E}_{\ell},\partial]=0,$ $\displaystyle\alpha^{\prime(1)}_{E}(a;l)$
$\displaystyle=R_{\nabla^{\prime
E}}(a,\ell)+[\partial,\omega^{\prime(2)}_{L}](a,\ell)=0,$
$\displaystyle\alpha^{\prime(2)}_{E}(a,b;l)$
$\displaystyle=d_{L}^{\nabla^{\prime
E}}\omega^{\prime(2)}_{L}(a,b,\ell)+[\partial,\omega^{\prime(3)}_{L}](a,b,\ell)=0,$
$\displaystyle\alpha^{\prime(3)}_{E}(a,b,c;l)$
$\displaystyle=d^{\nabla^{\prime
E}}\omega_{L}^{\prime(3)}(a,b,c,\ell)+\omega_{L}^{\prime(2)}\wedge\omega_{L}^{\prime(2)}(a,b,c,\ell)=0,$

for any $a,b,c\in\Gamma(A)$ and any $\ell\in\Gamma(L)$ projecting down to
$l\in\Gamma(L/A)$. That is to say, the Atiyah cocycle $\alpha_{E}^{\prime}$
associated to the $L$-superconnection
$D^{\prime}_{L}=\partial+d^{\nabla^{\prime
E}}+\omega_{E}^{\prime(2)}\wedge\cdot+\omega_{E}^{\prime(3)}\wedge\cdot$ is
identically zero, so $D_{L}$ is $A$-compatible. By Theorem 2.12, it means that
the Atiyah class $[\alpha^{\prime}_{E}]=[\alpha_{E}]$ vanishes, which proves
the claim of the proposition. Notice that any other choice of splittings
$\sigma_{0},\sigma_{1},\sigma_{2}$ does not change this result.∎

######  Remark 3.7.

Notice that the Atiyah cocycle $\mathrm{at}^{\nu}$ being a coboundary was
instrumental in proving Equation (83), and thus Equation (84) too. Using this
identity, we then could prove that $\alpha^{\prime(3)}_{E}=0$ too so, to prove
that $[\alpha_{E}]=0$ we crucially used the fact that $[\mathrm{at}^{\nu}]=0$.

######  Remark 3.8.

From the graded geometric perspective, since $\varphi^{(1)}$, $\varphi^{(2)}$
and $\varphi^{(3)}$ identically vanish when restricted to $A$, they
canonically induce elements
$\phi^{(0)}\in\Gamma(A^{\circ}\otimes\mathrm{End}_{0}(E))$,
$\phi^{(1)}\in\Omega^{1}(A,A^{\circ}\otimes\mathrm{End}_{-1}(E))$ and
$\phi^{(2)}\in\Omega^{2}(A,A^{\circ}\otimes\mathrm{End}_{-2}(E))$,
respectively. Their definition has been made so that their sum
$\phi=\phi^{(0)}+\phi^{(1)}+\phi^{(2)}$ satisfies $\alpha_{E}=s(\phi)$,
showing explicitly that $\alpha_{E}$ is a coboundary.

##  4 Relationship with Atiyah classes of differential graded vector bundles

This section is devoted to generalizing a result obtained for the normal
complex in [Liao, 2023], to representations up to homotopy.

###  4.1 Generalities about graded geometry and $Q$-connections

A graded manifold $\mathcal{M}$ is a locally ringed space whose structure
sheaf $\mathcal{C}^{\infty}(\mathcal{M})$ is locally of the form
$\mathcal{C}^{\infty}(U)\otimes\widehat{S}(V^{*})$, where $U\subset M$ is an
open subset of a given smooth manifold $M$ – called the _base_ or _body_ – and
$\widehat{S}(V^{*})$ stands for the formal power series over the dual of a
graded vector space $V=\bigoplus_{i\in\mathbb{Z}}V_{i}$. We refer to [Fairon,
2017, Kotov and Salnikov, 2024] for an explanation of why one should use
formal power series instead of mere polynomials (which are not local rings).
We say that $\mathcal{M}$ is positively (resp. negatively) graded if the
formal power series over $V^{*}$ are positively (resp. negatively) graded,
namely, if $V$ is a negatively (resp. positively) graded vector space. If the
graded manifold is (non-) positively or (non-) negatively graded, which will
be the case in the present section, using formal power series is equivalent to
using polynomials.

Following [Mehta, 2006, Remark 2.2.1], one should mainly consider graded
vector spaces of finite dimension, such that all but a finite number of the
$V_{i}$ are zero, and those which are not are finite dimensional. This choice
allows performing tensor product operations in the category of graded vector
spaces. This assumption is in line with the choice of graded vector bundles
made in Section 2. Batchelor’s theorem establishes that every positively
(resp. negatively) graded manifold is locally isomorphic to a split negatively
(resp. positively) graded vector bundle over a smooth manifold. For a
generalization of Batchelor’s statement to gradings over the integers, we
refer to [Kotov and Salnikov, 2024]. While in Section 2 we worked with (split,
finite dimensional) graded vector bundles over smooth manifolds, we refer to
[Mehta, 2009, Mehta et al., 2015] for details about defining graded vector
bundles over a graded manifold $\mathcal{M}$ in full generality. We will not
need those apart from settling the conventions.

######  Definition 4.1.

A _differential graded (dg) manifold_ or _$Q$ -manifold_ is a graded manifold
$\mathcal{M}$ equipped with a cohomological degree $+1$ vector field $Q$, i.e.
such that $[Q,Q]=0$.

######  Definition 4.2.

[Aschieri et al., 2021] Let $\mathcal{E}$ be a graded vector bundle over the
dg manifold $(\mathcal{M},Q)$. A _dg connection_ (or _$Q$ -connection_) on
$\mathcal{E}$ is a degree $+1$ vector field
$Q_{\mathcal{E}}\in\mathfrak{X}(\mathcal{E^{*}})$ satisfying the following
conditions:

  1. 1.

$Q_{\mathcal{E}}$ preserves
$\Gamma(\mathcal{E})\subset\mathcal{C}^{\infty}(\mathcal{E}^{*})$,

  2. 2.

$Q_{\mathcal{E}}$ projects to $Q$, i.e $Q_{\mathcal{E}}|_{\mathcal{M}}=Q$.

The curvature of a dg connection $Q_{\mathcal{E}}$ on $\mathcal{E}$ is the
degree 2 vector field on $\mathcal{E^{*}}$ given by:

$R_{Q_{\mathcal{E}}}=Q_{\mathcal{E}}^{2}=\frac{1}{2}[Q_{\mathcal{E}},Q_{\mathcal{E}}].$

######  Lemma 4.3.

[Mehta et al., 2015] Let $Q_{\mathcal{E}}$ be a $Q$-connection on
$\mathcal{E}$. If the curvature of $Q_{\mathcal{E}}$ is zero then
$(\mathcal{E},Q_{\mathcal{E}})$ is a _differential graded (dg) vector bundle_
– or _$Q$ -bundle_ – i.e. a vector bundle is the category of dg manifolds.

A Lie algebroid $A$ can be realized as a differential graded manifold by
shifting the degree of the fibers by $-1$. The corresponding graded manifold
being denoted $A[1]$ is modelled on the vector bundle $\pi:A[1]\to M$. It is
also known that the Lie algebroid differential corresponds to a cohomological
vector field of degree $+1$. Moreover, the space $\Omega(A,E)$ of $A$-forms
valued in $E$ can be seen as the space of sections of the pull-back graded
vector bundle $\mathcal{E}=\pi^{*}E\to A[1]$ over the dg manifold $A[1]$. By
duality [Mehta, 2009], the sheaf of sections of $\mathcal{E}\to A[1]$ –
denoted $\Gamma_{A[1]}(\mathcal{E})$ – is the space of linear functions on the
dual graded vector bundle $\mathcal{E}^{*}=\pi^{*}E^{*}\to A[1]$, denoted
$\mathcal{C}^{\infty}_{lin}(\mathcal{E}^{*})$. Definition 2.2 of an
$A$-superconnection on $E$ can now be understood from the differential graded
geometric language.

######  Lemma 4.4.

The data of an $A$-superconnection on the graded vector bundle $E$ is
equivalent to the data of a dg connection on the dg vector bundle
$\mathcal{E}=\pi^{*}E\to A[1]$.

###### Proof.

$\Longrightarrow$ An $A$-superconnection
$D_{A}\colon\Omega(A,E)\to\Omega(A,E)$ of total degree $+1$ is understood as a
linear vector field on $\mathcal{E}^{*}$, i.e. a degree $+1$ vector field
$D_{A}$ on $\mathcal{E}^{*}$ preserving
$\mathcal{C}^{\infty}_{lin}(\mathcal{E}^{*})$. Moreover, this vector field is
compatible with the Lie algebroid differential $d_{A}$ in the sense that the
canonical projection $\mathcal{P}:\mathcal{E}^{*}\to A[1]$ is a morphism of
differential graded manifolds:

$\mathcal{P}^{*}\circ d_{A}=D_{A}\circ\mathcal{P}^{*}.$

The $A$-superconnection $D_{A}$ is thus a $Q$-connection on $\mathcal{E}$.

$\Longleftarrow$ Conversely, choose a $Q$-connection on $\mathcal{E}$. Under
the identification

$\Gamma(\mathcal{E})\simeq\Omega(A)\otimes\Gamma(E)$

the dg-connection $Q_{\mathcal{E}}$ is determined by its action on
$\Gamma(E)$. Since it is a degree 1 operator with respect to the total degree
we have that the restriction on $\Gamma(E_{i})$ is a map

$\displaystyle
Q_{\mathcal{E}}\colon\Gamma(E_{i})\longrightarrow\bigoplus_{k=0}^{\mathrm{rk}(A)}\Omega^{k}(A)\otimes\Gamma(E_{i-k+1}).$

We write $Q_{\mathcal{E}}$ in term of components:

$Q_{\mathcal{E}}=\sum_{k=0}^{\mathrm{rk}(A)}Q^{(k)}_{\mathcal{E}},$

where
$Q^{(k)}_{\mathcal{E}}\colon\Gamma(E_{\bullet})\to\Omega^{k}(A)\otimes\Gamma(E_{\bullet-k+1})$.
Observe that the component
$Q^{(0)}_{\mathcal{E}}\colon\Gamma(E_{\bullet})\to\Gamma(E_{\bullet})$ is a
vector bundle morphism $\partial:\Gamma(E_{\bullet})\to\Gamma(E_{\bullet+1})$,
while the component
$Q^{(1)}_{\mathcal{E}}\colon\Gamma(E_{\bullet})\to\Omega^{1}(A)\otimes\Gamma(E_{\bullet})$
is the exterior covariant derivative associated to an $A$-connection on $E$.

Introducing local coordinates $\\{x^{i},\xi^{\alpha}\\}$ on $A[1]$ and fiber
coordinates $\\{s_{i}^{l}\\}_{1\leq l\leq r_{i}}$ on $E$, where
$r_{i}=\text{rk}(E_{i})$, then $Q_{\mathcal{E}}$ is given by:

$Q_{\mathcal{E}}=d_{A}+\Gamma_{l}^{p}s_{i}^{l}\frac{\partial}{\partial
s_{i}^{p}}=d_{A}+(\Gamma_{\alpha
l}^{p}\xi^{\alpha}s_{i}^{l}+\Gamma_{\beta\gamma
l}^{p}\xi^{\beta}\xi^{\gamma}s_{i-1}^{l}+\underbrace{\cdots}_{\text{higher
degree terms}})\frac{\partial}{\partial s_{i}^{p}},$ (88)

where $\Gamma^{p}_{l}\in C^{\infty}(A[1])\simeq\Omega(A)$ has degree
$|\Gamma^{p}_{l}|=1+|s_{p}|-|s_{l}|$ and is a polynomial of $\xi$ variables,
as shown on the right-hand side. The sum of all terms of polynomial degree $k$
define a connection $k$-form on $E$. Thus we indee obtain the data of an
$A$-superconnection on $E$. ∎

######  Remark 4.5.

If $E$ is a vector bundle over $M$ concentrated in degree zero, terms of total
degree other than one vanish and (88) reduces to

$Q_{\mathcal{E}}=d_{A}+\Gamma_{\alpha
l}^{k}\xi^{\alpha}s^{l}\frac{\partial}{\partial s^{k}},$

which is the local expression of the covariant exterior derivative associated
to an $A$-connection on $E$.

Definition 2.3 of a representation up to homotopy of a Lie algebroid
$A\overset{\pi}{\longrightarrow}M$ can also be understood in the differential
graded geometric picture. A homotopy $A$-module structure on $E$ is an
$A$-superconnection $D_{A}\colon\Omega(A,E)\to\Omega(A,E)$ squaring to zero.
The homological vector field $D_{A}$ is thus a $Q$-connection on
$\mathcal{E}$, with vanishing curvature. By Lemma 4.3, it makes
$\mathcal{E}^{*}$ a differential graded vector bundle over $A[1]$, which is
equivalent to $\mathcal{E}$ being a differential graded vector bundle over
$A[1]$. This equivalence follows from $E^{*}$ being a representation up to
homotopy of $A$ isomorphic to $E$ [Abad and Crainic, 2011]. As a consequence
of this, representations up to homotopy of $A$ correspond to flat
$Q$-connections on $\mathcal{E}$.

######  Proposition 4.6.

[Mehta, 2014, Lemma 4.4] Homotopy $A$-module structures on a (split, finite
dimensional) graded vector bundle $E\to M$ are in one-to-one correspondence
with differential graded vector bundle structures on $\mathcal{E}=\pi^{*}E\to
A[1]$.

###  4.2 Atiyah classes of dg vector bundles and homological perturbation

Differential graded (dg) Lie algebroids are dg vector bundles whose
homological vector field is linear (it preserves the arity of functions) and
which are equipped with a compatible Lie algebroid structure:

######  Definition 4.7.

[Mehta, 2006, Definition 2.4.26] A dg vector bundle
$(\mathcal{L},\mathcal{Q})$ over a dg manifold $(\mathcal{M},Q)$ is a dg Lie
algebroid if the two following conditions are satisfied:

  1. 1.

linearity: $\mathcal{Q}$ preserves fiberwise linear functions on
$\mathcal{L}$444This linearity condition implies that $\mathcal{Q}$ is
invariant under the (de)suspension operator so that for any $j\in\mathbb{Z}$,
it canonically induces a linear homological vector field $\mathcal{Q}[j]$
(itself) on $\mathcal{M}[j]$.:

$\mathcal{Q}(\mathcal{L}^{*})\subset\mathcal{L}^{*},$

  2. 2.

compatibility: on $\mathcal{L}[1]$, $\mathcal{Q}[1]$ and the Chevalley-
Eilenberg differential $d_{\mathcal{L}}$ commute:

$[d_{\mathcal{L}},\mathcal{Q}[1]]=0.$

Let $(\mathcal{L},\mathcal{Q})$ be a dg Lie algebroid, defined over a dg
manifold $(\mathcal{M},Q)$, and let $(\mathcal{E},Q_{\mathcal{E}})$ be a dg
vector bundle over $\mathcal{M}$. We denote the sections of the graded vector
bundle $\mathcal{E}$ over $\mathcal{M}$ as
$\Gamma_{\mathcal{M}}(\mathcal{E})$. A $\mathcal{L}$-connection on
$\mathcal{E}$ is a degree 0 map
$\nabla:\Gamma_{\mathcal{M}}(\mathcal{L})\times\Gamma_{\mathcal{M}}(\mathcal{E})\to\Gamma_{\mathcal{M}}(\mathcal{L})$
satisfying the usual identities:

$\displaystyle\nabla_{fX}(e)$ $\displaystyle=f\nabla_{X}(e),$
$\displaystyle\nabla_{X}(fe)$
$\displaystyle=\rho(X)(f)\,e+(-1)^{|f||X|}f\nabla_{X}(e),$

for every $f\in\mathcal{C}^{\infty}(\mathcal{M})$,
$X\in\Gamma_{\mathcal{M}}(\mathcal{L})$ and
$e\in\Gamma_{\mathcal{M}}(\mathcal{E})$. The following Definition and
Proposition were first introduced in [Mehta et al., 2015].

######  Definition 4.8.

Given a $\mathcal{L}$-connection on $\mathcal{E}$, the _Atiyah cocycle_
associated to $\nabla$ is the bundle map
$\mathrm{At}_{\mathcal{E}}\in\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})$
defined by:

$\mathrm{At}_{\mathcal{E}}(X,e)=\mathcal{Q}_{\mathcal{E}}(\nabla_{X}(e))-(-1)^{|X|}\nabla_{X}(\mathcal{Q}_{\mathcal{E}}(e))-\nabla_{\mathcal{Q}(X)}(e).$

######  Proposition 4.9.

Let $\mathfrak{Q}=\mathcal{Q}+[\mathcal{Q}_{\mathcal{E}},.\,]$ be the induced
homological vector field on the graded manifold
$\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})$. Then

  1. 1.

$\mathrm{At}_{\mathcal{E}}$ is a $\mathfrak{Q}$-cocycle of degree $1$;

  2. 2.

the cohomology class of $\mathrm{At}_{\mathcal{E}}$ does not depend on the
choice of connection $\nabla$.

Let $A\hookrightarrow L$ be a Lie pair, and denote by $\mathcal{L}$ the pull-
back Lie algebroid through $\pi:A[1]\to M$ of the Lie algebroid $L$
[Mackenzie, 2005], namely $\mathcal{L}=\pi^{!}L$ is a graded vector bundle
over $A[1]$. This graded manifold, originally introduced by Stiénon,
Vitagliano and Xu [Stiénon et al., 2022], turns out to be a dg Lie algebroid,
whose homological vector field we denote by $\mathcal{Q}$. Using these data,
the authors in [Stiénon et al., 2022] establish a bijection between
$\Omega^{\bullet}(A,A^{\circ}\otimes\mathrm{End}(L/A))$ and
$\Gamma_{A[1]}(\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{L}))$, inducing a
map sending the Atiyah class of the Bott representation to the Atiyah class of
the dg vector bundle $\mathcal{L}\to A[1]$.

In [Liao, 2023], the author simplifies the proof of the former by summoning a
more straightforward homological perturbation theoretic argument relying on a
splitting of the following diagram

${0}$${A[1]}$${L}$${0}$${0}$${0}$${L/A}$${0}$$\scriptstyle{\iota\circ
sp}$$\scriptstyle{p}$$\scriptstyle{p}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{0}$

In the present section, we will generalize the result in [Liao, 2023], valid
for the normal complex and the Bott representation, to a wide class of Lie
algebroid representations of the Lie algebroid $A$. Namely, those
representations $K$ who are resolved by regular homotopy $A$-modules
$(E,\partial)$, introduced in Section 2.4:

${\ldots}$${E_{-2}}$${E_{-1}}$${E_{0}}$${0}$${\ldots}$${0}$${0}$${K}$${0}$$\scriptstyle{\partial}$$\scriptstyle{\phi}$$\scriptstyle{\partial}$$\scriptstyle{\phi}$$\scriptstyle{\partial}$$\scriptstyle{\phi}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{0}$
(89)

Recall that a _contraction_ – or _strong deformation retract_ – consists of
two chain complexes $(V,d_{V})$ and $(W,d_{W})$, together with two chain maps
$\phi:V\to W$ and $\sigma:W\to V$, and a degree $-1$ map of vector spaces
$\theta:V\to V$, such that:

  1. 1.

$W$ is a _retract_ of $V$, so $\phi\circ\sigma=\mathrm{id}_{W}$,

  2. 2.

$\theta$ is a _chain homotopy_ between $\mathrm{id}_{V}$ and
$\sigma\circ\phi$, namely:

$\mathrm{id}_{V}-\sigma\circ\phi=[d_{V},\theta]\,,$

  3. 3.

the _side conditions_ hold:

$\theta^{2}=0\,,\qquad\theta\circ\sigma=0\,\qquad\phi\circ\theta=0.$

We draw these data under the compact form

${\big{(}W,d_{W}\big{)}}$${\big{(}V,d_{V}\big{)}}$$\scriptstyle{\sigma}$$\scriptstyle{\phi}$$\scriptstyle{\theta}$

and use it as a shorthand notation for contraction data.

In the following, we denote $\nabla^{K}$ the flat Lie algebroid $A$-connection
on $K$ inducing the $A$-module structure on $K$ (with associated exterior
covariant derivative $d_{A}^{\nabla^{K}}$), and $D_{A}$ the flat
$A$-superconnection on $E$ inducing the homotopy $A$-structure, so that the
compatibility condition (38) holds. We denote $d^{K}$ the exterior covariant
derivative associated to the $A$-module structure on
$A^{\circ}\otimes\mathrm{End}(K)$. We denote $\mathcal{E}=\pi^{*}E$ the pull-
back bundle of $E$ along the projection $\pi:A[1]\to M$. We will show that we
have the contraction data

${\hskip
36.98866pt\left(\Omega\big{(}A,A^{\circ}\otimes\mathrm{End}(K)\big{)},d^{K}\right)}$${\left(\Gamma_{A[1]}\big{(}\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})\big{)},\mathcal{Q}+[D_{A},.\,]\right)\hskip
99.58464pt}$

which is such that, whenever $K=L/A$ and $E=A[1]\oplus L$ – so
$\mathcal{E}\simeq\mathcal{L}$ – we recover the contraction data presented in
[Liao, 2023].

Recall that the chain complex $(E,\partial)$ being a finite length resolution
of the vector bundle $K$, there exists a section $\sigma:K_{\bullet}\to
E_{\bullet}$ and a map $\theta:E_{\bullet}\to E_{\bullet-1}$, such that
$\phi:(E_{\bullet},\partial)\to(K_{\bullet},0)$ is a homotopy equivalence with
homotopy inverse $\sigma$ and chain homotopy $\theta$ (see Proposition 2.14).
Denote $\mathcal{K}=\pi^{*}K$ the pull-back bundle of $K$ through the
projection $\pi:A[1]\to M$. The morphisms $\partial$, $\theta$, $\phi$ and
$\sigma$ can be pulled back too, inducing graded vector bundle morphisms (over
$A[1]$) satisfying similar properties as the former with respect to
$\mathcal{K}$ and $\mathcal{E}$. We also denote them $\partial$, $\theta$,
$\phi$ and $\sigma$, except that they now become
$\mathcal{C}^{\infty}(A[1])\simeq\Omega(A)$-linear. It is straightforward to
see that we have the following contraction data

${\left(\Gamma_{A[1]}(\mathcal{K}),0\right)}$${\left(\Gamma_{A[1]}(\mathcal{E}),\partial\right)}$$\scriptstyle{\sigma}$$\scriptstyle{\phi}$$\scriptstyle{\theta}$

Now we only need proving similar results to [Liao, 2023, Lemma 2.6, Theorem
2.7], in the present context. Set

$d=D_{A}-\partial\qquad\text{and}\qquad F^{q}(E)=\bigoplus_{i\geq
q}\Gamma_{A[1]}(\mathcal{E}_{-i})\quad\text{for all $q$.}$

Let also $\varsigma,\varphi,\vartheta$ be the following perturbed contraction
operators:

$\vartheta=\theta_{d}=\sum_{k=0}^{\infty}\theta(-d\theta)^{k}\,,\quad\varsigma=\sigma_{d}=\sum_{k=0}^{\infty}(-\theta
d)^{k}\sigma\,,\quad\text{and}\quad\varphi=\phi_{d}=\sum_{k=0}^{\infty}\phi(-d\theta)^{k}.$

######  Lemma 4.10.

The operator $d$ is a perturbation of $(\Gamma_{A[1]}(\mathcal{E}),\partial)$
and satisfies the property:

$d\circ\theta\left(F^{q}(E)\right)\subset F^{q+1}(E),\quad\text{for all $q$}.$

###### Proof.

This is straightforward because $D_{A}$ contains $\partial$ by definition, see
Equation (6). ∎

######  Theorem 4.11.

The operator $d=D_{A}-\partial$ is a small perturbation of the contraction
$(\Gamma_{A[1]}(\mathcal{E}),\partial,\theta)$ over $\mathbb{R}$. The
perturbed contraction

${\left(\Gamma_{A[1]}(\mathcal{K}),d_{A}^{\nabla^{K}}\right)}$${\left(\Gamma_{A[1]}(\mathcal{E}),D_{A}\right)}$$\scriptstyle{\varsigma}$$\scriptstyle{\varphi}$$\scriptstyle{\vartheta}$

forms a contraction data over $(\mathcal{C}^{\infty}(A[1]),d_{A})$.

###### Proof.

The perturbed operator associated to the zero differential on the chain
complex $(\Gamma_{A[1]}(\mathcal{K}),\delta=0)$ is given by

$\delta_{d}=\sum_{k=0}^{\infty}\phi d(-\theta d)^{k}\sigma=\phi D_{A}\sigma.$

The last equality can be explained because the bottom chain complex spaces in
(89) vanish except at degree 0. Since $\phi d(-\theta
d)^{k}\sigma=\phi(-d\theta)^{k}d\sigma$, and $(d\theta)(F^{q}(E))\subset
F^{q+1}(E)$, we deduce that the only non-vanishing term, after projection
under $\phi$, can only be $\phi d\sigma=\phi D_{A}\sigma$.

We only need to show by direct computations that $\vartheta$, $\varsigma$ and
$\varphi$ are $\mathcal{C}^{\infty}(A[1])$-linear, and that $\delta_{d}$
coincides with $d_{A}^{\nabla^{K}}$. Let us start with $\vartheta$. Since
$\theta^{2}=0$, we have (by induction):

$\displaystyle\theta(d\theta)^{k}(f\cdot\lambda)$
$\displaystyle=\theta\bigg{(}(-1)^{|f|}d_{A}f\theta(d\theta)^{k-1}(\lambda)+f(d\theta)^{k}(\lambda)\bigg{)}$
$\displaystyle=(-1)^{2|f|+1}d_{A}f\underbrace{\theta^{2}}_{=\,0}(d\theta)^{k-1}(\lambda)+(-1)^{|f|}f\theta(d\theta)^{k}(\lambda)$
$\displaystyle=(-1)^{|f|}f\theta(d\theta)^{k}(\lambda),$

for every $f\in\Omega(A)$ and $\lambda\in\Gamma(\mathcal{E})$. This proves the
$\mathcal{C^{\infty}}(A[1])$-linearity of $\vartheta$.

Let us turn to $\varsigma$. By definition of the homotopy operator $\theta$,
we have $\theta\sigma=0$, so for every $f\in\Omega(A)$ and
$\kappa\in\Gamma_{A[1]}(\mathcal{K})$, on the one hand:

$\displaystyle\theta d\sigma(f\cdot\kappa)$
$\displaystyle=\theta(d(f\cdot\sigma(\kappa)))$
$\displaystyle=\theta(d_{A}f\sigma(\kappa))+\theta((-1)^{|f|}fd\sigma(\kappa))$
$\displaystyle=(-1)^{|f|+1}d_{A}f\underbrace{\theta\sigma}_{=\,0}(\kappa)+f\theta
d\sigma(\kappa)$ $\displaystyle=f\theta d\sigma(\kappa),$

and on the other hand, since $\theta^{2}=0$,

$\displaystyle(\theta d)^{2}\sigma(f\cdot\kappa)$ $\displaystyle=\theta
d(f\theta d\sigma(\kappa))$ $\displaystyle=\theta(d_{A}f\theta
d\sigma(\kappa)+(-1)^{|f|}fd\theta d\sigma(\kappa))$
$\displaystyle=(-1)^{|f|+1}d_{A}f\underbrace{\theta^{2}}_{=\,0}d\sigma(\kappa)+f(\theta
d)^{2}\sigma(\kappa)$ $\displaystyle=f(\theta d)^{2}\sigma(\kappa).$

More generally, by induction:

$(\theta
d)^{k}\sigma(f\cdot\kappa)=(-1)^{|f|+1}d_{A}f\underbrace{\theta^{2}}_{=\,0}d(\theta
d)^{k-2}\sigma(\kappa)+f(\theta d)^{k}\sigma(\kappa)=f(\theta
d)^{k}\sigma(\kappa).$

This proves the $\mathcal{C^{\infty}}(A[1])$-linearity of $\varsigma$.

Finally, again by definition of the homotopy operator $\theta$, we have
$\phi\theta=0$, and so we have, for every $f\in\Omega(A)$,

$\displaystyle\phi d\theta(f\cdot\lambda)$ $\displaystyle=(-1)^{|f|}\phi
d(f\theta(\lambda))$
$\displaystyle=(-1)^{|f|}\phi(d_{A}(f)\theta(\lambda))+\phi(fd\theta(\lambda))$
$\displaystyle=(-1)^{|f|}d_{A}(f)\underbrace{\phi\theta}_{=\,0}(\lambda)+f\phi
d\theta(\lambda)$ $\displaystyle=f\phi d\theta(\lambda).$

Since $\theta^{2}=0$, it is

$\displaystyle\phi(d\theta)^{2}(f\cdot\lambda)$ $\displaystyle=(-1)^{|f|}\phi
d\theta(d_{A}(f)\theta(\lambda))+\phi d\theta(fd\theta(\lambda))$
$\displaystyle=(-1)^{2|f|+1}\phi
d(d_{A}f\underbrace{\theta^{2}}_{=\,0}(\lambda))+(-1)^{|f|}\phi d(f\theta
d\theta(\lambda))$ $\displaystyle=(-1)^{|f|}\phi(d_{A}f\theta
d\theta(\lambda))+\phi(f(d\theta)^{2}(\lambda))$
$\displaystyle=(-1)^{|f|}d_{A}f\underbrace{\phi\theta}_{=\,0}d\theta(\lambda)+\phi(f(d\theta)^{2}(\lambda))$
$\displaystyle=f\phi(d\theta)^{2}(\lambda),$

and more generally, by induction, it is

$\phi(d\theta)^{k}(f\cdot\lambda)=(-1)^{|f|}d_{A}f\underbrace{\phi\theta}_{=\,0}(d\theta)^{k-1}(\lambda)+f\phi(d\theta)^{k}(\lambda)=f\phi(d\theta)^{k}(\lambda).$

This proves the $\mathcal{C^{\infty}}(A[1])$-linearity of $\varphi$.

Denote by $d_{A}^{\nabla}$ the exterior covariant derivative associated to the
Lie algebroid $A$-connection $\nabla$ on $E$, that is:
$d_{A}^{\nabla}=D_{A}^{(1)}$. The compatibility condition (38) then reads

$\phi\circ d_{A}^{\nabla}=d_{A}^{\nabla^{K}}\circ\phi.$ (90)

For every $\kappa\in\Gamma_{A[1]}(\mathcal{K})$, $D_{A}\sigma(\kappa)$ has
components in $F^{-1}(E)$, but only the component in $E_{0}$ is non-trivially
projected down to $K$ as the projections of the other components vanish. The
only possibly non-vanishing component is then precisely
$d_{A}^{\nabla}\sigma(\kappa)$. Thus, by Equation (90), we have

$\delta_{d}(\kappa)=\phi D_{A}\sigma(\kappa)=\phi
d_{A}^{\nabla}\sigma(\kappa)=d_{A}^{\nabla^{K}}\phi\sigma(\kappa)=d_{A}^{\nabla^{K}}\kappa.$

Since both $\phi D_{A}\sigma$ and $d_{A}^{\nabla^{K}}$ satisfy the Leibniz
rule with respect to $\mathcal{C^{\infty}}(A[1])=\Omega(A)$, we deduce that
$\delta_{d}=d_{A}^{\nabla^{K}}$.∎

Theorem 4.11 together with [Liao, 2023, Proposition A.9], imply:

######  Corollary 4.12.

The following forms a contraction data over
$(\mathcal{C}^{\infty}(A[1]),d_{A})$:

${\hskip
22.76228pt\left(\Gamma_{A[1]}\big{(}\pi^{*}(A^{\circ})\otimes\mathrm{End}(\mathcal{K})\big{)},d^{K}\right)}$${\left(\Gamma_{A[1]}\big{(}\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})\big{)},\mathcal{Q}+[D_{A},.\,]\right)\hskip
99.58464pt}$

The space on the left-hand side being canonically isomorphic to
$\Omega\big{(}A,A^{\circ}\otimes\mathrm{End}(K)\big{)}$, we see that the space
of sections of $\pi^{*}(A^{\circ})\otimes\mathrm{End}(\mathcal{K})$ over
$A[1]$ is isomorphic to
$\Omega^{1}\big{(}A,A^{\circ}\otimes\mathrm{End}(K)\big{)}$, namely where
belongs the Atiyah cocycle of the representation $K$ with respect to the Lie
algebroid pair $A\hookrightarrow L$. On the other hand, the space of sections
of $\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})$ over $A[1]$ is where
lives the Atiyah cocycle of the dg vector bundle $\mathcal{E}\to A[1]$ with
respect to an $\mathcal{L}$-connection (in the sense of [Mehta et al., 2015]).
Hence, the isomorphism induced by the contraction data at the level of
cohomology sends the former Atiyah class to the latter, and vice versa. The
main results of this section are gathered in the following statement.

######  Theorem 4.13.

Let $A\hookrightarrow L$ be a Lie pair and let $(E,\partial)$ be a regular
representation up to homotopy of $A$. Assume $(E,\partial)$ is a resolution of
a Lie algebroid representation $K$ of $A$. Then we have the following quasi-
isomorphisms

$\left(\widehat{\Omega}(E),s\right)\simeq\left(\Omega\big{(}A,A^{\circ}\otimes\mathrm{End}(K)\big{)},d^{K}\right)\simeq\left(\Gamma_{A[1]}\big{(}\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})\big{)},\mathcal{Q}+[D_{A},.\,]\right).$

###### Proof.

The first isomorphism is Theorem 2.15 while the second is Corollary 4.12. ∎

We should then be able to recover the result about the Atiyah class of the
normal complex presented in [Liao, 2023] as an application of the statement
above to the setting of Section 3.2. First, choosing a splitting of the short
exact sequence

${0}$${A}$${L}$${L/A}$${0,}$$\scriptstyle{\iota}$$\scriptstyle{\tau}$$\scriptstyle{p}$$\scriptstyle{\sigma}$
(91)

we have the contraction data

${\left(\Gamma_{A[1]}\big{(}\pi^{*}(L/A)\big{)},0\right)}$${\left(\Gamma_{A[1]}(\mathcal{E}),\widetilde{\iota}\right)}$$\scriptstyle{\sigma}$$\scriptstyle{p}$$\scriptstyle{\widetilde{\tau}}$

where $\widetilde{\iota}:A[1]\to A\to L$ is the suspension of the inclusion
$\iota$ and $\widetilde{\tau}:L\to A\to A[1]$ is the desuspension of the
projection $\tau$. Theorem 4.11 then establishes the contraction data

${\left(\Gamma_{A[1]}\big{(}\pi^{*}(L/A)\big{)},d_{A}^{Bott}\right)}$${\left(\Gamma_{A[1]}(\mathcal{E}),D_{A}\right)}$$\scriptstyle{\varsigma}$$\scriptstyle{p}$$\scriptstyle{\widetilde{\tau}}$

where $\varsigma=\sigma-\widetilde{\tau}D_{A}\sigma$. From this, the right-
hand side of Theorem 4.13 reads

$\left(\Omega\big{(}A,A^{\circ}\otimes\mathrm{End}(L/A)\big{)},d^{L/A}\right)\simeq\left(\Gamma_{A[1]}\big{(}\mathcal{L}^{*}\otimes\mathrm{End}(\mathcal{E})\big{)},\mathcal{Q}+[D_{A},.\,]\right),$

where $d^{L/A}$ is understood as the exterior covariant derivative associated
to the representation $A^{\circ}\otimes\mathrm{End}(L/A)$. This, together with
the following Lemma, establishes the result on the Atiyah class of the normal
complex from [Liao, 2023] as a particular case of Theorem 4.13.

######  Lemma 4.14.

Let $A\hookrightarrow L$ be a Lie pair and let $E=A[1]\oplus L$ be the normal
complex. The dg vector bundle $(\mathcal{L},\mathcal{Q})$ is dg isomorphic to
the dg vector bundle $(\mathcal{E},D_{A})$.

###### Proof.

For the sake of completeness, let us use the notations of [Liao, 2023, Remark
2.5]. Choose a trivialization of $L$ and a local frame $e_{1},\ldots,e_{r}$ of
$A$, extended to a local frame
$e_{1},\ldots,e_{r},e_{r+1},\ldots,e_{r+r^{\prime}}$ of $L$, that in turn
induces a local dual frame of $L^{*}$ denoted $e^{1},\ldots,e^{r+r^{\prime}}$.
Choosing a splitting of the sequence (91), let
$\frac{\partial}{\partial\eta^{1}},\ldots,\frac{\partial}{\partial\eta^{r}}$
be the local frame of $A[1]$ obtained from $e_{1},\ldots,e_{r}$ through the
homotopy map $\widetilde{\tau}$, namely
$\frac{\partial}{\partial\eta^{i}}=\widetilde{\tau}(e_{i})$ and
$\iota\left(\frac{\partial}{\partial\eta^{i}}\right)=e_{i}$. This choice of
notation comes from the identification $TA[1]^{\text{vertical}}\simeq A[1]$,
so that $\eta^{1},\ldots,\eta^{r}$ is a local frame of $A[1]^{*}$ such that
$\frac{\partial}{\partial\eta^{j}}(\eta^{i})=\delta^{i}_{j}$.

Pick up a $TM$-connection on $A$ – say $\nabla$, giving an $L$-connection on
$A$ which in turn induces a basic connection $\nabla^{bas}$ (see Definition
3.1 and the preceding paragraph). This connection allows to write the
$A$-superconnection on $E$ with respect to the fiber coordinates under a form
that explicitly involves the basic connection. More precisely, the connection
1-form $\omega^{(1)}$ associated to $\nabla$ is expressed in local coordinates
as

$\omega^{(1)}=\sum_{\mu=1}^{n}\sum_{k,l=1}^{r}\omega_{\mu
k}^{l}dx^{\mu}e_{l}\otimes e^{k}+\sum_{k,l=1}^{r}\omega_{\mu
k}^{l}\,dx^{\mu}\frac{\partial}{\partial\eta^{l}}\otimes\eta^{k}.$

The dual connection $\nabla^{*}$, acting on $E^{*}$, is then associated to the
following dual connection 1-form:

$\omega^{(1)*}=-\sum_{\mu=1}^{n}\sum_{k,l=1}^{r}\omega_{\mu
k}^{l}dx^{\mu}e^{k}\otimes e_{l}+\sum_{k,l=1}^{r}\omega_{\mu
k}^{l}\,dx^{\mu}\eta^{k}\otimes\frac{\partial}{\partial\eta^{l}}.$

With respect to the chosen trivialization of $L$, the basic connection
$\nabla^{bas}$ induced by $\nabla$ induces a connection 1-form $\omega^{bas}$
whose expression in local coordinates is
$\omega_{ij}^{bas,k}=\sum_{\mu=1}^{n}\rho_{j}^{\mu}\omega_{\mu
i}^{k}+c_{ij}^{k}$, where the indices range from 1 to $r$ for $i$, and from 1
to $r+r^{\prime}$ for $j,k$. Then, the Lie algebroid derivative $d_{A}$ can be
recast under the form

$d_{A}=\sum_{i=1}^{r}\eta^{i}\nabla^{*}_{\rho(e_{i})}-\frac{1}{2}\sum_{i,j,k=1}^{r}T_{\nabla^{bas}}(e_{i},e_{j})^{k}\eta^{i}\eta^{j}\frac{\partial}{\partial\eta^{k}}.$

In particular see Equation (4.5) in [Chatzistavrakidis and Jonke, 2023],
reminding that $\nabla^{bas}$ is the opposite connection of the induced
connection $\overset{\bullet}{\nabla}$.

With respect to the choice of connection $\nabla$ in the given local
trivialization, the other components of $D_{A}$ are given by:

$\displaystyle D_{A}\Big{(}\frac{\partial}{\partial\eta^{m}}\Big{)}$
$\displaystyle=\sum_{i,k=1}^{r}\omega_{im}^{bas,k}\eta^{i}\frac{\partial}{\partial\eta^{k}}+e_{m},$
$\displaystyle D_{A}\left(e_{l}\right)$
$\displaystyle=\frac{1}{2}\sum_{i,j,k=1}^{r}R^{bas}(e_{i},e_{j})(e_{l})^{k}\eta^{i}\eta^{j}\frac{\partial}{\partial\eta^{k}}+\sum_{k=1}^{r+r^{\prime}}\sum_{i=1}^{r}\omega_{il}^{bas,k}\eta^{i}\otimes
e_{k}.$

for every $1\leq m\leq r$ and $1\leq l\leq r+r^{\prime}$. Both formulas are
obtained by hand computations, and one can compare them to Equations (15)-(16)
of [Liao, 2023] to observe that $c_{ij}^{k}$ (resp. $\rho_{l}(c_{ij}^{k})$) in
the latter becomes $\omega_{ij}^{bas,k}$ (resp. $(R^{bas}_{ij})_{l}^{k}$) in
the former.

In other words, the local expression of the homological vector field as given
in [Liao, 2023, Remark 2.5] corresponds to taking the $A$-superconnection on
$E$ induced by the standard flat $TM$-connection on $A$ with respect to the
chosen trivialisation of $L$, i.e. the unique connection whose connection
1-form vanishes when evaluated on the local frame. Thus, the expressions of
$\mathcal{Q}$ in[Liao, 2023, Remark 2.5] and that of $D_{A}$ here only differ
by a choice of $TM$-connection on $A$, which, regarding the theory of
representations up to homotopy, are essentially indistinguishable [Abad and
Crainic, 2011]. ∎

## References

  * [Abad and Crainic, 2011] Abad, C. A. and Crainic, M. (2011).  Representations up to homotopy of Lie algebroids.  J. Reine Angew. Math., 2012(663):91–126.
  * [Aschieri et al., 2021] Aschieri, P., Bonechi, F., and Deser, A. (2021).  On Curvature and Torsion in Courant Algebroids.  Ann. Henri Poincaré, 22(7):2475–2496.
  * [Batakidis and Voglaire, 2018] Batakidis, P. and Voglaire, Y. (2018).  Atiyah classes and dg-Lie algebroids for matched pairs.  J. Geom. Phys., 123:156–172.
  * [Blaom, 2006] Blaom, A. (2006).  Geometric structures as deformed infinitesimal symmetries.  Trans. Amer. Math. Soc., 358(8):3651–3671.
  * [Chatzistavrakidis and Jonke, 2023] Chatzistavrakidis, A. and Jonke, L. (2023).  Basic curvature and the Atiyah cocycle in gauge theory.
  * [Chen et al., 2019] Chen, Z., Lang, H., and Xiang, M. (2019).  Atiyah classes of strongly homotopy Lie pairs.  Algebra Colloq., 26(02):195–230.
  * [Chen et al., 2016] Chen, Z., Stiénon, M., and Xu, P. (2016).  From Atiyah classes to homotopy Leibniz algebras.  Commun. Math. Phys., 341(1):309–349.
  * [Crainic and Fernandes, 2005] Crainic, M. and Fernandes, R. L. (2005).  Secondary Characteristic Classes of Lie Algebroids.  In Quantum Field Theory and Noncommutative Geometry, pages 157–176. Springer, Berlin, Heidelberg.
  * [Fairon, 2017] Fairon, M. (2017).  Introduction to graded geometry.  Eur. J. Math., 3(2):208–222.
  * [Kotov and Salnikov, 2024] Kotov, A. and Salnikov, V. (2024).  The category of z-graded manifolds: What happens if you do not stay positive.  Differ. Geom. Appl., 93:102109.
  * [Laurent-Gengoux et al., 2020] Laurent-Gengoux, C., Lavau, S., and Strobl, T. (2020).  The universal Lie infinity-algebroid of a singular foliation.  Doc. Math., 25:1571–1652.
  * [Liao, 2023] Liao, H.-Y. (2023).  Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs.  Commun. Math. Phys., 404(2):701–734.
  * [Mackenzie, 2005] Mackenzie, K. C. H. (2005).  General Theory of Lie Groupoids and Lie Algebroids.  London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge.
  * [Mehta, 2006] Mehta, R. A. (2006).  Supergroupoids, Double Structures, and Equivariant Cohomology.  PhD thesis, University of California, Berkeley.
  * [Mehta, 2009] Mehta, R. A. (2009).  Q-algebroids and their cohomology.  J. Symplectic Geom., 7(3):263–293.
  * [Mehta, 2014] Mehta, R. A. (2014).  Lie algebroid modules and representations up to homotopy.  Indag. Math., 25(5):1122–1134.
  * [Mehta et al., 2015] Mehta, R. A., Stiénon, M., and Xu, P. (2015).  The Atiyah class of a dg-vector bundle.  C. R. Math., 353(4):357–362.
  * [Quillen, 1985] Quillen, D. (1985).  Superconnections and the Chern character.  Topology, 24(1):89–95.
  * [Stiénon et al., 2022] Stiénon, M., Vitagliano, L., and Xu, P. (2022).  A-infinity algebras from Lie pairs.
  * [Stiénon and Xu, 2020] Stiénon, M. and Xu, P. (2020).  Fedosov dg manifolds associated with Lie pairs.  Math. Ann., 378(1):729–762.
  * [Weibel, 1994] Weibel, C. A. (1994).  An Introduction to Homological Algebra.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.
  * [Zambon, 2022] Zambon, M. (2022).  Singular subalgebroids.  Ann. Inst. Fourier, 72(6):2109–2190.