# Homotopy conformal algebras

,  Anupam Sahoo  Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, West Bengal, India<EMAIL_ADDRESS>and  Apurba Das
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The notion of conformal algebras was introduced by Victor G. Kac using the
axiomatic description of the operator product expansion of chiral fields in
conformal field theory. The structure theory, representations and cohomology
of Lie and associative conformal algebras are extensively studied in the
literature. In this paper, we first introduce $A_{\infty}$-conformal algebras
as the homotopy analogue of associative conformal algebras, provide some
equivalent descriptions and prove the homotopy transfer theorem. We
characterize some $A_{\infty}$-conformal algebras in terms of Hochschild
cohomology classes of associative conformal algebras. Next, we introduce
associative conformal $2$-algebras as the categorification of associative
conformal algebras. We show that the category of associative conformal
$2$-algebras and the category of $2$-term $A_{\infty}$-conformal algebras are
equivalent. Finally, we consider $L_{\infty}$-conformal algebras and find
their relations with $A_{\infty}$-conformal algebras.

2020 MSC classifications: 16E40, 16S99, 18N25, 18N40.

Keywords: Conformal algebras, $A_{\infty}$-conformal algebras, Conformal
$2$-algebras, $L_{\infty}$-conformal algebras.

###### Contents

  1. 1 Introduction
  2. 2 Associative conformal algebras and their cohomology
  3. 3 Homotopy associative conformal algebras
  4. 4 Characterizations of some homotopy associative conformal algebras
  5. 5 Associative conformal 2-algebras
  6. 6 Homotopy Lie conformal algebras

##  1\. Introduction

###  1.1. Conformal algebras

The concept of conformal algebras was first introduced by V. G. Kac as a
useful tool to study vertex algebras [10], [11]. Conformal algebras encode the
axiomatic description of the operator product expansion of chiral fields in
conformal field theory. They appear naturally in the study of formal
distribution Lie algebras. They also have connections with infinite-
dimensional Lie algebras satisfying the locality property. Hence the theory of
finite conformal algebras is useful to provide a classification of some
infinite-dimensional Lie algebras. Lie conformal algebras also have
applications in two-dimensional conformal field theory and string theory [10],
[25]. In the last twenty-five years, the structure theory and representations
of Lie conformal algebras and associative conformal algebras are extensively
studied by several authors [2], [5], [21], [14], [23], [24], [7].

The notion of cohomology theory in the context of operator product expansion
was first considered in [13] for vertex algebras and conformal field theories.
Subsequently, the cohomology theory of conformal algebras was systematically
described by Bakalov, Kac and Voronov [3]. The cohomology of Lie conformal
algebras generalizes the Chevalley-Eilenberg cohomology of usual Lie algebras
and the cohomology of associative conformal algebras generalizes the classical
Hochschild cohomology of associative algebras. These cohomologies describe the
deformation of respective conformal algebras. See [25], [4], [15], [8] for
more details about the cohomology of conformal algebras and their
applications. We will follow the conventions of [25] while considering the
cohomology (although the cohomology is the same with [3]).

###  1.2. Homotopy algebras and 2-algebras

The first instance of homotopy algebras arises in the work of Stasheff in the
recognization of topological loop spaces [27]. Since its inception,
$A_{\infty}$-algebras (also called strongly homotopy associative algebras)
have found important applications in homotopy theory, topology and geometry
[22], [9]. Among other flexibilities, $A_{\infty}$-algebras satisfy the
homotopy transfer theorem whereas usual associative algebras do not. The
concept of $L_{\infty}$-algebras (which are the Lie analogue of
$A_{\infty}$-algebras) was introduced by Lada, Markl and Stasheff [18], [17].
They have close connections with deformation theory, higher differential
geometry and mathematical physics [6], [16], [20]. On the other hand, in [1]
Baez and Crans introduced the notion of Lie $2$-algebras as the
categorification of Lie algebras and find a relationship with the
Zamolodchikov tetrahedron equation. They showed that the category of Lie
$2$-algebras is equivalent to the category of $2$-term $L_{\infty}$-algebras.
In the same article, the authors observed that skeletal $L_{\infty}$-algebras
are characterized by the third cocycles of the Chevalley-Eilenberg complex of
Lie algebras, and strict $L_{\infty}$-algebras are characterized by crossed
modules of Lie algebras. These results are fundamental in connections with Lie
algebras, $L_{\infty}$-algebras and Lie $2$-algebras. All these results are
valid for algebras of other types. In particular, one can also consider
$2$-term $A_{\infty}$-algebras and associative $2$-algebras and show their
categorical equivalence. The standard skew-symmetrization functor from the
category of associative algebras to the category of Lie algebras has a
generalization to the context of homotopy algebras. That is, there is a
commutative diagram

${\mathrm{associative~{}algebras~{}}}$${2\text{-term
}A_{\infty}\text{-algebras}\simeq\text{associative
}2\text{-algebras}}$${A_{\infty}\text{-algebras}}$${\mathrm{Lie~{}algebras}}$${2\text{-term
}L_{\infty}\text{-algebras}\simeq\text{Lie }2\text{-algebras
}}$${L_{\infty}\text{-algebras}.}$

###  1.3. Bridges between conformal algebras, homotopy algebras and 2-algebras

Our aim in this paper is to find bridges between conformal algebras, homotopy
algebras and $2$-algebras. To do so, we first introduce $A_{\infty}$-conformal
algebras as the conformal analogue of $A_{\infty}$-algebras. We also define
representations of an $A_{\infty}$-conformal algebra. Given a graded
$\mathbb{C}[\partial]$-module $\mathcal{A}$, we construct a graded Lie algebra
whose Maurer-Cartan elements correspond to $A_{\infty}$-conformal algebra
structures on $\mathcal{A}$. This characterization allows us to define the
cohomology of an $A_{\infty}$-conformal algebra with coefficients in a
representation. Our cohomology of an $A_{\infty}$-conformal algebra
generalizes the Hochschild cohomology of associative conformal algebras. Next,
we prove (various forms of) the homotopy transfer theorem for
$A_{\infty}$-conformal algebras which shows that the category of
$A_{\infty}$-conformal algebras is much flexible than the category of
associative conformal algebras. We also consider $2$-term
$A_{\infty}$-conformal algebras and characterize some important classes of
$2$-term $A_{\infty}$-conformal algebras in terms of Hochschild cohomology of
associative conformal algebras. We denote the category of $2$-term
$A_{\infty}$-conformal algebras by ${\bf 2A_{\infty}conf}$. Next, we consider
associative conformal $2$-algebras as the categorification of associative
conformal algebras. The collection of all associative conformal 2-algebras and
homomorphisms between them forms a category, denoted by conf2Alg. We show that
the category conf2Alg of associative conformal $2$-algebras and the category
${\bf 2A_{\infty}conf}$ of $2$-term $A_{\infty}$-conformal algebras are
equivalent.

In the Lie context, it is worth mentioning that the concepts of $2$-term
$L_{\infty}$-conformal algebras and Lie conformal $2$-algebras are recently
considered by Zhang [30]. The author also showed that the category of $2$-term
$L_{\infty}$-conformal algebras and the category of Lie conformal $2$-algebras
are equivalent. However, he didn’t consider the full concept of
$L_{\infty}$-conformal algebras. In this paper, we also fill up this gap by
introducing $L_{\infty}$-conformal algebras as the homotopy analogue of Lie
conformal algebras. The notion of Zhang is then a particular case of our
$L_{\infty}$-conformal algebras. We show that a suitable skew-symmetrization
of an $A_{\infty}$-conformal algebra yields an $L_{\infty}$-conformal algebra.
Hence we obtain the following diagram that generalizes the above diagram in
the conformal context

${\mathrm{associative~{}conf.~{}algebras~{}}}$${\begin{subarray}{c}2\text{-term
}A_{\infty}\text{-conf. ~{}algebras}\\\ \simeq~{}\text{associative~{}conf.
}2\text{-algebras}\end{subarray}}$${A_{\infty}\text{-conf.~{}algebras}}$${\mathrm{Lie~{}conf.~{}algebras}}$${\begin{subarray}{c}2\text{-term
}L_{\infty}\text{-conf.~{}algebras}\\\ \simeq~{}\text{Lie~{}conf.
}2\text{-algebras }\end{subarray}}$${L_{\infty}\text{-conf.~{}algebras}.}$

###  1.4. Organization of the paper

The paper is organized as follows. In section 2, we recall associative
conformal algebras and their Hochschild cohomology with coefficients in a
conformal bimodule. In section 3, we introduce the notion of
$A_{\infty}$-conformal algebras which are the homotopy analogue of associative
conformal algebras. We also consider representations of an
$A_{\infty}$-conformal algebra and introduce the cohomology of an
$A_{\infty}$-conformal algebra with coefficients in a representation. In
section 4, we mainly consider $2$-term $A_{\infty}$-conformal algebras and
classify some particular classes of such algebras. The notion of conformal
associative $2$-algebras is introduced in section 5. We also show that the
categories ${\bf 2A_{\infty}conf}$ and ${\bf conf2Alg}$ are equivalent.
Finally, in section 6, we recall Lie conformal algebras and introduce
$L_{\infty}$-conformal algebras, and show that a suitable skew-symmetrization
of an $A_{\infty}$-conformal algebra yields an $L_{\infty}$-conformal algebra.

### Notations

All vector spaces, linear and multilinear maps, and tensor products are over
the field $\mathbb{C}$ of complex numbers.

##  2\. Associative conformal algebras and their cohomology

In this section, we recall associative conformal algebras, conformal bimodules
and the cohomology theory of associative conformal algebras. Our main
referances are [3], [4], [15].

######  2.1 Definition.

An associative conformal algebra is a $\mathbb{C}[\partial]$-module $A$
equipped with a $\mathbb{C}$-linear map (called the $\lambda$-multiplication)
$\cdot_{\lambda}{\cdot}:A\otimes A\to A[\lambda]$, $a\otimes b\mapsto
a_{\lambda}b$ satisfying the conformal sesquilinearity condition:

$\displaystyle(\partial a)_{\lambda}b=-\lambda
a_{\lambda}b\quad\mathrm{and}\quad a_{\lambda}(\partial
b)=(\partial+\lambda)a_{\lambda}b,$ (1)

and the following conformal associativity:
$a_{\lambda}(b_{\mu}c)=(a_{\lambda}b)_{\lambda+\mu}c$, for all $a,b,c\in A$.

An associative conformal algebra as above may be simply denoted by $A$ when
the $\lambda$-multiplication map is clear from the context.

Let $A$ be an associative conformal algebra with the $\lambda$-multiplication
$\cdot_{\lambda}\cdot$. For each $j\geq 0$, we define the $j$-th product on
$A$ by a $\mathbb{C}$-linear map $\cdot_{(j)}\cdot:A\otimes A\rightarrow A$
that satisfies

$\displaystyle a_{\lambda}b=\sum_{j\geq
0}\frac{\lambda^{j}}{j!}a_{(j)}b,\text{ for }a,b\in A.$ (2)

Since $a_{\lambda}b$ is an $A$-valued polynomial in $\lambda$, there are
finitely many $j$’s (that depends on $a$ and $b$ both) for which $a_{(j)}b\neq
0$. With the correspondence (2) between the $\lambda$-multiplication and
$j$-th products, the conformal sesquilinearity condition (1) of the
$\lambda$-multiplication is equivalent to the conditions

$\displaystyle(\partial a)_{(j)}b=-ja_{(j-1)}b,\quad a_{(j)}(\partial
b)=\partial(a_{(j)}b)+ja_{(j-1)}b,\text{ for }a,b\in A\text{ and }j\geq 0.$

Moreover, the conformal associativity is equivalent to

$\displaystyle
a_{(j)}(b_{(k)}c)=\sum_{p=0}^{j}\binom{j}{p}(a_{(p)}b)_{(j+k-p)}c,\text{ for
all }a,b,c\in A\text{ and }j,k\geq 0.$ (3)

Associative conformal algebras appear naturally in the context of formal
distribution associative algebras [3]. Another class of examples are current
conformal algebras associated with associative algebras. More precisely, let
$A$ be an associative algebra. Then the $\mathbb{C}[\partial]$-module
$\mathrm{Cur}A=\mathbb{C}[\partial]\otimes A$ carries an associative conformal
algebra structure with the $\lambda$-multiplication $a_{\lambda}b:=ab$, for
all $a,b\in A$.

Let $M$ be any $\mathbb{C}[\partial]$-module. A conformal linear endomorphism
of $M$ is a $\mathbb{C}$-linear map $f:M\rightarrow M[\lambda],~{}m\mapsto
f_{\lambda}(m)$ that satisfies $f_{\lambda}(\partial
m)=(\partial+\lambda)f_{\lambda}(m)$, for all $m\in M$. The space of all
conformal linear endomorphisms of $M$ is denoted by $\mathrm{Cend}(M)$. Note
that $\mathrm{Cend}(M)$ has canonical structure of a
$\mathbb{C}[\partial]$-module given by $(\partial f)_{\lambda}(m):=-\lambda
f_{\lambda}(m)$, for all $m\in M$. Then $\mathrm{Cend}(M)$ is an associative
conformal algebra with the $\lambda$-multiplication

$\displaystyle(f_{\lambda}g)_{\mu}(m):=f_{\lambda}(g_{\mu-\lambda}(m)),\text{
for }f,g\in\mathrm{Cend}(M),m\in M.$

######  2.2 Definition.

Let $A$ be an associative conformal algebra. A conformal $A$-bimodule is a
$\mathbb{C}[\partial]$-module $M$ equipped with two $\mathbb{C}$-linear maps
(called the left and right $\lambda$-actions, respectively) $A\otimes M\to
M[\lambda]$, $a\otimes m\mapsto a_{\lambda}m$ and $M\otimes A\to M[\lambda]$,
$m\otimes a\mapsto m_{\lambda}a$ satisfying the conformal sesquilinearity
conditions

$\displaystyle(\partial a)_{\lambda}m=-\lambda a_{\lambda}m,\quad
a_{\lambda}(\partial m)=(\partial+\lambda)a_{\lambda}m,$
$\displaystyle(\partial m)_{\lambda}a=-\lambda m_{\lambda}a,\quad
m_{\lambda}(\partial a)=(\partial+\lambda)m_{\lambda}a,$

and the following bimodule conditions

$\displaystyle a_{\lambda}(b_{\mu}m)=(a_{\lambda}b)_{\lambda+\mu}m,\quad
a_{\lambda}(m_{\mu}b)=(a_{\lambda}m)_{\lambda+\mu}b~{}~{}~{}\mathrm{and}~{}~{}~{}m_{\lambda}(a_{\mu}b)=(m_{\lambda}a)_{\lambda+\mu}b,\text{
for all }a,b\in A,m\in M.$

It follows from the above definition that any associative conformal algebra
$A$ is itself a conformal $A$-bimodule, where both the left and right
$\lambda$-actions are given by the $\lambda$-multiplication. This is called
the adjoint conformal $A$-bimodule.

In the following, we recall the Hochschild cohomology of associative conformal
algebras introduced in [3]. This cohomology is the conformal analogue of the
standard Hochschild cohomology theory of associative algebras. Let $A$ be an
associative conformal algebra and $M$ be a conformal $A$-bimodule. Let $n\geq
1$ be a natural number. A $\mathbb{C}$-linear map $\varphi:A^{\otimes n}\to
M[\lambda_{1},\ldots,\lambda_{n-1}]$, $a_{1}\otimes\cdots\otimes
a_{n}\mapsto\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{n})$ is
called a conformal sesquilinear map if $\varphi$ satisfies

$\displaystyle\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,\partial
a_{i},\ldots,a_{n})=~{}$
$\displaystyle-\lambda_{i}\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{n}),\mathrm{~{}for~{}}i=1,2,\ldots,n-1,$
$\displaystyle\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{n-1},\partial
a_{n})=~{}$
$\displaystyle(\partial+\lambda_{1}+\cdots+\lambda_{n-1})\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{n}).$

We denote the set of all such conformal sesquilinear maps by
$\mathrm{Hom}_{cs}(A^{\otimes n},M[\lambda_{1},\ldots,\lambda_{n-1}])$. Note
that, for $n=1$, we have

$\displaystyle\mathrm{Hom}_{cs}(A,M)=\mathrm{Hom}_{\mathbb{C}[\partial]}(A,M),$

the set of all $\mathbb{C}[\partial]$-module homomorphisms from $A$ to $M$. We
are now ready to define the Hochschild cochain complex of the associative
conformal algebra $A$ with coefficients in the conformal $A$-bimodule $M$.
For each $n\geq 0$, the space of $n$-cochains $C^{n}(A,M)$ is given by

$C^{n}(A,M)=\begin{cases}M/{\partial M}&\text{if }n=0,\\\
\mathrm{Hom}_{cs}(A^{\otimes n},M[\lambda_{1},\ldots,\lambda_{n-1}])&\text{if
}n\geq 1.\end{cases}$

There is a map $\delta:C^{n}(A,M)\to C^{n+1}(A,M)$ given by

$\displaystyle\delta(m+\partial{M})(a)=(a_{-\lambda-\partial}m-m_{\lambda}a)~{}\\!\vline_{\lambda=0},\mathrm{~{}for~{}}m+\partial{M}\in
M/{\partial{M}}\mathrm{~{}and~{}}a\in A,$

and

$\displaystyle(\delta{\varphi})_{\lambda_{1},\ldots,\lambda_{n}}(a_{1},\ldots,a_{n+1})=$
$\displaystyle~{}{a_{1}}_{\lambda_{1}}(\varphi_{\lambda_{2},\ldots,\lambda{n}}(a_{2},\ldots,a_{n+1}))$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n}{(-1)^{i}~{}\varphi_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\lambda_{i+1},\lambda_{i+2},\ldots,\lambda_{n}}(a_{1},\ldots,a_{i-1},{a_{i}}_{\lambda_{i}}a_{i+1},a_{i+2},\ldots,a_{n+1})}$
$\displaystyle+$
$\displaystyle(-1)^{n+1}({\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{n})})_{{\lambda_{1}+\cdots+\lambda_{n}}}a_{n+1},$

for $\varphi\in{C^{n\geq 1}(A,M)}=\mathrm{Hom}_{cs}(A^{\otimes
n},M[\lambda_{1},\ldots,\lambda_{n-1}])$ and $a_{1},\ldots,a_{n+1}\in A$. One
can easily verify that $\delta$ is a differential, that is, $\delta^{2}=0$. In
other words, $\\{C^{\bullet}(A,M),\delta\\}$ is a cochain complex. Let
$Z^{n}(A,M)=\\{\varphi\in C^{n}(A,M)~{}|~{}\delta{\varphi}=0\\}$ be the space
of all $n$-cocycles and $B^{n}(A,M)=\\{\delta{\varphi}~{}|~{}\varphi\in
C^{n-1}(A,M)\\}$ be the space of all $n$-coboundaries. The quotients

$\displaystyle{H}^{n}(A,M):={\frac{{Z}^{n}(A,M)}{{{B}^{n}(A,M)}},~{}\mathrm{for}~{}n\geq
0}$

are called the Hochschild cohomology groups of the associative conformal
algebra $A$ with coefficients in the conformal $A$-bimodule $M$.

##  3\. Homotopy associative conformal algebras

In this section, we introduce $A_{\infty}$-conformal algebras as the homotopy
analogue of associative conformal algebras. Given a graded
$\mathbb{C}[\partial]$-module $\mathcal{A}$, we also construct a graded Lie
algebra whose Maurer-Cartan elements correspond to $A_{\infty}$-conformal
algebra structures on $\mathcal{A}$. Next, we define representations of an
$A_{\infty}$-conformal algebra and introduce the cohomology with coefficients
in a representation. Finally, we prove homotopy transfer theorems for
$A_{\infty}$-conformal algebras.

######  3.1 Definition.

([27]) An $A_{\infty}$-algebra is a pair $(\mathcal{A},\\{\mu_{k}\\}_{k\geq
1})$ consisting of a graded vector space
$\mathcal{A}=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}$ equipped with a
collection of graded linear maps $\\{\mu_{k}:\mathcal{A}^{\otimes
k}\to\mathcal{A}\\}_{k\geq 1}$ with $\mathrm{deg}(\mu_{k})=k-2$ for $k\geq 1$,
satisfying the following set of identities:

$\displaystyle\sum_{k+l=n+1}\sum_{i=1}^{k}{(-1)^{i(l+1)+l(|a_{1}|+\cdots+|a_{i-1}|)}~{}\mu_{k}(a_{1},\ldots,a_{i-1},\mu_{l}(a_{i},\ldots,a_{i+l-1}),a_{i+1},\ldots,a_{n})}=0,$

for all $n\geq 1$ and homogeneous elements $a_{1},\ldots,a_{n}\in\mathcal{A}$.

Note that, in an arbitrary $A_{\infty}$-algebra, the degree $0$ multiplication
map $\mu_{2}:\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{A}$ in general
doesn’t satisfy the usual associativity. However, it does satisfy the
associativity up to a coherent homotopy. For this reason,
$A_{\infty}$-algebras are also called strongly homotopy associative algebras.
Any associative algebra can be regarded as an $A_{\infty}$-algebra whose
underlying graded vector space is concentrated in degree $0$.

Let $\mathcal{A}=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}$ be a graded vector
space. Then for any $k\geq 1$, the space
$\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]$ can be equipped with a graded
vector space structure by

$\displaystyle\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}[\lambda_{1},\ldots,\lambda_{k-1}].$

######  3.2 Definition.

An $A_{\infty}$-conformal algebra (or a strongly homotopy associative
conformal algebra) is a graded $\mathbb{C}[\partial]$-module
$\mathcal{A}=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}$ equipped with a
collection $\\{\mu_{k}:\mathcal{A}^{\otimes
k}\to\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ of graded
$\mathbb{C}$-linear maps with $\mathrm{deg}(\mu_{k})=k-2$ for $k\geq 1$,
satisfying the following conditions:

\- each $\mu_{k}$ is conformal sesquilinear in the sense that

$\displaystyle(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,\partial{a_{i}},\ldots,a_{k})=~{}$
$\displaystyle-\lambda_{i}(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k}),\mathrm{~{}for~{}}i=1,\ldots,k-1,$
$\displaystyle(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k-1},\partial{a_{k}})=~{}$
$\displaystyle(\partial+\lambda_{1}+\cdots+\lambda_{k-1})(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k}),$

\- the following set of identities are hold:

$\displaystyle\sum_{k+l=n+1}$
$\displaystyle\sum_{i=1}^{k}(-1)^{i(l+1)+l(|a_{1}|+\cdots+|a_{i-1}|)}$ (4)
$\displaystyle(\mu_{k})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\cdots+\lambda_{i+l-1},\ldots,\lambda_{n-1}}(a_{1},\ldots,a_{i-1},(\mu_{l})_{\lambda_{i},\ldots,\lambda_{i+l-2}}(a_{i},\ldots,a_{i+l-1}),a_{i+l},\ldots,a_{n})=0,$

for any $n\geq 1$ and homogeneous elements $a_{1},\ldots,a_{n}\in\mathcal{A}$.

The collection $\\{\mu_{k}:\mathcal{A}^{\otimes
k}\to\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ of maps are
called the structure maps of the $A_{\infty}$-conformal algebra. We often
denote an $A_{\infty}$-conformal algebra as above by $\mathcal{A}$ when the
structure maps are clear from the context.

The above definition of an $A_{\infty}$-conformal algebra has the following
consequences. Note that, for $n=1$, the identity (4) is equivalent to
$(\mu_{1})^{2}=0$. In other words, the $\mathbb{C}[\partial]$-linear degree
$-1$ map $\mu_{1}:\mathcal{A}\rightarrow\mathcal{A}$ makes
$(\mathcal{A},\mu_{1})$ into a chain complex in the category of
$\mathbb{C}[\partial]$-modules. Next, for $n=2$, the identity (4) is same as

$\displaystyle\mu_{1}\big{(}(\mu_{2})_{\lambda}(a,b)\big{)}=(\mu_{2})_{\lambda}\big{(}\mu_{1}(a),b\big{)}+(-1)^{|a|}(\mu_{2})_{\lambda}\big{(}a,\mu_{1}(b)\big{)},$

for all homogeneous elements $a,b\in\mathcal{A}$. It says that $\mu_{1}$ is a
graded derivation for the degree zero $\lambda$-multiplication
$\mu_{2}:\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{A}[\lambda]$.
Similarly, for $n=3$, we have

$\displaystyle(\mu_{2})_{\lambda}\big{(}a,(\mu_{2})_{\mu}(b,c)\big{)}-(\mu_{2})_{\lambda+\mu}\big{(}(\mu_{2})_{\lambda}(a,b),c\big{)}$
$\displaystyle=-\big{\\{}\mu_{1}\big{(}(\mu_{3})_{\lambda,\mu}(a,b,c)\big{)}+(\mu_{3})_{\lambda,\mu}\big{(}\mu_{1}(a),b,c\big{)}+(-1)^{|a|}(\mu_{3})_{\lambda,\mu}(a,\mu_{1}(b),c)$
$\displaystyle\qquad\qquad+(-1)^{|a|+|b|}(\mu_{3})_{\lambda,\mu}(a,b,\mu_{1}(c))\big{\\}},$

for all homogeneous elements $a,b,c\in\mathcal{A}$. This shows that the
$\lambda$-multiplication $\mu_{2}$ doesn’t satisfy (in general) the conformal
associativity. However, it holds up to an exact term of the conformal
sesquilinear map $\mu_{3}$. For this reason, an $A_{\infty}$-conformal algebra
can be realized as the homotopy analogue of associative conformal algebras.
Note that, for higher values of $n$, we have higher homotopy analogues of the
conformal associativity.

Let $\mathcal{A}$ be an $A_{\infty}$-conformal algebra with the structure maps
$\\{\mu_{k}:\mathcal{A}^{\otimes
k}\rightarrow\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$. For
any $k\geq 1$ and a $(k-1)$ tuple $J=(j_{1},\ldots,j_{k-1})$ of non-negative
integers, we define a graded $\mathbb{C}$-linear map
$(\mu_{k})_{(j_{1},\ldots,j_{k-1})}:\mathcal{A}^{\otimes
k}\rightarrow\mathcal{A}$ with
$\mathrm{deg}\big{(}(\mu_{k})_{(j_{1},\ldots,j_{k-1})}\big{)}=k-2$, by

$\displaystyle(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k})=\sum_{(j_{1},\ldots,j_{k-1})\geq
0}\frac{\lambda_{1}^{j_{1}}\cdots\lambda_{k-1}^{j_{k-1}}}{j_{1}!\cdots
j_{k-1}!}(\mu_{k})_{(j_{1},\ldots,j_{k-1})}(a_{1},\ldots,a_{k}),$

for any homogeneous elements $a_{1},\ldots,a_{k}\in\mathcal{A}$. We call them
$J$-products, which generalize the $j$-products (2) from the non-homotopic
case. Since $(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k})$
lies in the polynomial space $\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]$,
it follows that $(\mu_{k})_{(j_{1},\ldots,j_{k-1})}(a_{1},\ldots,a_{k})\neq 0$
only for finitely many tuples $(j_{1},\ldots,j_{k-1})$’s that depends on
$\mu_{k}$ and the inputs $a_{1},\ldots,a_{k}$. The conformal sesquilinearity
of $\mu_{k}$ can be equivalently expressed in terms of the $J$-products as

$\displaystyle(\mu_{k})_{(j_{1},\ldots,j_{k-1})}(a_{1},\ldots,\partial
a_{i},\ldots,a_{k})=~{}$ $\displaystyle-
j_{1}(\mu_{k})_{(j_{1},\ldots,j_{i}-1,\ldots,j_{k-1})}(a_{1},\ldots,a_{k}),\text{
for }1\leq i\leq k-1,$
$\displaystyle(\mu_{k})_{(j_{1},\ldots,j_{k-1})}(a_{1},\ldots,a_{k-1},\partial
a_{k})=~{}$
$\displaystyle\partial\big{(}(\mu_{k})_{(j_{1},\ldots,j_{k-1})}(a_{1},\ldots,a_{k})\big{)}$
$\displaystyle+\sum_{i=1}^{k-1}j_{i}(\mu_{k})_{(j_{1},\ldots,j_{i}-1,\ldots,j_{k-1})}(a_{1},\ldots,a_{k}),$

for any $(j_{1},\ldots,j_{k-1})\geq 0$ and $a_{1},\ldots,a_{k}\in\mathcal{A}$.
Finally, the $A_{\infty}$-conformal associativity conditions (4) are
equivalent to

$\displaystyle\sum_{k+l=n+1}$
$\displaystyle\sum_{i=1}^{k}(-1)^{i(l+1)+l(|a_{1}|+\cdots+|a_{i-1}|)}\sum_{\begin{subarray}{c}0\leq
q_{i}\leq p_{i}\\\ \vdots\\\ 0\leq q_{i+l-2}\leq
p_{i+l-2}\end{subarray}}\binom{p_{i}}{q_{i}}\cdots\binom{p_{i+l-2}}{q_{i+l-2}}$
(5)
$\displaystyle(\mu_{k})_{(p_{1},\ldots,p_{i-1},p_{i}-q_{i}+\cdots+p_{i+l-2}-q_{i+l-2}+p_{i+l-1},p_{i+l},\ldots,p_{n-1})}\big{(}a_{1},\ldots,a_{i-1},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(\mu_{l})_{(q_{i},\ldots,q_{i+l-2})}(a_{i},\ldots,a_{i+l-1}),\ldots,a_{n}\big{)}=0,$

for any $p_{1},\ldots,p_{n-1}\geq 0$, homogeneous elements
$a_{1},\ldots,a_{n}\in\mathcal{A}$ and $n\geq 1$. These relations generalize
(3).

######  3.3 Remark.

Any associative conformal algebra $(A,\cdot_{\lambda}\cdot)$ can be viewed as
an $A_{\infty}$-conformal algebra $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$,
where $\mathcal{A}=A$ concentrated in degree $0$,

$\displaystyle(\mu_{2})_{\lambda}(a,b)=a_{\lambda}b~{}~{}~{}\text{ and
}~{}~{}~{}\mu_{k}=0,\mathrm{~{}for~{}}k\neq 2.$

######  3.4 Example.

A differential graded associative conformal algebra is a graded
$\mathbb{C}[\partial]$-module
$\mathcal{A}=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}$ equipped with a
$\mathbb{C}[\partial]$-module homomorphism
$d:\mathcal{A}\rightarrow\mathcal{A}$ of degree $-1$ that satisfies $d^{2}=0$
and a degree $0$ conformal sesquilinear map
$\cdot_{\lambda}\cdot:{\mathcal{A}\otimes\mathcal{A}}\to{\mathcal{A}[\lambda]}$,
$a\otimes b\mapsto{a_{\lambda}b}$ that satisfies

$\displaystyle
a_{\lambda}(b_{\mu}c)=(a_{\lambda}b)_{\lambda+\mu}c~{}~{}~{}\text{ and
}~{}~{}~{}d(a_{\lambda}b)=(da)_{\lambda}b+a_{\lambda}(db),\mathrm{~{}for~{}all}~{}a,b,c\in\mathcal{A}.$

A differential graded associative conformal algebra
$(\mathcal{A},d,\cdot_{\lambda}\cdot)$ is an $A_{\infty}$-conformal algebra
$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ in which $\mu_{1}=d$,
$(\mu_{2})_{\lambda}=\cdot_{\lambda}\cdot$ and $\mu_{k}=0$ for $k\geq 3$.

######  3.5 Example.

Let $A$ be an associative conformal algebra. Then the graded
$\mathbb{C}[\partial]$-module
$\mathcal{A}=\underbrace{A}_{\mathrm{deg}0}\oplus\underbrace{A}_{\mathrm{deg}1}$
carries an $A_{\infty}$-conformal algebra structure with the
$\lambda$-multiplications $\\{\mu_{k}:\mathcal{A}^{\otimes
k}\rightarrow\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ given
by

$\displaystyle\mu_{1}=\mathrm{id}:\mathcal{A}_{1}\rightarrow\mathcal{A}_{0},\quad\mu_{2}:\mathcal{A}_{i}\otimes\mathcal{A}_{j}\rightarrow\mathcal{A}_{i+j}[\lambda],~{}a\otimes
b\mapsto a_{\lambda}b~{}\text{ for }0\leq i,j,i+j\leq 1,\text{ and
}\mu_{k}=0\text{ for }k\geq 3.$

The above example can be generalized as follows. Let $A$ and $B$ be two
associative conformal algebras and $f:A\rightarrow B$ be a morphism of
associative conformal algebras (i.e. $f$ is a $\mathbb{C}[\partial]$-linear
map satisfying $f(a_{\lambda}b)=f(a)_{\lambda}f(b)$, for $a,b\in A$). Then the
graded $\mathbb{C}[\partial]$-module
$\mathcal{A}=\underbrace{A}_{\mathrm{deg}0}\oplus\underbrace{\mathrm{ker}f}_{\mathrm{deg}1}$
inherits an $A_{\infty}$-conformal algebra structure with the
$\lambda$-multiplications $\\{\mu_{k}:\mathcal{A}^{\otimes
k}\rightarrow\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ given
by

$\displaystyle\mu_{1}=i:\mathcal{A}_{1}\hookrightarrow\mathcal{A}_{0},\quad\mu_{2}:\mathcal{A}_{i}\otimes\mathcal{A}_{j}\rightarrow\mathcal{A}_{i+j}[\lambda],~{}a\otimes
b\mapsto a_{\lambda}b\text{ for }0\leq i,j,i+j\leq 1,~{}\text{ and
}\mu_{k}=0\text{ for }k\geq 3.$

######  3.6 Example.

Let $A$ be an associative conformal algebra, $M$ be a conformal $A$-module and
let $\varphi:A\otimes A\rightarrow M[\lambda]$ be any conformal sesquilinear
map (i.e. $\varphi\in C^{2}(A,M)$). Then the graded
$\mathbb{C}[\partial]$-module
$\mathcal{A}=\underbrace{A}_{\mathrm{deg}0}\oplus\underbrace{M}_{\mathrm{deg}1}$
has an $A_{\infty}$-conformal algebra with the maps
$\\{\mu_{k}:\mathcal{A}^{\otimes
k}\rightarrow\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ given
by

$\displaystyle\mu_{1}=0:\mathcal{A}_{1}\rightarrow\mathcal{A}_{0},\qquad\begin{cases}(\mu_{2})_{\lambda}(a,b):=a_{\lambda}b,\\\
(\mu_{2})_{\lambda}(a,m):=a_{\lambda}m,\\\
(\mu_{2})_{\lambda}(m,a):=m_{\lambda}a,\end{cases}$
$\displaystyle(\mu_{3})_{\lambda,\mu}(a,b,c):=a_{\lambda}\varphi_{\mu}(b,c)-\varphi_{\lambda+\mu}(a_{\lambda}b,c)+\varphi_{\lambda}(a,b_{\mu}c)-\varphi_{\lambda}(a,b)_{\lambda+\mu}c,$

for $a,b,c\in A,m\in M$, and $\mu_{k}=0$ for $k\geq 4$.

The concept of $A_{\infty}$-conformal algebras are better understood when we
shift the degree of the underlying graded $\mathbb{C}[\partial]$-module. Let
$\mathcal{A}=\oplus_{i\in\mathbb{Z}}\mathcal{A}_{i}$ be a graded
$\mathbb{C}[\partial]$-module. We consider the graded
$\mathbb{C}[\partial]$-module
$\mathcal{A}[-1]=\oplus_{i\in\mathbb{Z}}(\mathcal{A}[-1])_{i}$, where
$\mathcal{A}[-1]_{i}=\mathcal{A}_{i-1}$. Note that a graded
$\mathbb{C}$-linear map $\mu_{k}:\mathcal{A}^{\otimes
k}\to\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]$ with deg$(\mu_{k})=k-2$ is
equivalent to having a graded $\mathbb{C}$-linear map
$\rho_{k}:(\mathcal{A}[-1])^{\otimes
k}\to(\mathcal{A}[-1])[\lambda_{1},\ldots,\lambda_{k-1}]$ with
deg$(\rho_{k})=-1$. The correspondence is given by

$\displaystyle\rho_{k}:=(-1)^{\frac{k(k-1)}{2}}~{}s\circ\mu_{k}\circ(s^{-1})^{\otimes
k},$

where $s:\mathcal{A}\to\mathcal{A}[-1]$ is the degree $+1$ map that identifies
$\mathcal{A}$ with $\mathcal{A}[-1]$, and
$s^{-1}:\mathcal{A}[-1]\to\mathcal{A}$ is the degree $-1$ map inverse to $s$.
The sign $(-1)^{\frac{k(k-1)}{2}}$ is a consequence of the Koszul sign
convention. See [17] for more details about the Koszul sign. We are now ready
to define the concept of $A_{\infty}[1]$-conformal algebras and their relation
with Definition 3.2.

######  3.7 Definition.

An $A_{\infty}[1]$-conformal algebra is a graded $\mathbb{C}[\partial]$-module
$\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}_{i}$ together with a
collection $\\{\rho_{k}:\mathcal{V}^{\otimes
k}\to\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ of degree
$-1$ graded $\mathbb{C}$-linear maps that satisfy the following conditions:

\- each $\rho_{k}$ is conformal sesquilinear,

\- for any $n\geq 1$ and homogenous elements
$v_{1},\ldots,v_{n}\in\mathcal{V}$,

$\displaystyle\sum_{k+l=n+1}$
$\displaystyle\sum_{i=1}^{k}(-1)^{|v_{1}|+\cdots+|v_{i-1}|}$ (6)
$\displaystyle(\rho_{k})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\cdots+\lambda_{i+l-1},\ldots,\lambda_{n-1}}(v_{1},\ldots,v_{i-1},(\rho_{l})_{\lambda_{i},\ldots,\lambda_{i+l-2}}(v_{i},\ldots,v_{i+l-1}),v_{i+l},\ldots,v_{n})=0.$

It is easy to see that $A_{\infty}[1]$-conformal algebras are easier to keep
in mind than $A_{\infty}$-conformal algebras because of the fixed degree of
the structure maps and the signs in the defining identities. However, they are
equivalent up to a degree shift.

######  3.8 Proposition.

An $A_{\infty}$-conformal algebra structure on a graded
$\mathbb{C}[\partial]$-module $\mathcal{A}$ is equivalent to an
$A_{\infty}[1]$-conformal algebra structure on the graded
$\mathbb{C}[\partial]$-module $\mathcal{A}[-1]$. More precisely,
$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is an $A_{\infty}$-conformal algebra
if and only if $(\mathcal{A}[-1],\\{\rho_{k}\\}_{k\geq 1})$ is an
$A_{\infty}[1]$-conformal algebra.

Maurer-Cartan characterization and cohomology of $A_{\infty}$-conformal
algebras. In the following, we show that $A_{\infty}$-conformal algebra
structures on a graded $\mathbb{C}[\partial]$-module can be equivalently
described by Maurer-Cartan elements in a suitable graded Lie algebra. Let
$\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}_{i}$ be a graded
$\mathbb{C}[\partial]$-module. For each $k\geq 1$, let
$\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])$ be the set of all degree
$n$ conformal sesquilinear maps from $\mathcal{V}^{\otimes k}$ to
$\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]$. That is,

$\displaystyle\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])=\\{\varphi\in\mathrm{Hom}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])~{}|\mathrm{~{}deg}(\varphi)=n\\}.$

We define a graded vector space
$C^{\bullet}_{cs}(\mathcal{V})=\oplus_{n\in\mathbb{Z}}C^{n}_{cs}(\mathcal{V})$,
where

$\displaystyle C^{n}_{cs}(\mathcal{V})=\oplus_{k\geq
1}\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]).$

Thus, an element $\rho\in C^{n}_{cs}(\mathcal{V})$ is given by a sum
$\rho=\sum_{k\geq 1}\rho_{k}$, where
$\rho_{k}\in\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])$ for any $k\geq 1$. Note
that the graded vector space
$C^{\bullet}_{cs}(\mathcal{V})=\oplus_{n\in\mathbb{Z}}C^{n}_{cs}(\mathcal{V})$
inherits a degree $0$ bracket given by

$\displaystyle\llbracket\sum_{k\geq 1}\rho_{k},\sum_{l\geq
1}\varrho_{l}\rrbracket=\sum_{p\geq
1}\sum_{k+l=p+1}(\rho_{k}\diamond\varrho_{l}-(-1)^{mn}\varrho_{l}\diamond\rho_{k}),$

where

$\displaystyle(\rho_{k}\diamond\varrho_{l}$
$\displaystyle)_{\lambda_{1},\ldots,\lambda_{p-1}}(v_{1},\ldots,v_{p})=\sum_{i=1}^{k}(-1)^{m(|v_{1}|+\cdots+|v_{i-1}|)}$
(7)
$\displaystyle(\rho_{k})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\cdots+\lambda_{i+l-1},\ldots,\lambda_{p-1}}(v_{1},\ldots,v_{i-1},(\varrho_{l})_{\lambda_{i},\ldots,\lambda_{i+l-2}}(v_{i},\ldots,v_{i+l-1}),v_{i+1},\ldots,v_{p}),$

for $\rho=\sum_{k\geq 1}\rho_{k}\in C^{n}_{cs}(\mathcal{V})$ and
$\varrho=\sum_{l\geq 1}\varrho_{l}\in C^{m}_{cs}(\mathcal{V})$. Then we have
the following.

######  3.9 Proposition.

With the above notations,
$(C^{\bullet}_{cs}(\mathcal{V}),\llbracket~{},~{}\rrbracket)$ is a graded Lie
algebra.

###### Proof.

For any $\rho_{k}\in\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])$,
$\varrho_{l}\in\mathrm{Hom}^{m}_{cs}(\mathcal{V}^{\otimes
l},\mathcal{V}[\lambda_{1},\ldots,\lambda_{l-1}])$ and $1\leq i\leq k$, we
define an element
$\rho_{k}\diamond_{i}\varrho_{l}\in\mathrm{Hom}^{n+m}_{cs}(\mathcal{V}^{\otimes(k+l-1)},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k+l-2}])$
by

$\displaystyle(\rho_{k}\diamond_{i}\varrho_{l}$
$\displaystyle)_{\lambda_{1},\ldots,\lambda_{k+l-2}}(v_{1},\ldots,v_{k+l-1})=(-1)^{m(|v_{1}|+\cdots+|v_{i-1}|)}$
$\displaystyle(\rho_{k})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\cdots+\lambda_{i+l-1},\ldots,\lambda_{k+l-2}}(v_{1},\ldots,v_{i-1},(\varrho_{l})_{\lambda_{i},\ldots,\lambda_{i+l-2}}(v_{i},\ldots,v_{i+l-1}),v_{i+1},\ldots,v_{k+l-1}),$

Then we have

$\displaystyle\llbracket\rho_{k},\varrho_{l}\rrbracket=\rho_{k}\diamond\varrho_{l}-(-1)^{mn}\varrho_{l}\diamond\rho_{k}=\sum_{i=1}^{k}\rho_{k}\diamond_{i}\varrho_{l}-(-1)^{mn}\sum_{i=1}^{l}\varrho_{l}\diamond_{i}\rho_{k}.$
(8)

Moreover, for any elements
$\rho_{k}\in\mathrm{Hom}^{n}_{cs}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])$,
$\varrho_{l}\in\mathrm{Hom}^{m}_{cs}(\mathcal{V}^{\otimes
l},\mathcal{V}[\lambda_{1},\ldots,\lambda_{l-1}])$ and
$\psi_{s}\in\mathrm{Hom}^{p}_{cs}(\mathcal{V}^{\otimes
s},\mathcal{V}[\lambda_{1},\ldots,\lambda_{s-1}])$, it is easy to verify that
the $\diamond_{i}$ operations satisfy the following operad-like identities
(see also [8] for the non-graded case)

$\displaystyle(\rho_{k}\diamond_{i}\varrho_{l})\diamond_{i+j-1}\psi_{s}=~{}$
$\displaystyle\rho_{k}\diamond_{i}(\varrho_{l}\diamond_{j}\psi_{s}),\text{ for
}1\leq i\leq k,~{}1\leq j\leq l,$
$\displaystyle(\rho_{k}\diamond_{i}\varrho_{l})\diamond_{j+l-1}\psi_{s}=~{}$
$\displaystyle(\rho_{k}\diamond_{j}\varrho_{l})\diamond_{i}\psi_{s},\text{ for
}1\leq i<j\leq k.$

As a consequence, the graded skew-symmetric bracket (8) satisfies the graded
Jacobi identity:

$\displaystyle\llbracket\rho_{k},\llbracket\varrho_{l},\psi_{s}\rrbracket\rrbracket=\llbracket\llbracket\rho_{k},\varrho_{l}\rrbracket,\psi_{s}\rrbracket+(-1)^{mn}\llbracket\varrho_{l},\llbracket\rho_{k},\psi_{s}\rrbracket\rrbracket.$

Finally, if $\rho=\sum_{k\geq 1}\rho_{k}\in C^{n}_{cs}(\mathcal{V})$,
$\varrho=\sum_{l\geq 1}\varrho_{l}\in C^{m}_{cs}(\mathcal{V})$ and
$\psi=\sum_{s\geq 1}\psi_{s}\in C^{p}_{cs}(\mathcal{V})$, then we have

$\displaystyle\llbracket\rho,\llbracket\varrho,\psi\rrbracket\rrbracket=~{}$
$\displaystyle\sum_{r\geq
1}\sum_{k+l+s=r+2}\llbracket\rho_{k},\llbracket\varrho_{l},\psi_{s}\rrbracket\rrbracket$
$\displaystyle=~{}$ $\displaystyle\sum_{r\geq
1}\sum_{k+l+s=r+2}\big{(}\llbracket\llbracket\rho_{k},\varrho_{l}\rrbracket,\psi_{s}\rrbracket+(-1)^{mn}\llbracket\varrho_{l},\llbracket\rho_{k},\psi_{s}\rrbracket\rrbracket\big{)}$
$\displaystyle=~{}$
$\displaystyle\llbracket\llbracket\rho,\varrho\rrbracket,\psi\rrbracket+(-1)^{mn}\llbracket\varrho,\llbracket\rho,\psi\rrbracket\rrbracket,$

which verifies the graded Jacobi identity for general elements. ∎

######  3.10 Theorem.

Let $\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}_{i}$ be a graded
$\mathbb{C}[\partial]$-module. A collection of degree $-1$ conformal
sesquilinear maps $\\{\rho_{k}:\mathcal{V}^{\otimes
k}\to\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ makes the
pair $(\mathcal{V},\\{\rho_{k}\\}_{k\geq 1})$ into an
$A_{\infty}[1]$-conformal algebra if and only if the element $\rho=\sum_{k\geq
1}\rho_{k}\in C_{cs}^{-1}(\mathcal{V})$ is a Maurer-Cartan element in the
graded Lie algebra
$(C^{\bullet}_{cs}(\mathcal{V}),\llbracket~{},~{}\rrbracket)$.

###### Proof.

We observe that

$\displaystyle\llbracket\rho,\rho\rrbracket=\llbracket\sum_{k\geq
1}\rho_{k},\sum_{l\geq 1}\rho_{l}\rrbracket=~{}$ $\displaystyle\sum_{n\geq
1}\sum_{k+l=n+1}(\rho_{k}\diamond\rho_{l}-(-)^{1}\rho_{l}\diamond\rho_{k})$
$\displaystyle=~{}$ $\displaystyle\sum_{n\geq
1}\big{(}2\sum_{k+l=n+1}\rho_{k}\diamond\rho_{l}\big{)}.$

This shows that $\rho=\sum_{k\geq 1}\rho_{k}$ is a Maurer-Cartan element in
the graded Lie algebra
$(C^{\bullet}_{cs}(\mathcal{V}),\llbracket~{},~{}\rrbracket)$ if and only if
$\sum_{k+l=n+1}\rho_{k}\diamond\rho_{l}=0$ for all $n\geq 1$. Hence the result
follows by using (7). ∎

Thus, by combining Proposition 3.8 and Theorem 3.10, we conclude that the
following statements are equivalent:

  1. (1)

$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is an $A_{\infty}$-conformal algebra,

  2. (2)

$(\mathcal{A}[-1],\\{\rho_{k}\\}_{k\geq 1})$ is an $A_{\infty}[1]$-conformal
algebra,

  3. (3)

$\rho=\sum_{k\geq 1}\rho_{k}$ is a Maurer-Cartan element in the graded Lie
algebra $(C^{\bullet}_{cs}(\mathcal{A}[-1]),[\\![~{},~{}]\\!])$.

Let $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ be an $A_{\infty}$-conformal
algebra. Consider the corresponding Maurer-Cartan element $\rho=\sum_{k\geq
1}\rho_{k}$ in the graded Lie algebra
$(C^{\bullet}_{cs}(\mathcal{A}[-1]),\llbracket~{},~{}\rrbracket)$. For each
$n\in\mathbb{Z}$, we define the space of $n$-cochains by
$C^{n}(\mathcal{A},\mathcal{A}):=C^{-(n-1)}_{cs}(\mathcal{A}[-1])$. Thus, an
element $\varphi\in C^{n}(\mathcal{A},\mathcal{A})$ is given by a sum
$\varphi=\sum_{l\geq 1}\varphi_{l}$, where
$\varphi_{l}\in\mathrm{Hom}_{cs}^{-(n-1)}(\mathcal{A}[-1]^{\otimes
l},\mathcal{A}[-1][\lambda_{1},\ldots,\lambda_{l-1}])$ for any $l\geq 1$. We
define a map $\delta_{\rho}:C^{n}(\mathcal{A},\mathcal{A})\rightarrow
C^{n+1}(\mathcal{A},\mathcal{A})$ by

$\displaystyle\delta_{\rho}(\varphi):=(-1)^{n-1}\llbracket\rho,\varphi\rrbracket=(-1)^{n-1}\sum_{p\geq
1}\sum_{k+l=p+1}\big{(}\rho_{k}\diamond\varphi_{l}-(-1)^{n-1}\varphi_{l}\diamond\rho_{k}\big{)},\text{
for }\varphi=\sum_{l\geq 1}\varphi_{l}\in C^{n}(\mathcal{A},\mathcal{A}).$

Since $\rho$ is a Maurer-Cartan element (i.e.
$\llbracket\rho,\rho\rrbracket=0$), it follows that $(\delta_{\rho})^{2}=0$.
Hence $\\{C^{\bullet}(\mathcal{A},\mathcal{A}),\delta_{\rho}\\}$ is a cochain
complex. The corresponding cohomology groups are called the cohomology of the
$A_{\infty}$-conformal algebra $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$.

In the following, we introduce representations of an $A_{\infty}$-conformal
algebra and define cohomology with coefficients in a representation. We first
need the following notation. Let $\mathcal{A}$ and $\mathcal{M}$ be two graded
$\mathbb{C}[\partial]$-modules. For any $k\geq 1$, let $\mathcal{A}^{k-1,1}$
be the direct sum of all possible tensor powers of $\mathcal{A}$ and
$\mathcal{M}$ in which $\mathcal{A}$ appears $k-1$ times and $\mathcal{M}$
appears exactly once. For instance,

$\displaystyle\mathcal{A}^{1,1}=(\mathcal{A}\otimes\mathcal{M})\oplus(\mathcal{M}\otimes\mathcal{A})\quad\mathrm{and}\quad\mathcal{A}^{2,1}=(\mathcal{A}\otimes\mathcal{A}\otimes\mathcal{M})\oplus(\mathcal{A}\otimes\mathcal{M}\otimes\mathcal{A})\oplus(\mathcal{M}\otimes\mathcal{A}\otimes\mathcal{A}).$

######  3.11 Definition.

Let $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ be an $A_{\infty}$-conformal
algebra. A representation of $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is a
graded $\mathbb{C}[\partial]$-module
$\mathcal{M}=\oplus_{i\in\mathbb{Z}}\mathcal{M}_{i}$ equipped with a
collection
$\\{\eta_{k}:\mathcal{A}^{k-1,1}\to\mathcal{M}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq
1}$ of graded conformal sesquilinear maps with $\mathrm{deg}(\eta_{k})=k-2$
for any $k\geq 1$, such that the identities in (4) are hold when exactly one
of $a_{1},\ldots,a_{n}$ is from $\mathcal{M}$ and the corresponding $\mu_{k}$
or $\mu_{l}$ is replaced by $\eta_{k}$ or $\eta_{l}$.

It follows from the above definition that any $A_{\infty}$-conformal algebra
$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is a representation of itself, where
$\eta_{k}=\mu_{k}$ for any $k\geq 1$. This is called the adjoint
representation.

Let $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ be an $A_{\infty}$-conformal
algebra and $(\mathcal{M},\\{\eta_{k}\\}_{k\geq 1})$ be a representation of
it. Then it can be checked that the direct sum graded
$\mathbb{C}[\partial]$-module $\mathcal{A}\oplus\mathcal{M}$ inherits an
$A_{\infty}$-conformal algebra with the structure maps
$\\{\theta_{k}:(\mathcal{A}\oplus\mathcal{M})^{\otimes
k}\rightarrow(\mathcal{A}\oplus\mathcal{M})[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq
1}$ given by

$\displaystyle(\theta_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}$
$\displaystyle\big{(}(a_{1},m_{1}),\ldots,(a_{k},m_{k})\big{)}:=$
$\displaystyle\big{(}(\mu_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k}),~{}\sum_{i=1}^{k}(\eta_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{i-1},m_{i},a_{i+1},\ldots,a_{k})\big{)},$

for $(a_{1},m_{1}),\ldots,(a_{k},m_{k})\in\mathcal{A}\oplus\mathcal{M}$. This
is called the semidirect product $A_{\infty}$-conformal algebra, often denoted
by $\mathcal{A}\ltimes\mathcal{M}$. Let $\Delta\in
C^{-1}_{cs}((\mathcal{A}\oplus\mathcal{M})[-1])$ be the Maurer-Cartan element
in the graded Lie algebra
$\big{(}C^{\bullet}_{cs}((\mathcal{A}\oplus\mathcal{M})[-1]),\llbracket~{},~{}\rrbracket\big{)}$
corresponding to the semidirect product $A_{\infty}$-conformal algebra
$(\mathcal{A}\oplus\mathcal{M},\\{\theta_{k}\\}_{k\geq 1})$. Explicitly,
$\Delta$ is given by

$\displaystyle\Delta=\sum_{k\geq 1}\Delta_{k},~{}\text{ where
}\Delta_{k}=(-1)^{\frac{k(k-1)}{2}}~{}s\circ\theta_{k}\circ(s^{-1})^{\otimes
k}\text{ for any }k\geq 1.$

The element $\Delta$ induces the cochain complex
$\\{C^{\bullet}(\mathcal{A}\oplus\mathcal{M},\mathcal{A}\oplus\mathcal{M}),\delta_{\Delta}\\}$
defining the cohomology of the semidirect product $A_{\infty}$-conformal
algebra. For any $n\in\mathbb{Z}$, let $C^{n}(\mathcal{A},\mathcal{M})\subset
C^{n}(\mathcal{A}\oplus\mathcal{M},\mathcal{A}\oplus\mathcal{M})$ be the
subspace given by

$\displaystyle C^{n}(\mathcal{A},\mathcal{M}):=\\{\varphi=\sum_{l\geq
1}\varphi_{l}\in
C^{n}(\mathcal{A}\oplus\mathcal{M},\mathcal{A}\oplus\mathcal{M})~{}|~{}(\varphi_{l})|_{\mathcal{A}[-1]^{\otimes
l}}\subset\mathcal{M}[-1][\lambda_{1},\ldots,\lambda_{l-1}]\text{ for any
}l\geq 1\\}.$

Note that, an element $\varphi\in C^{n}(\mathcal{A},\mathcal{M})$ can be
realized by a sum $\varphi=\sum_{l\geq 1}\varphi_{l}$, where

$\displaystyle\varphi_{l}\in\mathrm{Hom}_{cs}^{-(n-1)}(\mathcal{A}[-1]^{\otimes
l},\mathcal{M}[-1][\lambda_{1},\ldots,\lambda_{l-1}]),\text{ for any }l\geq
1.$

Then it follows from the explicit description of $\delta_{\Delta}$ that
$\delta_{\Delta}\big{(}C^{n}(\mathcal{A},\mathcal{M})\big{)}\subset
C^{n+1}(\mathcal{A},\mathcal{M})$ for all $n$. Hence $\delta_{\Delta}$
restricts to a differential (denoted by $\delta_{\rho}$ if no confusion
arises) $\delta_{\rho}:C^{n}(\mathcal{A},\mathcal{M})\rightarrow
C^{n+1}(\mathcal{A},\mathcal{M})$. In other words,
$\\{C^{\bullet}(\mathcal{A},\mathcal{M}),\delta_{\rho}\\}$ is a cochain
complex. The corresponding cohomology groups are called the cohomology of the
$A_{\infty}$-conformal algebra $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ with
coefficients in the representation $(\mathcal{M},\\{\eta_{k}\\}_{k\geq 1})$.

######  3.12 Remark.

When $(\mathcal{M},\\{\eta_{k}\\}_{k\geq 1})=(\mathcal{A},\\{\mu_{k}\\}_{k\geq
1})$ is the adjoint representation, the above cohomology coincides with the
cohomology of the $A_{\infty}$-conformal algebra
$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ introduced earlier.

Homotopy transfer theorems. Here we will prove the homotopy transfer theorems
for $A_{\infty}$-conformal algebras. As we have observed earlier that
$A_{\infty}[1]$-conformal algebras are easier to work, we will first prove the
homotopy transfer theorems for $A_{\infty}[1]$-conformal algebras. Motivation
for our results comes from the following equivalent description of
$A_{\infty}[1]$-conformal algebras.

Let $(\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}^{i},\rho_{1})$ be a chain
complex of $\mathbb{C}[\partial]$-modules. For any $k\geq 2$, we define a map
$\partial_{\rho_{1}}:\mathrm{Hom}_{cs}^{p}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])\rightarrow\mathrm{Hom}_{cs}^{p-1}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}])$ by

$\displaystyle\partial_{\rho_{1}}(\rho_{k})=\llbracket\rho_{1},\rho_{k}\rrbracket=\rho_{1}\diamond\rho_{k}-(-1)^{p}\rho_{k}\diamond\rho_{1}=\rho_{1}\diamond_{1}\rho_{k}-(-1)^{p}\sum_{i=1}^{k}\rho_{k}\diamond_{i}\rho_{1},$

for $\rho_{k}\in\mathrm{Hom}_{cs}^{p}(\mathcal{V}^{\otimes
k},\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]).$ With the above notations,
the identities (6) for $n\geq 2$ can be equivalently written as

$\displaystyle\partial_{\rho_{1}}(\rho_{n})=\begin{cases}0&\text{ if }n\geq
2,\\\ -\sum_{\begin{subarray}{c}k+l=n+1\\\
k,l>1\end{subarray}}\rho_{k}\diamond\rho_{l}=-\sum_{\begin{subarray}{c}k+l=n+1\\\
k,l>1\end{subarray}}\sum_{i=1}^{k}\rho_{k}\diamond_{i}\rho_{l}&\text{ if
}n\geq 3.\end{cases}$ (9)

Thus, an $A_{\infty}[1]$-conformal algebra can be realized as a chain complex
$(\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}^{i},\rho_{1})$ of
$\mathbb{C}[\partial]$-modules equipped with a collection of degree $-1$
conformal sesquilinear maps $\\{\rho_{k}:\mathcal{V}^{\otimes
k}\rightarrow\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 2}$ that
satisfy the identities (9).

Let $(\mathcal{V}=\oplus_{i\in\mathbb{Z}}\mathcal{V}_{i},\rho_{1})$ and
$(\mathcal{W}=\oplus_{i\in\mathbb{Z}}\mathcal{W}_{i},\theta_{1})$ be two chain
complexes of $\mathbb{C}[\partial]$-modules. We say that the complex
$(\mathcal{W},\theta_{1})$ is a deformation retract of
$(\mathcal{V},\rho_{1})$ if there exist chain maps
$p:(\mathcal{V},\rho_{1})\rightarrow(\mathcal{W},\theta_{1})$ and
$i:(\mathcal{W},\theta_{1})\rightarrow(\mathcal{V},\rho_{1})$ between
complexes of $\mathbb{C}[\partial]$-modules and a degree $+1$ chain homotopy
$h:\mathcal{V}\rightarrow\mathcal{V}$,

(12)

satisfying

$\displaystyle\mathrm{id}_{\mathcal{V}}-ip=\rho_{1}h+h\rho_{1}~{}\text{ and
}~{}pi=\mathrm{id}_{\mathcal{W}}.$

In this case, the full data described in (12) is often called contraction
data.

Next, suppose that the complex $(\mathcal{V},\rho_{1})$ carries a degree $-1$
$\lambda$-multiplication
$\rho_{2}:\mathcal{V}\otimes\mathcal{V}\rightarrow\mathcal{V}[\lambda]$ which
makes $(\mathcal{V},\rho_{1},\rho_{2})$ into an $A_{\infty}[1]$-conformal
algebra (here $\rho_{k}=0$ for $k\geq 3$). In other words, $\rho_{2}$ makes
the triple $(\mathcal{V},\rho_{1},\rho_{2})$ into a shifted differential
graded associative conformal algebra. We now define a degree $-1$ conformal
sesquilinear map
$\theta_{2}:\mathcal{W}\otimes\mathcal{W}\rightarrow\mathcal{W}[\lambda]$ by

$\displaystyle(\theta_{2})_{\lambda}(x,y):=p\big{(}(\rho_{2})_{\lambda}(i(x),i(y))\big{)},\text{
for }x,y\in\mathcal{W}.$

Then we have the following straightforward observation.

######  3.13 Lemma.

The map $\theta_{2}\in\mathrm{Hom}_{cs}^{-1}(\mathcal{W}^{\otimes
2},\mathcal{W}[\lambda])$ satisfies $\partial_{\theta_{1}}(\theta_{2})=0$.
Moreover, for any $x,y,z\in\mathcal{W}$, we have

$\displaystyle(\theta_{2})_{\lambda+\mu}\big{(}(\theta_{2})_{\lambda}(x,y),z\big{)}+(-1)^{|x|}(\theta_{2})_{\lambda}\big{(}x,(\theta_{2})_{\mu}(y,z)\big{)}=-(\partial_{\theta_{1}}(\theta_{3}))_{\lambda,\mu}(x,y,z),$
(13)

where $\theta_{3}:\mathcal{W}^{\otimes 3}\rightarrow\mathcal{W}[\lambda,\mu]$
is the degree $-1$ conformal sesquilinear map given by

$\displaystyle(\theta_{3})_{\lambda,\mu}(x,y,z)=p(\mu_{2})_{\lambda+\mu}\big{(}h(\mu_{2})_{\lambda}(i(x),i(y)),i(z)\big{)}+(-1)^{|x|}p(\mu_{2})_{\lambda}\big{(}i(x),h(\mu_{2})_{\lambda}(i(y),i(z))\big{)}.$

It follows from the above lemma that $\theta_{2}$ need not satisfy the shifted
conformal associativity condition (in general). That it,
$(\mathcal{W},\theta_{1},\theta_{2})$ need not be an $A_{\infty}[1]$-conformal
algebra. However, it does satisfy if the maps $i$ and $p$ are inverses to each
other. Further, the obstruction for the shifted conformal associativity of
$\theta_{2}$ is given by $-\partial_{\theta_{1}}(\theta_{3})$. Observe that
the identity (13) is equivalent to the condition
$\theta_{2}\diamond\theta_{2}=-\partial_{\theta_{1}}(\theta_{3})$, which is
nothing but the condition (9) for $n=3$. In fact, in the following, we will
construct a sequence $\\{\theta_{k}:\mathcal{W}^{\otimes
k}\rightarrow\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 3}$ of
degree $-1$ conformal sesquilinear maps satisfying (9). Moreover, for $k=3$,
the construction coincides with $\theta_{3}$ as defined in the above lemma.
Our method for the constructions of higher $\theta_{k}$’s is similar to the
classical case of $A_{\infty}$-algebras [19, 12]. For that, we first define a
sequence $\\{\rho_{k}:\mathcal{V}^{\otimes
k}\rightarrow\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 2}$ of
maps inductively as follows. We define inductively as

$\displaystyle\rho_{k}=\sum_{\begin{subarray}{c}i+j=k\\\ i,j\geq
1\end{subarray}}\rho_{2}\big{(}(h\circ\rho_{i})\otimes(h\circ\rho_{j})\big{)}$

with the convention that $h\circ\rho_{1}=\mathrm{id}$. Alternatively, the maps
$\rho_{k}$’s can be defined in terms of the planar binary trees with $k$
leaves. For any planar binary tree $T$ with $k$ leaves ($k\geq 3$), there
exist unique planar binary trees $T_{1}$ and $T_{2}$ such that $T=T_{1}\vee
T_{2}$, where $\vee$ stands for the grafting of trees. For such $T$, we
associate a conformal sesquilinear map $\rho_{T}:\mathcal{V}^{\otimes
k}\rightarrow\mathcal{V}[\lambda_{1},\ldots,\lambda_{k-1}]$ by
$\rho_{T}=\rho_{2}\big{(}(h\circ\rho_{T_{1}})\otimes(h\circ\rho_{T_{2}})\big{)}$
with the convention that $h\circ\rho_{|}=\mathrm{id}$, where $|$ is the unique
planar tree with one leaf. Finally, we define

$\displaystyle\rho_{k}:=\sum_{T\in\mathsf{PBT}_{(k)}}\rho_{T},$

where $\mathsf{PBT}_{(k)}$ is the set of all isomorphism classes of planar
binary trees with $k$ leaves. Using the map $\rho_{k}$, we now define
$\theta_{k}:\mathcal{W}^{\otimes
k}\rightarrow\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]$ by
$\theta_{k}:=p\circ\rho_{k}\circ i^{\otimes k}$, for $k\geq 3$.

######  3.14 Theorem.

(Homotopy transfer theorem) Let (12) be a contraction data. If
$(\mathcal{V},\rho_{1},\rho_{2})$ is an $A_{\infty}[1]$-conformal algebra then
$(\mathcal{W},\\{\theta_{k}\\}_{k\geq 1})$ is an $A_{\infty}[1]$-conformal
algebra.

The proof of the above theorem is similar to the classical case [19, 12]. The
above homotopy transfer theorem can be further generalized. More precisely, we
will now show that if the chain complex $(\mathcal{V},\rho_{1})$ inherits a
full $A_{\infty}[1]$-conformal algebra structure, then it can be transferred
to $(\mathcal{W},\theta_{1})$. In this case, one need to use planar trees (not
necessarily binary). For any planar tree $T$ with $k$ leaves, there exist
unique planar trees $T_{1},\ldots,T_{l}$ such that $T=T_{1}\vee\cdots\vee
T_{l}$. Then we define
$\rho_{T}=\rho_{l}\big{(}(h\circ\rho_{T_{1}})\otimes\cdots\otimes(h\circ\rho_{T_{l}})\big{)}$,
and

$\displaystyle\rho_{k}:=\sum_{T\in\mathsf{PB}_{(k)}}\rho_{T},$

where $\mathsf{PB}_{(k)}$ is the set of all isomorphism classes of planar
trees with $k$ leaves. Finally, we define the map
$\theta_{k}:\mathcal{W}^{\otimes
k}\rightarrow\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]$
$\theta_{k}:=p\circ\rho_{k}\circ i^{\otimes k}$, for $k\geq 3$.

######  3.15 Theorem.

(Homotopy transfer theorem) Let (12) be a contraction data. If
$(\mathcal{V},\\{\rho_{k}\\}_{k\geq 1})$ is an $A_{\infty}[1]$-conformal
algebra, then this structure transfers to an $A_{\infty}[1]$-conformal algebra
structure on $(\mathcal{W},\theta_{1})$.

Recall that an $A_{\infty}$-conformal algebra is equivalent to an
$A_{\infty}[1]$-conformal algebra on the shifted graded
$\mathbb{C}[\partial]$-module (Proposition 3.8). On the other hand, if there
is a deformation retract of chain complexes, then it induces a deformation
retract of the shifted chain complexes. Combining these results, we can now
prove homotopy transfer theorem for $A_{\infty}$-conformal algebras.

######  3.16 Theorem.

(Homotopy transfer theorem) Let
$\textstyle{(\mathcal{A},\mu_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{p}$$\textstyle{(\mathcal{H},\nu_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$
be a given contraction data. If $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is an
$A_{\infty}$-conformal algebra, then it can be transferred to an
$A_{\infty}$-conformal algebra structure on $(\mathcal{H},\nu_{1})$.

##  4\. Characterizations of some homotopy associative conformal algebras

In this section, we consider those $A_{\infty}$-conformal algebras whose
underlying graded $\mathbb{C}[\partial]$-module is concentrated only in two
degrees. In particular, we study ‘skeletal’ $A_{\infty}$-conformal algebras.
We show that skeletal $A_{\infty}$-conformal algebras can be characterized by
third cocycles of associative conformal algebras.

A $2$-term $A_{\infty}$-conformal algebra is an $A_{\infty}$-conformal algebra
$(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ whose underlying graded
$\mathbb{C}[\partial]$-module $\mathcal{A}$ is concentrated in degrees $0$ and
$1$, that is, $\mathcal{A}=\mathcal{A}_{0}\oplus\mathcal{A}_{1}$. In that
case, we have $\mu_{k}=0$ for $k>3$. If we write explicitly the higher
associative conformal identities (4), we obtain the following.

######  4.1 Definition.

A $2$-term $A_{\infty}$-conformal algebra is a graded
$\mathbb{C}[\partial]$-module
$\mathcal{A}=\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ equipped with

\- a $\mathbb{C}[\partial]$-module homomorphism
$\beta:\mathcal{A}_{1}\to\mathcal{A}_{0}$,

\- a $\mathbb{C}$-linear conformal sesquilinear map
$\mu_{2}:\mathcal{A}_{i}\otimes\mathcal{A}_{j}\to\mathcal{A}_{i+j}[\lambda]$,
for $0\leq i,j,i+j\leq 1$,

\- a $\mathbb{C}$-linear conformal sesquilinear map
$\mu_{3}:\mathcal{A}_{0}\otimes\mathcal{A}_{0}\otimes\mathcal{A}_{0}\to\mathcal{A}_{1}[\lambda_{1},\lambda_{2}]$
subject to the following conditions:

  1. (i)

$(\mu_{2})_{\lambda}(m,n)=0$,

  2. (ii)

$\beta((\mu_{2})_{\lambda}(a,m))=(\mu_{2})_{\lambda}(a,\beta m)$,

  3. (iii)

$\beta((\mu_{2})_{\lambda}(m,a))=(\mu_{2})_{\lambda}(\beta m,a)$,

  4. (iv)

$(\mu_{2})_{\lambda}(\beta m,n)=(\mu_{2})_{\lambda}(m,\beta n)$,

  5. (v)

$\beta((\mu_{3})_{\lambda_{1},\lambda_{2}}(a,b,c))=(\mu_{2})_{\lambda_{1}+\lambda_{2}}((\mu_{2})_{\lambda_{1}}(a,b),c)-(\mu_{2})_{\lambda_{1}}(a,(\mu_{2})_{\lambda_{2}}(b,c))$,

  6. (vi)

$(\mu_{3})_{\lambda_{1},\lambda_{2}}(a,b,\beta
m)=(\mu_{2})_{\lambda_{1}+\lambda_{2}}((\mu_{2})_{\lambda_{1}}(a,b),m)-(\mu_{2})_{\lambda_{1}}(a,(\mu_{2})_{\lambda_{2}}(b,m))$,

  7. (vii)

$(\mu_{3})_{\lambda_{1},\lambda_{2}}(a,\beta
m,c)=(\mu_{2})_{\lambda_{1}+\lambda_{2}}((\mu_{2})_{\lambda_{1}}(a,m),c)-(\mu_{2})_{\lambda_{1}}(a,(\mu_{2})_{\lambda_{1}}(m,c))$,

  8. (viii)

$(\mu_{3})_{\lambda_{1},\lambda_{2}}(\beta
m,b,c)=(\mu_{2})_{\lambda_{1}+\lambda_{2}}((\mu_{2})_{\lambda_{1}}(m,b),c)-(\mu_{2})_{\lambda_{1}}(m,(\mu_{2})_{\lambda_{1}}(b,c))$,

  9. (ix)

$(\mu_{3})_{\lambda_{1}+\lambda_{2},\lambda_{3}}((\mu_{2})_{\lambda_{1}}(a,b),c,d)-(\mu_{3})_{\lambda_{1},\lambda_{2}+\lambda_{3}}(a,(\mu_{2})_{\lambda_{2}}(b,c),d)+(\mu_{3})_{\lambda_{1},\lambda_{2}}(a,b,(\mu_{2})_{\lambda_{3}}(c,d))\\\
\quad=(\mu_{2})_{\lambda_{1}+\lambda_{2}+\lambda_{3}}((\mu_{3})_{\lambda_{1},\lambda_{2}}(a,b,c),d)+(\mu_{2})_{\lambda_{1}}(a,(\mu_{3})_{\lambda_{2},\lambda_{3}}(b,c,d))$,

for all $a,b,c,d\in\mathcal{A}_{0}$ and $m,n\in\mathcal{A}_{1}$. We denote a
$2$-term $A_{\infty}$-conformal algebra as above simply by
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$.

Let $(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ and
$(\mathcal{A}^{\prime}_{1}\xrightarrow{\beta^{\prime}}\mathcal{A}^{\prime}_{0},\mu^{\prime}_{2},\mu^{\prime}_{3})$
be two $2$-term $A_{\infty}$-conformal algebras. A homomorphism of $2$-term
$A_{\infty}$-conformal algebras from
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ to
$(\mathcal{A}^{\prime}_{1}\xrightarrow{\beta^{\prime}}\mathcal{A}^{\prime}_{0},\mu^{\prime}_{2},\mu^{\prime}_{3})$
is given by a triple $(f_{0},f_{1},f_{2})$ consisting of
$\mathbb{C}[\partial]$-module maps
$f_{0}:\mathcal{A}_{0}\rightarrow\mathcal{A}^{\prime}_{0}$ and
$f_{1}:\mathcal{A}_{1}\rightarrow\mathcal{A}^{\prime}_{1}$, and a conformal
sesquilinear map
$f_{2}:\mathcal{A}_{0}\otimes\mathcal{A}_{0}\rightarrow\mathcal{A}_{1}^{\prime}[\lambda],$
$a\otimes b\mapsto(f_{2})_{\lambda}(a,b)$ such that the following identities
are hold: for any $a,b,c\in\mathcal{A}_{0}$ and $m\in\mathcal{A}_{1}$,

$\displaystyle\beta^{\prime}\big{(}(f_{2})_{\lambda}(a,b)\big{)}=~{}$
$\displaystyle
f_{0}\big{(}(\mu_{2})_{\lambda}(a,b)\big{)}-(\mu^{\prime}_{2})_{\lambda}\big{(}f_{0}(a),f_{0}(b)\big{)},$
$\displaystyle(f_{2})_{\lambda}(a,\beta m)=~{}$ $\displaystyle
f_{1}\big{(}(\mu_{2})_{\lambda}(a,m)\big{)}-(\mu^{\prime}_{2})_{\lambda}\big{(}f_{0}(a),f_{1}(m)\big{)},$
$\displaystyle(f_{2})_{\lambda}(\beta m,a)=~{}$ $\displaystyle
f_{1}\big{(}(\mu_{2})_{\lambda}(m,a)\big{)}-(\mu^{\prime}_{2})_{\lambda}\big{(}f_{1}(m),f_{0}(a)\big{)},$
$\displaystyle(f_{2})_{\lambda}\big{(}a,(\mu_{2})_{\mu}(b,c)\big{)}-$
$\displaystyle(f_{2})_{\lambda+\mu}\big{(}(\mu_{2})_{\lambda}(a,b),c\big{)}+(\mu^{\prime}_{2})_{\lambda+\mu}\big{(}(f_{2})_{\lambda}(a,b),f_{0}(c)\big{)}-(\mu_{2}^{\prime})_{\lambda}\big{(}f_{0}(a),(\mu_{2})_{\mu}(b,c)\big{)}$
$\displaystyle=~{}$
$\displaystyle(\mu^{\prime}_{3})_{\lambda,\mu}\big{(}f_{0}(a),f_{0}(b),f_{0}(c)\big{)}-f_{1}\big{(}(\mu_{3})_{\lambda,\mu}(a,b,c)\big{)}.$

Suppose $f=(f_{0},f_{1},f_{2})$ is a homomorphism of $2$-term
$A_{\infty}$-conformal algebras from
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ to
$(\mathcal{A}^{\prime}_{1}\xrightarrow{\beta^{\prime}}\mathcal{A}^{\prime}_{0},\mu^{\prime}_{2},\mu^{\prime}_{3})$,
and $g=(g_{0},g_{1},g_{2})$ is a homomorphism from $2$-term
$A_{\infty}$-conformal algebras from
$(\mathcal{A}^{\prime}_{1}\xrightarrow{\beta^{\prime}}\mathcal{A}^{\prime}_{0},\mu^{\prime}_{2},\mu^{\prime}_{3})$
to
$(\mathcal{A}^{\prime\prime}_{1}\xrightarrow{\beta^{\prime\prime}}\mathcal{A}^{\prime\prime}_{0},\mu^{\prime\prime}_{2},\mu^{\prime\prime}_{3})$.
Their composition is the homomorphism

$\displaystyle(g\circ f):=~{}$ $\displaystyle\big{(}g_{0}\circ
f_{0},g_{1}\circ f_{1},(g\circ f)_{2}\big{)},\text{ where }$
$\displaystyle\big{(}(g\circ f)_{2}\big{)}_{\lambda}(a,b)=~{}$
$\displaystyle(g_{2})_{\lambda}\big{(}f_{0}(a),f_{0}(b)\big{)}+g_{1}\big{(}(f_{2})_{\lambda}(a,b)\big{)},\text{
for }a,b\in\mathcal{A}_{0}.$

The collection of all $2$-term $A_{\infty}$-conformal algebras and
homomorphisms between them form a category, denoted by ${\bf
2A_{\infty}conf}$.

Skeletal $A_{\infty}$-conformal algebras. Here we focus on skeletal
$A_{\infty}$-conformal algebras which are a particular type of $2$-term
$A_{\infty}$-conformal algebras. We also characterize skeletal
$A_{\infty}$-conformal algebras by third cocycles of associative conformal
algebras.

######  4.2 Definition.

A $2$-term $A_{\infty}$-conformal algebra
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ is
called skeletal if $\beta=0$.

Let $(\mathcal{A}_{1}\xrightarrow{0}\mathcal{A}_{0},\mu_{2},\mu_{3})$ and
$(\mathcal{A}_{1}\xrightarrow{0}\mathcal{A}_{0},\mu_{2}^{\prime},\mu_{3}^{\prime})$
be two skeletal $A_{\infty}$-conformal algebras with the same underlying
graded $\mathbb{C}[\partial]$-module. They are said to be equivalent if
$\mu_{2}=\mu_{2}^{\prime}$ and there exists a conformal sesquilinear map
$\sigma:\mathcal{A}_{0}\otimes\mathcal{A}_{0}\to\mathcal{A}_{1}[\lambda]$ such
that

$\displaystyle(\mu_{3})^{\prime}_{\lambda_{1},\lambda_{2}}(a,b,c)=~{}$
$\displaystyle(\mu_{3})_{\lambda_{1},\lambda_{2}}(a,b,c)+(\mu_{2})_{\lambda_{1}}(a,\sigma_{\lambda_{2}}(b,c))-\sigma_{\lambda_{1}+\lambda_{2}}((\mu_{2})_{\lambda_{1}}(a,b),c)$
(14)
$\displaystyle-\sigma_{\lambda_{1}}(a,(\mu_{2})_{\lambda_{2}}(b,c))-(\mu_{2})_{\lambda_{1}+\lambda_{2}}(\sigma_{\lambda_{1}}(a,b),c),$

for all $a,b,c\in\mathcal{A}_{0}$. With these, we can characterize
(equivalence classes of) skeletal $A_{\infty}$-conformal algebras.

######  4.3 Theorem.

There is a one-to-one correspondence between skeletal $A_{\infty}$-conformal
algebras and tuples $(A,M,\Theta)$ in which $A$ is an associative conformal
algebra, $M$ is a conformal $A$-bimodule and $\Theta\in Z^{3}(A,M)$ is a
$3$-cocycle.

Moreover, this extends to a one-to-one correspondence between the equivalence
classes of skeletal $A_{\infty}$-conformal algebras and tuples
$(A,M,[\Theta])$, where $[\Theta]\in H^{3}(A,M)$.

We will prove a more general result for which the above theorem is a
particular case. An $n$-term $A_{\infty}$-conformal algebra is an
$A_{\infty}$-conformal algebra $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ whose
uderlying graded $\mathbb{C}[\partial]$-module $\mathcal{A}$ is concentrated
in degrees $0,1,\ldots,n-1$. That is,

$\displaystyle\mathcal{A}=\mathcal{A}_{0}\oplus\mathcal{A}_{1}\oplus\cdots\oplus\mathcal{A}_{n-1}.$

An $n$-term $A_{\infty}$-conformal algebra
$(\mathcal{A}_{0}\oplus\mathcal{A}_{1}\oplus\cdots\oplus\mathcal{A}_{n-1},\\{\mu_{k}\\}_{k\geq
1})$ is said to be skeletal if it that has only two nonzero terms
$\mathcal{A}_{0}$ and $\mathcal{A}_{n-1}$ with the zero differential
$(\mathrm{i.e.}~{}\mu_{1}=0)$. For the degree reason, the maps $\mu_{k}$’s are
nonzero only for $k=2$ and $n+1$. It also turns out from the identities (4)
that $\mathcal{A}_{0}$ is an associative conformal algebra with the
$\lambda$-multiplication $a_{\lambda}b:=(\mu_{2})_{\lambda}(a,b)$, for
$a,b\in\mathcal{A}_{0}$. Moreover, $\mathcal{A}_{n-1}$ is a conformal
$\mathcal{A}_{0}$-bimodule with the left and right $\lambda$-actions are
respectively given by

$\displaystyle a_{\lambda}m:=(\mu_{2})_{\lambda}(a,m)~{}~{}~{}\text{ and
}~{}~{}~{}m_{\lambda}a:=(\mu_{2})_{\lambda}(m,a),\text{ for
}a\in\mathcal{A}_{0},m\in\mathcal{A}_{n-1}.$

A skeletal $n$-term $A_{\infty}$-conformal algebra may be simply denoted by
$(\mathcal{A}_{n-1}\xrightarrow{0}0\to\cdots\xrightarrow{0}\mathcal{A}_{0},\mu_{2},\mu_{n+1})$.

Let
$(\mathcal{A}_{n-1}\xrightarrow{0}0\to\cdots\xrightarrow{0}\mathcal{A}_{0},\mu_{2},\mu_{n+1})$
and
$(\mathcal{A}_{n-1}\xrightarrow{0}0\to\cdots\xrightarrow{0}\mathcal{A}_{0},\mu^{\prime}_{2},\mu^{\prime}_{n+1})$
be two skeletal $n$-term $A_{\infty}$-conformal algebras with the same
underlying graded $\mathbb{C}[\partial]$-module. They are said to be
equivalent if $\mu_{2}=\mu_{2}^{\prime}$ and there exists a conformal
sesquilinear map $\sigma:(\mathcal{A}_{0})^{\otimes
n}\to\mathcal{A}_{n-1}[\lambda_{1},\ldots,\lambda_{n-1}]$ that satisfies

$\displaystyle(\mu_{n+1}^{\prime})_{\lambda_{1},\ldots,\lambda_{n}}(a_{1},\ldots,a_{n+1})=(\mu_{n+1})_{\lambda_{1},\ldots,\lambda_{n}}(a_{1},\ldots,a_{n+1})+(\delta\sigma)_{\lambda_{1},\ldots,\lambda_{n}}(a_{1},\ldots,a_{n+1}),$
(15)

for all $a_{1},\ldots,a_{n+1}\in\mathcal{A}_{0}$. Here $\delta$ is the
Hochschild coboundary operator of the associative conformal algebra
$\mathcal{A}_{0}$ with coefficients in the conformal
$\mathcal{A}_{0}$-bimodule $\mathcal{A}_{n-1}$. With the above notations, we
have the following generalization of Theorem 4.3.

######  4.4 Theorem.

There is a one-to-one correspondence between skeletal $n$-term
$A_{\infty}$-conformal algebras and tuples $(A,M,\Theta)$ in which $A$ is an
associative conformal algebra, $M$ is a conformal $A$-bimodule and $\Theta\in
Z^{n+1}(A,M)$ is a $(n+1)$-cocycle of the associative conformal algebra $A$
with coefficients in the conformal $A$-bimodule $M$. Further, this extends to
a one-to-one correspondence between the equivalence classes of skeletal
$n$-term $A_{\infty}$-conformal algebras and tuples $(A,M,[\Theta])$, where
$[\Theta]\in H^{n+1}(A,M)$.

###### Proof.

Let
$(\mathcal{A}_{n-1}\xrightarrow{0}0\to\cdots\xrightarrow{0}\mathcal{A}_{0},\mu_{2},\mu_{n+1})$
be a skeletal $n$-term $A_{\infty}$-conformal algebra. Then it follows from
the definition of an $A_{\infty}$-conformal algebra that the conformal
sesquilinear map
$\mu_{2}:\mathcal{A}_{0}\otimes\mathcal{A}_{0}\to\mathcal{A}_{0}[\lambda]$
makes the $\mathbb{C}[\partial]$-module $\mathcal{A}_{0}$ into an associative
conformal algebra. Moreover, the conformal sesquilinear maps
$\mu_{2}:\mathcal{A}_{0}\otimes\mathcal{A}_{n-1}\to\mathcal{A}_{n-1}[\lambda]$
and
$\mu_{2}:\mathcal{A}_{n-1}\otimes\mathcal{A}_{0}\to\mathcal{A}_{n-1}[\lambda]$
makes the $\mathbb{C}[\partial]$-module $\mathcal{A}_{n-1}$ into a conformal
$\mathcal{A}_{0}$-bimodule. Further, it follows from the higher associative
conformal identities $(\ref{a-inf-conf})$ that

$\displaystyle\sum_{k+l=n+3}\sum_{i=1}^{k}(-1)^{i(l+1)+l(|a_{1}|+\cdots+|a_{i-1}|)}$
(16)
$\displaystyle(\mu_{k})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\cdots+\lambda_{i+l-1},\ldots,\lambda_{n+1}}(a_{1},\ldots,a_{i-1},(\mu_{l})_{\lambda_{i},\ldots,\lambda_{i+l-2}}(a_{i},\ldots,a_{i+l-1}),a_{i+l},\ldots,a_{n+2})=0,$

for all $a_{1},\ldots,a_{n+2}\in\mathcal{A}_{0}$. The non-zero terms in the
above summation occur when $k=2$ (then $l=n+1$) and $k=n+1$ (then $l=2$).
Thus, $(\ref{equation-n})$ is equivalent to

$\displaystyle(-1)^{n+2}$
$\displaystyle(\mu_{2})_{\lambda_{1}+\cdots+\lambda_{n+1}}\big{(}(\mu_{n+1})_{\lambda_{1},\ldots,\lambda_{n}}(a_{1},\ldots,a_{n+1}),a_{n+2}\big{)}+(\mu_{2})_{\lambda_{1}}\big{(}a_{1},(\mu_{n+1})_{\lambda_{1},\ldots,\lambda_{n+1}}(a_{2},\ldots,a_{n+2})\big{)}$
$\displaystyle+\sum_{i=1}^{n+1}(-1)^{3i}(\mu_{n+1})_{\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i}+\lambda_{i+l},\ldots,\lambda_{n+1}}\big{(}a_{1},\ldots,a_{i-1},(\mu_{2})_{\lambda_{i}}(a_{i},a_{i+1}),\ldots,a_{n+2}\big{)}=0.$

This is same as
$(\delta\mu_{n+1})_{\lambda_{1},\ldots,\lambda_{n+1}}(a_{1},\ldots,a_{n+2})=0$.
In other words, $\mu_{n+1}\in Z^{n+1}(\mathcal{A}_{0},\mathcal{A}_{n-1})$ is a
$(n+1)$-cocycle of the associative conformal algebra $\mathcal{A}_{0}$ with
coefficients in the conformal $\mathcal{A}_{0}$-bimodule $\mathcal{A}_{n-1}$.

Conversely, let $(A,M,\Theta)$ be a triple consisting of an associative
conformal algebra $A$, a conformal $A$-bimodule $M$ and a $(n+1)$-cocycle
$\Theta\in Z^{n+1}(A,M)$. Then it is straightforward to verify that the triple
$(M\xrightarrow{0}0\rightarrow\cdots\xrightarrow{0}A,\mu_{2},\mu_{n+1}=\Theta)$
is a skeletal $n$-term $A_{\infty}$-conformal algebra, where $\mu_{2}$ is
given by the $\lambda$-multiplication of $A$, and left and right
$\lambda$-actions of $A$ on $M$.

The final part about the correspondence between equivalence classes of
skeletal $n$-term $A_{\infty}$-conformal algebras and tuples $(A,M,[\Theta])$
follows from the identity (15). ∎

##  5\. Associative conformal 2-algebras

In this section, we introduce the notion of associative conformal
$2$-algebras. They can be thought of as the categorification of associative
conformal algebras. Our main result in this section shows that the category of
associative conformal $2$-algebras and the category of $2$-term
$A_{\infty}$-conformal algebras are equivalent.

Let ${\bf{Vect}^{\mathbb{C}[\partial]}}$ be the category of all
$\mathbb{C}[\partial]$-modules and $\mathbb{C}[\partial]$-linear maps. A
$2$-vector space in the category of $\mathbb{C}[\partial]$-modules is a
category internal to the category ${\bf{Vect}^{\mathbb{C}[\partial]}}$. Thus,
it follows that a $2$-vector space in the category of
$\mathbb{C}[\partial]$-modules is a category $C=(C_{1}\rightrightarrows
C_{0})$ with $\mathbb{C}[\partial]$-module of objects $C_{0}$ and
$\mathbb{C}[\partial]$-module of morphisms $C_{1}$ such that the source and
target maps $s,t:C_{1}\to C_{0}$, the identity-assigning map $i:C_{0}\to
C_{1}$ and the composition map $C_{1}\times_{C_{0}}C_{1}\to C_{1}$ are all
$\mathbb{C}[\partial]$-linear maps. We write the images of the object-
inclusion map as $i(a)=1_{a}$, for $a\in A$. A morphism between $2$-vector
spaces in the category of $\mathbb{C}[\partial]$-modules is a functor internal
to the category ${\bf{Vect}^{\mathbb{C}[\partial]}}$. Thus, a morphism
$F:C\rightarrow C^{\prime}$ between $2$-vector spaces in the category of
$\mathbb{C}[\partial]$-modules is given by a pair $F=(F_{0},F_{1})$ of
$\mathbb{C}[\partial]$-linear maps $F_{0}:C_{0}\rightarrow C^{\prime}_{0}$ and
$F_{1}:C_{1}\rightarrow C^{\prime}_{1}$ commuting with all the structure maps.

The notion of $2$-vector spaces in the category of
$\mathbb{C}[\partial]$-modules are closely related to $2$-term complexes of
$\mathbb{C}[\partial]$-modules. Let $C=(C_{1}\rightrightarrows C_{0})$ be a
$2$-vector space in the category of $\mathbb{C}[\partial]$-modules. Then
$\mathrm{ker}(s)\xrightarrow{t}C_{0}$ is a $2$-term complex of
$\mathbb{C}[\partial]$-modules. On the other hand, if
$\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0}$ is a complex of
$\mathbb{C}[\partial]$-modules then

$\displaystyle
C=(\mathcal{A}_{0}\oplus\mathcal{A}_{1}\rightrightarrows\mathcal{A}_{0})$ (17)

is a $2$-vector space in the category of $\mathbb{C}[\partial]$-modules, where
the source, target and object inclusion are respectively given by

$\displaystyle s(a,h)=a,\quad t(a,h)=a+\beta
h~{}~{}\mathrm{~{}and~{}}~{}~{}i(a)=(a,0).$

A morphism between $2$-term complexes induces a morphism between the
corresponding $2$-vector spaces in the category of
$\mathbb{C}[\partial]$-modules.

######  5.1 Definition.

An associative conformal $2$-algebra is a triple $(C,\pi,\mathbb{A})$
consisting of a $2$-vector space $C$ in the category of
$\mathbb{C}[\partial]$-modules with a conformal sesquilinear functor
$\pi:C\otimes C\to C[\lambda]$, $a\otimes b\mapsto a_{\lambda}b$ and a
conformal sesquilinear natural isomorphism (called the conformal associator)

$\displaystyle\mathbb{A}_{a,b,c}:a_{\lambda}(b_{\mu}c)\longrightarrow(a_{\lambda}b)_{\lambda+\mu}c$

satisfying the following pentagonal identity:

$\textstyle{a_{\lambda}(b_{\mu}(c_{\nu}d))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{b,c,d}}$$\scriptstyle{\mathbb{A}_{a.b,c_{\nu}d}}$$\textstyle{a_{\lambda}((b_{\mu}c)_{\mu+\nu}d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{a,b_{\mu}c,d}}$$\textstyle{(a_{\lambda}b)_{\lambda+\mu}(c_{\nu}d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{a_{\lambda}b,c,d}}$$\textstyle{(a_{\lambda}(b_{\mu}c))_{\lambda+\mu+\nu}d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{a,b,c}}$$\textstyle{((a_{\lambda}b)_{\lambda+\mu}c)_{\lambda+\mu+\nu}d}$

for all $a,b,c,d\in C_{0}$.

######  5.2 Definition.

Let $(C,\pi,\mathbb{A})$ and $(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})$
be two associative conformal $2$-algebras. A homomorphism of associative
conformal $2$-algebras from $(C,\pi,\mathbb{A})$ to
$(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})$ is a functor
$F=(F_{0},F_{1}):C\rightarrow C^{\prime}$ between the underlying $2$-vector
spaces in the category of $\mathbb{C}[\partial]$-modules and a conformal
sesquilinear natural isomorphism

$\displaystyle\mathbb{F}_{a,b}:F_{0}(a)^{\prime}_{\lambda}F_{0}(b)\rightarrow
F_{0}(a_{\lambda}b)$

that makes the following diagram commutative:

$\textstyle{F_{0}(a)^{\prime}_{\lambda}(F_{0}(b)^{\prime}_{\mu}F_{0}(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}^{\prime}_{F(a),F(b),F(c)}}$$\scriptstyle{\mathbb{F}_{b,c}}$$\textstyle{(F_{0}(a)^{\prime}_{\lambda}F_{0}(b))^{\prime}_{\lambda+\mu}F_{0}(c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{F}_{a,b}}$$\textstyle{F_{0}(a)^{\prime}_{\lambda}F_{0}(b_{\mu}c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{F}_{a,b_{\mu}c}}$$\textstyle{F_{0}(a_{\lambda}b)^{\prime}_{\lambda+\mu}F_{0}(c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{F}_{a_{\lambda}b,c}}$$\textstyle{F_{0}(a_{\lambda}(b_{\mu}c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{a,b,c}}$$\textstyle{F_{0}((a_{\lambda}b)_{\lambda+\mu}c)}$

for all $a,b,c\in C_{0}$. We denote a homomorphism as above simply by the pair
$(F=(F_{0},F_{1}),\mathbb{F})$.

Let $(C,\pi,\mathbb{A})$, $(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})$ and
$(C^{\prime\prime},\pi^{\prime\prime},\mathbb{A}^{\prime\prime})$ be three
associative conformal $2$-algebras. Suppose
$(F=(F_{0},F_{1}),\mathbb{F}):(C,\pi,\mathbb{A})\to(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})$
and
$(G=(G_{0},G_{1}),\mathbb{G}):(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})\to(C^{\prime\prime},\pi^{\prime\prime},\mathbb{A}^{\prime\prime})$
are homomorphisms between associative conformal $2$-algebras. Then their
composition is given by $(G\circ F=(G_{0}\circ F_{0},G_{1}\circ
F_{1}),\mathbb{G}\diamond\mathbb{F})$, where $\mathbb{G}\diamond\mathbb{F}$ is
the ordinary composition

$\displaystyle(G_{0}\circ F_{0})(a)^{\prime\prime}_{\lambda}(G_{0}\circ
F_{0})(b)\xrightarrow{\mathbb{G}_{F_{0}(a),F_{0}(b)}}G_{0}(F_{0}(a)^{\prime}_{\lambda}F_{0}(b))\xrightarrow{\mathbb{F}_{a,b}}(G_{0}\circ
F_{0})(a_{\lambda}b),\text{ for }a,b\in C_{0}.$

For any associative conformal $2$-algebra $(C,\pi,\mathbb{A})$, the identity
homomorphism is given by the identity functor $\mathrm{id}_{C}:C\to C$ with
the identity natural isomorphism $\mathbbm{id}_{a,b}:a_{\lambda}b\rightarrow
a_{\lambda}b$, for all $a,b\in C_{0}$. With the above definitions and
notations, associative conformal $2$-algebras and homomorphisms between them
form a category. We denote this category by conf2Alg. We will now prove the
main result of the present section.

######  5.3 Theorem.

The categories ${\bf 2A_{\infty}conf}$ and conf2Alg are equivalent.

###### Proof.

Here we only sketch the proof. Let
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ be a
$2$-term $A_{\infty}$-conformal algebra. We consider the $2$-vector space
$C=(\mathcal{A}_{0}\oplus\mathcal{A}_{1}\rightrightarrows\mathcal{A}_{0})$ in
the category of $\mathbb{C}[\partial]$-modules given in (17). We now define a
conformal sesquilinear functor $\pi:C\otimes C\to
C[\lambda],(a,h)\otimes(b,k)\mapsto(a,h)_{\lambda}(b,k)$ by

$\displaystyle(a,h)_{\lambda}(b,k)=((\mu_{2})_{\lambda}(a,b),(\mu_{2})_{\lambda}(a,k)+(\mu_{2})_{\lambda}(h,b)+(\mu_{2})_{\lambda}(dh,k)),$

for $(a,h),(b,k)\in\mathcal{A}_{0}\oplus\mathcal{A}_{1}$. We define the
conformal associator by

$\displaystyle\mathbb{A}_{a,b,c}=\big{(}a_{\lambda}(b_{\mu}c),(\mu_{3})_{\lambda,\mu}(a,b,c)\big{)},\mathrm{~{}for~{}}a,b,c\in
A_{0}.$ (18)

Then it follows from the condition (v) of Definition 4.1 that
$\mathbb{A}_{a,b,c}$ is a morphism from source $a_{\lambda}(b_{\mu}c)$ to
target $(a_{\lambda}b)_{\lambda+\mu}c$. It is easy to check that
$\mathcal{A}_{a,b,c}$ is a natural isomorphism. Finally, a direct computation
shows that the conformal associator (18) satisfies the pentagonal identity.
Thus, we obtain an associative conformal $2$-algebra
$(C=(\mathcal{A}_{0}\oplus\mathcal{A}_{1}\rightrightarrows\mathcal{A}_{0}),\pi,\mathbb{A})$.

Let $(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ and
$(\mathcal{A}_{1}^{\prime}\xrightarrow{\beta^{\prime}}\mathcal{A}_{0}^{\prime},\mu_{2}^{\prime},\mu_{3}^{\prime})$
be $2$-term $A_{\infty}$-conformal algebras and $(f_{0},f_{1},f_{2})$ be a
homomorphism between them. Let
$(C=(\mathcal{A}_{0}\oplus\mathcal{A}_{1}\rightrightarrows\mathcal{A}_{0}),\pi,\mathbb{A})$
and
$(C^{\prime}=(\mathcal{A}_{0}^{\prime}\oplus\mathcal{A}_{1}^{\prime}\rightrightarrows\mathcal{A}_{0}^{\prime}),\pi^{\prime},\mathbb{A}^{\prime})$
be the corresponding associative conformal $2$-algebras. We define a functor
$F:C\to C^{\prime}$ by $F=(F_{0},F_{1}):=(f_{0},f_{0}\oplus f_{1})$ and a
natural isomorphism $\mathbb{F}_{a,b}:F_{0}(a)^{\prime}_{\lambda}F_{0}(b)\to
F_{0}(a_{\lambda}b)$ by
$\mathbb{F}_{a,b}=(f_{0}(a)_{\lambda}f_{0}(b),(f_{2})_{\lambda}(a,b))$, for
all $a,b\in\mathcal{A}_{0}$. Then it is easy to check that
$(F=(F_{0},F_{1}),\mathbb{F})$ is a homomorphism of associative conformal
$2$-algebras. The above constructions gives rise to a functor $S:{\bf
2A_{\infty}conf}\to{\bf conf2Alg}$.

Conversely, let $(C=(C_{1}\rightrightarrows C_{0}),\pi,\mathbb{A})$ be an
associative conformal $2$-algebra. We consider the two term chain complex
$\mathcal{A}_{1}=\mathrm{ker}(s)\xrightarrow{t}C_{0}=\mathcal{A}_{0}$ and
define conformal sesquilinear maps
$\mu_{2}:\mathcal{A}_{i}\otimes\mathcal{A}_{j}\to\mathcal{A}_{i+j}[\lambda]$
and
$\mu_{3}:\mathcal{A}_{0}\otimes\mathcal{A}_{0}\otimes\mathcal{A}_{0}\to\mathcal{A}_{1}[\lambda,\mu]$
by

$\displaystyle(\mu_{2})_{\lambda}(a,b)=a_{\lambda}b,~{}(\mu_{2})_{\lambda}(a,h)={1_{a}}_{\lambda}h,~{}(\mu_{2})_{\lambda}(h,a)=h_{\lambda}{1_{a}}$
$\displaystyle(\mu_{3})_{\lambda,\mu}(a,b,c)=\mathrm{pr}_{2}\big{(}\mathcal{A}_{a,b,c}:a_{\lambda}(b_{\mu}c)\to(a_{\lambda}b)_{\lambda+\mu}c\big{)},$

where $\mathrm{pr}_{2}:C_{0}\oplus C_{1}\to C_{1}$ is the projection onto the
second factor. Then one can easily verify that
$(\mathcal{A}_{1}=\mathrm{ker}(s)\xrightarrow{t}C_{0}=\mathcal{A}_{0},\mu_{2},\mu_{3})$
is a $2$-term $A_{\infty}$-conformal algebra. Let $(C,\pi,\mathbb{A})$ and
$(C^{\prime},\pi^{\prime},\mathbb{A}^{\prime})$ be two associative conformal
$2$-algebras and $(F=(F_{0},F_{1}),\mathbb{F})$ be a homomorphism between
them. Let $(\mathrm{ker}(s)\xrightarrow{t}C_{0},\mu_{2},\mu_{3})$ and
$(\mathrm{ker}(s^{\prime})\xrightarrow{t^{\prime}}C_{0}^{\prime},\mu_{2}^{\prime},\mu_{3}^{\prime})$
be the corresponding $2$-term $A_{\infty}$-conformal algebras. We define
$\mathbb{C}[\partial]$-linear maps $f_{0}:C_{0}\to C_{0}^{\prime}$,
$f_{1}:\mathrm{ker}(s)\to\mathrm{ker}(s^{\prime})$ and a conformal
sesquilinear map $f_{2}:C_{0}\otimes
C_{0}\to\mathrm{ker}(s^{\prime})[\lambda]$ by $f_{0}=F_{0}$,
$f_{1}=F_{1}\vline_{\mathrm{ker}(s)}$ and
$f_{2}(a,b)=\mathbb{F}_{a,b}-1_{s(\mathbb{F}_{a,b})}$, for $a,b\in
C_{0}=\mathcal{A}_{0}$. It is easy to see that $(f_{0},f_{1},f_{2})$ gives
rise to a morphism between $2$-term $A_{\infty}$-conformal algebras. Thus, we
obtain a functor $T:{\bf conf2Alg}\to{\bf 2A_{\infty}conf}$.

We are now left to show that there are natural isomorphisms $\Lambda:T\circ
S\implies 1_{{\bf 2A_{\infty}conf}}$ and $\Upsilon:S\circ T\implies 1_{{\bf
conf2Alg}}$. Let
$(\mathcal{A}_{1}\xrightarrow{\beta}\mathcal{A}_{0},\mu_{2},\mu_{3})$ be a
given $2$-term $A_{\infty}$-conformal algebra. If we apply $T\circ S$ to this
$2$-term $A_{\infty}$-conformal algebra, we obtain the same $2$-term
$A_{\infty}$-conformal algebra. Thus, $T\circ S$ is the identity functor
$1_{{\bf 2A_{\infty}conf}}$ on the category ${\bf 2A_{\infty}conf}$. Hence we
can take $\Lambda$ to be the identity natural isomorphism. Next, let
$(C=(C_{1}\rightrightarrows C_{0}),\pi,\mathbb{A})$ be an associative
conformal $2$-algebra. If we apply $T$, we obtain the $2$-term
$A_{\infty}$-conformal algebra
$(\mathrm{ker}(s)\xrightarrow{t}C_{0},\mu_{2},\mu_{3})$. Further, applying
$S$, we obtain the associative conformal $2$-algebra
$\big{(}C^{\prime}=(C_{0}\oplus\mathrm{ker}(s)\rightrightarrows
C_{0}),\pi^{\prime},\mathbb{A}^{\prime}\big{)}$. We define a functor
$\Upsilon_{C}=\big{(}(\Upsilon_{C})_{0},(\Upsilon_{C})_{1}\big{)}:C^{\prime}\rightarrow
C$ by

$\displaystyle(\Upsilon_{C})_{0}(a)=a~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}(\Upsilon_{C})_{1}(a,m)=1_{a}+m,\text{ for }a\in C_{0}\text{ and
}(a,m)\in C_{0}\oplus\mathrm{ker}(s).$

We also take the identity natural isomorphism
$(\mathbbm{id}_{C})_{a,b}:(\Upsilon_{C})_{0}(a)_{\lambda}(\Upsilon_{C})_{0}(b)\rightarrow(\Upsilon_{C})_{0}(a^{\prime}_{\lambda}b)=(\Upsilon_{C})_{0}(a_{\lambda}b)$,
for $a,b\in C_{0}$. It can be checked that
$\big{(}\Upsilon_{C}=((\Upsilon_{C})_{0},(\Upsilon_{C})_{1}),\mathbbm{id}_{C}\big{)}$
is an isomorphism of associative conformal $2$-algebras from $C^{\prime}$ to
$C$. This construction yields a natural isomorphism $\Upsilon:S\circ T\implies
1_{{\bf conf2Alg}}$. Hence the proof follows. ∎

##  6\. Homotopy Lie conformal algebras

In this section, we first recall Lie conformal algebras. Then we introduce and
study $L_{\infty}$-conformal algebras which are the homotopy analogue of Lie
conformal algebras. We show that a suitable skew-symmetrization of an
$A_{\infty}$-conformal algebra gives rise to an $L_{\infty}$-conformal algebra
structure.

######  6.1 Definition.

A Lie conformal algebra is a pair $(L,[\cdot_{\lambda}\cdot])$ consisting of a
$\mathbb{C}[\partial]$-module $L$ with a $\mathbb{C}$-linear map (called the
$\lambda$-bracket) $[\cdot_{\lambda}\cdot]:L\otimes L\to
L[\lambda],~{}x\otimes y\mapsto[x_{\lambda}y]$ satisfying the conformal
sesquilinearity condition:

$\displaystyle[\partial x_{\lambda}y]=-\lambda[x_{\lambda}y]~{}~{}\text{ and
}~{}~{}[x_{\lambda}\partial y]=(\lambda+\partial)[x_{\lambda}y]$

and the following identities:

\- skew-symmetry: $[x_{\lambda}y]=-[y_{-\lambda-\partial}x]$,

\- conformal Jacobi identity:
$[x_{\lambda}[y_{\mu}z]]=[[x_{\lambda}y]_{\lambda+\mu}z]+[y_{\mu}[x_{\lambda}z]]$.

Let $L=\mathbb{C}[\partial]l$ be the $\mathbb{C}[\partial]$-module with one
free generator $l$. Define the $\lambda$-bracket
$[\cdot_{\lambda}\cdot]:L\otimes L\to L[\lambda]$ by

$\displaystyle[l_{\lambda}l]=(\partial+2\lambda)l.$

Then $(L,[\cdot_{\lambda}\cdot])$ is a Lie conformal algebra, called the
Virasoro Lie conformal algebra and denoted by Vir.

Let $(\mathfrak{g},[~{},~{}])$ be a Lie algebra. The current Lie conformal
algebra associated to $\mathfrak{g}$ is given by
$\mathrm{Cur~{}}\mathfrak{g}=\mathbb{C}[\partial]\otimes\mathfrak{g}$ with the
$\lambda$-bracket

$\displaystyle[x_{\lambda}y]:=[x,y],\mathrm{~{}for~{}all~{}}x,y\in\mathfrak{g}.$

It has been observed by D′Andrea and Kac [2] that the Virasoro Lie conformal
algebra and current Lie conformal algebras $\mathrm{Cur~{}}\mathfrak{g}$
(where $\mathfrak{g}$ is finite-dimensional simple Lie algebra) exhaust all
finite simple Lie conformal algebras.

######  6.2 Remark.

([3]) Let $(A,\cdot_{\lambda}\cdot)$ be an associative conformal algebra. Then
the $\mathbb{C}[\partial]$-module $A$ equipped with the $\lambda$-bracket
$[a_{\lambda}b]:=a_{\lambda}b-b_{-\partial-\lambda}a$, for $a,b\in A$, is a
Lie conformal algebra. This is called the skew-symmetrization of the
associative conformal algebra $(A,\cdot_{\lambda}\cdot)$.

######  6.3 Definition.

Let $(L,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra. A conformal
$L$-module is a pair $(M,\cdot_{\lambda}\cdot)$ in which $M$ is a
$\mathbb{C}[\partial]$-module and

$\displaystyle\cdot_{\lambda}\cdot:L\otimes M\to M[\lambda],~{}x\otimes
v\mapsto x_{\lambda}v$

is a $\mathbb{C}$-linear map satisfying the following properties: for all
$x,y\in L$ and $v\in M$,

$\displaystyle(\partial
x)_{\lambda}v=-\lambda(x_{\lambda}v),~{}~{}~{}x_{\lambda}(\partial
v)=(\lambda+\partial)x_{\lambda}v,$ $\displaystyle
x_{\lambda}(y_{\mu}v)-y_{\mu}(x_{\lambda}v)=[x_{\lambda}y]_{\lambda+\mu}v.$

It follows that any Lie conformal algebra is a conformal module over itself,
called the adjoint module.

In the following, we recall the graded Lie algebra whose Maurer-Cartan
elements correspond to Lie conformal algebra structures on a given
$\mathbb{C}[\partial]$-module [26, 28] (see also [29]). Let $L$ be a
$\mathbb{C}[\partial]$-module. A conformal sesquilinear map

$\displaystyle\varphi:L^{\otimes n}\to
L[\lambda_{1},\ldots,\lambda_{n-1}],~{}x_{1}\otimes\cdots\otimes
x_{n}\mapsto\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(x_{1},\ldots,x_{n})$

is said to be skew-symmetric [25] if it is skew-symmetric with respect to
simultaneous permutations of the $x_{i}$’s and the $\lambda_{i}$’s in the
sense that, for any permutation $\sigma\in\mathbb{S}_{n}$, we have

$\displaystyle\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(x_{1},\ldots,x_{n})=\mathrm{sgn}(\sigma)~{}\varphi_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(n-1)},x_{\sigma(n)})\big{|}_{\lambda_{n}\mapsto\lambda_{n}^{\dagger}}.$
(19)

for all $x_{1},\ldots,x_{n}\in L$. The notation on the right-hand side of (19)
simply means that $\lambda_{n}$ is replaced by
$\lambda_{n}^{\dagger}=-\sum_{i=1}^{n-1}\lambda_{i}-\partial$, if it occurs
and $\partial$ is moved to the left. We denote the set of all such skew-
symmetric conformal sesquilinear maps by
$\mathrm{Hom}_{cs}(\wedge^{n}L,L[\lambda_{1},\ldots,\lambda_{n-1}])$. Define a
graded vector space

$\displaystyle\mathfrak{g}={\oplus}_{n\geq
1}\mathrm{Hom}_{cs}(\wedge^{n}L,L[\lambda_{1},\ldots,\lambda_{n-1}]).$

For any
$\varphi\in\mathrm{Hom}_{cs}(\wedge^{n}L,L[\lambda_{1},\ldots,\lambda_{n-1}])$
and
$\psi\in\mathrm{Hom}_{cs}(\wedge^{m}L,L[\lambda_{1},\ldots,\lambda_{m-1}])$,
we define a bracket
$[\varphi,\psi]_{\mathrm{CNR}}\in\mathrm{Hom}_{cs}(\wedge^{m+n-1}L,L[\lambda_{1},\ldots,\lambda_{m+n-2}])$
by

$\displaystyle[\varphi,\psi]_{\mathrm{CNR}}=\varphi\diamond\psi-(-)^{(m-1)(n-1)}\psi\diamond\varphi,\mathrm{~{}where}$
$\displaystyle(\varphi\diamond\psi$
$\displaystyle)_{\lambda_{1},\ldots,\lambda_{m+n-2}}(x_{1},\ldots,x_{m+n-1})=\sum_{\sigma\in\mathrm{Sh}(m,n-1)}\mathrm{sgn}{(\sigma)}~{}$
(20)
$\displaystyle\varphi_{\lambda_{\sigma}^{\dagger},\lambda_{\sigma(m+1)},\ldots,\lambda_{\sigma(m+n-2)}}(\psi_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(m-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(m)}),x_{\sigma(m+1)},\ldots,x_{\sigma(m+n-1)}),$

for $x_{1},\ldots,x_{m+n-1}\in L$. Note that $\sigma\in\mathrm{Sh}(m,n-1)$
implies that $\sigma(m)=m+n-1$ or $\sigma(m+n-1)=m+n-1$. If $\sigma(m)=m+n-1$,
we use the notation
$\lambda_{\sigma}^{\dagger}=-\partial-(\lambda_{\sigma(m+1)}+\cdots+\lambda_{\sigma(m+n-1)})$
and if $\sigma(m+n-1)=m+n-1$, we use the notation
$\lambda_{\sigma}^{\dagger}=\lambda_{\sigma(1)}+\cdots+\lambda_{\sigma(m)}$ in
the above expression (20). With this, it is not hard to verify that the
product $\diamond$ makes the graded vector space $\mathfrak{g}$ into a degree
$-1$ graded pre-Lie algebra. Hence the bracket $[~{},~{}]_{\mathrm{CNR}}$ is a
degree $-1$ graded Lie bracket on $\mathfrak{g}$. In other words, the shifted
graded vector space $\mathfrak{g}[1]={\oplus}_{n\geq
0}\mathrm{Hom}_{cs}(\wedge^{n+1}L,L[\lambda_{1},\ldots,\lambda_{n}])$ is a
graded Lie algebra. For $\pi\in\mathrm{Hom}_{cs}(\wedge^{2}L,L[\lambda])$, we
have

$\displaystyle([\pi,\pi]_{\mathrm{CNR}})_{\lambda,\mu}(x,y,z)=2(\pi_{\lambda+\mu}(\pi_{\lambda}(x,y),z)-\pi_{-\partial-\mu}(\pi_{\lambda}(x,z),y)+\pi_{-\partial-\lambda}(\pi_{\mu}(y,z),x)),$

for $x,y,z\in L$. This shows that $\pi$ is a Maurer-Cartan element in the
graded Lie algebra $(\mathfrak{g}[1],[~{},~{}]_{\mathrm{CNR}})$ if and only if
the $\lambda$-bracket $[\cdot_{\lambda}\cdot]:L\otimes L\to
L[\lambda],~{}[x_{\lambda}y]:=\pi_{\lambda}(x,y)$ defines a Lie conformal
algebra structure on $L$. This characterization allows us to express the known
coboundary operator of a Lie conformal algebra ([25]) in terms of the graded
Lie bracket $[~{},~{}]_{\mathrm{CNR}}$.

Let $(L,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra with the Maurer-
Cartan element $\pi\in\mathrm{Hom}_{cs}(\wedge^{2}L,L[\lambda])$ in the graded
Lie algebra $(\mathfrak{g}[1],[~{},~{}]_{\mathrm{CNR}})$. For each $n\geq 0$,
we define the $n$-th cochain group $C^{n}(L,L)$ by

$C^{n}(L,L)=\begin{cases}L/{\partial L}&\text{if }n=0,\\\
\mathrm{Hom}_{cs}(\wedge^{n}L,L[\lambda_{1},\ldots,\lambda_{n-1}])&\text{if
}n\geq 1.\end{cases}$

Then there is a map $\delta_{\pi}:C^{n}(L,L)\to C^{n+1}(L,L)$, for $n\geq 0$,
given by

$\displaystyle\delta_{\pi}(x+\partial L)(y)=~{}$
$\displaystyle[y_{-\lambda-\partial}x]\big{|}_{\lambda=0},\mathrm{~{}for~{}}x+\partial
L\in{L}/{\partial L},$ $\displaystyle\delta_{\pi}(\varphi)=~{}$
$\displaystyle(-1)^{n-1}[\pi,\varphi]_{\mathrm{CNR}},\mathrm{~{}for~{}}\varphi\in
C^{n\geq 1}(L,L).$

Since $[\pi,\pi]_{\mathrm{CNR}}=0$, it follows that $(\delta_{\pi})^{2}=0$. In
other words, $\\{C^{\bullet}(L,L),\delta_{\pi}\\}$ is a cochain complex. The
corresponding cohomology groups are called the cohomology of the Lie conformal
algebra $(L,[\cdot_{\lambda}\cdot])$ with coefficients in the adjoint module.

The above construction can be easily generalized when we have an arbitrary
conformal module. Let $(L,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra
and $(M,\cdot_{\lambda}\cdot)$ be any conformal module over it. Consider the
$\mathbb{C}[\partial]$-module $L\oplus M$ with the $\lambda$-bracket
$[\cdot_{\lambda}\cdot]^{\ltimes}:(L\oplus M)\otimes(L\oplus
M)\rightarrow(L\oplus M)[\lambda]$ given by

$\displaystyle[(x,u)_{\lambda}(y,v)]^{\ltimes}:=\big{(}[x_{\lambda}y],x_{\lambda}v-y_{-\partial-\lambda}u\big{)},\text{
for }(x,u),(y,v)\in L\oplus M.$

This is a Lie conformal algebra, called the semidirect product. Let
$\pi_{\ltimes}\in\mathrm{Hom}_{cs}\big{(}\wedge^{2}(L\oplus M),(L\oplus
M)[\lambda]\big{)}$ be the Maurer-Cartan element corresponding to the
semidirect product. Then $\pi_{\ltimes}$ yields the coboundary operator
$\delta_{\pi_{\lambda}}$ of the cochain complex $\\{C^{\bullet}(L\oplus
M,L\oplus M),\delta_{\pi_{\lambda}}\\}$ defining the cohomology of the
semidirect product Lie conformal algebra with coefficients in the adjoint
module. For each $n\geq 0$, we consider a space $C^{n}(L,M)$ by

$\displaystyle C^{0}(L,M)={M}/{\partial M}\quad\mathrm{and}\quad C^{n\geq
1}(L,M)=\mathrm{Hom}_{cs}(\wedge^{n}L,M[\lambda_{1},\ldots,\lambda_{n-1}]).$

It is easy to see that the coboundary map $\delta_{\pi_{\ltimes}}$ restricts
to a coboundary map (denoted by $\delta_{\pi}$)
$\delta_{\pi}:C^{n}(L,M)\rightarrow C^{n+1}(L,M)$. In other words,
$\\{C^{\bullet}(L,M),\delta_{\pi}\\}$ is a cochain complex. The map
$\delta_{\pi}$ is explicitly given by $\delta_{\pi}(v+\partial
M)(x)=(x_{-\lambda-\partial}v)\big{|}_{\lambda=0}$, for $v+\partial M\in
C^{0}(L,M)$, and

$\displaystyle(\delta_{\pi}\varphi)_{\lambda_{1},\ldots,\lambda_{n}}(x_{1},\ldots,x_{n+1})$
$\displaystyle=\sum_{i=1}^{n}(-1)^{i+1}~{}{x_{i}}_{\lambda_{i}}\varphi_{\lambda_{1},\ldots,\widehat{\lambda_{i}},\ldots,\lambda_{n}}(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+(-1)^{n}~{}{x_{n+1}}_{-\partial-\sum_{l=1}^{n}\lambda_{l}}\varphi_{\lambda_{1},\ldots,\lambda_{n-1}}(x_{1},\ldots,x_{n})$
$\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}~{}\varphi_{-\partial-\sum_{\begin{subarray}{c}l=1\\\ l\neq
i,j\end{subarray}}^{n}\lambda_{l},\lambda_{1},\ldots,\widehat{\lambda_{i}},\ldots,\widehat{\lambda_{j}},\ldots,\lambda_{n}}([{x_{i}}_{\lambda_{i}}x_{j}],x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+n+1}~{}\varphi_{-\partial-\sum_{\begin{subarray}{c}l=1\\\
l\neq
i\end{subarray}}^{n}\lambda_{l},\lambda_{1},\ldots,\widehat{\lambda_{i}},\ldots,\lambda_{n-1}}\big{(}[{x_{i}}_{\lambda_{i}}x_{n+1}],x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n}\big{)},$

for $\varphi\in C^{n}(L,M)$ and $x_{1},\ldots,x_{n+1}\in L$. This is precisely
the one given in [25]. The cohomology groups of the cochain complex
$\\{C^{\bullet}(L,M),\delta_{\pi}\\}$ are called the cohomology of the Lie
conformal algebra $L$ with coefficients in the conformal $L$-module $M$.

In the following, we introduce $L_{\infty}$-conformal algebras and find a
relation with $A_{\infty}$-conformal algebras introduced earlier. We first
recall the notion of $L_{\infty}$-algebras.

######  6.4 Definition.

([17, 18]) An $L_{\infty}$-algebra is a pair $(\mathcal{L},\\{l_{k}\\}_{k\geq
1})$ consisting of a graded vector space
$\mathcal{L}=\oplus_{i\in\mathbb{Z}}\mathcal{L}_{i}$ equipped with a
collection of graded linear maps $\\{l_{k}:\mathcal{L}^{\otimes
k}\to\mathcal{L}\\}_{k\geq 1}$ with $\mathrm{deg}(l_{k})=k-2$ for $k\geq 1$,
that satisfies the following conditions:

\- each $l_{k}$ is graded skew-symmetric in the sense that

$\displaystyle
l_{k}(x_{1},\ldots,x_{k})=\mathrm{sgn}(\sigma)\epsilon(\sigma)~{}l_{k}(x_{\sigma(1)},\ldots,x_{\sigma(k)}),\mathrm{~{}for~{}all~{}}\sigma\in\mathbb{S}_{k},$

where $\epsilon(\sigma)$ is the Koszul sign that appears in the graded
context,

\- higher Jacobi identities: for $n\geq 1$ and homogeneous elements
$x_{1},\ldots,x_{n}\in\mathcal{L}$,

$\displaystyle\sum_{p+q=n+1}\sum_{\sigma\in\mathrm{Sh}(q,n-q)}\mathrm{sgn}(\sigma)\epsilon(\sigma)(-1)^{q(p-1)}~{}l_{p}(l_{q}(x_{\sigma(1)},\ldots,x_{\sigma(q)}),x_{\sigma(q+1)},\ldots,x_{\sigma(n)})=0.$

Any Lie algebras, graded Lie algebras and differential graded Lie algebras are
particular cases of $L_{\infty}$-algebras. More precisely, a differential
graded Lie algebra $(\mathfrak{g},[~{},~{}],d)$ is an $L_{\infty}$-algebra
$(\mathfrak{g},\\{l_{k}\\}_{k\geq 1})$, where $l_{1}=d,~{}l_{2}=[~{},~{}]$ and
$l_{k}=0$ for $k\geq 3$.

######  6.5 Definition.

An $L_{\infty}$-conformal algebra (also called a strongly homotopy Lie
conformal algebra) is a graded $\mathbb{C}[\partial]$-module
$\mathcal{L}=\oplus_{i\in\mathbb{Z}}\mathcal{L}_{i}$ with a collection
$\\{l_{k}:\mathcal{L}^{\otimes
k}\to\mathcal{L}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ of graded
$\mathbb{C}$-linear maps (called the $\lambda$-multiplications) with
$\mathrm{deg}(l_{k})=k-2$ for $k\geq 1$, subject to the following conditions:

\- each $l_{k}$ is conformal sesquilinear,

\- each $l_{k}$ is graded skew-symmetric in the sense that

$\displaystyle(l_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(x_{1},\ldots,x_{k})=\mathrm{sgn}(\sigma)\epsilon(\sigma)~{}(l_{k})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(k-1)},x_{\sigma(k)})\big{|}_{\lambda_{k}\mapsto\lambda_{k}^{\dagger}},\text{
for }\sigma\in\mathbb{S}_{k},$

\- higher conformal Jacobi idntities: for any $n\geq 1$ and homogeneous
$x_{1},\ldots,x_{n}\in\mathcal{L}$,

$\displaystyle\sum_{p+q=n+1}$
$\displaystyle\sum_{\sigma\in\mathrm{Sh}(q,n-q)}\mathrm{sgn}(\sigma)\epsilon(\sigma)(-1)^{q(p-1)}$
(21)
$\displaystyle(l_{p})_{\lambda^{\dagger}_{\sigma},\lambda_{\sigma(q+1)},\ldots,\lambda_{\sigma(n-1)}}\big{(}(l_{q})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(q-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(q)}),x_{\sigma(q+1)},\ldots,x_{\sigma(n)}\big{)}=0.$

Note that $\sigma\in\mathrm{Sh}(q,n-q)$ implies that either $\sigma(q)=n$ or
$\sigma(n)=n$. When $\sigma(q)=n$, we use the notation
$\lambda_{\sigma}^{\dagger}=-\partial-(\lambda_{\sigma(q+1)}+\cdots+\lambda_{\sigma(n)})$
and when $\sigma(n)=n$, we use the notation
$\lambda_{\sigma}^{\dagger}=\lambda_{\sigma(1)}+\cdots+\lambda_{\sigma(q)}$ in
the above identities.

Lie conformal algebras and (differential) graded Lie conformal algebras are
examples of $L_{\infty}$-conformal algebras. The notion of $2$-term
$L_{\infty}$-conformal algebras considered in [30] and examples therein are
particular case of $L_{\infty}$-conformal algebras defined above. Similar to
$A_{\infty}$-conformal algebras, the concept of $L_{\infty}$-conformal
algebras has a simple interpretation when we shift the degree. We first
consider the following definition.

######  6.6 Definition.

An $L_{\infty}[1]$-conformal algebera is a pair
$(\mathcal{W},\\{\varrho_{k}\\}_{k\geq 1})$ that consists of a graded
$\mathbb{C}[\partial]$-module
$\mathcal{W}=\oplus_{i\in\mathbb{Z}}\mathcal{W}_{i}$ with a collection
$\\{\varrho_{k}:\mathcal{W}^{\otimes
k}\to\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ of graded
$\mathbb{C}$-linear maps with deg$(\varrho_{k})=-1$ subject to the following
conditions:

\- each $\varrho_{k}$ is conformal sesquilinear,

\- each $\varrho_{k}$ is graded symmetric, i.e.

$\displaystyle(\varrho_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(x_{1},\ldots,x_{k})=\epsilon(\sigma)~{}(\varrho_{k})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(k-1)},x_{\sigma(k)})\big{|}_{\lambda_{k}\mapsto\lambda_{k}^{\dagger}},\text{
for }\sigma\in\mathbb{S}_{k},$ (22)

\- shifted higher conformal Jacobi identities: for any $n\geq 1$ and
homogeneous elements $w_{1},\ldots,w_{n}\in\mathcal{W}$,

$\displaystyle\sum_{p+q=n+1}$
$\displaystyle\sum_{\sigma\in\mathrm{Sh}(q,n-q)}\epsilon(\sigma)$
$\displaystyle(\varrho_{p})_{\lambda_{\sigma}^{\dagger},\lambda_{\sigma(q+1)},\ldots,\lambda_{\sigma(n-1)}}\big{(}(\varrho_{q})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(q-1)}}(x_{\sigma(1)},\ldots,x_{\sigma(q)}),x_{\sigma(q+1)},\ldots,x_{\sigma(n)}\big{)}=0.$

The concept of $L_{\infty}[1]$-conformal algebra is the conformal analogue of
an $L_{\infty}[1]$-algebra considered by Lada and Markl [17]. The following
result is a straightforward generalization of a result proved in [17].

######  6.7 Proposition.

Let $\mathcal{L}$ be a graded $\mathbb{C}[\partial]$-module. Then an
$L_{\infty}$-conformal algebra structure on $\mathcal{L}$ is equivalent to
having an $L_{\infty}[1]$-conformal algebra structure on the graded
$\mathbb{C}[\partial]$-module $\mathcal{L}[-1]$. Explicitly,
$(\mathcal{L},\\{l_{k}\\}_{k\geq 1})$ is an $L_{\infty}$-conformal algebra if
and only if $(\mathcal{L}[-1],\\{\varrho_{k}\\}_{k\geq 1})$ is an
$L_{\infty}[1]$-conformal algebra, where

$\displaystyle\varrho_{k}=(-1)^{\frac{k(k-1)}{2}}~{}s\circ
l_{k}\circ(s^{-1})^{\otimes k},~{}\text{ for }k\geq 1.$

Let $\mathcal{W}=\oplus_{i\in\mathbb{Z}}\mathcal{W}_{i}$ be a graded
$\mathbb{C}[\partial]$-module. Let $n\in\mathbb{Z}$ be an integer. For each
$k\geq 1$, let

$\displaystyle\mathrm{Hom}^{n}_{cs}(S^{k}\mathcal{W},\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}])$

be the set of all conformal sesquilinear maps from $\mathcal{W}^{\otimes k}$
to $\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]$ that are graded symmetric
and of degree $n$. That is,

$\displaystyle\mathrm{Hom}^{n}_{cs}(S^{k}\mathcal{W},\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}])=\\{\varphi\in\mathrm{Hom}^{n}_{cs}(\mathcal{W}^{\otimes
k},\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}])~{}|~{}\varphi\mathrm{~{}is~{}graded~{}symmetric}\\}.$

Define
$C^{\bullet}_{cs}(S\mathcal{W})=\oplus_{n\in\mathbb{Z}}C^{n}_{cs}(S\mathcal{W})$,
where $C^{n}_{cs}(S\mathcal{W})=\oplus_{k\geq
1}\mathrm{Hom}^{n}_{cs}(S^{k}\mathcal{W},\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}]).$
Thus, an element $\varrho\in C^{n}_{cs}(S\mathcal{W})$ is of the form
$\varrho=\sum_{k\geq 1}\varrho_{k}$, where
$\varrho_{k}\in\mathrm{Hom}^{n}_{cs}(S^{k}\mathcal{W},\mathcal{W}[\lambda_{1},\ldots,\lambda_{k-1}])$
for all $k\geq 1$. For $\varrho=\sum_{k\geq 1}\varrho_{k}\in
C^{n}_{cs}(S\mathcal{W})$ and $\tau=\sum_{l\geq 1}\tau_{l}\in
C^{m}_{cs}(S\mathcal{W})$, we define

$\displaystyle\\{\\![\sum_{k\geq 1}\varrho_{k},\sum_{l\geq
1}\tau_{l}]\\!\\}=\sum_{p\geq
1}\sum_{k+l=p+1}(\varrho_{k}~{}\overline{\diamond}~{}\tau_{l}-(-1)^{mn}~{}\tau_{l}~{}\overline{\diamond}~{}\varrho_{k}),$
(23)

where

$\displaystyle(\varrho_{k}$
$\displaystyle~{}\overline{\diamond}~{}\tau_{l})_{\lambda_{1},\ldots,\lambda_{p-1}}(w_{1},\ldots,w_{p})$
$\displaystyle=\sum_{\sigma\in\mathrm{Sh}(l,k-1)}\epsilon(\sigma)~{}(\varrho_{k})_{\lambda_{\sigma}^{\dagger},\lambda_{\sigma(l+1)},\ldots,\lambda_{\sigma(p-1)}}\big{(}(\tau_{l})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(l-1)}}(w_{\sigma(1)},\ldots,w_{\sigma(l)}),w_{\sigma(l+1)},\ldots,w_{\sigma(p)}\big{)}.$

Then $(C^{\bullet}_{cs}(S\mathcal{W}),\\{\\![~{},~{}]\\!\\})$ is a graded Lie
algebra. Moreover, we have the following result.

######  6.8 Theorem.

Let $\mathcal{L}$ be a graded $\mathbb{C}[\partial]$-module. Then an
$L_{\infty}$-conformal algebra structure on $\mathcal{L}$ is equivalent to the
existence of a Maurer-Cartan element in the graded Lie algebra
$(C^{\bullet}_{cs}(S\mathcal{W}),\\{\\![~{},~{}]\\!\\})$, where
$\mathcal{W}=\mathcal{L}[-1]$.

###### Proof.

We have seen earlier that an $L_{\infty}$-conformal algebra structure on
$\mathcal{L}$ is equivalent to an $\mathcal{L}_{\infty}[1]$-conformal algebra
structure on $\mathcal{W}=\mathcal{L}[-1].$ Next, it follows from (23) that an
$L_{\infty}[1]$-conformal algebra structure on $\mathcal{W}$ is equivalent to
a Maurer-Cartan element in the graded Lie algebra
$(C^{\bullet}_{cs}(S\mathcal{W}),\\{\\![~{},~{}]\\!\\})$. Hence the result
follows. ∎

In [17] Lada and Markl showed that the skew-symmetrization of an
$A_{\infty}$-algebra gives rise to an $L_{\infty}$-algebra. On the other hand,
an associative conformal algebra gives a Lie conformal algebra by a suitable
skew-symmetrization (cf. Remark 6.2). Unifying both the above results, we have
the following.

######  6.9 Theorem.

Let $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ be an $A_{\infty}$-conformal
algebra. Then the underlying graded $\mathbb{C}[\partial]$-module
$\mathcal{A}$ carries an $L_{\infty}$-conformal algebra structure with the
$\lambda$-multiplications $\\{l_{k}:\mathcal{A}^{\otimes
k}\to\mathcal{A}[\lambda_{1},\ldots,\lambda_{k-1}]\\}_{k\geq 1}$ given by

$\displaystyle(l_{k})_{\lambda_{1},\ldots,\lambda_{k-1}}(a_{1},\ldots,a_{k})=\sum_{\sigma\in\mathbb{S}_{k}}\mathrm{sgn}(\sigma)\epsilon(\sigma)~{}(\mu_{k})_{\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k)}}(a_{\sigma(1)},\ldots,a_{\sigma(k)})\big{|}_{\lambda_{k}\mapsto\lambda_{k}^{\dagger}},\mathrm{~{}for~{}}a_{1},\ldots,a_{k}\in\mathcal{A}.$

###### Proof.

Since $(\mathcal{A},\\{\mu_{k}\\}_{k\geq 1})$ is an $A_{\infty}$-conformal
algebra, it follows from Proposition 3.8 that
$(\mathcal{A}[-1],\\{\rho_{k}\\}_{k\geq 1})$ is an $A_{\infty}[1]$-conformal
algebra. Hence $\rho=\sum_{k\geq 1}\rho_{k}\in C^{-1}_{cs}(\mathcal{A}[-1])$
is a Maurer-Cartan element in the graded Lie algebra
$(C^{\bullet}_{cs}(\mathcal{A}[-1]),\llbracket~{},~{}\rrbracket)$.

On the other hand, a symmetrization of conformal sesquilinear maps yields a
morphism $\mathfrak{S}:C^{\bullet}_{cs}(\mathcal{A}[-1])\rightarrow
C^{\bullet}_{cs}(S\mathcal{A}[-1])$ of graded vector spaces that preserves the
corresponding graded Lie brackets. Further, we have
$\mathfrak{S}(\rho)=\varrho$, where $\varrho=\sum_{k\geq 1}\varrho_{k}$ with
$\varrho_{k}=(-1)^{\frac{k(k-1)}{2}}~{}s\circ l_{k}\circ(s^{-1})^{\otimes k}$,
for $k\geq 1$. This shows that

$\displaystyle\\{\\![\varrho,\varrho]\\!\\}=\\{\\![\mathfrak{S}(\rho),\mathfrak{S}(\rho)]\\!\\}=\mathfrak{S}\llbracket\rho,\rho\rrbracket=0.$

Hence $\varrho=\sum_{k\geq 1}\varrho_{k}$ defines an $L_{\infty}[1]$-conformal
algebra structure on the graded vector space $\mathcal{A}[-1]$. In other
words, $\\{l_{k}\\}_{k\geq 1}$ gives rise to an $L_{\infty}$-conformal algebra
structure on $\mathcal{A}$. ∎

Conflict of interest statement. There is no conflict of interest.

Data Availability Statement. Data sharing is not applicable to this article as
no new data were created or analyzed in this study.

Acknowledgements. Both authors thank Jiefeng Liu for his comments on the
earlier version of the paper. Anupam Sahoo would like to thank CSIR,
Government of India for funding the PhD fellowship. Both authors are greatful
to Department of Mathematics, IIT Kharagpur for providing the beautiful
academic atmosphere where the research has been carried out.

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