}}$ and
the CE differential $d^{\mathfrak{g}}$, i.e.,

$At(\nabla_{B\mathfrak{g}}):=\nabla_{B\mathfrak{g}}\circ
d^{\mathfrak{g}}+d^{\mathfrak{g}}\circ\nabla_{B\mathfrak{g}},$

where we have extended $\nabla_{B\mathfrak{g}}$ to the de Rham complex
$\Omega^{*}_{B\mathfrak{g}}\otimes\mathfrak{g}[1]$. The differential on
$\Omega^{1}_{B\mathfrak{g}}\otimes\mathfrak{g}[1]\cong
C^{*}(\mathfrak{g})\otimes\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}[1]$ is
given by the CE differential of the $L_{\infty}$ module
$\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}[1]$, and we again denote it as
$d^{\mathfrak{g}}$.

######  Lemma 29.

Viewed as a linear map
$C^{*}(\mathfrak{g})\otimes\mathfrak{g}[1]\to\Omega^{1}_{B\mathfrak{g}}\otimes\mathfrak{g}[1]$,
the class $At(\nabla_{B\mathfrak{g}})$ is $d^{\mathfrak{g}}$-closed and
$\nabla_{B\mathfrak{g}}$-flat.

###### Proof.

The differential $d^{\mathfrak{g}}$ extends naturally from
$C^{*}(\mathfrak{g})\otimes M$ to $\Omega^{1}_{B\mathfrak{g}}\otimes M$ for
any $L_{\infty}$ module $M$. For linear maps, we have that

$d^{\mathfrak{g}}\big{(}At(\nabla_{B\mathfrak{g}})\big{)}=d^{\mathfrak{g}}\circ
At(\nabla_{B\mathfrak{g}})-At(\nabla_{B\mathfrak{g}})\circ
d^{\mathfrak{g}}=[d^{\mathfrak{g}},[d^{\mathfrak{g}},\nabla_{B\mathfrak{g}}]],$

which vanishes by Jacobi identity.

The connection $\nabla_{B\mathfrak{g}}$ extends from
$C^{*}(\mathfrak{g})\otimes\mathfrak{g}[1]$ to $C^{*}(\mathfrak{g})\otimes{\rm
End}(\mathfrak{g}[1])$, and can be shown flat by local computation. So

$\nabla_{B\mathfrak{g}}\big{(}At(\nabla_{B\mathfrak{g}})\big{)}=\nabla_{B\mathfrak{g}}\circ
At(\nabla_{B\mathfrak{g}})-At(\nabla_{B\mathfrak{g}})\circ\nabla_{B\mathfrak{g}}=0.$

∎

From the analysis of linear structure,

$At(\nabla_{B\mathfrak{g}})\in
C^{*}(\mathfrak{g})\otimes\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}[1]\cong
C^{*}(\mathfrak{g})\otimes\mathfrak{g}^{\vee}[-1]\otimes{\rm
End}\,\mathfrak{g}[1].$

###  4.1. Atiyah class of Lie algebroids

Now we consider a geometrically deformed $L_{\infty}$ algebras with the
underlying space $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{v}$ as before. A
local computation on basis $\\{1\otimes y_{i},1\otimes v_{a}|i=1,\cdots,{\rm
dim}(M),\,a=1,\cdots,{\rm rk}(V)\\}\subset
C^{*}(\mathfrak{g})\otimes\mathfrak{g}[1]$ produces the homogeneous components
of $At(\nabla_{B\mathfrak{g}})$ as a $C^{*}(\mathfrak{g})$-linear map which
takes value in
$C^{*}(\mathfrak{g})\otimes\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}[1]$.

Notice that

$\displaystyle d^{\mathfrak{g}}(1\otimes y_{i})$ $\displaystyle=$
$\displaystyle 1\otimes\nabla y_{i}+\sum_{n\geq 1}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}\otimes(l_{n+1})_{i_{1}i_{2}\cdots i_{n}i}\,^{j}y_{j}$
$\displaystyle+\sum_{n\geq 1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
y^{i_{n-1}}v^{b}\otimes(\rho_{n+1})_{i_{1}i_{2}\cdots i_{n-1}ib}\,^{j}y_{j}$
$\displaystyle+\sum_{n\geq 2}\frac{1}{2!}\frac{1}{(n-2)!}y^{i_{1}}\cdots
y^{i_{n-2}}v^{a}v^{b}\otimes(\mu_{n+1})_{i_{1}i_{2}\cdots
i_{n-2}iab}\,^{c}v_{c}$ $\displaystyle+\sum_{n\geq
0}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}v^{a}\otimes(l^{\prime}_{n+2})_{i_{1}i_{2}\cdots
i_{n}ia}\,^{b}v_{b},$

so

$\displaystyle\nabla_{B\mathfrak{g}}\circ d^{\mathfrak{g}}(1\otimes y_{i})$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
Dy^{i_{n}}\otimes(l_{n+1})_{i_{1}i_{2}\cdots i_{n}i}\,^{j}y_{j}$
$\displaystyle+\sum_{n\geq 2}\frac{1}{(n-2)!}y^{i_{1}}\cdots
Dy^{i_{n-1}}v^{b}\otimes(\rho_{n+1})_{i_{1}i_{2}\cdots i_{n-1}ib}\,^{j}y_{j}$
$\displaystyle+\sum_{n\geq 1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
y^{i_{n-1}}Dv^{b}\otimes(\rho_{n+1})_{i_{1}i_{2}\cdots i_{n-1}ib}\,^{j}y_{j}$
$\displaystyle+\sum_{n\geq 3}\frac{1}{2!}\frac{1}{(n-3)!}y^{i_{1}}\cdots
Dy^{i_{n-2}}v^{a}v^{b}\otimes(\mu_{n+1})_{i_{1}i_{2}\cdots
i_{n-2}iab}\,^{c}v_{c}$ $\displaystyle+\sum_{n\geq
2}\frac{1}{(n-2)!}y^{i_{1}}\cdots
y^{i_{n-2}}Dv^{a}v^{b}\otimes(\mu_{n+1})_{i_{1}i_{2}\cdots
i_{n-2}iab}\,^{c}v_{c}$ $\displaystyle+\sum_{n\geq
1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
Dy^{i_{n}}v^{a}\otimes(l^{\prime}_{n+2})_{i_{1}i_{2}\cdots
i_{n}ia}\,^{b}v_{b}$ $\displaystyle+\sum_{n\geq 0}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}Dv^{a}\otimes(l^{\prime}_{n+2})_{i_{1}i_{2}\cdots
i_{n}ia}\,^{b}v_{b}.$

This gives the component

$\displaystyle At(\nabla_{B\mathfrak{g}})(1\otimes y_{i})$ $\displaystyle=$
$\displaystyle Dy^{j}\otimes
l_{2}(y_{j},y_{i})+Dy^{j}v^{b}\otimes\rho_{3}(y_{j},y_{i},v_{b})+Dv^{b}\otimes\rho_{2}(y_{i},v_{b})$
$\displaystyle+\frac{1}{2!}Dy^{j}v^{a}v^{b}\otimes\mu_{4}(y_{j},y_{i},v_{a},v_{b})+Dv^{a}v^{b}\otimes\mu_{3}(y_{i},v_{a},v_{b})$
$\displaystyle+Dy^{j}v^{a}\otimes
l^{\prime}_{3}(y_{j},y_{i},v_{a})+Dv^{a}\otimes
l^{\prime}_{2}(y_{i},v_{a})+\cdots,$

where “$\cdots$” stands for terms with dependence on $\\{y^{i}\\}_{i=1}^{{\rm
dim}(M)}$.

Similarly, by

$\displaystyle d^{\mathfrak{g}}(1\otimes v_{a})$ $\displaystyle=$
$\displaystyle 1\otimes\nabla^{\prime}v_{a}+\sum_{n\geq
1}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}\otimes(l^{\prime}_{n+1})_{i_{1}i_{2}\cdots i_{n}a}\,^{b}v_{b}$
$\displaystyle+\sum_{n\geq 0}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}v^{b}\otimes(\mu_{n+2})_{i_{1}i_{2}\cdots i_{n}ba}\,^{c}v_{c}$
$\displaystyle+1\otimes\rho(v_{a})+\sum_{n\geq 1}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}\otimes(\rho_{n+1})_{i_{1}i_{2}\cdots i_{n}a}\,^{j}y_{j}$

and

$\displaystyle\nabla_{B\mathfrak{g}}\circ d^{\mathfrak{g}}(1\otimes v_{a})$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
Dy^{i_{n}}\otimes(l^{\prime}_{n+1})_{i_{1}i_{2}\cdots i_{n}a}\,^{b}v_{b}$
$\displaystyle+\sum_{n\geq 0}\frac{1}{(n-1)!}y^{i_{1}}\cdots
Dy^{i_{n}}v^{b}\otimes(\mu_{n+2})_{i_{1}i_{2}\cdots i_{n}ba}\,^{c}v_{c}$
$\displaystyle+\sum_{n\geq 0}\frac{1}{n!}y^{i_{1}}\cdots
y^{i_{n}}Dv^{b}\otimes(\mu_{n+2})_{i_{1}i_{2}\cdots i_{n}ba}\,^{c}v_{c}$
$\displaystyle+\sum_{n\geq 1}\frac{1}{(n-1)!}y^{i_{1}}\cdots
Dy^{i_{n}}\otimes(\rho_{n+1})_{i_{1}i_{2}\cdots i_{n}a}\,^{j}y_{j}.$

So

$\displaystyle At(\nabla_{B\mathfrak{g}})(1\otimes v_{a})$ $\displaystyle=$
$\displaystyle Dy^{j}\otimes
l^{\prime}_{2}(y_{j},v_{a})+Dy^{j}v^{b}\otimes\mu_{3}(y_{j},v_{b},v_{a})$
$\displaystyle+Dv^{b}\otimes\mu_{2}(v_{b},v_{a})+Dy^{j}\otimes\rho_{2}(y_{j},v_{a})+\cdots.$

######  Proposition 30.

The constant part of Atiyah class defined on the geometrically deformed
$L_{\infty}$ algebra $\mathfrak{g}$ is given by

$\displaystyle At(\nabla_{B\mathfrak{g}})_{0}$ $\displaystyle=$ $\displaystyle
Dy^{i}\otimes\left(\begin{array}[]{cc}l_{2}(y_{i},-)&\rho_{1+1}(y_{i},-)\\\
0&l^{\prime}_{2}(y_{i},-)\end{array}\right)$
$\displaystyle+Dv^{a}\otimes\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
l^{\prime}_{2}(-,v_{a})&\mu_{2}(v_{a},-)\end{array}\right)$

as a linear map in ${\rm
Hom}(\mathfrak{g}[1],\mathfrak{g}^{\vee}[-1]\otimes\mathfrak{g}[1])$.
Particularly in $\delta^{-1}$ gauge, this is

$\displaystyle
Dy^{i}\otimes\left(\begin{array}[]{cc}\frac{1}{3}\iota_{y_{i}}R(-)+\frac{1}{3}\iota_{(-)}R(y_{i})&\nabla_{\rho(-)}(y_{i})-\check{\nabla}^{bas}_{(-)}(y_{i})\\\
0&\frac{1}{2}\iota_{y_{i}}R^{\prime}(-)\end{array}\right)$
$\displaystyle+Dv^{a}\otimes\left(\begin{array}[]{cc}\nabla_{\rho(v_{a})}-\check{\nabla}^{bas}_{v_{a}}&0\\\
\frac{1}{2}\iota_{(-)}R^{\prime}(v_{a})&\nabla^{\prime}_{\rho(v_{a})}-\hat{\nabla}^{bas}_{v_{a}}\end{array}\right).$

###  4.2. The Weil algebra

Consider $\mathfrak{g}$ being the curved $L_{\infty}$ algebra arising from the
geometric deformation on $\mathfrak{h}\oplus\mathfrak{v}$ as described above
in $\delta^{-1}$ gauge999In fact, we shall remain in $\delta^{-1}$ gauge till
the end of this section.. On $\mathcal{O}_{B\mathfrak{g}}\equiv
C^{*}(\mathfrak{g})$, the CE differential decomposes into
$(\delta+d^{\nabla}_{\geq 1}+d^{\nabla^{\prime}})+(d^{\rho}+d^{\mu})$.
Likewise, on $\Omega_{B\mathfrak{g}}$, the differential decomposes as
$(\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}})+(d^{\rho}+d^{\mu}+d^{DR}),$where the meaning of each
component should be clear.

######  Lemma 35.

On $\Omega^{\star\geq 1}_{B\mathfrak{g}}$, we have that

$d_{\geq
1}^{\nabla}=\nabla+\sum_{n>0}\frac{1}{n!}\big{(}(l_{n+1})^{i}\,_{i_{1},i_{2},\cdots,i_{n+1}}\,y^{i_{1}}\cdots
Dy^{i_{n+1}}\big{)}\frac{\partial}{\partial Dy^{i}},$ $\displaystyle
d^{\nabla^{\prime}}$ $\displaystyle=$
$\displaystyle\nabla^{\prime}+\sum_{n>0}\big{(}\frac{1}{(n-1)!}(l^{\prime}_{n+1})^{b}\,_{i_{1},i_{2},\cdots,i_{n+1},a}\,y^{i_{1}}\cdots
Dy^{i_{n}}v^{a}$
$\displaystyle+\frac{1}{n!}(l^{\prime}_{n+1})^{b}\,_{i_{1},i_{2},\cdots,i_{n},a}\,y^{i_{1}}\cdots
y^{i_{n}}Dv^{a}\big{)}\frac{\partial}{\partial Dv^{b}},$ $\displaystyle
d^{\rho}$ $\displaystyle=$
$\displaystyle-\sum_{n=0}\frac{1}{n!}\big{(}(\rho_{n+1})^{i}\,_{j_{1}\cdots
j_{n}a}y^{j_{1}}\cdots y^{j_{n}}Dv^{a}$
$\displaystyle\qquad+(\rho_{n+2})^{i}\,_{j_{1}\cdots j_{n+1}a}y^{j_{1}}\cdots
Dy^{j_{n+1}}v^{a}\big{)}\frac{\partial}{\partial Dy^{i}},$ $\displaystyle
d^{\mu}$ $\displaystyle=$
$\displaystyle-\sum_{n=0}\frac{1}{n!}\left((\mu_{n+2})^{a}\,_{j_{1}\cdots
j_{n}bc}y^{j_{1}}\cdots y^{j_{n}}Dv^{b}\wedge
v^{c}\right)\frac{\partial}{\partial Dv^{a}}$
$\displaystyle-\sum_{n=0}\frac{1}{n!}\frac{1}{2!}\left((\mu_{n+3})^{a}\,_{j_{1}\cdots
j_{n+1}bc}y^{j_{1}}\cdots Dy^{j_{n+1}}v^{b}\wedge
v^{c}\right)\frac{\partial}{\partial Dv^{a}}.$

Let $W$ denote

$\Gamma\big{(}\widehat{{\rm Sym}}(T^{\vee}_{M}[-1]\oplus V^{\vee}[-1]\oplus
V^{\vee}[-2])\big{)}.$

######  Lemma 36.

The exists an $\mathbb{R}$-linear special deformation retract

$\textstyle{\left(W,0\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{0}}$$\textstyle{\left(\Omega_{B\mathfrak{g}},\delta\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\scriptstyle{-\delta^{-1}}$

where $\iota_{0}$ sends elements in $W$ identically into the bi-degree $(0,0)$
component, and $p_{0}$ is the obvious projection.

###### Proof.

It is easy to check that the following holds,

$p_{0}\circ\iota_{0}=id,\quad\iota_{0}\circ
p_{0}=id-\delta\circ\delta^{-1}-\delta^{-1}\circ\delta,$
$\delta^{-1}\circ\iota_{0}=0=p_{0}\circ\delta^{-1},\quad\delta^{-1}\circ\delta^{-1}=0.$

∎

######  Proposition 37.

The exists an $\mathbb{R}$-linear special deformation retract

$\textstyle{\left(W,0\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\left(\Omega_{B\mathfrak{g}},\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}_{\geq
1}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{h}$

which sends a section from the left-and side to a $(\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}_{\geq 1})$-cocycle in $\Omega_{B\mathfrak{g}}$.

###### Proof.

Consider the special deformation retract as in Lemma 36, and consider a
perturbation of $\delta$ by $d_{\geq 1}\equiv d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}_{\geq 1}$. By the Homological Perturbation Lemma [13],
there exists a perturbed special deformation retract, with

$\iota=\sum_{n=0}^{\infty}(-\delta^{-1}\circ d_{\geq
1})^{n}\circ\iota_{0},\quad p=p_{0},$
$h=-\delta^{-1}\circ\sum_{n=0}^{\infty}(-d_{\geq 1}\circ\delta^{-1})^{n},$

such that $(\delta+d_{\geq 1})\circ\iota=0$ is satisfied. So the image of
$\iota$ is in the cocycle of $(\Omega_{B\mathfrak{g}},\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}_{\geq 1})$. ∎

It is useful to obtain a explicit form of the embedding $\iota$.

######  Proposition 38.

Let $\\{y^{i}\\}_{i=1}^{{\rm dim}(M)}$ be a local basis for
$\Gamma(T^{\vee}_{M})$ chosen as before. Given any $f\in W$, there exists a
unique element $\tilde{f}$ in $\Omega_{B\mathfrak{g}}$ such that

  * •

$\tilde{f}=f+\text{terms at least linear in $y^{i}$'s.}$, and

  * •

$(\delta+d^{\nabla}_{\geq 1}+d^{\nabla^{\prime}})\tilde{f}=0$.

Furthermore, $\iota(f)=\tilde{f}$.

###### Proof.

The existence of $\tilde{f}$ is through a recursive computation starting with
$\delta f=0$ and $\delta(\text{terms linear in
$y^{i}$'s})+(d^{\nabla}_{1}+d^{\nabla^{\prime}}_{1})f=0$, etc. Notice that
such $\tilde{f}$ is also unique, since $\delta$ is an isomorphism over
components

$\Gamma\big{(}\widehat{{\rm Sym}}^{\geq 1}(T^{\vee}_{M})\otimes\widehat{{\rm
Sym}}(T^{\vee}_{M}[-1]\oplus V^{\vee}[-1]\oplus V^{\vee}[-2])\big{)}.$

We claim that the map which sends $\forall f\in W$ to $\tilde{f}$ is precisely
$\iota$. By the explicit formula,

$\iota(f)=f+\sum_{n=1}^{\infty}(-\delta^{-1}\circ d_{\geq 1})^{n}(f),$

where the second part contains terms at least linearly dependent on $y^{i}$’s.
By the uniqueness, we have that $\iota(f)=\tilde{f}$. ∎

######  Example 39.

The Proposition 37 and 38 works for arbitrary gauge. In the $\delta^{-1}$
gauge, simplification for $\iota$ exists. Over the component

$\Gamma\big{(}\widehat{{\rm Sym}}(T^{\vee}_{M}\oplus T^{\vee}_{M}[-1]\oplus
V^{\vee}[-1]\oplus V^{\vee}[-2])\big{)},$

the operator $\delta^{-1}\circ d_{\geq
1}=\delta^{-1}\circ(d^{\nabla}_{1}+d^{\nabla^{\prime}}_{1})$. So

$\iota=\sum_{n=0}^{\infty}(-\delta^{-1}\circ(d^{\nabla}_{1}+d^{\nabla^{\prime}}_{1}))^{n}\circ\iota_{0}.$

In particular we have that

$f(x)-\partial_{i}f(x)y^{i}+\cdots,$
$Dy^{i}-\Gamma^{i}_{j}\,{}_{k}y^{j}Dy^{k}+\cdots,$ $v^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}y^{j}v^{b}+\cdots$

and

$Dv^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}y^{j}Dv^{b}+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}y^{m}Dy^{j}v^{b}+\cdots$

are flat sections to $\delta+d^{\nabla}_{\geq 1}+d^{\nabla^{\prime}}$, where
we have omitted the terms containing quadratic or higher powers of $y$. ∎

To conclude, for any $f\in W$, one computes that

$\tilde{f}=f-\bar{\nabla}_{i}fy^{i}+\text{terms at least quadratic in
$y^{i}$'s}.$

Using Proposition 37, we can induce a new differential $D$ on $W$ by

$p\circ(d^{\rho}+d^{\mu}+d^{DR})\circ\iota.$

For degree reason, to compute $D(f)$ it is only necessary to compute the image
of $f-\bar{\nabla}_{i}fy^{i}$ under $d^{\rho}_{\leq 2}+d^{\mu}_{\leq
3}+d^{DR}$, where $\bar{\nabla}$ is the connection on $W$ induced by $\nabla$
and $\nabla^{\prime}$ on $T_{M}$ and on $V$. The differential $D$ is at most
first order, and can be computed component-wise.

There is a known differential on the Weil algebra $W$ as described in [1, 2].
To compare the differential $D$ with the known differential, we denote by
horizontal differential $d_{//}:=\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}$ and vertical differential
$d_{\bot}:=d^{\rho}+d^{\mu}+d^{DR}$.

######  Proposition 40.

The horizontal cohomology algebra
$H_{//}(\Omega_{B\mathfrak{g}},d_{//}+d_{\bot})$ is isomorphic to $(W,D)$,
which is precisely the Weil algebra for Lie algebroid defined in [1, 2].

###### Proof.

We only need to verify the last statement. To compare, we denote the basis of
$T^{\vee}_{M}[-1]$, $V^{\vee}[-1]$ and $V^{\vee}[-2]$ by
$\\{z^{i}\\}_{i=1}^{{\rm dim}(M)}$, $\\{v^{a}\\}_{a=1}^{{\rm rk}(V)}$ and
$\\{w^{a}\\}_{a=1}^{{\rm rk}(V)}$ respectively, then compute the local
expressions of $D(z^{i})$, $D(v^{a})$ and $D(w^{a})$. The relevant
differentials are

$d^{\rho}=\rho^{i}\,_{a}v^{a}\frac{\partial}{\partial
y^{i}}-\left(\rho^{i}\,_{a}Dv^{a}+\rho^{i}_{ja}Dy^{j}v^{a}\right)\frac{\partial}{\partial
Dy^{i}},$ $d^{\mu}=\frac{1}{2}\mu^{a}_{bc}v^{b}\wedge
v^{c}\frac{\partial}{\partial v^{a}}-\left(\mu^{a}_{bc}Dv^{b}\wedge
v^{c}+\frac{1}{2!}\mu^{a}_{jbc}Dy^{j}v^{b}\wedge
v^{c}\right)\frac{\partial}{\partial Dv^{a}}.$

So under the composition map $p\circ d_{\bot}\circ\iota$ on $W$, one has that

$\displaystyle Dy^{i}$ $\displaystyle\mapsto$ $\displaystyle
Dy^{i}-\Gamma^{i}_{j}\,{}_{k}y^{j}Dy^{k}+\cdots$ $\displaystyle\mapsto$
$\displaystyle-\left(\rho^{i}\,_{a}Dv^{a}+\rho^{i}\,_{ja}Dy^{j}v^{a}\right)-\rho^{j}\,_{a}\Gamma^{i}_{j}\,{}_{k}v^{a}Dy^{k}$
$\displaystyle-\left(\rho^{i}\,_{a}Dv^{a}-[-(\Gamma)_{j}^{i}\,{}_{k}(\rho)^{k}\,_{a}+\partial_{j}(\rho^{i}\,_{a})+\rho^{i}\,_{b}(\Gamma^{\prime})_{j}^{b}\,{}_{a}]Dy^{j}v^{a}\right)$
$\displaystyle-\rho^{j}\,_{a}\Gamma^{i}_{j}\,{}_{k}v^{a}Dy^{k}+\cdots$
$\displaystyle\mapsto$
$\displaystyle-\rho^{i}\,_{a}Dv^{a}+\left(\partial_{j}(\rho^{i}\,_{a})+\rho^{i}\,_{b}(\Gamma^{\prime})_{j}^{b}\,{}_{a})Dy^{j}v^{a}\right),$

where the relation

$\rho^{i}\,_{ja}=(\Gamma)_{j}^{i}\,{}_{k}(\rho)^{k}\,_{a}-\partial_{j}(\rho^{i}\,_{a})-\rho^{i}\,_{b}(\Gamma^{\prime})_{j}^{b}\,{}_{a}$

has been used.

Similarly

$\displaystyle v^{a}$ $\displaystyle\mapsto$ $\displaystyle
v^{a}-\Gamma^{\prime a}_{j}\,{}_{b}y^{j}v^{b}+\cdots$ $\displaystyle\mapsto$
$\displaystyle Dv^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}Dy^{j}v^{b}+(\frac{1}{2}\mu^{a}_{bc}-\rho^{j}\,_{b}\Gamma^{\prime
a}_{j}\,{}_{c})v^{b}\wedge v^{c}+\cdots$ $\displaystyle\mapsto$ $\displaystyle
Dv^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}Dy^{j}v^{b}-\frac{1}{2}C_{bc}^{a}v^{b}\wedge v^{c},$

$\displaystyle Dv^{a}$ $\displaystyle\mapsto$ $\displaystyle
Dv^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}y^{j}Dv^{b}+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}y^{m}Dy^{j}v^{b}+\cdots$
$\displaystyle\mapsto$ $\displaystyle-\left(\mu^{a}_{bc}Dv^{b}\wedge
v^{c}+\frac{1}{2!}\mu^{a}_{jbc}Dy^{j}v^{b}\wedge
v^{c}\right)-\rho^{j}\,_{b}\Gamma^{\prime a}_{j}\,{}_{c}v^{b}\wedge Dv^{c}$
$\displaystyle+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}\rho^{m}\,_{c}v^{c}\wedge
Dy^{j}\wedge v^{b}$ $\displaystyle-\Gamma^{\prime
a}_{j}\,{}_{b}Dy^{j}Dv^{b}+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}Dy^{m}Dy^{j}v^{b}+\cdots$
$\displaystyle\mapsto$
$\displaystyle-[(\Gamma^{\prime})_{i}\,^{a}{}_{c}(\rho)^{i}\,_{b}-C_{bc}^{a}]Dv^{b}\wedge
v^{c}-\Gamma^{\prime a}_{j}\,{}_{b}Dy^{j}Dv^{b}$
$\displaystyle+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}Dy^{m}Dy^{j}v^{b}+\frac{1}{2!}R^{bas}\,_{bcj}\,{}^{a}Dy^{j}v^{b}\wedge
v^{c}.$

We have used the relation

$\mu^{a}_{jbc}=-(R^{bas}(v_{b},v_{c})y_{j})^{a}+\frac{1}{2}R^{\prime}_{ij}\,{}_{b}\,^{a}(\rho)_{c}\,^{i}-\frac{1}{2}R^{\prime}_{ij}\,{}_{c}\,^{a}(\rho)_{b}\,^{i},$

which follows from the definition of basic curvature [1]

$R^{bas}(v_{b},v_{c})y_{j}=\nabla^{\prime}_{j}[v_{b},v_{c}]-[\nabla^{\prime}_{j}v_{b},v_{c}]-[v_{b},\nabla^{\prime}_{j}v_{c}]+\nabla^{\prime}_{\nabla^{bas}_{v_{b}}y_{j}}v_{c}-\nabla^{\prime}_{\nabla^{bas}_{v_{c}}y_{j}}v_{b},$

(where

$\nabla^{bas}_{v_{b}}y_{j}=-\nabla_{j}\rho(v_{b})+\rho(\nabla^{\prime}_{j}v_{b})+\nabla_{\rho(v_{b})}y_{j}$

is the corresponding basic connection, as we shall recall at the beginning of
Section 4.4, see also [1]). The calculation is straight-forward using an
explicit formula for $\mu_{3}$ given in Appendix B. ∎

A more functorial approach of the result was given in [20], where a homotopy
functor $enh_{mod}$ from the category of Lie algebroid representations up to
homotopy to (dg-)vector bundles over $L_{\infty}$ spaces was constructed.
However, both the representations up to homotopy and the $L_{\infty}$
structures rely on some choices. Our result shows that, if one fixes the
connections on $T_{M}\oplus V$ once for all, a one-to-one correspondence can
be achieved on the nose. Moreover, by specifying different gauge conditions on
the $L_{\infty}$ structure that encodes the Lie algebroid $V$, it is possible
to induce more differentials on the Weil algebra $W$.

######  Remark 41.

Since the vertical differential on $\Omega_{B\mathfrak{g}}$ decomposes into
$d^{DR}$ and $d^{\mu}+d^{\rho}$, there is also a decomposition on the
differential $D$ over $W$, which we write as $D^{DR}+D^{\mu+\rho}$. In
particular, we have that

$D^{DR}(Dy^{i})=0,\quad D^{DR}(v^{a})=Dv^{a}-\Gamma^{\prime
a}_{j}\,{}_{b}Dy^{j}v^{b},$ $D^{DR}(Dv^{a})=-\Gamma^{\prime
a}_{j}\,{}_{b}Dy^{j}Dv^{b}+\frac{1}{2}R^{\prime}_{jm}\,{}^{a}\,_{b}Dy^{m}Dy^{j}v^{b}.$

∎

###  4.3. Atiyah-Chern class

The Atiyah class plays a fundamental role in the construction of Atiyah-Chern
classes through the Chern-Weil theory, which is defined as

$c_{k}:={\rm STr}\big{(}At(\nabla_{B\mathfrak{g}})^{\wedge
k}\big{)}\in\Omega^{k}_{B\mathfrak{g}},$

where ${\rm STr}$ stands for the super-trace.

Since $\Omega_{B\mathfrak{g}}$ is bi-graded (in the way described at the
beginning of Section 4.2), the $k$-th Atiyah-Chern class $c_{k}$ may exhibit
nontrivial (bi-)degree $(p,q)$ components for all $p,q\in\mathbb{Z}_{\geq 0}$
satisfying $p+q=k$ in $\Omega_{B\mathfrak{g}}$. The usual transgression can be
generalised to deal with mixed degrees here, which we now present. The
transgression enables us find a new representative for $c_{k}$ sitting in
degree $(0,k)$, which leads to a cocycle in the Weil algebra as shown in
Proposition 43.

######  Lemma 42.

Consider a bicomplex $(C^{\bullet,\star},d_{//},(-1)^{\bullet}d_{\bot})$
defined from a total complex with total differential $d=d_{//}+d_{\bot}$,
together with a cocycle $\mathbf{a}$ of total degree $k$. Suppose the
horizontal differential $d_{//}$ is acyclic on components $C^{\geq 1,\star}$,
then there exists a representative $\alpha$ of degree $(0,k)$ such that
$\mathbf{a}-\alpha$ belongs to $d$-coboundaries.

###### Proof.

Without loss of generality, we can assume that $a$ has nontrivial components
in degree $(p,k-p)$ for each $0\leq p\leq k$, which we denote by
$\mathbf{a}_{0,k},\mathbf{a}_{1,k-1},\cdots,\mathbf{a}_{k,0}$ respectively. So
we have, for $0\leq p\leq k-1$

$d_{//}\mathbf{a}_{k,0}=0,\quad
d_{\bot}\mathbf{a}_{k-p,p}+d_{//}\mathbf{a}_{k-p-1,p+1}=0,\quad
d_{\bot}\mathbf{a}_{0,k}=0.$

By the acyclicity of $d_{//}$ on $C^{\geq 1,\star}$, there exists elements
$b_{0,k-1},b_{1,k-2},\cdots,b_{k-1,0}$ such that for $0\leq p\leq k-2$

$d_{//}b_{k-1,0}=\mathbf{a}_{k,0},\quad
d_{//}b_{k-p-2,p+1}=\mathbf{a}_{k-p-1,p+1}-d_{\bot}b_{k-p-1,p}.$

That can be shown by an inductive method. For $1\leq i<n$, suppose that there
exists $b_{k-i-1,i}$ such that
$\mathbf{a}_{k-i,i}=d_{//}b_{k-i-1,i}+d_{\bot}b_{k-i,i-1}$ holds. When
$i=n<k$,

$d_{//}(\mathbf{a}_{k-n,n}-d_{\bot}b_{k-n,n-1})=-d_{\bot}\mathbf{a}_{k-n+1,n-1}+d_{\bot}d_{//}b_{k-n,n-1}=0$

and so there exists $b_{k-n-1,n}$ such that

$d_{//}b_{k-n-1,n}=\mathbf{a}_{k-n,n}-d_{\bot}b_{k-n,n-1}.$

Finally let $\alpha=\mathbf{a}_{0,k}-d_{\bot}b_{0,k-1}$, and check that

$\sum_{i=0}^{k}\mathbf{a}_{i,k-i}-\alpha=d(\sum_{i=0}^{k-1}b_{i,k-1-i}).$

∎

######  Proposition 43.

Consider $\mathfrak{g}$ being the curved $L_{\infty}$ algebra arising from the
geometric deformation on $\mathfrak{h}\oplus\mathfrak{v}$. Suppose $u_{0,k}$
is a total cocycle in $\Omega_{B\mathfrak{g}}$ concentrated in bi-degree
$(0,k)$, then its $y^{i}$-independent part is a cocycle in the Weil algebra
$(W,D)$.

###### Proof.

One checks that $u_{0,k}$ is both $d_{//}$-closed and $d_{\bot}$-closed. By
Proposition 37, $y^{i}$-independent part sits in the image of the projection
map $p:\Omega_{B\mathfrak{g}}\to W$. Since the Weil differential is induced by
$d_{\bot}$ from $\Omega_{B\mathfrak{g}}$, it is a cocycle. ∎

At first glance, this result may not seem particularly intriguing. It has been
established that the Weil algebra is quasi-isomorphic to the de Rham coochain
$(\Omega_{M},d)$ [2], meaning the only nontrivial classes land in Chern
characters of $T_{M}$. However, this finding holds significance as it serves
as an initial step towards developing an “equivariant” cohomology theory for
Lie algebroids. It becomes interesting to examine captivating coboundaries
within this context. To streamline the discussion, we will refer to the
transgressed elements by their corresponding images in $W$, without explicitly
stating this fact.

Similarly, it is interesting to explore the part of the Atiyah class that does
not depend on $\\{y^{i}\\}$, represented as
$At(\nabla_{B\mathfrak{g}})|_{y^{i}=0}$. This specific component, denoted by
more brackets compared to the constant component
($At(\nabla_{B\mathfrak{g}})_{0}$), contains certain information resembling
characters in a representation sense.

$At(\nabla_{B\mathfrak{g}})|_{y^{i}=0}=Dy^{i}\otimes\left(\begin{array}[]{cc}l_{2}(y_{i},-)&\rho_{1+1}(y_{i},-)\\\
0&l^{\prime}_{2}(y_{i},-)\end{array}\right)+Dv^{a}\otimes\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
l^{\prime}_{2}(-,v_{a})&\mu_{2}(v_{a},-)\end{array}\right)$
$+Dy^{i}v^{a}\otimes\left(\begin{array}[]{cc}\rho_{3}(y_{i},-,v_{a})&0\\\
l^{\prime}_{3}(y_{i},-,v_{a})&\mu_{3}(y_{i},v_{a},-)\end{array}\right)$
$+Dv^{a}v^{b}\otimes\left(\begin{array}[]{cc}0&0\\\
\mu_{3}(-,v_{a},v_{b})&0\end{array}\right)+Dy^{i}v^{a}v^{b}\otimes\left(\begin{array}[]{cc}0&0\\\
\frac{1}{2}\mu_{4}(y_{i},-,v_{a},v_{b})&0\end{array}\right)$

$=Dy^{i}\otimes\left(\begin{array}[]{cc}\frac{1}{3}dx^{m}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&\rho_{1+1}(y_{i},-)\\\
0&\frac{1}{2}dx^{m}R^{\prime}_{im}\end{array}\right)-Dv^{a}\otimes\left(\begin{array}[]{cc}\check{\nabla}^{bas}_{v_{a}}-\nabla_{\rho(v_{a})}&0\\\
-l^{\prime}_{2}(-,v_{a})&\hat{\nabla}^{bas}_{v_{a}}-\nabla^{\prime}_{\rho(v_{a})}\end{array}\right)$
$+Dy^{i}v^{a}\otimes\left(\begin{array}[]{cc}\rho_{3}(y_{i},-,v_{a})&0\\\
l^{\prime}_{3}(y_{i},-,v_{a})&\mu_{3}(y_{i},v_{a},-)\end{array}\right)$
$+Dy^{i}v^{a}v^{b}\otimes\left(\begin{array}[]{cc}0&0\\\
\frac{1}{2}\mu_{4}(y_{i},-,v_{a},v_{b})&0\end{array}\right).$

###  4.4. Modular class and the first Atiyah-Chern class

Recall the notion of basic connections associated to a Lie algebroid $V$
together with a vector bundle (representation up to homotopy) $T_{M}$ from
[1]:

$\hat{\nabla}^{bas}_{\alpha}\beta=\nabla^{\prime}_{\rho(\alpha)}\beta-\mu(\alpha,\beta)=[\alpha,\beta]+\nabla^{\prime}_{\rho(\beta)}\alpha$

and

$\check{\nabla}^{bas}_{\alpha}X=\nabla_{\rho(\alpha)}X-\nabla_{X}\rho(\alpha)+\rho\nabla^{\prime}_{X}\alpha=-\rho_{1+1}(X,\alpha)+\nabla_{\rho(\alpha)}X$

for all $X\in\Gamma(T_{M})$ and $\alpha,\beta\in\Gamma(V)$. So

$-\mu(\alpha,\beta)=\hat{\nabla}^{bas}_{\alpha}\beta-\nabla^{\prime}_{\rho(\alpha)}\beta,$
$-\rho_{1+1}(X,\alpha)=\check{\nabla}^{bas}_{\alpha}X-\nabla_{\rho(\alpha)}X.$

By [12, 14], there is a Chern-Weil type of construction for secondary
character classes from
$\hat{\nabla}^{bas}_{\alpha}\beta-\nabla^{\prime}_{\rho(\alpha)}\beta$ and
$\check{\nabla}^{bas}_{\alpha}X-\nabla_{\rho(\alpha)}X$ via Chern-Simons
transgression.

######  Proposition 44.

When $(\nabla,\nabla^{\prime})$ is a metric connection on $T_{M}[-1]\oplus V$,
the expression

${\rm
STr}\left(\hat{\nabla}^{bas}\oplus\check{\nabla}^{bas}-\nabla^{\prime}_{\rho(-)}\oplus\nabla_{\rho(-)}\right)$

computes the secondary character class $u_{1}$ in the Lie algebroid cohomology
$H^{1}(V)$.

The (linear) connection $\hat{\nabla}^{bas}\oplus\check{\nabla}^{bas}$ is
equivalent to a nonlinear flat connection on the superbundle $(V\oplus
T_{M}[-1],\rho)$ up to homotopy. The flatness is guaranteed by the “bracket”
parts of $\hat{\nabla}^{bas}$ and $\check{\nabla}^{bas}$, while the
compatibility with respect to the anchor map $\rho$ makes sure the two
components form a connection for $(V\oplus T_{M},\rho)$.

It is interesting to see whether higher secondary character classes
$u_{\frac{k+1}{2}}$ as defined in [14, 18] can be recovered by Atiyah class.

Consider now the transgression of $c_{1}$. We start with the $(1,0)$-component
of $c_{1}$, and apply the homotopy contraction101010Strictly speaking, the
homotopy contraction for $d_{//}\equiv\delta+d^{\nabla}_{\geq
1}+d^{\nabla^{\prime}}_{\geq 1}$ is $h$ as given in Proposition 37, which
includes additional linear operators with higher hweights. However, for the
leading weight computation, these corrections are considered irrelevant.
$\delta^{-1}$, which gives

$\delta^{-1}Dy^{i}dx^{m}{\rm
STr}\left(\begin{array}[]{cc}\frac{1}{3}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&\rho_{1+1}(y_{i},-)\\\
0&\frac{1}{2}R^{\prime}_{im}\end{array}\right)$ $=-y^{m}Dy^{i}{\rm
STr}\left(\begin{array}[]{cc}\frac{1}{3}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&0\\\
0&\frac{1}{2}R^{\prime}_{im}\end{array}\right).$

Upon applying $d_{\bot}$, the above expression

$\displaystyle\mapsto$ $\displaystyle(-Dy^{m}-\rho^{m}\,_{a}v^{a})\wedge
Dy^{i}{\rm
STr}\left(\begin{array}[]{cc}\frac{1}{3}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&0\\\
0&\frac{1}{2}R^{\prime}_{im}\end{array}\right)$ $\displaystyle=$
$\displaystyle(Dy^{i}\wedge Dy^{m}+\rho^{m}\,_{a}Dy^{i}v^{a}){\rm
STr}\left(\begin{array}[]{cc}\frac{1}{3}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&0\\\
0&\frac{1}{2}R^{\prime}_{im}\end{array}\right)$ $\displaystyle=$
$\displaystyle Dy^{i}\wedge
Dy^{m}\left(\frac{1}{3}R_{jm}\,{}_{i}\,^{j}\right)+\rho^{m}\,_{a}Dy^{i}v^{a}\left(\frac{1}{3}R_{jm}\,{}_{i}\,^{j}\right).$

The first term vanishes, since

$R_{j[mi]}\,^{j}=\frac{1}{2}(-R_{mij}\,^{j}-R_{ijm}\,^{j})-\frac{1}{2}R_{jim}\,^{j}=-\frac{1}{2}R_{ijm}\,^{j}-\frac{1}{2}R_{jim}\,^{j}=0.$

By Lemma 42, the new representative for the first Atiyah-Chern class is

$\displaystyle-\frac{1}{3}\rho^{m}\,_{a}Dy^{i}v^{a}R_{jm}\,{}_{i}\,^{j}+Dv^{a}{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
0&\mu_{2}(v_{a},-)\end{array}\right)$ $\displaystyle+Dy^{i}v^{a}{\rm
STr}\left(\begin{array}[]{cc}\rho_{3}(y_{i},-,v_{a})&0\\\
0&\mu_{3}(y_{i},v_{a},-)\end{array}\right).$

Let $C^{*}(V)$ denote the Chevalley-Eilenberg cochain for Lie algebroid $V$,
which may also be regarded as a subspace inside $W$, depending on the context.

######  Proposition 49.

If $\nabla^{\prime}$ is a metric connection on $V$, the first Atiyah-Chern
class transgresses to the image of modular class [16]

${\rm STr}\left(\begin{array}[]{cc}-(\check{\nabla}^{bas}-\nabla_{\rho})&0\\\
0&-(\hat{\nabla}^{bas}-\nabla^{\prime}_{\rho})\end{array}\right)\in C^{1}(V)$

under the de Rham map $D^{DR}$ in Weil algebra for Lie algebroid $V$.

###### Proof.

It remains to compute

$D^{DR}\left(v^{a}\cdot{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
0&\mu_{2}(v_{a},-)\end{array}\right)\right)$

and compare with expression given above.

According to the local form of $D^{DR}$ given in Remark 4.2, it is given by

$(Dv^{a}-\Gamma^{\prime a}_{j}\,{}_{b}Dy^{j}v^{b})\otimes\cdot{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
0&\mu_{2}(v_{a},-)\end{array}\right)$ $-Dy^{j}v^{a}{\rm
STr}\left(\begin{array}[]{cc}\nabla_{j}\rho_{1+1}(-,v_{a})-\rho_{1+1}(\nabla_{j}(-),v_{a})&0\\\
0&\nabla^{\prime}_{j}\mu_{2}(v_{a},-)-\mu_{2}(v_{a},\nabla_{j}^{\prime}(-))\end{array}\right).$

Note that

$\displaystyle-\Gamma^{\prime a}_{j}\,{}_{b}Dy^{j}v^{b}\otimes{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
0&\mu_{2}(v_{a},-)\end{array}\right)$ $\displaystyle=$ $\displaystyle
Dy^{j}v^{a}\otimes{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,\nabla^{\prime}_{j}v_{a})&0\\\
0&\mu_{2}(\nabla^{\prime}_{j}v_{a},-)\end{array}\right).$

So in total we have

(54) $\displaystyle Dv^{a}\otimes{\rm
STr}\left(\begin{array}[]{cc}\rho_{1+1}(-,v_{a})&0\\\
0&\mu_{2}(v_{a},-)\end{array}\right)$ (57) $\displaystyle-Dy^{j}v^{a}{\rm
STr}\left(\begin{array}[]{cc}\nabla_{j}(\rho_{1+1})(-,v_{a})&0\\\
0&\nabla^{\prime}_{j}(\mu_{2})(-,v_{a})\end{array}\right),$

where we have used $\nabla_{j}(\rho_{1+1})(-,v_{a})$ to denote

$\nabla_{j}\rho_{1+1}(-,v_{a})-\rho_{1+1}(\nabla_{j}(-),v_{a})-\rho_{1+1}(-,\nabla_{j}^{\prime}v_{a}),$

and likewise $\nabla^{\prime}_{j}(\mu_{2})(-,v_{a})$ to denote

$\nabla_{j}^{\prime}\mu_{2}(v_{a},-)-\mu_{2}(v_{a},\nabla^{\prime}_{j}(-))-\mu_{2}(\nabla_{j}^{\prime}v_{a},-).$

The second line in equation (57) gives111111For more details please refer to
equation (B) in Appendix B.:

$Dy^{k}v^{a}\left({\rm Tr}\rho_{3}(-,y_{k},v_{a})-{\rm
Tr}\mu_{3}(y_{k},v_{a},-)\right)-\frac{1}{3}Dy^{k}v^{a}\rho^{i}\,_{a}R_{jik}\,^{j}.$

The proposition follows.

∎

The first Atiyah-Chern class can be seen as a coboundary in the context of
$W$, which is not surprising. As the Weil algebra $W$ is quasi-isomorphic to
the de Rham cochain over $M$ [2], the modular class, being a geometric
invariant, is unlikely to persist in ordinary cohomology.

There are potential hope to recognise more Atiyah-Chern classes due to the
acyclicity of $d^{DR}$ in $\Omega_{B\mathfrak{g}}$, which might allow us to
relate those classes to more secondary invariants of Lie algebroids in
$C^{*}(V)\subset W$ as defined in [12, 14]. Nonetheless, given the absence of
a clear equivariant de Rham algebra construction in the current scenario, it
is uncertain whether such identifications can be established at this stage.

###  4.5. Action algebroids

We consider the action Lie algebroid coming from a Lie group action $G\to{\rm
Diff}(M)$. In that case, let $\mathfrak{v}$ be the Lie algebra of $G$, and the
bundle $V\equiv M\times\mathfrak{v}$ is flat. So $\\{l^{\prime}_{n}\\}_{n\geq
2}$ vanishes by the induction. Moreover,

$\mu(\alpha,\beta)=-[\alpha,\beta]^{\mathfrak{v}}$

is a constant map, hence all the higher brackets $\\{\mu_{2+n}\\}_{n\geq 1}$
vanish. As a result,

$At(\nabla_{B\mathfrak{g}})|_{y^{i}=0}=Dy^{i}\otimes\left(\begin{array}[]{cc}\frac{1}{3}dx^{m}\left(R_{im}\,{}_{k}\,^{j}+R_{km}\,{}_{i}\,^{j}\right)&\rho_{1+1}(y_{i},-)\\\
0&0\end{array}\right)$
$-Dv^{a}\otimes\left(\begin{array}[]{cc}\check{\nabla}^{bas}_{v_{a}}-\nabla_{\rho(v_{a})}&0\\\
0&\hat{\nabla}^{bas}_{v_{a}}-\nabla^{\prime}_{\rho(v_{a})}\end{array}\right)$
$+Dy^{i}v^{a}\otimes\left(\begin{array}[]{cc}\rho_{3}(y_{i},-,v_{a})&0\\\
0&0\end{array}\right).$

The component
$-Dv^{a}\otimes\left(\check{\nabla}^{bas}_{v_{a}}-\nabla_{\rho(v_{a})}\right)$
is precisely the moment map for the infinitesimal $\mathfrak{v}$-action on
bundle $T_{M}$, which was denoted by
$\mathcal{L}^{T_{M}}(\alpha)-\nabla_{\rho(\alpha)}$ for any
$\alpha\in\mathfrak{v}$ in [4], where $\mathcal{L}^{T_{M}}$ is the Lie
derivative on $T_{M}$.

######  Proposition 58.

The $k$-th Atiyah-Chern class of the action Lie algebroid has its constant
part given by

$(-1)^{k}{\rm
STr}\left(\begin{array}[]{cc}R(-)+\mathcal{L}^{T_{M}}_{(-)}(\alpha)-\nabla_{\rho(\alpha)}(-)&0\\\
0&[\alpha,-]\end{array}\right)^{k}$

in the Cartan sub-algebra
$\left(S^{*}(\mathfrak{v}^{\vee})\otimes\Omega_{M}\right)^{\mathfrak{v}}$ of
the BRST complex as in [23].

The proof is obtained through a direct computation, focusing solely on the
leading order contributions in powers of $v^{a}$’s and $y^{i}$’s, which
greatly simplifies the transgression. In this case, only $d_{\bot}=d^{DR}$
needs to be taken into account, and the contributions reside in
$S^{*}(\mathfrak{v})\otimes\Omega_{M}$. Remarkably, the result aligns with the
equivariant Chern character of $T_{M}\ominus V$ using the Chern-Weil
construction, leading to an immediate conclusion. Consequently, the Atiyah-
Chern classes of the geometrically deformed $L_{\infty}$ algebra
$\mathfrak{h}\oplus\mathfrak{v}$ serve as a generalization of the equivariant
Chern character for the action Lie algebroid case. It would be intriguing to
provide a comprehensive equivariant description of the de Rham algebra
$\Omega_{B(\mathfrak{h}\oplus\mathfrak{v})}$ for the general Lie algebroid
situation.

##  Appendix A Solving for $d^{\nabla}$ and $d^{\nabla^{\prime}}$

In this appendix, we show that the $L_{\infty}$ structures given in
Proposition 5 and 9 can be solved recursively from the quadratic equation on
the respective CE differentials. In Lemma 61 we give the explicit formula, and
the discussion leading to Proposition 62 shows that such structure is unique
up to a “gauge” condition. Then we show in Proposition 65 that any such
structure is homotopy equivalent to each other. Finally, we attach some
calculation results for the first couple of operations.

On $C^{*}(\mathfrak{h})$, the quadratic relation

(59) $\frac{1}{2}[d^{\nabla},d^{\nabla}]=0$

for $d^{\nabla}$ decomposes as follows. For $n\geq 2$,

(59.n) $-[\delta,d^{\nabla}_{n}]=\frac{1}{2}\sum_{1\leq k\leq
n-1}[d_{k}^{\nabla},d^{\nabla}_{n-k}].$

For consistency, we also have

(59.1) $-[\delta,d^{\nabla}_{1}]=0.$

For the set of equations $\\{(\ref{appeqn1}.n)\\}_{n=1}^{\infty}$, the left-
hand side of each equation is $\delta$-exact. If the right-hand side is not
$\delta$-closed, such equation can not be solved. We say that equation
$(\ref{appeqn1}.n)$ is compatible if its right-hand side commutes with
$\delta$.

######  Lemma 60.

Let $n$ be a positive integer. Suppose equation $(\ref{appeqn1}.k)$ is
solvable for each $k\leq n$, then equation $(\ref{appeqn1}.n+1)$ is
compatible.

###### Proof.

Consider $\delta$ acting on the right-hand side of $(\ref{appeqn1}.n+1)$:

$\sum_{1\leq k\leq
n}\frac{1}{2}[\delta,[d_{k}^{\nabla},d^{\nabla}_{1+n-k}]]=\frac{1}{2}\sum_{1\leq
k\leq
n}[[\delta,d_{k}^{\nabla}],d^{\nabla}_{1+n-k}]-[d_{k}^{\nabla},[\delta,d^{\nabla}_{1+n-k}]].$

Using $(\ref{appeqn1}.k)$ for all $1\leq k\leq n$, we have that

$-\frac{1}{4}\sum_{1\leq m<k\leq
n}[[d_{m}^{\nabla},d_{k-m}^{\nabla}],d^{\nabla}_{1+n-k}]+\frac{1}{4}\sum_{1\leq
k<m\leq n}[d_{k}^{\nabla},[d_{m-k}^{\nabla},d^{\nabla}_{1+n-m}]].$

The summation can be re-expressed into a more symmetric form

$\sum_{s+t+l=n+1}-\frac{1}{4}[[d_{s}^{\nabla},d_{t}^{\nabla}],d^{\nabla}_{l}]+\frac{1}{4}[d_{s}^{\nabla},[d_{t}^{\nabla},d^{\nabla}_{l}]].$

Note that for a fixed (unordered) positive partition of $n+1$, the number of
assignments of $s,t,l$ varies due to the symmetry of each partition. For
example, if we have $n+1=a+a+b\,(a\neq b)$, then we shall have in total three
terms for $[[d_{s}^{\nabla},d_{t}^{\nabla}],d^{\nabla}_{l}]$, corresponding to
$s=b$, $t=b$ and $l=b$ respectively. So we look into each case individually.

  1. (1)

Case $s=t=l$;

  2. (2)

Case two of $s,t,l$ are equal;

  3. (3)

Case $s\neq t\neq l$.

In each case, the contributions cancel among themselves due to the Jacobi
identity of differential operators under composition. ∎

######  Lemma 61.

Assume that equations $\\{(\ref{appeqn1}.k)\\}_{k=1}^{n}$ are solvable by
$\\{d^{\nabla}_{k}\\}_{k=1}^{n}$, if equation $(\ref{appeqn1}.n+1)$ is
compatible, then there exists a derivation $d^{\nabla}_{n+1}$ solving equation
$(\ref{appeqn1}.n+1)$, under which the generators $\\{y^{i}\\}_{i=1}^{{\rm
dim}(M)}$ of $C^{*}(\mathfrak{h})$ have homogeneous images of hweight $(n+1)$
given by

$d^{\nabla}_{n+1}:y^{i}\mapsto-\frac{1}{2}\sum_{1\leq k\leq
n}\delta^{-1}\left([d^{\nabla}_{k},d^{\nabla}_{n+1-k}](y^{i})\right),\forall
i.$

###### Proof.

A choice of Levi-Civita connection $\nabla$ induces a connection on
$C^{*}(\mathfrak{h})$, which is again denoted by $\nabla$. It is known that
$d^{\nabla}_{1}=\nabla$, which is locally written (in compatible coordinates)
as

$d^{\nabla}_{1}=d_{M}+\Gamma^{i}_{jk}dx^{j}y^{k}\frac{\partial}{\partial
y^{i}}.$

For $k\geq 2$, each $d^{\nabla}_{k}$ is $\Omega_{M}$-linear, and is entirely
determined by the image of $y^{i}$ for each $i$. We claim that

$d^{\nabla}_{n+1}(y^{i})=-\frac{1}{2}\sum_{1\leq k\leq
n}\delta^{-1}\left([d^{\nabla}_{k},d^{\nabla}_{n+1-k}](y^{i})\right)$

provides a solution for $d^{\nabla}_{n+1}(y^{i})$, $\forall i$, and hence a
solution for $d^{\nabla}_{n+1}$.

Let $\delta$ act on both sides of the above relation. The left-hand side is
precisely $-[\delta,d^{\nabla}_{n+1}](y^{i})$. The right-hand side is

$\frac{1}{2}\sum_{1\leq k\leq
n-1}\delta\delta^{-1}\left([d_{k}^{\nabla},d^{\nabla}_{n-k}](y^{i})\right)\equiv\frac{1}{2}\sum_{1\leq
k\leq
n-1}(id-\delta^{-1}\delta)\left([d_{k}^{\nabla},d^{\nabla}_{n-k}](y^{i})\right).$

But the last term vanishes, which follows from

$\sum_{1\leq k\leq
n-1}\delta\left([d_{k}^{\nabla},d^{\nabla}_{n-k}](y^{i})\right)=\sum_{1\leq
k\leq
n-1}[\delta,[d_{k}^{\nabla},d^{\nabla}_{n-k}]](y^{i})+[d_{k}^{\nabla},d^{\nabla}_{n-k}](\delta
y^{i})$

and the compatibility assumption. ∎

In the following discussion, we shall denote by $\delta^{-1}(d^{\nabla}_{n})$
the operator

$\sum_{i=1}^{{\rm
dim}(M)}\delta^{-1}\big{(}d^{\nabla}_{n}(y^{i})\big{)}\frac{\partial}{\partial
y^{i}}\in\Gamma\big{(}{\rm Sym}^{n}(T_{M}^{\vee})\otimes T_{M}\big{)}\subset
C^{*}(\mathfrak{h})\otimes\mathfrak{h}[1]$

for $n\geq 2$.

######  Proposition 62.

The differential $d^{\nabla}\equiv\delta+\nabla+\sum_{n\geq 2}d^{\nabla}_{n}$
be solved recursively and uniquely from the equation $d^{\nabla}\circ
d^{\nabla}=0$ for any fixed Levi-Civita connection $\nabla$, subjecting to the
gauge condition $\delta^{-1}(d^{\nabla}_{n})=0$ for all $n\geq 2$.

###### Proof.

The induction starts with the relation for $d^{\nabla}_{1}\equiv\nabla$, which
states the torsion-free condition in the initial step. The construction for
$d^{\nabla}_{n+1}(y^{i})$ uses a gauge choice that
$\delta^{-1}d^{\nabla}_{n}(y^{i})=0$ for all $n\geq 2$, $i=1,\cdots,{\rm
dim}(M)$. Indeed, any solution $\tilde{d}^{\nabla}_{n+1}(y^{i})$ satisfying
the same gauge condition and the same initial condition
$\tilde{d}^{\nabla}_{1}\equiv\nabla$ has to be given by Lemma 61, since the
operator $\delta$ is invertible on images of $\delta^{-1}$, and so the
relation

$\delta(\tilde{d}^{\nabla}_{n+1}(y^{i}))=\delta(d^{\nabla}_{n+1}(y^{i})),\quad
i=1,\cdots,{\rm dim}(M)$

enforced by equation $(\ref{appeqn1}.n+1)$ implies that
$\tilde{d}^{\nabla}_{n+1}=d^{\nabla}_{n+1}$. ∎

Indeed, the $\delta^{-1}$ gauge needs not to be the unique choice. Without
much efforts, one obtains the following generalisation.

######  Corollary 63.

For any $\phi\in\Gamma\big{(}\widehat{\rm Sym}^{\geq 3}(T_{M}^{\vee})\otimes
T_{M}\big{)}$, and for any Levi-Civita connection $\nabla$ on $T_{M}$, there
exists a unique differential $d^{\nabla,\phi}$ satisfying the quadratic
equation (59) such that

$d^{\nabla,\phi}\equiv\delta+\nabla+\sum_{n\geq 2}d^{\nabla,\phi},$

and

$\delta^{-1}(d^{\nabla,\phi}_{\geq 2})=\phi.$

###### Proof.

Consider the recursive solution

$d^{\nabla,\phi}_{n+1}(y^{i})=-\frac{1}{2}\sum_{1\leq k\leq
n}\delta^{-1}\left([d^{\nabla,\phi}_{k},d^{\nabla,\phi}_{n+1-k}](y^{i})\right)+\delta\phi_{n+2}(y^{i}),$

for $i=1,\cdots,{\rm dim}(M)$, where $\phi_{n+2}$ is the homogeneous component
of $\phi$ in $\Gamma\big{(}{\rm Sym}^{n+2}(T_{M}^{\vee})\otimes T_{M}\big{)}$.
∎

######  Lemma 64.

Given a family of connections $\nabla^{t}$ on $T_{M}$ and a family of gauge
conditions $\phi^{t}\in\Gamma\big{(}\widehat{\rm Sym}^{\geq
3}(T_{M}^{\vee})\otimes T_{M}\big{)}$, both parameterised by the standard
$n$-simplex $\Delta^{n}$, there exists a flat (over $\Delta^{n}$,) family of
CE differentials $d(t)$ on $C^{*}(\mathfrak{h})$, such that
$d_{1}(t)=\nabla^{t}$ and $\delta^{-1}(\sum_{m\geq 2}d_{m}(t))=\phi^{t}$.

###### Proof.

Consider the parameterised differential

$d(t)\equiv\delta+\sum_{m\geq 1}d_{m}(t)=\delta+d_{1}^{\nabla^{t}}+\cdots$

and the gauge condition

$\delta^{-1}(d(t))=\phi^{t}.$

By previous results, $d(t)$ exists uniquely. It remains to construct a flat
connection $d_{t}+F(t)$, of which $d(t)$ is a flat section, with

$F(t)\equiv\sum_{m\geq 2}F_{m}(t),\quad
F_{m}(t)\in\Omega_{\Delta^{n}}\otimes_{\mathbb{R}}\Gamma\big{(}{\rm
Sym}^{m}(T_{M}^{\vee})\otimes T_{M}\big{)}.$

A recursive computation can be done by the flat section equation

$[d_{t}+\sum_{k\geq 2}F_{k}(t),\delta+d_{1}(t)+\sum_{m\geq 2}d_{m}(t)]=0,$

and the solution is unique (considering that $\delta$ is invertible on zero-
forms over $M$). In particular,

$F_{2}(t)=-\delta^{-1}\big{(}[d_{t},\nabla^{t}]\big{)}=dt^{i}\wedge\delta^{-1}\big{(}\frac{d}{dt^{i}}\nabla^{t}\big{)}.$

Then one needs to check the flatness

$[d_{t},\sum_{m\geq 2}F_{m}(t)]+\frac{1}{2}[\sum_{m\geq 2}F_{m}(t),\sum_{k\geq
2}F_{k}(t)]=0,$

which is equivalent to the equation

$\displaystyle\left[\delta,\big{[}d_{t},\sum_{m\geq
2}F_{m}(t)\big{]}+\frac{1}{2}\big{[}\sum_{m\geq 2}F_{m}(t),\sum_{k\geq
2}F_{k}(t)\big{]}\right]=0$ $\displaystyle=$
$\displaystyle-\left[d_{t}+\sum_{k\geq 2}F_{k}(t),\big{[}\delta,\sum_{m\geq
2}F_{m}(t)\big{]}\right]$ $\displaystyle=$
$\displaystyle\left[d_{t}+\sum_{k\geq 2}F_{k}(t),\big{[}\sum_{l\geq
1}d_{l}(t),d_{t}+\sum_{m\geq 2}F_{m}(t)\big{]}\right]$ $\displaystyle=$
$\displaystyle-\left[\sum_{l\geq 1}d_{l}(t),\big{[}d_{t},\sum_{m\geq
2}F_{m}(t)\big{]}+\frac{1}{2}\big{[}\sum_{m\geq 2}F_{m}(t),\sum_{k\geq
2}F_{k}(t)\big{]}\right].$

In homogeneous weight components, the equation can be solved recursively, as
long as the initial condition

$[d_{t},F_{2}(t)]=0$

is satisfied. The latter can be checked explicitly. ∎

######  Proposition 65.

Let $d^{\nabla}$ be a CE differential as solved as in Proposition 62
subjecting to the condition that $\delta^{-1}(d_{\geq 2}^{\nabla})=0$, and let
$d^{\prime}=\delta+d^{\prime}_{\geq 1}$ be a CE differential satisfying the
following $\delta^{-1}$ gauge condition $\delta^{-1}(d^{\prime}_{\geq
2})=\phi$ for any $\phi\in\Gamma\big{(}\widehat{\rm Sym}^{\geq
3}(T_{M}^{\vee})\otimes T_{M}\big{)}$. Then there exists a flat family of
differential $d(t)$ such that $d(t=0)=d^{\nabla}$ and $d(t=1)=d^{\prime}$.

###### Proof.

Consider the $\Delta^{1}$ family $\nabla+t(\nabla^{\prime}-\nabla)$ of
connections, and consider the parameterised gauge $\delta^{-1}(d(t))=t\phi$,
the result follows from Lemma 64.∎

For the differential on $C^{*}(\mathfrak{h}\oplus\mathfrak{v})$ encoding the
$L_{\infty}$ structure of the semi-direct product, we also need to compute
$d^{\nabla^{\prime}}\equiv d^{\nabla^{\prime}}_{1}+\sum_{n\geq
2}d^{\nabla^{\prime}}_{n}$, where $d^{\nabla^{\prime}}_{1}$, upon acting on
$\mathfrak{v}^{\vee}[-1]$, is the (de Rham extended) dual connection coming
from a chosen affine connection $\nabla^{\prime}$ on vector bundle $V$. Now
one can show that by picking local basis $\\{v^{a}\\}_{a=1}^{{\rm rk}(V)}$ for
$V^{\vee}$, $d^{\nabla^{\prime}}$ can be solved uniquely from the recursive
formula

$\displaystyle d^{\nabla^{\prime}}_{n+1}:v^{a}\mapsto$ $\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{1\leq k\leq
n}\delta^{-1}\left([d^{\nabla^{\prime}}_{k},d^{\nabla^{\prime}}_{n+1-k}](v^{a})\right)$
$\displaystyle-$ $\displaystyle\sum_{1\leq k\leq
n-1}\delta^{-1}\left(d^{\nabla}_{k}\circ
d^{\nabla^{\prime}}_{n+1-k}(v^{a})\right),\forall a,$

up to a gauge condition on $\delta^{-1}d^{\nabla^{\prime}}_{n}(v^{a})$ for all
$n\geq 2$. Furthermore, there exists a family version parallelising Lemma 64
and Proposition 65, which are omitted here.

Local computations for lower components of $d^{\nabla}$ and
$d^{\nabla^{\prime}}$ are given here. The curvature 2-forms are locally
$\frac{1}{2}R_{ij}\,{}_{m}\,^{n}dx^{i}\wedge dx^{j}$ on $T_{M}$ and
$\frac{1}{2}R^{\prime}_{ij}\,{}_{a}\,^{b}dx^{i}\wedge dx^{j}$ on $V$. When
necessary, we use the asterisked notation for the dual Christoffel symbols and
curvatures. Suppose the first gauge condition is given by

$\delta^{-1}(d^{\nabla}_{2})=\frac{1}{3!}(\phi_{3})^{i}\,_{jkl}y^{j}y^{k}y^{l}\frac{\partial}{\partial
y^{i}},$
$\delta^{-1}(d^{\nabla^{\prime}}_{2})=\frac{1}{2!}(\psi_{3})^{a}\,_{bjk}y^{j}y^{k}v^{b}\frac{\partial}{\partial
v^{a}},$

where $\phi_{3}\in\Gamma\big{(}{\rm Sym}^{3}(T_{M}^{\vee})\otimes
T_{M}\big{)}$ and $\psi_{3}\in\Gamma\big{(}{\rm Sym}^{2}(T_{M}^{\vee})\otimes
V^{\vee}[-1]\otimes V[1]\big{)}$.

$d^{\nabla}_{1}(y^{i})=(\Gamma^{*})^{i}\,_{k}y^{k},$

$d^{\nabla}_{2}(y^{i})=-\frac{1}{3}dx^{m}R^{*}_{km}\,{}^{i}\,_{j}y^{j}y^{k}+\frac{1}{2!}(\phi_{3})^{i}\,_{mjk}dx^{m}y^{j}y^{k},$

$d^{\nabla^{\prime}}_{1}(v^{a})=(\Gamma^{\prime*})^{a}\,_{b}v^{b},$

$d^{\nabla^{\prime}}_{2}(v^{a})=-\frac{1}{2}dx^{m}R^{\prime*}_{im}\,{}^{a}\,_{b}v^{b}y^{i}+(\psi_{3})^{a}\,_{bjk}dx^{j}y^{k}v^{b}.$

Dually, for any $X,Y\in\Gamma(T_{M})$ and $\alpha\in\Gamma(V)$,

$l_{1}(Y)=\nabla Y,$ $l^{\prime}_{1}(\alpha)=\nabla^{\prime}\alpha,$
$\displaystyle l_{2}(X,Y)$ $\displaystyle\equiv$ $\displaystyle
l_{2}(y_{j},y_{k})X^{j}Y^{k}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{3}R_{jmk}\,^{i}+\frac{1}{3}R_{kmj}\,^{i}+(\phi_{3})_{mjk}\,^{i}\right)dx^{m}y_{i}X^{j}Y^{k}$
$\displaystyle=$
$\displaystyle\frac{1}{3}\big{(}\iota_{X}RY+\iota_{Y}RX\big{)}+(\phi_{3})_{mjk}\,^{i}dx^{m}y_{i}X^{j}Y^{k},$
$\displaystyle l^{\prime}_{2}(Y,\alpha)$ $\displaystyle=$ $\displaystyle
Y^{i}\alpha^{a}\left(\frac{1}{2}R^{\prime}_{im}\,{}_{a}\,^{b}+(\psi_{3})_{aim}\,^{b}\right)dx^{m}v_{b}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\iota_{Y}R^{\prime}\alpha+(\psi_{3})_{aim}\,^{b}dx^{m}v_{b}Y^{i}\alpha^{a}.$

To recover the results in $\delta^{-1}$ gauge, one can simply set
$\phi_{3}=\psi_{3}=0$.

##  Appendix B Lower brackets from $d^{\rho}$ and $d^{\mu}$

The recursion starts with $\rho_{1}$ and $\mu_{2}$. Defining

$\rho_{1}(v_{a})\equiv(\rho)_{a}\,^{i}y_{i},\qquad
d^{\rho}_{1}(y^{i})\equiv(\rho)^{i}\,_{a}\cdot v^{a},$

one observes that $(\rho)^{i}\,_{a}$ is an odd transformation since the whole
structure is $(\Omega_{M},d)$-linear(, as opposed to $\mathbb{R}$-linear or
$C^{\infty}_{M}$-linear). Like the local connection form, we have that
$\rho=-(\rho^{*})^{t}$. One helpful way to understand this is to consider the
superconnection $\left(\begin{array}[]{cc}\nabla&\rho\\\
0&\nabla^{\prime}\end{array}\right)$ on $T_{M}\oplus V[1]$, where the dual of
$\rho_{1}$ operation is determined by the dual connection. Solving the
equation

$-[\delta,d^{\rho}_{1+1}](y^{i})=[d^{\nabla}_{1},d^{\rho}_{1}](y^{i})+d^{\nabla^{\prime}}_{1}(d^{\rho}_{1}(y^{i}))$

and dualizing the result, one obtains

$\rho_{1+1}(X,\alpha)=-\rho(\nabla^{\prime}_{X}(\alpha))+\nabla_{X}(\rho(\alpha)).$

For the next order operation,

$-[\delta,d^{\rho}_{1+2}](y^{i})=[d^{\nabla}_{1},d^{\rho}_{2}](y^{i})+[d^{\nabla}_{2},d^{\rho}_{1}](y^{i})+d^{\nabla^{\prime}}_{2}(d^{\rho}_{1}(y^{i}))+d^{\nabla^{\prime}}_{1}(d^{\rho}_{2}(y^{i})),$

gives that

$\displaystyle d^{\rho}_{1+2}(y^{i})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}y^{j}y^{k}v^{a}\big{(}\partial_{k}(\rho_{2})^{i}\,_{ja}+(\rho_{2})^{i}\,_{na}(\Gamma^{*})^{n}\,_{jk}+(\rho_{2})^{i}\,_{jb}(\Gamma^{\prime*})^{b}\,_{ka}$
$\displaystyle-(\Gamma^{*})^{i}\,_{kn}(\rho_{2})^{n}_{ja}+\frac{1}{3}(R^{*})_{nj}\,^{i}\,{}_{k}\rho^{n}\,_{a}-(\phi_{3})^{i}\,_{jnk}\rho^{n}\,_{a}+(\rho_{1})^{i}\,_{b}(\psi_{3})^{b}\,_{ajk}\big{)}.$

The existence of symmetric product $y^{j}$ and $y^{k}$ will automatically
symmetrise the corresponding indices. So after taking duals,

$\displaystyle\rho(y_{j},y_{k},v_{a})$ $\displaystyle=$
$\displaystyle-\nabla_{\\{k}\rho_{2}(y_{j\\}},v_{a})+\rho_{2}(\nabla_{\\{j}y_{k\\}},v_{a})$
$\displaystyle+\rho_{2}(y_{\\{j},\nabla^{\prime}_{k\\}}v_{a})-\frac{1}{3}R_{n\\{j}\,{}_{k\\}}\,^{i}y_{i}\rho_{a}\,^{n}$
$\displaystyle-(\phi_{3})_{jnk}\,^{i}\rho_{a}\,^{n}y_{i}+\rho_{b}\,^{i}(\psi_{3})_{ajk}\,^{b}y_{i},$

where we have used the notation $\\{-,-,\cdots\\}$ to denote the
symmetrisation with respect to the corresponding indices.

In the computation of the first Atiyah-Chern class in Section 4.4, the
following result (only works in $\delta^{-1}$ gauge) is helpful.

$\displaystyle\nabla_{k}\rho_{2}(y_{j},v_{a})-\rho_{2}(\nabla_{k}y_{j},v_{a})-\rho_{2}(y_{j},\nabla^{\prime}_{k}v_{a})$
$\displaystyle=$
$\displaystyle\nabla_{\\{k}\rho_{2}(y_{j\\}},v_{a})-\rho_{2}(\nabla_{\\{k}y_{j\\}},v_{a})-\rho_{2}(y_{\\{j},\nabla^{\prime}_{k\\}}v_{a})+\nabla_{[k}\rho_{2}(y_{j]},v_{a})$
$\displaystyle-\rho_{2}(\nabla_{[k}y_{j]},v_{a})-\rho_{2}(y_{[j},\nabla^{\prime}_{k]}v_{a})$
$\displaystyle=$
$\displaystyle-\rho_{jka}\,^{i}y_{i}-\frac{1}{3}R_{n\\{j}\,{}_{k\\}}\,^{i}y_{i}\rho_{a}\,^{n}-\frac{1}{2}R^{\prime}_{kja}\,{}^{c}\rho_{c}\,^{i}y_{i}+\frac{1}{2}\rho_{a}\,^{n}R_{kjn}\,^{i}y_{i}$
$\displaystyle=$
$\displaystyle-\rho(y_{j},y_{k},v_{a})+\frac{1}{3}R(\rho(v_{a}),y_{j})y_{k}-\frac{2}{3}R(\rho(v_{a}),y_{k})y_{j}-\frac{1}{2}\rho(R^{\prime}_{kj}v_{a}).$

For $d^{\mu}$, the first nontrivial recursive computation gives

$-[\delta,d^{\mu}_{2+1}](v^{a})=[d^{\nabla^{\prime}}_{1},d^{\mu}_{2}](v^{a})+d^{\rho}_{1}(d^{\nabla^{\prime}}_{2}(v^{a})),$

so

$\displaystyle d^{\mu}_{2+1}(v^{a})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}y^{k}v^{b}\wedge
v^{c}\big{(}\partial_{k}(\mu^{a}_{bc})+\mu^{a}_{dc}(\Gamma^{\prime*})_{k}\,^{d}\,{}_{b}+\mu^{a}_{bd}(\Gamma^{\prime*})_{k}\,^{d}\,{}_{c}$
$\displaystyle-(\Gamma^{\prime*})_{k}\,^{a}\,{}_{d}\mu^{d}_{bc}-R^{\prime*}_{ik}\,{}^{a}\,_{b}(\rho^{*})^{i}\,_{c}+2(\psi_{3})^{a}\,_{bki}(\rho^{*})^{i}\,_{c}\big{)}.$

Dually this gives

$\displaystyle\mu_{2+1}(y_{k},v_{b},v_{c})$ $\displaystyle=$
$\displaystyle-\nabla^{\prime}_{k}\mu(v_{b},v_{c})+\mu(\nabla^{\prime}_{k}v_{b},v_{c})+\mu(v_{b},\nabla^{\prime}_{k}v_{c})$
$\displaystyle+\frac{1}{2}R^{\prime}(\rho(v_{c}),y_{k})v_{b}-\frac{1}{2}R^{\prime}(\rho(v_{b}),y_{k})v_{c}$
$\displaystyle+(\psi_{3})_{bki}\,^{a}(\rho)_{c}\,^{i}v_{a}-(\psi_{3})_{cki}\,^{a}(\rho)_{b}\,^{i}v_{a}.$

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