# On the L∞ structure of Poisson gauge theory

O. Abla  CCNH - Universidade Federal do ABC, 09210-580, Santo André, SP,
Brazil.  V. G. Kupriyanov  CMCC - Universidade Federal do ABC, 09210-580,
Santo André, SP, Brazil.  Phisics Department, Tomsk State University, 634050,
Tomsk, Russia M. Kurkov  INFN-Sezione di Napoli, Complesso Universitario di
Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy. Dipartimento di
Fisica “E. Pancini”, Università di Napoli Federico II, Complesso Universitario
di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy.

###### Abstract

The Poisson gauge theory is a semi-classical limit of full non-commutative
gauge theory. In this work we construct an L${}_{\infty}^{full}$ algebra which
governs both the action of gauge symmetries and the dynamics of the Poisson
gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and
prove that they satisfy the corresponding homotopy relations. On the one hand,
it provides new explicit non-trivial examples of L∞ algebras. On the other
hand, it can be used as a starting point for bootstrapping the full non-
commutative gauge theory. The first few brackets of such a theory are
constructed explicitly in the text. In addition we show that the derivation
properties of $\ell$-brackets on L${}_{\infty}^{full}$ with respect to the
truncated product on the exterior algebra are satisfied only for the canonical
non-commutativity. In general, L${}_{\infty}^{full}$ does not have a structure
of P∞ algebra.

##  1 Introduction

Higher algebraic structures such as L∞ and A∞ algebras [1, 2, 3], exhibit a
growing interest, motivated by important applications in physics and
mathematics. In the physical literature L∞ algebras appeared for the first
time in closed string field theory [2] as generalized gauge symmetries. In
such theories the commutator of two gauge transformations is again a gauge
transformation, however, with a field dependent gauge parameter. This
circumstance weakens the closure constraint, thereby suggesting to generalize
the very notion of a Lie algebra. An appropriate generalization is given by
the L∞ algebras where one has not only a two-bracket (the commutator) but more
general multilinear $n$-brackets $\ell_{n}$ with $n$ inputs. These brackets
should satisfy the L∞ relations also known as higher Jacobi identities. In
particular, the usual Jacobi identity for the two-bracket can be violated by
“total derivative” terms, thus allowing a mild form of non-associativity. The
same may happen with the Leibniz rule. The price to pay is the presence of the
higher brackets.

The L∞ algebras do not show up in the string field theory only. They also
naturally arise in “conventional” gauge theories, like Chern-Simons (CS) and
Yang-Mills (YM) theories [4], see [5] for review. In these cases the structure
is considerably truncated, and only a finite number of higher products and
relations is non-trivial. At the classical level L∞ structure contains all
necessary information about the gauge theory, including the gauge symmetry,
the field equations and the Noether identities [6]. According to the
conjecture formulated in [5], every consistent gauge theory should be governed
by an underlying L∞ algebra.

One of the most recent approaches to non-commutative deformations of gauge
theories was developed in [7]-[10]. Being based on the use of L∞ algebras,
this formalism allows to deal with a non-constant NC-parameter $\Theta(x)$.
Technically the approach follows the bootstrap logic. A given undeformed gauge
theory defines the initial lower order brackets $\ell_{1}$ and $\ell_{2}$. For
instance, for $U(1)$ gauge theory we set,

$\ell_{1}(f):={\mathsf{d}}f,\quad\mbox{and}\qquad\ell_{1}(A):={\mathsf{d}}A\,,$
(1)

which defines the undeformed gauge variation, $\delta^{0}_{f}A={\mathsf{d}}f$,
and the coresponding field strength, $F_{0}={\mathsf{d}}A$. The non-
commutative deformation is encoded in the theory by identifying the two-
bracket with a star commutator of two gauge parameters,

$\ell_{2}^{NC}(f,g):=i[f,g]_{\star},$ (2)

for some given star product,

$f\star g=f\,g+\frac{i\hbar}{2}\,\\{f,g\\}+{\cal
O}\left(\hbar^{2}\right)\,,\qquad\mbox{with,}\qquad\\{f,g\\}=\Theta^{ij}(x)\,\partial_{i}f\,\partial_{j}g\,.$
(3)

After that, one has to build a complete set of $\ell$-brackets of an L∞
algebra corresponding to the deformed theory. The L∞ identities comprising the
initially given $\ell$-brackets turn into the equations for definition of the
additional brackets, like $\ell_{2}^{NC}(f,A)$, $\ell_{2}^{NC}(A,A)$,
$\ell_{3}^{NC}(f,A,A)$ and etc. These brackets provide the deformed gauge
variations,

$\delta_{f}^{NC}A={\mathsf{d}}f+\ell_{2}^{NC}(f,A)+\dots,$ (4)

closing the NC gauge algebra,

$[\delta_{f}^{NC},\delta_{g}^{NC}]=\delta_{-i[f,g]_{\star}}^{NC},$ (5)

and the deformed field strength,

${\cal F}_{NC}={\mathsf{d}}A-\ell_{2}^{NC}(A,A)/2+\dots,$ (6)

which transforms in a covariant way upon the deformed gauge transformations

$\delta_{f}^{{NC}}{\cal F}_{NC}=-i[{\cal F}_{NC},f]_{\star}.$ (7)

In other words, for a given undeformed gauge theory and a given star product,
imposing the guiding principle of an underlying L∞ algebra, we can
algebraically bootstrap all necessary corrections, rendering the deformed
field strength and the deformed gauge transformations, which close the non-
commutative gauge algebra. We stress that L∞-formalism is a powerful tool for
the construction of perturbative (order by order in $\Theta$) deformations [9,
10]. Though, to get an explicit all order expressions normally one needs to
invoke additional considerations.

It is a technically complicated task to elaborate the above construction in
full generality. For example, the expression for the star product and star
commutator is very complex and is known in a precise form only in a few number
of examples of non-constant $\Theta$. Therefore from now on we adopt some
simplifications. Instead of dealing with the full non-commutative gauge
algebra one may work with its semi-classical limit, which is given by the
Poisson gauge algebra [11]. In this algebra the commutator of two gauge
transformations with gauge parameters $f$ and $g$ is a gauge transformation
corresponding to the Poisson bracket of these parameters111Remind that the
Poisson bracket is the semi-classical limit of the star commutator,
$\\{f,g\\}=\lim_{\hbar\to 0}[f,g]_{\star}/i\hbar$.,

$[\delta_{f},\delta_{g}]A=\delta_{\\{f,g\\}}A\,.$ (8)

A dynamical field theoretical model having the Poisson gauge algebra (8) as a
corresponding algebra of gauge symmetries is called the Poisson gauge theory
[12]. This model is designed to investigate the semi-classical features of the
full non-commutative gauge theory with coordinate dependent non-commutativity
$\Theta^{ij}(x)$.

It is remarkable, that there is another way to build the Poisson gauge theory,
which does not rely explicitly on the L∞-structures. In particular, in [11] it
was proposed an approach to the construction of (almost)-Poisson gauge
transformations and the corresponding gauge algebra, based on the symplectic
embeddings of (almost)-Poisson structures. The further generalization of the
symplectic embedding formalism to the construction of the Poisson gauge theory
was elaborated in [12]. The two approaches for non-commutative gauge theories,
mentioned above, can be seen as complementary. As it has already been
mentioned the homotopy algebras provide the perturbative in $\Theta$
corrections. Once we know the structure of these corrections we may employ the
symplectic embedding framework which is especially good for obtaining explicit
all-order in $\Theta$ form of the deformed constructions.

The aim of the present work is to esteblish the direct connection between the
two approaches to Poisson gauge theories: the formulation in terms of
L∞-algebras [7]-[10] and the symplectic embedding formalism [11, 12]. More
precisely, we will construct the L${}_{\infty}^{full}$ algebra corresponding
to a generic Poisson gauge theory [12]. A part of this work has already been
done in [11], where the algebra L${}_{\infty}^{gauge}$ describing the action
of Poisson gauge symmetries on the gauge field, was constructed. The new
results of the present research include the description of a dynamical sector
of the Poisson gauge theory in terms of L∞ algebras and investigation of the
corresponding P∞ structure.

The paper is organized as follows. In Sec. 2 we briefly review the
construction of the Poisson gauge theory in the symplectic embedding
formalism. We start Sec. 3 providing the basic definitions of L∞ algebra.
After that, in Proposition 32, we represent the L∞-relations in the form,
which is convenient for our purposes. The main result is formulated and proven
in Proposition 57. The explicit examples of $\mathfrak{su}(2)$-like structure
and $\kappa$-Minkowski deformations are analyzed Sec. 4. In Sec. 5 we discuss
the corresponding P∞ structure. The last section is dedicated to the explicit
construction of the lower order $\ell$-brackets corresponding to the full non-
commutative gauge theory.

##  2 Poisson gauge theory

The aim of this section is to review the construction of the field theoretical
model which has a Poisson gauge algebra (8) as a corresponding algebra of
gauge symmetries. We start with the definition of the Poisson gauge
transformations $\delta_{f}A$ which should satisfy the following two
conditions: the closure of the algebra (8) and the correct commutative limit
to the standard $U(1)$ gauge transformations, $\lim_{\Theta\to
0}\delta_{f}A=\partial f$. If $\Theta^{ij}$ is constant, one may easily see
that the straightforward expression, $\delta^{can}_{f}A=\partial f+\\{A,f\\}$,
satisfies (8). However, for non-constant $\Theta^{ij}(x)$ the standard Leibniz
rule with respect to the partial derivative is violated,
$\partial\\{f,g\\}\neq\\{\partial f,g\\}+\\{f,\partial g\\}$, therefore the
same expression will not close the algebra (8) anymore. To overcome this
problem one has to modify the expression for the gauge transformations
introducing the corrections proportional to the derivatives of the non-
commutativity $\partial\Theta^{ij}(x)$ which would compensate the violation of
the Leibniz rule.

The problem with the violation of the Leibniz rule can be solved in an
extended space [11]. To each coordinate $x^{i}$ we introduce a conjugate
variable $p_{i}$, in such a way that the corresponding Poisson brackets,

$\\{x^{i},x^{j}\\}=\Theta^{ij}(x)\,,\qquad\\{x^{i},p_{j}\\}=\gamma^{i}_{j}(x,p)\,,\qquad\\{p_{i},p_{j}\\}=0\,,$
(9)

should satisfy the Jacobi identity. For constant $\Theta^{ij}$ one finds,
$\gamma^{i}_{j}(x,p)=\delta^{i}_{j}$, so the Poisson bracket,
$\\{f(x),p_{i}\\}=\partial_{i}f(x)$, is just a partial derivative of this
function. In case if $\Theta^{ij}(x)$ is not constant the expression for
$\gamma^{i}_{j}(x,p)$ is more complicated. The Jacobi identity for the algebra
(9) implies the partial differential equation on the matrix
$\gamma^{i}_{j}(x,p)$,

$\gamma^{l}_{b}\,\partial^{b}_{p}\gamma_{a}^{k}-\gamma^{k}_{b}\,\partial^{b}_{p}\gamma^{l}_{a}+\Theta^{lm}\,\partial_{m}\gamma_{a}^{k}-\Theta^{km}\,\partial_{m}\gamma_{a}^{l}-\gamma^{m}_{a}\,\partial_{m}\Theta^{lk}=0\,,$
(10)

with $\partial_{m}=\partial/\partial x^{m}$ and
$\partial^{b}_{p}=\partial/\partial p_{b}$. Perturbative solution is given by,

$\displaystyle\gamma^{k}_{a}(x,p)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\gamma_{a}^{k(n)}$ $\displaystyle=$
$\displaystyle\delta^{k}_{a}-\frac{1}{2}\,\partial_{a}\Theta^{kb}p_{b}-\frac{1}{12}\left(2\,\Theta^{cm}\partial_{a}\partial_{m}\Theta^{bk}+\partial_{a}\Theta^{bm}\partial_{m}\Theta^{kc}\right)p_{b}p_{c}+{\cal
O}(\Theta^{3})\,.$

The recursive formula for the construction of the functions,
$\gamma_{a}^{k(n)}=\gamma_{a}^{k|j_{1}\dots j_{n}}(x)\,p_{j_{1}}\dots
p_{j_{n}}$, can be found in [13].

Instead of working with standard partial derivatives $\partial_{i}$ in the
definition of the gauge transformation one may introduce the ‘twisted’
derivative of function $f(x)$ defined in terms of Poisson bracket with
$p$-variable,
$\partial^{\tau}_{i}f(x):=\\{f(x),p_{i}\\}=\gamma^{l}_{i}(x,p)\partial_{l}f(x)$.
The Jacobi identity for the Poisson brackets (9) and the fact that
$\\{p_{i},p_{j}\\}=0$ imply two important properties of ‘twisted’ derivatives:
they commute,
$\partial^{\tau}_{i}\partial^{\tau}_{j}=\partial^{\tau}_{j}\partial^{\tau}_{i}$,
and satisfy the Leibniz rule,
$\partial^{\tau}_{a}\\{f,g\\}=\\{\partial^{\tau}_{a}f,g\\}+\\{f,\partial^{\tau}_{a}g\\}$.
The price to pay however is that the expression $\\{f(x),p_{i}\\}$ depends
also on the auxiliary non-physical $p$-variables. It turns out that the
auxiliary variables can be eliminated in the consistent way by introducing the
constraints, $p_{a}=A_{a}(x)$. In [11] it was demonstrated that the gauge
transformations given by,

$\delta_{f}A_{a}=\gamma^{l}_{a}(A)\,\partial_{l}f(x)+\\{A_{a}(x),f(x)\\}\,,$
(12)

where, $\gamma^{l}_{a}(A):=\gamma^{l}_{a}(x,p)|_{p_{a}=A_{a}(x)}$, close the
algebra (8) and have the desired commutative limit.

To be consistent with the semi-classical limit the gauge variation of the
matter field $\psi$ is defined as, $\delta_{f}\psi=\\{\psi,f\\}$. The gauge
covariant derivative of matter field is the object which should satisfy the
following two conditions: it should transform covariantly under the gauge
transformation,

$\displaystyle\delta_{f}{\cal D}_{a}(\psi)=\\{{\cal D}_{a}(\psi),f\\}\,,$ (13)

and reproduce the standard derivative in the commutative limit,
$\lim_{\Theta\to 0}{\cal D}_{a}(\psi)=\partial_{a}\psi\,.$ Such an object can
be constructed following [12]:

${\cal
D}_{a}(\psi)=\rho_{a}^{i}(A)\left(\gamma_{i}^{l}(A)\,\partial_{l}\psi+\\{A_{a},\psi\\}\right)\,,$
(14)

where the matrix, $\rho_{a}^{i}(A):=\rho^{i}_{a}(x,p)|_{p_{a}=A_{a}(x)}$,
with,

$\displaystyle\gamma^{j}_{b}\,\partial^{b}_{p}\,\rho_{a}^{i}+\rho_{a}^{b}\,\partial^{i}_{p}\gamma^{j}_{b}+\Theta^{jb}\,\partial_{b}\rho_{a}^{i}=0\,.$
(15)

Perturbative solution of this equation reads,

$\displaystyle\rho^{i}_{a}(x,p)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\rho_{a}^{i(n)}$ $\displaystyle=$
$\displaystyle\delta^{i}_{a}-\frac{1}{2}\,\partial_{a}\Theta^{ib}p_{b}+\frac{1}{6}\left(2\,\Theta^{cm}\partial_{a}\partial_{m}\Theta^{ib}-\partial_{a}\Theta^{bm}\partial_{m}\Theta^{ic}\right)p_{b}p_{c}+{\cal
O}(\Theta^{3})\,.$

For constant $\Theta^{ij}$ one finds just, $\rho^{i}_{a}(x,p)=\delta^{i}_{a}$.

The commutator relation for the covariant derivatives,

$\displaystyle\left[{\cal D}_{a},{\cal D}_{b}\right]=\\{{\cal
F}_{ab},\,\cdot\,\\}+\left({\cal F}_{ad}\,{\Lambda}_{b}{}^{de}-{\cal
F}_{bd}\,{\Lambda}_{a}{}^{de}\right){\cal D}_{e}\,,$ (17)

where,
${\Lambda}_{b}{}^{de}=\left(\rho^{-1}\right)_{j}^{d}\left(\partial^{j}_{A}\rho_{b}^{m}-\partial^{m}_{A}\rho_{b}^{j}\right)\left(\rho^{-1}\right)_{m}^{e}\,,$
defines the Poisson field strength:

$\displaystyle{\cal
F}_{ab}:=\frac{1}{2}\left(\rho^{c}_{a}\rho^{d}_{b}-\rho^{c}_{b}\rho^{d}_{a}\right)\left(\gamma_{c}^{m}\,\partial_{m}A_{d}+\\{A_{c},A_{d}\\}\right)\,,$
(18)

which transforms covariantly,

$\delta_{f}{\cal F}_{ab}=\\{{\cal F}_{ab},f\\}\,.$ (19)

and reproduces the abelian field strength in the commutative limit,

${\lim_{\Theta\to 0}\,{\cal F}_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}\,.}$
(20)

Now we turn to the consistent gauge covariant deformation of the equations of
motion.Taking into account (19) and (13) the straightforward candidate reads,

${\cal E}_{S}^{b}:={\cal D}_{a}\left({\cal F}^{ab}\right)=0\,,$ (21)

where the subscript $S$ stands for “straightforward”. This quantity transforms
covariantly, $\delta_{f}{\cal E}_{S}^{b}=\\{{\cal E}_{S}^{b},f\\}$, and
reproduces the first pare of Maxwell equations in the commutative limit,
$\lim_{\Theta\to 0}{\cal E}_{S}^{b}=\partial_{a}F^{ab}_{0}$. These two are
essential requirements that we should impose on the equations of motion of the
deformed theory. However, they are not the only existing. When we start
working with the equations of motion it is not always guaranteed that these
equations can be obtained from the action principle.

The alternative way of obtaining of the deformed equations of motion from the
corresponding deformed Lagrangian was proposed in [14]. Basically the idea is
the following, having in hands the gauge covariant Poisson field strength (18)
and the covariant derivative (14) one constructs the gauge covariant
deformation of the standard Lagrangian,

${\cal L}_{g}=-\frac{1}{4}\,{\cal F}_{ab}\,{\cal
F}^{ab}\,,\qquad\mbox{with,}\qquad\delta_{f}{\cal L}_{g}=\\{{\cal
L}_{g},f\\}\,.$ (22)

Then introducing an appropriate measure $\mu(x)$, such that for any two
Schwartz functions $f$ and $g$ holds,

$\int\\!d^{N}x\,\mu(x)\,\\{f,g\\}=0\,\qquad\Leftrightarrow\qquad\partial_{l}\left(\mu(x)\,\Theta^{lk}(x)\right)=0\,,$
(23)

one constructs the gauge invariant action,

$S_{g}=\int\\!d^{N}x\,\mu(x)\,{\cal L}_{g}\,,\qquad\mbox{such
that,}\qquad\delta_{f}S_{g}\equiv 0\,.$ (24)

Note that in general the measure function $\mu(x)$ is not constant, moreover
in $\kappa$-Minkowski case, see e.g. [15, 16], it does not tend to the trivial
measure in the commutative limit. Therefore the action (24) does not have a
desired commutative limit, $\lim_{\Theta\to 0}\mu(x)\,{\cal L}\neq{\cal
L}^{0}$. This issue was addressed in [15] by introducing the gauge covariant
factor in the Lagrangian density which would “absorb” the non-trivial measure.
Another possibility is to rescale the original non-commutativity in such a way
that the measure becomes trivial, see [16] for more details.

The Euler-Lagrange equations following from the action (24),

${\cal E}^{b}_{EL}:=\frac{\delta S_{g}}{\delta A_{b}}=0\,,$ (25)

by construction are gauge covariant, in a sense that the solutions are mapped
onto the solutions under the gauge transformations. The subscript $EL$ stands
for the “Euler-Lagrange”. Without going into the tedious calculations which
can be found in [12] we write here,

$\displaystyle{\cal E}^{b}_{EL}=\rho_{c}^{b}\left[\mu\,{\cal D}_{a}\left({\cal
F}^{ac}\right)+\frac{\mu}{2}\,{\cal F}^{cb}\,\Lambda_{b}{}^{de}\,{\cal
F}_{de}-\mu\,{\cal F}^{db}\,\Lambda_{b}{}^{ce}\,{\cal
F}_{de}+\left(\rho^{-1}\right)^{c}_{k}{\cal
F}^{ab}\partial_{i}\left(\mu\,\rho_{a}^{l}\,\rho_{b}^{k}\,\gamma^{i}_{l}\right)\right]\,,$
(26)

where,

$\displaystyle\Lambda_{b}{}^{de}(A)=\left(\rho^{-1}\right)_{j}^{d}\left(\partial^{j}_{A}\rho_{b}^{m}(A)-\partial^{m}_{A}\rho_{b}^{j}(A)\right)\left(\rho^{-1}\right)_{m}^{e}\,.$
(27)

We stress that the whole theory is completely determined by the two matrices,
the matrix $\gamma^{l}_{a}(A)$ which should obey by the equation (10) and the
matrix $\rho_{a}^{i}(A)$ satisfying the equation (15).

It is worth mentioning here that for some specific choices of the non-
commutativity parameter the equations of motion constructed according to (21)
and (26) are equivalent. In particular, for the $su(2)$-like Poisson
structure, $\Theta^{ab}(x)=2\,\alpha\,\varepsilon^{ab}{}_{c}\,x^{c}$, the
integration measure is constant $\mu(x)=1$ and the additional terms in (26)
vanish in such a way that,222In this particular situation the matrix
$\rho_{a}^{b}(A)$ plays the role of the Lagrangian multiplier in the sense of
non-Lagrangian systems [17]. The original equations of motion, ${\cal
E}^{a}_{S}=0$, are non-Lagrangian, however the multiplication by the non-
degenerate matrix $\rho_{a}^{b}(A)$ transforms them into a equivalent set of
Euler-Lagrange equations (25) for the action (24).

${\cal E}^{b}_{EL}=\rho_{a}^{b}\,{\cal E}^{a}_{S}\,.$ (28)

On the other hand, for the $\kappa$-Minkowski non-commutativity the
integration measure $\mu(x)$ is non-trivial [16], so the relation (28) does
not hold. In the Section 4 we will discuss these two examples in more details.

##  3 Relation to L∞ algebras

In this section we discuss the relation of the described above field
theoretical model with the L∞ algebras. We will construct an explicit form of
the L${}_{\infty}^{full}$ algebra which governs both kinematical and dynamical
sectors of the theory.

###  3.1 Basic definitions and identities

The definition of the L∞ algebra can be resumed as follows. An L∞-algebra is a
$\mathbb{Z}$-graded vector space $V=\bigoplus_{k\in\mathbb{Z}}\,V_{k}$
together with a sequence of graded antisymmetric multilinear maps,

$\displaystyle\ell_{n}:V^{\otimes n}\to V\ ,\quad v_{1}\wedge\dots\wedge
v_{n}\longmapsto\ell_{n}(v_{1},\dots,v_{n})\,,$

called $\ell$-brackets. The property of graded antisymmetry menas

$\ell_{n}(\dots,v,v^{\prime},\dots)=-(-1)^{|v|\,|v^{\prime}|}\,\ell_{n}(\dots,v^{\prime},v,\dots)\
,$ (29)

where we denote the degree of a homogeneous element $v\in V$ by $|v|$. The
$\ell$-bracket is a map of degree $|\ell_{n}|=2-n$, implying that

$\big{|}\ell_{n}(v_{1},\dots,v_{n})\big{|}=2-n+\sum_{j=1}^{n}\,|v_{j}|\ .$

The brackets $\ell_{n}$ should satisfy an (infinite) tower of identities
${\cal J}_{n}(v_{1},\dots,v_{n})=0$ for each $n\geq 1$, called homotopy
relations, with

$\displaystyle{\cal J}_{n}(v_{1},\dots,v_{n})$ $\displaystyle:=$
$\displaystyle\sum^{n}_{i=1}\,(-1)^{i\,(n-i)}\ \sum_{\sigma\in{\rm
Sh}_{i,n-i}}\,\chi(\sigma;v_{1},\dots,v_{n})$
$\displaystyle\qquad\times\ell_{n+1-i}\big{(}\ell_{i}(v_{\sigma(1)},\dots,v_{\sigma(i)}),v_{\sigma(i+1)},\dots,v_{\sigma(n)}\big{)}\
,$

where, for each $i=1,\dots,n$, the second sum runs over $(i,n-i)$-shuffled
permutations $\sigma\in S_{n}$ of degree $n$ which are restricted as

$\sigma(1)<\dots<\sigma(i)\qquad\mbox{and}\qquad\sigma(i+1)<\dots<\sigma(n)\
.$

The Koszul sign $\chi(\sigma;v_{1},\dots,v_{n})=\pm\,1$ is determined from the
grading by

$\displaystyle v_{\sigma(1)}\wedge\cdots\wedge
v_{\sigma(n)}=\chi(\sigma;v_{1},\dots,v_{n})\ v_{1}\wedge\cdots\wedge v_{n}\
.$

In particular, the first relation ${\cal J}_{1}(v):=\ell_{1}(\ell_{1}(v))=0$
signifies that $\ell_{1}$ is a differential making $(V,\ell_{1})$ into a
cochain complex, while the second identity,

${\cal
J}_{2}(v_{1},v_{2}):=\ell_{1}\big{(}\ell_{2}(v_{1},v_{2})\big{)}-\ell_{2}\big{(}\ell_{1}(v_{1}),v_{2}\big{)}-(-1)^{|v_{1}|}\,\ell_{2}\big{(}v_{1},\ell_{1}(v_{2})\big{)}\,,$

implies that $\ell_{1}$ is a (graded) derivation of the bracket $\ell_{2}$,
i.e. $\ell_{2}$ is a cochain map. The identity with three entries
schematically written as ${\cal
J}_{3}:=\ell_{1}\ell_{3}+\ell_{2}\ell_{2}+\ell_{3}\ell_{1}=0$ means that the
bracket $\ell_{2}$ obeys the Jacobi identity up to exact in $\ell_{1}$ terms,
i.e. $\ell_{2}$ induces a (graded) Lie bracket on the cohomology of
$\ell_{1}$.

In this work we are dealing with the structure of a three-term L∞-algebra
describing the cochain complex,

$\displaystyle V_{0}\xrightarrow{\ \ \ell_{1}\ \ }V_{1}\xrightarrow{\ \
\ell_{1}\ \ }V_{2}$ (31)

corresponding to the underlying graded vector space

$V=V_{0}\oplus V_{1}\oplus V_{2}\ .$

Physically, $V_{0}$ corresponds to the space of gauge parameters or the zero-
forms $f$. The subspace $V_{1}=\\{A\\}\cup\\{\varphi\\}$ contains the gauge
fields represented by the $1$-forms, $A=A_{a}{\mathsf{d}}x^{a}$, and the
matter fields $\varphi$ which are $0$-forms. And finally,
$V_{2}=\\{E^{A}\\}\cup\\{E^{\varphi}\\}\cup\\{{\cal E}_{EL}\\}$ is the space
of covariant objects. The subspace of $2$-forms $\\{E^{A}\\}$ contains the
field strength, ${\cal F}(A)={\cal
F}_{ab}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}$, while the subspace of
$1$-forms $\\{E^{\varphi}\\}$ contains covariant derivatives of the matter
fields, ${\cal D}(\varphi)={\cal D}(\varphi)_{a}\,{\mathsf{d}}x^{a}$ and
$\\{{\cal E}_{EL}\\}$ is the subspace of the equations of motion.

In what follows we will need the explicit form of some identities which is
given by the following Proposition.

###### Proposition 32

The L∞ relations, ${\cal J}_{n}(f,g,A^{\otimes n-2})=0$, ${\cal
J}_{n}(f,A^{\otimes n-1})=0$, and ${\cal J}_{n}(f,\varphi,A^{\otimes n-2})=0$
are given explicitly by,

$\displaystyle{\cal J}_{n}(f,g,A^{\otimes n-2})$ $\displaystyle=$
$\displaystyle\ell_{1}\left(\ell_{n}(f,g,A^{\otimes
n-2})\right)-(-1)^{n}\ell_{n}(\ell_{1}(f),g,A^{\otimes n-2})$
$\displaystyle-(-1)^{n}\ell_{n}(f,\ell_{1}(g),A^{\otimes
n-2})-(-1)^{n}(n-2)\,\ell_{n}\left(f,g,\ell_{1}(A),A^{\otimes n-3}\right)$
$\displaystyle+\sum_{i=2}^{n-1}(-1)^{i(n-i)}\left[\binom{n-2}{i-2}\,\ell_{n-i+1}\left(\ell_{i}(f,g,A^{\otimes
i-2}),A^{\otimes n-i}\right)\right.$
$\displaystyle-(-1)^{i}\binom{n-2}{i-1}\,\ell_{n-i+1}\left(\ell_{i}(f,A^{\otimes
i-1}),g,A^{\otimes n-i-1}\right)$
$\displaystyle+\left.(-1)^{i}\binom{n-2}{i-1}\,\ell_{n-i+1}\left(\ell_{i}(g,A^{\otimes
i-1}),f,A^{\otimes n-i-1}\right)\right]\,,$ $\displaystyle{\cal
J}_{n}(f,A^{\otimes n-1})$ $\displaystyle=$
$\displaystyle\ell_{1}\left(\ell_{n}(f,A^{\otimes
n-1})\right)-(-1)^{n}\ell_{n}(\ell_{1}(f),A^{\otimes n-1})$
$\displaystyle-(-1)^{n}(n-1)\,\ell_{n}(f,\ell_{1}(A),A^{\otimes n-1})$
$\displaystyle+\sum_{i=2}^{n-1}(-1)^{i(n-i)}\left[\binom{n-1}{i-1}\,\ell_{n-i+1}\left(\ell_{i}(f,A^{\otimes
i-1}),A^{\otimes n-i}\right)\right.$
$\displaystyle+\left.(-1)^{i}\binom{n-1}{i}\,\ell_{n-i+1}\left(\ell_{i}(A^{\otimes
i}),f,A^{\otimes n-i-1}\right)\right]\,,$

and

$\displaystyle{\cal J}_{n}(f,\varphi,A^{\otimes n-2})$ $\displaystyle=$
$\displaystyle\ell_{1}\left(\ell_{n}(f,\varphi,A^{\otimes
n-2})\right)-(-1)^{n}\ell_{n}(\ell_{1}(f),\varphi,A^{\otimes n-2})$
$\displaystyle-(-1)^{n}\ell_{n}(f,\ell_{1}(\varphi),A^{\otimes
n-2})-(-1)^{n}(n-2)\,\ell_{n}\left(f,\varphi,\ell_{1}(A),A^{\otimes
n-3}\right)$
$\displaystyle+\sum_{i=2}^{n-1}(-1)^{i(n-i)}\left[\binom{n-2}{i-2}\,\ell_{n-i+1}\left(\ell_{i}(f,\varphi,A^{\otimes
i-2}),A^{\otimes n-i}\right)\right.$
$\displaystyle+\binom{n-2}{i-1}\,\ell_{n-i+1}\left(\ell_{i}(f,A^{\otimes
i-1}),\varphi,A^{\otimes n-i-1}\right)$
$\displaystyle+\left.(-1)^{i}\binom{n-2}{i-1}\,\ell_{n-i+1}\left(\ell_{i}(\varphi,A^{\otimes
i-1}),f,A^{\otimes n-i-1}\right)\right]\,.$

Proof. Let us first discuss the relations with two gauge parameters $f$ and
$g$ and $(n-2)$ gauge fields $A$. The sum in (3.1) is taken over inequivalent
splittings of elements. For fixed $n$ and $i$ there are
$\binom{n}{i}=n!/(i!(n-i)!)$ splittings of elements, so the sum
$\sum_{\sigma\in{\rm Sh}_{i,n-i}}$ contains $\binom{n}{i}$ elements, see [5]
for more details. If $i=n$ the contribution to (3.1) is
$\ell_{1}(\ell_{n}(f,g,A^{\otimes n-2}))$, and when $i=1$ the conributions are
$\ell_{n}(\ell_{1}(f),g,A^{\otimes n-2})$, $\ell_{n}(f,\ell_{1}(g)A^{\otimes
n-2})$ and also $n-2$ terms of the type
$\ell_{n}\left(f,g,\ell_{1}(A),A^{\otimes n-3}\right)$. In the case $i\neq n$
and $i\neq 1$, taking into account that we have $n-2$ identical entries, the
sum $\sum_{\sigma\in{\rm Sh}_{i,n-i}}$ will contain $\binom{n-2}{i-2}$
contributions of the type $\ell_{n-i+1}(\ell_{i}(f,g,A^{\otimes
i-2}),A^{\otimes n-i})$, $\binom{n-2}{i-1}$ contributions of the type
$\ell_{n-i+1}\left(\ell_{i}(f,A^{\otimes i-1}),g,A^{\otimes n-i-1}\right)$,
the same number of the terms $\ell_{n-i+1}\left(\ell_{i}(g,A^{\otimes
i-1}),f,A^{\otimes n-i-1}\right)$, and also $\binom{n-2}{i}$ elements like
$\ell_{n-i+1}(\ell_{i}(A^{\otimes i}),f,g,A^{\otimes n-i-2})$. The latter
however can be set to zero since we are working with the off-shell closure
condition for the gauge algebra, see the comment after the eq. (37). All
together,

$\binom{n-2}{i-2}+\binom{n-2}{i-1}+\binom{n-2}{i-1}+\binom{n-2}{i}=\binom{n}{i}\,,$

which is exactly what we need. The sign $(-1)^{i-1}$ in the fourth and the
fifth lines appeared because the element $g$ which originally was staying in
the second slot was moved to the $(i+1)$-th slot.

The expression for the identities with one gauge parameter ${\cal
J}_{n}(f,A^{\otimes n-1})$ can be obtained in the same manner. For $i=n$ the
contribution to the sum (3.1) is $\ell_{1}(\ell_{n}(f,g,A^{\otimes n-2}))$,
when $i=1$ the corresponding terms are $\ell_{n}(\ell_{1}(f),A^{\otimes n-1})$
and $(n-1)\ell_{n}(f,\ell_{1}(A),A^{\otimes n-2})$ multiplied by the sign
factor. In case if $i\neq n$ and $i\neq 1$, the fact that there are $n-1$
identical elements, implies that the sum $\sum_{\sigma\in{\rm Sh}_{i,n-i}}$
will contain $\binom{n-1}{i-1}$ contributions of the type
$\ell_{n-i+1}(\ell_{i}(f,A^{\otimes i-1}),A^{\otimes n-i})$ and also
$\binom{n-1}{i}$ elements $\ell_{n-i+1}(\ell_{i}(A^{\otimes i}),f,A^{\otimes
n-i-1})$. All together it gives,
$\binom{n-1}{i-1}+\binom{n-1}{i}=\binom{n}{i}\,,$ as expected. The sign
$(-1)^{i}$ in the last line appeared because the element $f$ which originally
was staying in the first slot appears now in the $(i+1)$-th slot.

The derivation of the expression for ${\cal J}_{n}(f,\varphi,A^{\otimes n-2})$
is absolutely analogous to ${\cal J}_{n}(f,g,A^{\otimes n-2})$ with only
difference that the element $\varphi$ has degree one and therefore its
permutations with the elements $A$ do not create a sign factor. $\Box$

###  3.2 L∞ description of Poisson gauge theory

Now we may proceed to the definition of the corresponding $\ell$-brackets
comparing the constructions of the Poisson gauge theory with the corresponding
expressions in the L∞-formalism. We start with the gauge variations of the
gauge fields $A$ which are defined in the L∞ framework as,

$\displaystyle\delta_{f}A$ $\displaystyle:=$
$\displaystyle\sum^{\infty}_{n=0}{1\over n!}(-1)^{n(n-1)\over
2}\,\ell_{n+1}(f,A,\dots,A)$ (36) $\displaystyle=$
$\displaystyle{\mathsf{d}}f+\ell_{2}(f,A)-\frac{1}{2}\ell_{3}(f,A,A)+\dots\,.$

The homotopy relations with two gauge parameters, ${\cal
J}_{n+2}(f,g,A^{\otimes n})=0$, imply the gauge closure condition [3, 5],

$\displaystyle[\delta_{f},\delta_{g}]A=\delta_{[\\![f,g]\\!]_{A}}A+\delta^{T}_{C(f,g,E,A)}A\
,$ (37)

where,

$\displaystyle[\\![f,g]\\!]_{A}:=-\sum_{n=0}^{\infty}\,\frac{1}{n!}\,(-1)^{n(n-1)\over
2}\,\ell_{n+2}\left(f,g,A^{\otimes
n}\right)=-\ell_{2}(f,g)-\ell_{3}(f,g,A)+\dots\,,$

and the object $C(f,g,E,A)$ is constructed from the brackets of the type
$\ell_{n+3}(f,g,E,A^{\otimes n})$ with $E=0$ being the equations of motion.
The term $\delta^{T}_{C(f,g,E,A)}A$ accounts for the on-shell gauge closure.
In our case we have the off-shell closure, so we just set,
$\ell_{n+3}(f,g,E,A^{\otimes n})=0$, as it was already mentioned in the proof
of Proposition 32. The homotopy relations with three gauge parameters, ${\cal
J}_{n+3}(f,g,h,A^{\otimes n})=0$, imply the Jacobi identity for the
corresponding gauge algebra,
$[\delta_{h},[\delta_{f},\delta_{g}]]+\mbox{cycl.}(f,g,h)=0$.

The Poisson gauge transformations closing tha algebra (8) are determined
according to (12),

$\delta_{f}A_{a}=\gamma^{k}_{a}(A)\,\partial_{k}f+\\{A_{a},f\\}\,,$ (38)

where the matrix

$\gamma_{a}^{k}(A)=\sum_{n=0}^{\infty}\gamma_{a}^{k(n)}\,,\qquad\gamma_{a}^{k(0)}=\delta_{a}^{k}\,,\qquad\gamma_{a}^{k(n)}=\gamma_{a}^{k|j_{1}\dots
j_{n}}(x)\,A_{j_{1}}\dots A_{j_{n}}\,,$ (39)

should satisfy the equation (10). Comparing (36) to (38) and (37) to (8) we
conclude that the only non-vanishing brackets coming from this sector read
[11],

$\displaystyle\ell_{1}(f)$ $\displaystyle=$
$\displaystyle{\mathsf{d}}f\,,\qquad\ell_{2}(f,g)=-\\{f,g\\}\,,$ (40)
$\displaystyle\ell_{2}(f,A)$ $\displaystyle=$
$\displaystyle\left(\gamma^{k(1)}_{i}\,\partial_{k}f+\\{A_{i},f\\}\right){\mathsf{d}}x^{i}\,,$
$\displaystyle\ell_{m}(f,A^{\otimes m-1})$ $\displaystyle=$
$\displaystyle(m-1)!(-1)^{\frac{(m-1)\,(m-2)}{2}}\,\gamma_{i}^{k(m-1)}\partial_{k}f\,{\mathsf{d}}x^{i}\,,\qquad
m\geq 3\,.$

Since the tensor $\gamma_{a}^{k|j_{1}\dots j_{n}}(x)$ by construction is
symmetric in the indices $j_{1},\dots,j_{n}$ the off-diagonal elements read,

$\ell_{m}(f,A^{(1)},\dots,A^{(m-1)})=(m-1)!(-1)^{\frac{(m-1)\,(m-2)}{2}}\,\gamma_{i}^{k|j_{1}\dots
j_{m-1}}A^{(1)}_{j_{1}}\dots
A^{(m-1)}_{j_{m-1}}\partial_{k}f\,{\mathsf{d}}x^{i}\,.$ (41)

To prove that these brackets indeed determine an L∞ algebra we have to show
that the corresponding homotopy relations ${\cal J}_{n+2}(f,g,A^{\otimes
n})=0$, and ${\cal J}_{n+3}(f,g,h,A^{\otimes n})=0$, hold true. Since the only
non-vanishing bracket with two gauge parameters is,
$\ell_{2}(f,g)=-\\{f,g\\}$, the only non-trivial L∞ relation with three gauge
parameters to be checked reads,

${\cal
J}_{3}(f,g,h):=\ell_{2}(\ell_{2}(f,g),h)+\ell_{2}(\ell_{2}(g,h),f)+\ell_{2}(\ell_{2}(h,f),g)=0\,.$

It holds true as a consequence of the Jacobi identity for the Poisson bracket.
The situation with the brackets $\ell_{m}(f,A^{\otimes m-1})$ and the
identities with two gauge parameters is more complicated, we will prove it
carefully in the Proposition 57.

The gauge variation of the matter field $\varphi$ can be encoded in terms of
the brackets $\ell_{n+2}(f,\varphi,A^{\otimes n})$ as,

$\displaystyle\delta_{f}\varphi$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}{1\over n!}(-1)^{n(n+1)\over
2}\,\ell_{n+2}(f,\varphi,A,\dots,A)$ $\displaystyle=$
$\displaystyle\ell_{2}(f,\phi)-\ell_{3}(f,\varphi,A)-{1\over
2}\ell_{4}(f,\varphi,A,A)+\dots\,.$

Then the homotopy relations, ${\cal J}_{n+3}(f,g,\varphi,A^{\otimes n})=0$,
imply the gauge closure condition,

$[\delta_{f},\delta_{g}]\varphi=\delta_{[\\![f,g]\\!]_{A}}\varphi\,.$

In our case, $\delta_{f}\varphi=\\{\varphi,f\\}\,.$ So, the additional non-
vanishing bracket is, $\ell_{2}(f,\varphi)=-\\{f,\varphi\\}$. Again the only
non-trivial L∞ relation with two gauge parameters and one matter field, ${\cal
J}_{3}(f,g,\varphi)=0$, is satisfied because of the Jacobi identity for the
Poisson bracket.

From now on we take into account the brackets which belong to the subspace of
the covariant objects $V_{2}$. If we define the field strength as the
following combination of $\ell$-brackets,

$\displaystyle{\cal F}(A)=\sum^{\infty}_{n=1}\;{1\over
n!}(-1)^{\frac{n(n-1)}{2}}\,\ell_{n}(A^{\otimes
n})={\mathsf{d}}A-\frac{1}{2}\,\ell_{2}(A,A)+\dots\,,$ (43)

then the L∞ relations with one gauge parameter, ${\cal J}_{n+1}(f,A^{\otimes
n})=0$, imply the gauge covariance of such a determined object,

$\displaystyle\delta_{f}\mathcal{F}=\sum_{n=0}^{\infty}\,\frac{1}{n!}\,(-1)^{n(n-1)\over
2}\,\ell_{n+2}(f,\mathcal{F},A,\dots,A)=\ell_{2}(f,{\cal F})+\ell_{3}(f,{\cal
F},A)+\dots\ .$ (44)

To obtain the expression for the corresponding $\ell$-brackets one has to
compare these expressions to the corresponding formulas for the Poisson field
strength (18) and the gauge covariance condition. The Poisson field strength
(18) can be expressed as,

${\cal
F}=\frac{1}{2}\left(P_{ab}{}^{cd}\left(A\right)\,\partial_{c}A_{d}+R_{ab}{}^{cd}\left(A\right)\,\left\\{A_{c},A_{d}\right\\}\right){\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,,$
(45)

where,

$P_{ab}{}^{cd}=2\,\gamma^{c}_{l}R_{ab}{}^{ld}\,,\qquad\mbox{and}\qquad
R_{ab}{}^{cd}=\frac{1}{2}\left(\rho^{c}_{a}\rho^{d}_{b}-\rho^{c}_{b}\rho^{d}_{a}\right)\,.$
(46)

The definition (46) and the equation (15) imply that the tensors
$P_{ab}{}^{cd}$ and $R_{ab}{}^{cd}$ satisfy the equations [12],

$\gamma^{k}_{l}\,\partial^{l}_{A}P_{ab}{}^{cd}+\Theta^{kl}\,\partial_{l}P_{ab}{}^{cd}+P_{ab}{}^{cl}\,\partial^{d}_{A}\gamma^{k}_{l}+P_{ab}{}^{ld}\,\partial_{l}\Theta^{ck}+2\,R_{ab}{}^{ld}\,\partial_{m}\gamma^{k}_{l}\,\Theta^{mc}=0\,,$
(47)

and

$\gamma^{k}_{l}\,\partial^{l}_{A}R_{ab}{}^{cd}+\Theta^{kl}\,\partial_{l}R_{ab}{}^{cd}+R_{ab}{}^{cl}\,\partial^{d}_{A}\gamma^{k}_{l}+R_{ab}{}^{ld}\,\partial^{c}_{A}\gamma^{k}_{l}=0\,,$
(48)

correspondingly. We may write,

$\displaystyle P_{ab}{}^{cd}(A)=\sum_{n=0}^{\infty}P_{ab}{}^{cd(n)},\qquad
P_{ab}{}^{cd(n)}=P_{ab}{}^{cd|i_{1}\dots i_{n}}(x)\,A_{i_{1}}\dots
A_{i_{n}}\,,$ (49) $\displaystyle
R_{ab}{}^{cd}(A)=\sum_{n=0}^{\infty}R_{ab}{}^{cd(n)}\,,\qquad
R_{ab}{}^{cd(n)}=R_{ab}{}^{cd|i_{1}\dots i_{n}}(x)\,A_{i_{1}}\dots
A_{i_{n}}\,.$

Comparing (43) to (45) and (44) to (19) one writes the following set of non-
vanishing brackets,

$\displaystyle\ell_{1}(A)$ $\displaystyle=$
$\displaystyle{\mathsf{d}}A\,,\qquad\ell_{2}(f,E^{A})=-\\{f,E^{A}\\}\,,$ (50)
$\displaystyle\ell_{n}\left(A^{\otimes n}\right)$ $\displaystyle=$
$\displaystyle\frac{n!}{2}\,(-1)^{\frac{n(n-1)}{2}}\left(P_{ab}{}^{cd(n-1)}\,\partial_{c}A_{d}+R_{ab}{}^{cd(n-2)}\left\\{A_{c},A_{d}\right\\}\right){\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,,\,\,\,n>1\,.$

The off-diagonal elements can be written as,

$\displaystyle\ell_{n}\left(A^{(1)},\dots,A^{(n)}\right)$ $\displaystyle=$
$\displaystyle\frac{(n-1)!}{2}\,(-1)^{\frac{n(n-1)}{2}}\left[\sum_{\sigma\in{\rm
Sh}_{1,n-1}}\partial_{c}A^{(\sigma(1))}_{d}\,P_{ab}{}^{cd|i_{2},\dots,i_{n}}A^{(\sigma(2))}_{i_{2}},\dots,A^{(\sigma(n))}_{i_{n}}\right.$
$\displaystyle\left.+\frac{2}{n-1}\sum_{\sigma\in{\rm
Sh}_{2,n-2}}\left\\{A^{(\sigma(1))}_{c},A^{(\sigma(2))}_{d}\right\\}R_{ab}{}^{cd|i_{3},\dots,i_{n}}A^{(\sigma(3))}_{i_{3}},\dots,A^{(\sigma(n))}_{i_{n}}\right]{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.$

Apart from the homotopy relations, ${\cal J}_{n+1}(f,A^{\otimes n})=0$, which
will be checked in the Proposition 57, the brackets
$\ell_{2}(f,E^{A})=-\\{f,E^{A}\\}$ also enters the L∞ relation with two gauge
parameters and one element $E^{A}\in V_{2}$, that is, ${\cal
J}_{3}(f,g,E^{A})=0$. The latter however are trivially satisfied because of
the Jacobi identity for the Poisson bracket, just like it happened with,
${\cal J}_{3}(f,g,h)=0$.

One more ingredient of the Poisson gauge theory that we need to express in
terms of $\ell$-brackets in the L∞ formalism is the gauge covariant derivative
of matter field (14) which we write as,

${\cal
D}(\varphi)=\left(\pi_{a}^{i}\left(A\right)\partial_{i}\varphi+\rho_{a}^{i}\left(A\right)\left\\{A_{i},\varphi\right\\}\right){\mathsf{d}}x^{a}\,,$
(52)

where, $\pi_{a}^{i}=\rho_{a}^{l}\gamma_{l}^{i}$, and the coefficient function,

$\rho_{a}^{i}(A)=\sum_{n=0}^{\infty}\rho_{a}^{i(n)}\,,\qquad\rho_{a}^{i(0)}=\delta_{a}^{i}\,,\qquad\rho_{a}^{i(n)}=\rho_{a}^{i|j_{1}\dots
j_{n}}(x)\,A_{j_{1}}\dots A_{j_{n}}\,,$ (53)

should satisfy the equation (15). In the L∞ approach one may introduce the
covariant derivative as a combination of the brackets involving the matter
field and the gauge fields $\ell_{n+1}(\varphi,A^{\otimes n})$ as,

$\displaystyle{\cal D}(\varphi)=\sum_{n=0}{1\over n!}(-1)^{n(n+1)\over
2}\,\ell_{n+1}(\varphi,A^{\otimes
n})={\mathsf{d}}\varphi+\ell_{2}(\varphi,A)+\dots\,.$ (54)

Provided that the following brackets vanish,
$\ell_{n+3}(f,\varphi,E_{A},A^{\otimes n})=0$, which can be assumed here
without any contradiction, the homotopy relations, ${\cal
J}_{n+2}(f,\varphi,A^{\otimes n})=0$, imply the gauge covariance of the object
given by (54), i.e.,

$\displaystyle\delta_{f}{\cal D}(\varphi)=\sum_{n=0}^{\infty}{1\over
n!}(-1)^{n(n+1)\over 2}\,\ell_{n+2}(f,{\cal D}(\varphi),A^{\otimes
n})=\ell_{2}(f,{\cal D}(\varphi))-\ell_{3}(f,{\cal D}(\varphi),A)+\dots\,.$
(55)

Comparing (52) to (54) and (14) to (55) we find an additional set of non-
vanishing brackets,

$\displaystyle\ell_{1}(\varphi)$ $\displaystyle=$
$\displaystyle{\mathsf{d}}\varphi\,,\qquad\ell_{2}(f,E^{\varphi})=-\\{f,E^{\varphi}\\}\,,$
(56) $\displaystyle\ell_{n+1}\left(\varphi,A^{\otimes n}\right)$
$\displaystyle=$ $\displaystyle
n!\,(-1)^{\frac{n(n+1)}{2}}\left(\pi_{a}^{i(n)}\,\partial_{i}\varphi+\rho_{a}^{i(n-1)}\,\left\\{A_{i},\varphi\right\\}\right){\mathsf{d}}x^{a}\,,\,\,\,n>1\,.$

Just like before, the homotopy relation with two gauge parameters and one
element $E^{\varphi}$ from the subspace $V_{2}$, ${\cal
J}_{3}(f,g,E^{\varphi})=0$, is satisfied because of the Jacobi identity for
the Poisson bracket. The rest is to prove that the L∞ relations, ${\cal
J}_{n+2}(f,\varphi,A^{\otimes n})=0$, hold true.

###### Proposition 57

Let the equations (10) on the matrix function,
$\gamma_{k}^{l}(A)=\sum_{n=0}^{\infty}\gamma_{k}^{l(n)},$ and (15) on,
$\rho_{a}^{i}(A)=\sum_{n=0}^{\infty}\rho_{a}^{i(n)},$ are satisfied. Then
there exist a three term L∞ structure denoted by L${}_{\infty}^{full}$ which
induces the Poisson gauge theory [12]. It is concentrated on $V_{0}\oplus
V_{1}\oplus V_{2}$ with, $V_{0}$, being the space of gauge parameters $f,g\in
C^{\infty}(M)$, the component, $V_{1}=C^{\infty}(M)\oplus\Omega^{1}(M)$, is
the space of matter fields $\varphi\in C^{\infty}(M)$ and gauge fields
$A\in\Omega^{1}(M)$ and $V_{2}=\Omega^{1}(M)\oplus\Omega^{2}(M)$ is the space
of the covariant objects $E^{\varphi}\in\Omega^{1}(M)$ and
$E^{A}\in\Omega^{2}(M)$. The diagonal non-vanishing brackets read,

$\displaystyle\ell_{1}(v)$ $\displaystyle=$
$\displaystyle{\mathsf{d}}v\,,\qquad\mbox{for}\qquad v=f,A,\varphi\,,$ (58)
$\displaystyle\ell_{2}(f,u)$ $\displaystyle=$
$\displaystyle-\\{f,u\\}\,,\qquad\mbox{for}\qquad u=g,\varphi,E\,,$
$\displaystyle\ell_{2}(f,A)$ $\displaystyle=$
$\displaystyle\left(\gamma^{k(1)}_{a}\,\partial_{k}f+\\{A_{a},f\\}\right){\mathsf{d}}x^{a}\,,$
$\displaystyle\ell_{n+1}(f,A^{\otimes n})$ $\displaystyle=$ $\displaystyle
n!(-1)^{\frac{n\,(n-1)}{2}}\,\gamma_{a}^{k(n)}\partial_{k}f\,{\mathsf{d}}x^{a}\,,\qquad
n\geq 2\,.$ $\displaystyle\ell_{n}\left(A^{\otimes n}\right)$ $\displaystyle=$
$\displaystyle\frac{n!}{2}\,(-1)^{\frac{n(n-1)}{2}}\left(P_{ab}{}^{cd(n-1)}\,\partial_{c}A_{d}+R_{ab}{}^{cd(n-2)}\,\left\\{A_{c},A_{d}\right\\}\right){\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,,\,\,\,n>1\,.$
$\displaystyle\ell_{n+1}\left(\varphi,A^{\otimes n}\right)$ $\displaystyle=$
$\displaystyle
n!\,(-1)^{\frac{n(n+1)}{2}}\left(\pi_{a}^{i(n)}\,\partial_{i}\varphi+\rho_{a}^{i(n-1)}\,\left\\{A_{i},\varphi\right\\}\right){\mathsf{d}}x^{a}\,,\,\,\,n>1\,,$

where

$\displaystyle R_{ab}{}^{cd}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\rho^{c}_{a}\rho^{d}_{b}-\rho^{c}_{b}\rho^{d}_{a}\right)=\sum_{n=0}^{\infty}R_{ab}{}^{cd(n)}\,,$
(59) $\displaystyle P_{ab}{}^{cd}$ $\displaystyle=$ $\displaystyle
2\,\gamma^{c}_{l}R_{ab}{}^{ld}=\sum_{n=0}^{\infty}P_{ab}{}^{cd(n)}\,,\qquad\mbox{and}\qquad\pi_{a}^{i}=\rho_{a}^{l}\gamma_{l}^{i}=\sum_{n=0}^{\infty}\pi_{a}^{i(n)}\,.$

The graded antisymmetry is imposed by the definition (29).

Proof. To prove it we need to show that all possible homotopy relations
involving the above brackets are satisfied. The L∞ relations with one, two and
three entries can be checked explicitly, see [10] and also consideration above
in the text concerning, ${\cal J}_{3}(f,g,u)=0$. Taking into account the
consideration of the degree of the brackets the only non-trivial higher
relations to be checked are, ${\cal J}_{n}(f,g,A^{\otimes n-2})=0$, ${\cal
J}_{n}(f,A^{\otimes n-1})=0$, and ${\cal J}_{n}(f,\varphi,A^{\otimes n-2})=0$,
and were given explicitly by the Proposition 32.

We start with the L∞ identities ${\cal J}_{n+2}(f,g,A^{\otimes n})=0$333The
original proof can be found in [11], here we provide the proof for
completeness. Let us note that only the bracket $\ell_{2}(f,A)$ contains the
Poisson bracket while $\ell_{m+1}(f,A^{\otimes m})$, with $m>1$ do not. So, it
makes sense to write separately all terms in (32) which contain the brackets
$\ell_{2}$. We write separately contributions for $i=2$ and $i=n+1$. Taking
into account that in the Poisson case $\ell_{n+2}(f,g,A^{\otimes n})=0$ for
$n\geq 1$, we write the left hand side of ${\cal J}_{n+2}(f,g,A^{\otimes
n})=0$ as,

$\displaystyle(-1)^{n+1}\ell_{n+2}(\ell_{1}(f),g,A^{\otimes
n})+(-1)^{n+1}\ell_{n+2}(f,\ell_{1}(g),A^{\otimes
n})+\ell_{n+1}\left(\ell_{i}(f,g),A^{\otimes n}\right)$ (60)
$\displaystyle-n\,\ell_{n+1}\left(\ell_{2}(f,A),g,A^{\otimes
n-1}\right)+n\,\ell_{n+1}\left(\ell_{2}(g,A),f,A^{\otimes n-1}\right)+$
$\displaystyle-\ell_{2}\left(\ell_{n+1}(f,A^{\otimes
n}),g\right)+\ell_{2}\left(\ell_{n+1}(g,A^{\otimes n}),f\right)$
$\displaystyle+\sum_{i=3}^{n}(-1)^{i(n-i)+i-1}\mbox{$\binom{n}{i-1}$}\left[\ell_{n-i+3}\left(\ell_{i}(f,A^{\otimes
i-1}),g,A^{\otimes n-i+1}\right)-\ell_{n-i+3}\left(\ell_{i}(g,A^{\otimes
i-1}),f,A^{\otimes n-i+1}\right)\right]\,.$

We substitute (40) in (60) and after careful simplification one may see that
all terms containing the Poisson brackets like $\\{f,A_{m}\\}$ or
$\\{\partial_{m}f,g\\}$ will cancel each other. The rest of the terms can be
reorganized as (in particular we set $i=m+1$ in the sum),

$\displaystyle{\cal J}_{n+2}(f,g,A^{\otimes n})$ $\displaystyle=$
$\displaystyle
n!\,(-1)^{\frac{n(n-1)}{2}}\left[(n+1)\left(\gamma_{i}^{k|li_{1}\dots
i_{n}}-\gamma_{i}^{l|ki_{1}\dots i_{n}}\right)+\gamma_{j}^{l|i_{1}\dots
i_{n}}\,\gamma_{i}^{k|j}-\gamma_{j}^{k|i_{1}\dots
i_{n}}\,\gamma_{i}^{l|j}\right.+$
$\displaystyle-\Theta^{kj}\,\partial_{j}\gamma_{i}^{l|i_{1}\dots
i_{n}}+\Theta^{lj}\,\partial_{j}\gamma_{i}^{k|i_{1}\dots
i_{n}}-\gamma_{i}^{j|i_{1}\dots i_{n}}\,\partial_{j}\Theta^{lk}+$
$\displaystyle\sum_{m=1}^{n-1}\left.(n-m+1)\left(\gamma_{j}^{l|i_{1}\dots
i_{m}}\,\gamma_{i}^{k|ji_{m+1}\dots i_{n}}-\gamma_{j}^{k|i_{1}\dots
i_{m}}\,\gamma_{i}^{l|ji_{m+1}\dots
i_{n}}\right)\right]\partial_{k}f\,\partial_{l}g\,A_{i_{1}}\dots A_{i_{n}}\,.$

On the other hand, substituting the decomposition (39) in the equation (10)
one obtains,

$\displaystyle\gamma^{k|l}_{i}-\gamma^{l|k}_{i}-\partial_{i}\Theta^{lk}+$ (62)
$\displaystyle\left[2\left(\gamma_{i}^{k|li_{1}}-\gamma_{i}^{l|ki_{1}}\right)+\gamma_{j}^{l|i_{1}}\gamma_{i}^{k|j}-\gamma_{j}^{k|i_{1}}\gamma_{i}^{l|j}-\Theta^{kj}\,\partial_{j}\gamma_{i}^{l|i_{1}}+\Theta^{lj}\,\partial_{j}\gamma_{i}^{k|i_{1}}-\gamma_{i}^{j|i_{1}}\,\partial_{j}\Theta^{lk}\right]A_{i_{1}}+$
$\displaystyle\sum_{n=2}^{\infty}\left[(n+1)\left(\gamma_{i}^{k|li_{1}\dots
i_{n}}-\gamma_{i}^{l|ki_{1}\dots i_{n}}\right)+\gamma_{j}^{l|i_{1}\dots
i_{n}}\,\gamma_{i}^{k|j}-\gamma_{j}^{k|i_{1}\dots
i_{n}}\,\gamma_{i}^{l|j}\right.+$
$\displaystyle-\Theta^{kj}\,\partial_{j}\gamma_{i}^{l|i_{1}\dots
i_{n}}+\Theta^{lj}\,\partial_{j}\gamma_{i}^{k|i_{1}\dots
i_{n}}-\gamma_{i}^{j|i_{1}\dots i_{n}}\,\partial_{j}\Theta^{lk}+$
$\displaystyle\sum_{m=1}^{n-1}\left.(n-m+1)\left(\gamma_{j}^{l|i_{1}\dots
i_{m}}\,\gamma_{i}^{k|ji_{m+1}\dots i_{n}}-\gamma_{j}^{k|i_{1}\dots
i_{m}}\,\gamma_{i}^{l|ji_{m+1}\dots i_{n}}\right)\right]A_{i_{1}}\dots
A_{i_{n}}=0\,.$

The above relation should hold for any gauge field $A$ so we get separately,

$\displaystyle\gamma^{k|l}_{i}-\gamma^{l|k}_{i}-\partial_{i}\Theta^{lk}=0\,,$
$\displaystyle\left[2\left(\gamma_{i}^{k|li_{1}}-\gamma_{i}^{l|ki_{1}}\right)+\gamma_{j}^{l|i_{1}}\gamma_{i}^{k|j}-\gamma_{j}^{k|i_{1}}\gamma_{i}^{l|j}-\Theta^{kj}\,\partial_{j}\gamma_{i}^{l|i_{1}}+\Theta^{lj}\,\partial_{j}\gamma_{i}^{k|i_{1}}-\gamma_{i}^{j|i_{1}}\,\partial_{j}\Theta^{lk}\right]A_{i_{1}}=0\,,$
$\displaystyle\sum_{n=2}^{\infty}\left[(n+1)\left(\gamma_{i}^{k|li_{1}\dots
i_{n}}-\gamma_{i}^{l|ki_{1}\dots i_{n}}\right)+\gamma_{j}^{l|i_{1}\dots
i_{n}}\,\gamma_{i}^{k|j}-\gamma_{j}^{k|i_{1}\dots
i_{n}}\,\gamma_{i}^{l|j}\right.+$
$\displaystyle-\Theta^{kj}\,\partial_{j}\gamma_{i}^{l|i_{1}\dots
i_{n}}+\Theta^{lj}\,\partial_{j}\gamma_{i}^{k|i_{1}\dots
i_{n}}-\gamma_{i}^{j|i_{1}\dots i_{n}}\,\partial_{j}\Theta^{lk}+$
$\displaystyle\sum_{m=1}^{n-1}\left.(n-m+1)\left(\gamma_{j}^{l|i_{1}\dots
i_{m}}\,\gamma_{i}^{k|ji_{m+1}\dots i_{n}}-\gamma_{j}^{k|i_{1}\dots
i_{m}}\,\gamma_{i}^{l|ji_{m+1}\dots i_{n}}\right)\right]A_{i_{1}}\dots
A_{i_{n}}=0\,.$

Which implies that the right hand side of (3.2) vanishes. Thus, the fact that
the matrix function $\gamma_{k}^{l}(A)$ satisfies the equation (10) implies
that the brackets (40) indeed satisfy the L∞ relations (32).

We follow the same logic to prove that the $\ell$-brackets defined in (50)
satisfy the L∞ relations, ${\cal J}_{n+1}(f,A^{\otimes n})=0$, given by (32).
Since the only non-vanishing bracket containing an element $E_{A}\in V_{2}$ is
$\ell_{2}(f,E_{A})=\\{f,E_{A}\\}$, the only contribution from the fourth line
of (32) appears when $i=n-1$. So for $n>2$ we have,

$\displaystyle{\cal J}_{n+1}(f,A^{\otimes n})$ $\displaystyle=$
$\displaystyle(-1)^{n}\ell_{n+1}\left(\ell_{1}(f),A^{\otimes(n)}\right)+\ell_{2}\left(\ell_{n}\left(A^{\otimes(n)}\right),f\right)+n\,\ell_{n}\left(\ell_{2}(f,A),A^{(n-1)}\right)+$
$\displaystyle\sum_{i=3}^{n}(-1)^{i(n-i+1)}\mbox{$\binom{n}{i-1}$}\ell_{n-i+2}\left(\ell_{i}\left(f,A^{\otimes(i-1)}\right),A^{\otimes(n-i+1)}\right)+\ell_{1}\left(\ell_{n+1}\left(f,A^{\otimes(n)}\right)\right)\,.$

To check (32) we will need the expression for off-diagonal elements of the
bracket $\ell_{n}(A^{\otimes n})$ given by (3.2). In particular,

$\displaystyle\ell_{n}\left(B,A^{\otimes(n-1)}\right)$ $\displaystyle=$
$\displaystyle(n-1)!\,(-1)^{\frac{n(n-1)}{2}}\left((n-1)\,P_{ab}{}^{cd(n-1)}(B,A)\,\partial_{c}A_{d}+P_{ab}{}^{cd(n-1)}(A)\,\partial_{c}B_{d}\right.$
(64)
$\displaystyle\left.+(n-2)R_{ab}{}^{cd(n-2)}\left(B,A\right)\,\left\\{A_{c},A_{d}\right\\}+2\,R_{ab}{}^{cd(n-2)}\left(A\right)\,\left\\{B_{c},A_{d}\right\\}\right)\,,$

where we use the notation,

$P_{ab}{}^{cd(k)}(B,A)=P_{ab}{}^{cd|i_{1}\dots i_{k}}B_{i_{1}}A_{i_{2}}\dots
A_{i_{k}}\,.$ (65)

We use (64) in (3.2) with, $B=\ell_{k+1}(f,A^{\otimes k})$, given by (50). The
resulting expression is huge, but different simplifications can be made. In
particular, all terms containing double Poisson brackets, like
$R_{ab}{}^{cd(n-2)}\\{f,\\{A_{c},A_{d}\\}\\}$, vanish because of the Jacobi
identity for the Poisson bracket. Some other terms cancel each other. Such
that the resulting expression after the careful simplification becomes,

$\displaystyle{\cal J}_{n+1}(f,A^{\otimes n})$ $\displaystyle=$ $\displaystyle
n!\,(-1)^{\frac{n(n-1)}{2}}\left\\{{\cal
U}_{ab}{}^{cdk(n-1)}\partial_{c}A_{d}\,\partial_{k}f+\sum_{m=0}^{n}P_{ab}{}^{cd(m)}\,\gamma_{d}^{k(n-m)}\partial_{c}\partial_{k}f+\right.$
$\displaystyle\left.\left[P_{ab}{}^{cd(n-1)}-2\sum_{m=0}^{n-1}\gamma_{l}^{c(m)}\,R_{ab}{}^{ld(n-m-1)}\right]\\{A_{d},\partial_{c}f\\}+{\cal
V}_{ab}{}^{cdk(n-2)}\\{A_{c},A_{d}\\}\,\partial_{k}f\right\\}\,.$

where the coefficients ${\cal U}_{ab}{}^{cdk(n)}$ and ${\cal
V}_{ab}{}^{cdk(n)}$ are given by,

$\displaystyle{\cal U}_{ab}{}^{cdk(0)}$ $\displaystyle=$ $\displaystyle
P_{ab}{}^{cd|k}+\delta_{a}^{c}\,\gamma_{b}^{k|d}-\delta_{b}^{c}\,\gamma_{a}^{k|d}+\delta_{b}^{d}\,\partial_{a}\Theta^{ck}-\delta_{a}^{d}\,\partial_{b}\Theta^{ck}\,,$
(67) $\displaystyle{\cal U}_{ab}{}^{cdk(1)}$ $\displaystyle=$
$\displaystyle\left[2\,P_{ab}{}^{cd|ki_{1}}+\gamma_{l}^{k|i_{1}}P_{ab}{}^{cd|l}+\Theta^{kl}\partial_{l}P_{ab}{}^{cd|i_{1}}+2\,\delta_{a}^{c}\,\gamma_{b}^{k|di_{1}}-2\,\delta_{b}^{c}\,\gamma_{a}^{k|di_{1}}+\right.$
$\displaystyle\left.P_{ab}{}^{cl|i_{1}}\gamma_{l}^{k|d}+P_{ab}{}^{ld|i_{1}}\partial_{l}\Theta^{ck}+\delta_{b}^{d}\partial_{q}\gamma_{a}^{k|i_{1}}\Theta^{qc}-\delta_{a}^{d}\partial_{q}\gamma_{b}^{k|i_{1}}\Theta^{qc}\right]A_{i_{1}}\,,$
$\displaystyle{\cal U}_{ab}{}^{cdk(n)}$ $\displaystyle=$
$\displaystyle\left[(n+1)\,P_{ab}{}^{cd|ki_{1}\dots
i_{n}}+\gamma_{l}^{k|i_{1}\dots
i_{n}}P_{ab}{}^{cd|l}+\Theta^{kl}\partial_{l}P_{ab}{}^{cd|i_{1}\dots
i_{n}}+\right.$ $\displaystyle(n+1)\,\delta_{a}^{c}\,\gamma_{b}^{k|di_{1}\dots
i_{n}}-(n+1)\,\delta_{b}^{c}\,\gamma_{a}^{k|di_{1}\dots
i_{n}}+P_{ab}{}^{cl|i_{1}\dots i_{n}}\gamma_{l}^{k|d}+$ $\displaystyle
P_{ab}{}^{ld|i_{1}\dots
i_{n}}\partial_{l}\Theta^{ck}+\delta_{b}^{d}\partial_{q}\gamma_{a}^{k|i_{1}\dots
i_{n}}\Theta^{qc}-\delta_{a}^{d}\partial_{q}\gamma_{b}^{k|i_{1}\dots
i_{n}}\Theta^{qc}+$
$\displaystyle\sum_{m=1}^{n-1}\left((n-m+1)\,\gamma_{l}^{k|i_{1}\dots
i_{m}}R_{ab}{}^{cd|li_{m+1}\dots i_{n}}+(n-m+1)\,P_{ab}{}^{cl|i_{1}\dots
i_{m}}\gamma_{l}^{k|di_{m+1}\dots i_{n}}\right.$
$\displaystyle\left.\left.+2\,R_{ab}{}^{ld|i_{1}\dots
i_{m}}\partial_{q}\gamma_{l}^{k|i_{m+1}\dots
i_{n}}\Theta^{qc}\right)\right]A_{i_{1}}\dots A_{i_{n}}\,.$

and

$\displaystyle{\cal V}_{ab}{}^{cdk(0)}$ $\displaystyle=$ $\displaystyle
R_{ab}{}^{cd|k}+\frac{1}{2}\left(\delta_{a}^{c}\,\gamma_{b}^{k|d}-\delta_{b}^{c}\,\gamma_{a}^{k|d}+\delta_{b}^{d}\,\gamma_{a}^{k|c}-\delta_{a}^{d}\,\gamma_{b}^{k|c}\right)\,,$
(68) $\displaystyle{\cal V}_{ab}{}^{cdk(1)}$ $\displaystyle=$
$\displaystyle\left[2\,R_{ab}{}^{cd|ki_{1}}+\gamma_{l}^{k|i_{1}}\,R_{ab}{}^{cd|l}+\Theta^{kl}\,\partial_{l}R_{ab}{}^{cd|i_{1}}+\right.$
$\displaystyle\left.\delta_{a}^{c}\,\gamma_{b}^{k|di_{1}}-\delta_{b}^{c}\,\gamma_{a}^{k|di_{1}}+\delta_{b}^{d}\,\gamma_{a}^{k|ci_{1}}-\delta_{a}^{d}\,\gamma_{b}^{k|ci_{1}}\right]A_{i_{1}}\,,$
$\displaystyle{\cal V}_{ab}{}^{cdk(n)}$ $\displaystyle=$
$\displaystyle\left[(n+1)\,R_{ab}{}^{cd|ki_{1}\dots
i_{n}}+\gamma_{l}^{k|i_{1}\dots
i_{n}}\,R_{ab}{}^{cd|l}+\Theta^{kl}\,\partial_{l}R_{ab}{}^{cd|i_{1}\dots
i_{n}}+\right.$
$\displaystyle\frac{n+1}{2}\left(\delta_{a}^{c}\,\gamma_{b}^{k|di_{1}\dots
i_{n}}-\delta_{b}^{c}\,\gamma_{a}^{k|di_{1}\dots
i_{n}}+\delta_{b}^{d}\,\gamma_{a}^{k|ci_{1}\dots
i_{n}}-\delta_{a}^{d}\,\gamma_{b}^{k|ci_{1}\dots i_{n}}\right)+$
$\displaystyle R_{ab}{}^{cl|i_{1}\dots
i_{n}}\,\gamma^{k|d}_{l}+R_{ab}{}^{ld|i_{1}\dots i_{n}}\,\gamma^{k|c}_{l}+$
$\displaystyle\sum_{m=1}^{n-1}(n-m+1)\left(\gamma_{l}^{k|i_{1}\dots
i_{m}}R_{ab}{}^{cd|li_{m+1}\dots i_{n}}+R_{ab}{}^{cl|i_{1}\dots
i_{m}}\,\gamma^{k|di_{m+1}\dots i_{n}}_{l}\right.$
$\displaystyle\left.\left.+R_{ab}{}^{ld|i_{1}\dots
i_{m}}\,\gamma^{k|ci_{m+1}\dots i_{n}}_{l}\right)\right]A_{i_{1}}\dots
A_{i_{n}}\,.$

Now we observe that substituting the decompositions for $R_{ab}{}^{cd}(A)$ and
$\gamma_{k}^{l}(A)$ in the equation (48) one may represent it as,

$\sum_{n=0}^{\infty}{\cal V}_{ab}{}^{cdk(n)}=0\,.$ (69)

Just like it happened with the equation (10) in (62). Each term ${\cal
V}_{ab}{}^{cdk(n)}$ is of the order $n$ in the gauge fields $A_{a}$. Since
(69) should hold for any gauge field $A_{a}$, it implies that,

${\cal V}_{ab}{}^{cdk(n)}=0\,,\qquad n=0,1,\dots$ (70)

The term in the square brackets in the second line of (3.2) is nothing but the
first of the eqs. (46) written in components, therefore it also vanishes. The
second term in the first line of (3.2) vanishes as a consequence of (46) and
the antisymmetry of the coefficient function $R_{ab}{}^{cd}(A)$ in the upper
two indices. Finally, it was shown in [14] that the equations (48) and (10)
imply that the coefficient function $P_{ab}{}^{cd}(A)$ defined by (46)
satisfies the equation (47). The left hand side of (47) can be written as
$\sum_{n=0}^{\infty}{\cal U}_{ab}{}^{cdk(n)}$, from where one finds,

${\cal U}_{ab}{}^{cdk(n)}=0\,,\qquad n=0,1,\dots\,.$ (71)

Thus we conclude that right hand side of (3.2) vanishes and consequently the
L∞ relations, ${\cal J}_{n+1}(f,A^{\otimes n})=0$ hold true.

The fact that the $\ell$-brackets defined in (56) satisfy the relations,
${\cal J}_{n+1}(f,A^{\otimes n})=0$, given explicitly in (32), can be proved
in the absolutely the same way, using the decompositions of the equation (15)
and the equation on the matrix function,
$\pi_{a}^{i}=\rho_{a}^{l}\gamma_{l}^{i}$. $\Box$

Now let us turn to the Maxwell-Poisson equations of motion. Aiming to arrive
at the cyclic L∞ algebras we work with the equations (25) obtained from the
action (24). Following [12] one may check that,

$\displaystyle\delta S_{g}=\int dx^{n}\,\delta A_{a}\cdot{\cal
E}^{a}_{EL}(A)\,,$ (72)

where, ${\cal E}^{a}_{EL}(A)$ is given by (26). Comparing this formula with
the corresponding expression in the L∞ formalism [5],

$\delta S=\langle\delta A,E_{A}\rangle\,,$ (73)

we identify the inner product on the L∞ algebra as,

$\langle A,E\rangle=\int dx^{n}\,A_{a}\,E^{a}\,,$ (74)

and the corresponding field equations become,

$\displaystyle{\cal E}_{EL}(A)=\sum^{\infty}_{n=1}\;{1\over
n!}(-1)^{\frac{n(n-1)}{2}}\,\ell^{\cal E}_{n}(A^{\otimes n})=\ell_{1}^{\cal
E}(A)-\frac{1}{2}\,\ell^{\cal E}_{2}(A,A)+\dots\,.$ (75)

In order to obtain explicit expressions for the brackets $\ell^{\cal
E}_{n}(A^{\otimes n})$ we rewrite the Maxwell-Poisson equations as follows:

${\cal E}^{q}_{EL}(A)=-\frac{\mu}{2}\,\frac{\partial\mathcal{F}_{ab}}{\partial
A_{q}}\,\mathcal{F}^{ab}+\frac{1}{2}\,\partial_{r}\left(\mu\,\frac{\partial\mathcal{F}_{ab}}{\partial(\partial_{r}A_{q})}\,\mathcal{F}^{ab}\right).$
(76)

Using the expression (45) for the field strength, one obtains the expansions:

$\mathcal{F}_{ab}=\sum_{n=1}^{\infty}\mathcal{F}_{ab}^{(n)},\quad\quad\frac{\partial\mathcal{F}_{ab}}{\partial
A_{q}}=\sum_{n=0}^{\infty}\mathcal{S}_{ab}^{q\,(n)},\quad\quad\frac{\partial\mathcal{F}_{ab}}{\partial(\partial_{r}A_{q})}=\sum_{n=0}^{\infty}\mathcal{Q}_{ab}^{rq\,(n)},$
(77)

where the quantities

$\displaystyle\mathcal{F}^{(1)}_{ab}$ $\displaystyle:=$
$\displaystyle\partial_{a}A_{b}-\partial_{b}A_{a},$
$\displaystyle\mathcal{F}^{(n)}_{ab}$ $\displaystyle:=$ $\displaystyle
P_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n-1}}A_{i_{1}}...A_{i_{n-1}}\,\partial_{c}A_{d}+R_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n-2}}A_{i_{1}}...A_{i_{n-2}}\\{A_{c},A_{d}\\},\quad
n\geq 2,$ $\displaystyle\mathcal{S}_{ab}^{q\,(0)}$ $\displaystyle:=$
$\displaystyle 0,$ $\displaystyle\mathcal{S}_{ab}^{q\,(1)}$ $\displaystyle:=$
$\displaystyle P_{ab~{}~{}}^{~{}~{}cd|q}\,\partial_{c}A_{d},$
$\displaystyle\mathcal{S}_{ab}^{q\,(n)}$ $\displaystyle:=$ $\displaystyle
n\,P_{ab~{}~{}}^{~{}~{}cd|q\,i_{2}i_{3}...i_{n}}A_{i_{2}}...A_{i_{n}}\,\partial_{c}A_{d}+(n-1)\,R_{ab~{}~{}}^{~{}~{}cd|q\,i_{2}i_{3}...i_{n-1}}A_{i_{2}}...A_{i_{n-1}}\\{A_{c},A_{d}\\},\quad
n\geq 2$ $\displaystyle\mathcal{Q}_{ab}^{rq\,(0)}$ $\displaystyle:=$
$\displaystyle\delta_{a}^{r}\delta_{b}^{q}-\delta_{b}^{r}\delta_{a}^{q},$
$\displaystyle\mathcal{Q}_{ab}^{rq\,(n)}$ $\displaystyle:=$ $\displaystyle
P_{ab~{}~{}}^{~{}~{}rq|i_{1}i_{2}i_{3}...i_{n}}A_{i_{1}}...A_{i_{n}}+2R_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n-1}}A_{i_{1}}...A_{i_{n-1}}\,\Theta^{pr}\,\partial_{p}A_{c},\quad
n\geq 1$ (78)

are $n$-linear in $A$. We remind that for any $n\in\mathbb{N}$ the coefficient
functions $P_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n}}$ and
$R_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n}}$ are defined in Eq. (49).

Substituting the expansions (78) in the equations of motion (76), we obtain

${\cal E}^{q}_{EL}(A)=\sum_{n=0}^{\infty}{\cal E}^{q\,(n)}_{EL},$ (79)

where the combinations

${\cal
E}^{q\,(n)}_{EL}=\sum_{k=0}^{n-1}\bigg{[}-\frac{\mu}{2}\,\mathcal{S}_{ab}^{q\,(k)}\,\mathcal{F}^{ab(n-k)}+\frac{1}{2}\,\partial_{r}\left(\mu\,\mathcal{Q}_{ab}^{rq\,(k)}\,\mathcal{F}^{ab\,(n-k)}\right)\bigg{]}$
(80)

are $n$-linear in $A$.

Confronting the decompositions (75) and (79) we obtain the required brackets:

$\ell^{\cal E}_{n}(A^{\otimes
n})=n!\,(-1)^{\frac{n(n-1)}{2}}\,\sum_{k=0}^{n-1}\bigg{[}-\frac{\mu}{2}\,\mathcal{S}_{ab}^{q\,(k)}\,\mathcal{F}^{ab(n-k)}+\frac{1}{2}\,\partial_{r}\left(\mu\,\mathcal{Q}_{ab}^{rq\,(k)}\,\mathcal{F}^{ab\,(n-k)}\right)\bigg{]}\mathrm{d}x^{q},$
(81)

where the quantities $\mathcal{F}^{(n)}_{ab}$, $\mathcal{S}_{ab}^{q\,(n)}$,
and $\mathcal{S}_{ab}^{q\,(n)}$ are defined in Eq. (78) in terms of the
coefficient functions $P_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n}}$ and
$R_{ab~{}~{}}^{~{}~{}cd|i_{1}i_{2}i_{3}...i_{n}}$, associated with the given
Poisson gauge theory. The fact that the action is gauge invariant,
$\delta_{f}S_{g}\equiv 0$, implies that the solutions of the field equations
(75) are mapped onto the solutions under the gauge transformations. In this
sense the equations (75) are gauge covariant. The latter can be translated to
the fact that the brackets (81) satisfy the corresponding homotopy relations
just like it was shown explicitly in the Proposition 57 for the brackets (50).

As a final remark we mention here that the important identities involving the
covariant objects from the space $V_{2}$, like the Bianchi identity for the
field strength, the commutator relation for the covariant derivatives or the
Noether identities for the field equations, also can be obtained in the L∞
formalism. However to this end we will need an additional non-empty subspace
of degree three - the space of identities. Then, from the non-vanishing
brackets of the form, $\ell_{n+1}({\cal F},A^{\otimes n})\in V_{3}$, one may
construct the object,

$\displaystyle{\mathsf{d}}_{A}{\cal F}=\sum_{n=0}^{\infty}{1\over
n!}(-1)^{n(n-1)\over 2}\,\ell_{n+1}({\cal F},A^{\otimes n})\ .$ (82)

The L∞ relations, ${\cal J}_{n}(A^{\otimes n})=0$, imply that,
${\mathsf{d}}_{A}{\cal F}\equiv 0$, that is we have the Bianchi identity for
the field strength ${\cal F}$, see [19, 20] for details. We leave the detailed
analysis of these identities and the corresponding L∞ structures for the
future research.

##  4 Examples.

In this section we illustrate our results, presenting explicit L∞ structures
for the particular choices of non-commutativity parameter, namely the
$\kappa$-Minkowski and the $\mathfrak{su}(2)$ structures. For the sake of
simplicity we abandon the matter fields, considering the gauge fields only.
The brackets of the structure $\ell_{n}\left(f,A^{\otimes n-1}\right)$, $f\in
V_{0}$, $A\in V_{1}$ have already been studied in [11], therefore all the
remaining nontrivial (i.e. noncommutativity-dependent) brackets have the
structure $\ell_{n}\left(A^{\otimes n}\right)$, $n>1$.

Our technical strategy is as follows. Starting from the known solutions for
$\gamma$ and $\rho$, we expand the quantity $\mathcal{F}_{ab}$ defined by (18)
in powers of $A$,

$\mathcal{F}_{ab}=\sum_{n=0}^{\infty}\mathcal{F}_{ab}^{(n)},$ (83)

where, by definition, $\mathcal{F}_{ab}^{(n)}$ is $n$-linear444i.e. upon the
rescaling $A\to\Omega A$, $\Omega\in\mathbb{R}$, the quantity
$\mathcal{F}_{ab}^{(n)}$ rescales as
$\mathcal{F}_{ab}^{(n)}\to\Omega^{n}\mathcal{F}_{ab}^{(n)}$ in $A$. It is
worth noticing that the combination

$P_{ab}{}^{cd(n-1)}\,\partial_{c}A_{d}+R_{ab}{}^{cd(n-2)}\,\left\\{A_{c},A_{d}\right\\},$

which enters in the Proposition 57 is nothing but $\mathcal{F}_{ab}^{(n)}$.
Thus we obtain explicit expressions for the $\ell_{n}\left(A^{\otimes
n}\right)$-brackets in the following form

$\ell_{n}\left(A^{\otimes
n}\right)=\frac{n!}{2}\,(-1)^{\frac{n(n-1)}{2}}\,\mathcal{F}_{ab}^{(n)}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}.$
(84)

### a. $\kappa$-Minkowski.

In the $N$-dimensional $\kappa$-Minkowski case the Poisson bi-vector is given
by

$\Theta^{ij}=2(\omega^{i}x^{j}-\omega^{j}x^{i}),$ (85)

where $\omega^{i}$, $i=1,...,N$ are deformation parameters555In the Euclidean
case $\omega^{i}$ can be transformed by a linear transformation of the
coordinates (rotation) to the form: $\omega^{i}=\delta^{iN}\kappa^{-2}$, see
[18].. The corresponding structure constants read:

$f^{aj}_{k}=2(\omega^{a}\delta^{j}_{k}-\omega^{j}\delta^{a}_{k}).$ (86)

For the forthcoming discussion it is convenient to introduce the quantity

$z:=\omega^{i}A_{i}.$ (87)

The matrices $\gamma$ and $\rho$ are given by [12, 16]:

$\gamma^{i}_{a}=\delta^{i}_{a}\,\xi(z)-\omega^{i}A_{a},$ (88)

and

$\rho^{i}_{a}=\delta^{i}_{a}\,\xi(z)-\omega^{i}A_{a}\,\xi^{\prime}(z),$ (89)

respectively, with $\xi^{\prime}(z)=d\xi/dz$. In these formulae the form
factor $\xi$ is defined as follows:

$\xi(z):=\sqrt{1+z^{2}}+z.$ (90)

Substituting this data in our general formula (18), we arrive at666This
formula is not new, it has already appeared in [16]. However, the present
parametrisation is the most convenient for the forthcoming expansion in powers
of $A$.

$\displaystyle\mathcal{F}_{ab}$ $\displaystyle=$
$\displaystyle\big{(}\delta^{j}_{b}\,\xi(z)-\omega^{j}A_{b}\,\xi^{\prime}(z)\big{)}\,\partial_{a}A_{j}$
(91) $\displaystyle+$
$\displaystyle\left(A_{q}\partial_{l}A_{j}\,f_{i}^{ql}+\frac{1}{2}\\{A_{i},A_{j}\\}\right)\Big{(}\delta_{a}^{i}\delta_{b}^{j}\cdot\big{(}2\,z\,\xi(z)+1\big{)}-\big{(}\delta_{a}^{i}\omega^{j}A_{b}+\delta_{b}^{j}\omega^{i}A_{a}\big{)}\cdot\big{(}z\,\xi(z)\big{)}^{\prime}\Big{)}$
$\displaystyle-$ $\displaystyle\Big{(}a\leftrightarrow b\Big{)}.$

The form factor $\xi$ exhibits the following Taylor expansion in powers of
$z$:

$\xi(z)=\sum_{n=0}^{\infty}\xi_{n}\,z^{n},$ (92)

where

$\xi_{n}=\left\\{\begin{array}[]{l}\frac{(-1)^{m-1}(2m)!}{4^{m}(m!)^{2}(2m-1)},\quad
n=2m,\quad m\in\mathbb{Z}_{+},\\\ 1,\quad n=1,\\\ 0,\quad n=2m+1,\quad
m\in\mathbb{N}.\end{array}\right.$ (93)

Substituting this expansion in Eq. (91), and taking into account its obvious
consequences,

$\displaystyle 2\,z\,\xi(z)+1$ $\displaystyle=$ $\displaystyle
1+\sum_{n=1}^{\infty}(2\,\xi_{n-1})\,z^{n},$
$\displaystyle\big{(}z\,\xi(z)\big{)}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\big{(}(n+1)\,\xi_{n}\big{)}\,z^{n},$ (94)

we obtain the expansion (83) with

$\displaystyle\mathcal{F}_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle\partial_{a}A_{b}-\partial_{b}A_{a},$
$\displaystyle\mathcal{F}_{ab}^{(2)}$ $\displaystyle=$
$\displaystyle\big{(}z\,\delta_{b}^{j}-\omega^{j}A_{b}\big{)}\,\partial_{a}A_{j}+A_{q}\partial_{l}A_{b}\,f_{a}^{ql}+\frac{1}{2}\\{A_{a},A_{b}\\}-(a\leftrightarrow
b),$ $\displaystyle\mathcal{F}_{ab}^{(n)}$ $\displaystyle=$
$\displaystyle\big{(}z\,\delta_{b}^{j}-(n-1)\,\omega^{j}A_{b}\big{)}\,\partial_{a}A_{j}\,z^{n-2}\,\xi_{n-1}+\left(A_{q}\partial_{l}A_{j}\,f_{i}^{ql}+\frac{1}{2}\\{A_{i},A_{j}\\}\right)\times$
(95) $\displaystyle\times$
$\displaystyle\big{[}2\,\delta_{a}^{i}\delta_{b}^{j}\,z-(n-2)(\delta_{a}^{i}\omega^{j}A_{b}+\delta_{b}^{j}\omega^{i}A_{a})\big{]}\,z^{n-3}\xi_{n-3}-(a\leftrightarrow
b),\quad n\geq 3.$

Therefore, according to Eq. (84), we arrive at the required L∞ brackets:

$\displaystyle\ell_{2}\left(A^{\otimes 2}\right)$ $\displaystyle=$
$\displaystyle-2\Big{[}\big{(}z\,\delta_{b}^{j}-\omega^{j}A_{b}\big{)}\,\partial_{a}A_{j}+A_{q}\partial_{l}A_{b}\,f_{a}^{ql}+\frac{1}{2}\\{A_{a},A_{b}\\}\Big{]}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b},$
$\displaystyle\ell_{n}\left(A^{\otimes n}\right)$ $\displaystyle=$
$\displaystyle{n!}\,(-1)^{\frac{n(n-1)}{2}}\,\Big{[}\big{(}z\,\delta_{b}^{j}-(n-1)\,\omega^{j}A_{b}\big{)}\,\partial_{a}A_{j}\,z^{n-2}\,\xi_{n-1}+\left(A_{q}\partial_{l}A_{j}\,f_{i}^{ql}+\frac{1}{2}\\{A_{i},A_{j}\\}\right)\times$
(96) $\displaystyle\times$
$\displaystyle\big{[}2\,\delta_{a}^{i}\delta_{b}^{j}\,z-(n-2)(\delta_{a}^{i}\omega^{j}A_{b}+\delta_{b}^{j}\omega^{i}A_{a})\big{]}\,z^{n-3}\xi_{n-3}\Big{]}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b},\quad
n\geq 3,$

where, we remind, the coefficients $\xi_{n}$ are defined by Eq. (93).

### b. $\mathfrak{su}(2)$-structure.

For the $\mathfrak{su}(2)$-case,

$\Theta^{jk}=f^{jk}_{l}\,x^{l},\quad\quad\varepsilon^{jk}_{~{}~{}\,l}:=\varepsilon^{jks}\,\delta_{sl},$
(97)

and hence,

$f^{jk}_{l}=2\lambda\,\varepsilon^{jk}_{~{}~{}\,l},$ (98)

where $\lambda$ is a deformation parameter. Solutions of the master equations
can be chosen as follows [14, 12]:

$\gamma^{k}_{a}=\big{[}1+{Z\,}\chi(Z)\big{]}\delta_{a}^{k}-\lambda^{2}\chi(Z)\,A_{a}A^{k}-\lambda\,\varepsilon^{kl}_{~{}~{}\,a}A_{l},\quad\quad
Z:=\lambda^{2}\,A_{i}A^{i},\quad\quad A^{j}:=\delta^{j\xi}A_{\xi}$ (99)

and

$\rho^{i}_{a}={\phi}(Z)\,\delta_{a}^{i}+\lambda^{2}\tau(Z)A^{i}A_{a}{-}\lambda\,{\beta}(Z)\varepsilon^{ik}_{~{}~{}\,a}A_{k},$
(100)

where, by definition,

$\displaystyle\chi(t)$ $\displaystyle=$
$\displaystyle\frac{1}{t}\cdot\big{(}\sqrt{t}\cot{\sqrt{t}}-1\big{)}$
$\displaystyle{\phi}(t)$ $\displaystyle=$
$\displaystyle\frac{\sin{2\sqrt{t}}}{2\sqrt{t}}$ $\displaystyle\tau(t)$
$\displaystyle=$
$\displaystyle-\frac{1}{t}\cdot\bigg{(}\frac{\sin{2\sqrt{t}}}{2\sqrt{t}}-1\bigg{)}$
$\displaystyle{\beta(t)}$ $\displaystyle{=}$
$\displaystyle{\frac{\big{(}\sin{\sqrt{t}}\big{)}^{2}}{t}}.$ (101)

Substituting these expressions in Eq. (45), and performing a few algebraic
simplifications, we arrive at the know expression for the deformed field
strength [14]:

$\mathcal{F}_{ab}=\partial_{c}\pi_{ab}^{~{}~{}\,c}+2\,\big{\\{}\rho_{ab}^{~{}~{}\,c},A_{c}\big{\\}},$
(102)

with

$\displaystyle\pi_{ab}^{~{}~{}\,c}$ $\displaystyle:=$
$\displaystyle\delta_{a}^{c}\,A_{b}\,\phi(Z)-\frac{\lambda}{2}\,\varepsilon_{abm}\,A^{m}\,A^{c}\,\beta(Z)+\frac{1}{2\lambda}\,\varepsilon_{ab}^{~{}~{}\,c}\,\Phi(Z)-(a\leftrightarrow
b),$ $\displaystyle\rho_{ab}^{~{}~{}\,c}$ $\displaystyle:=$
$\displaystyle\frac{1}{4}\,\delta_{b}^{c}\,A_{a}\,\beta(Z)-\frac{1}{8\lambda}\,\varepsilon_{ab}^{~{}~{}\,c}\,\Lambda(Z)-(a\leftrightarrow
b).$ (103)

In this formulae the form factors $\Lambda$ and $\Phi$ are defined in the
following way:

$\displaystyle\Lambda(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}u\,\beta(u),$ $\displaystyle\Phi(t)$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}u\,\phi(u)=t\cdot\beta(t).$ (104)

Substituting the Taylor series

$\displaystyle\beta(t)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\frac{(-1)^{m}\,4^{m+\frac{1}{2}}}{(2\,m+2)!}\,t^{m},\quad\quad\Lambda(t)=\sum_{m=1}^{\infty}\frac{(-1)^{m+1}\,4^{m-\frac{1}{2}}}{(2\,m)!\,m}\,t^{m},$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\frac{(-1)^{m}\,4^{m}}{(2m+1)!}\,t^{m},\quad\quad\quad\Phi(t)=\sum_{m=1}^{\infty}\frac{(-1)^{m+1}\,4^{m-1}}{(2\,m-1)!\,m}\,t^{m},$
(105)

in the definitions (103), we obtain the following expansions for
$\rho_{ab}^{~{}~{}\,c}$ and $\pi_{ab}^{~{}~{}\,c}$:

$\rho_{ab}^{~{}~{}\,c}=\sum_{n=1}^{\infty}\rho_{~{}ab}^{(n)\,c},\quad\quad\pi_{ab}^{~{}~{}\,c}=\sum_{n=1}^{\infty}\pi_{~{}ab}^{(n)\,c},$
(106)

where the quantities

$\displaystyle\rho_{~{}ab}^{(n)\,c}=\left\\{\begin{array}[]{l}\delta_{b}^{c}\,A_{a}\,\frac{(-1)^{m}\,4^{m-\frac{1}{2}}}{(2\,m+2)!}\,Z^{m}-(a\leftrightarrow
b),\quad n=2\,m+1,\quad m\in\mathbb{Z}_{+},\\\
\frac{1}{\lambda}\,\varepsilon_{ab}^{~{}~{}\,c}\,\frac{4^{m-2}\,(-1)^{m}}{(2\,m)!\,m}\,Z^{m}-(a\leftrightarrow
b),\quad n=2\,m,\quad m\in\mathbb{N},\end{array}\right.$ (109)

and

$\pi_{~{}ab}^{(n)\,c}=\left\\{\begin{array}[]{l}\delta_{a}^{c}\,A_{b}\,\frac{(-1)^{m}\,4^{m}}{(2\,m+1)!}\,Z^{m}-(a\leftrightarrow
b),\quad n=2\,m+1,\quad m\in\mathbb{Z}_{+},\\\
\big{(}-\lambda\,\varepsilon_{abp}\,A^{p}A^{c}+\frac{1}{\lambda}\varepsilon_{abc}\,Z\big{)}\,\frac{(-1)^{m-1}\,4^{m-1}}{(2m)!}\,Z^{m-1}-(a\leftrightarrow
b),\quad n=2\,m,\quad m\in\mathbb{N}.\end{array}\right.$ (110)

contain the $n$-th power of $A$. Substituting the expansions (106) in the
expression (102), we arrive at Eq. (83) with

$\mathcal{F}_{ab}^{(2m+1)}=\frac{(-1)^{m}\,4^{m}}{(2m+1)!}\partial_{a}(A_{b}Z^{m})+\frac{2}{\lambda}\,\frac{4^{m-2}\,(-1)^{m}}{(2\,m)!\,m}\,\varepsilon_{ab}^{~{}~{}\,c}\,\\{Z^{m},A_{c}\\}-(a\leftrightarrow
b),\quad m\in\mathbb{Z}_{+},$

and

$\displaystyle\mathcal{F}_{ab}^{(2m)}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{m-1}\,4^{m-1}}{(2m)!}\left[-\lambda\varepsilon_{abp}\partial_{c}(A^{p}A^{c}Z^{m-1})+\frac{1}{\lambda}\varepsilon_{ab}^{~{}~{}\,c}\partial_{c}(Z^{m})+\\{A_{a}Z^{m-1},A_{b}\\}\right]-(a\leftrightarrow
b),$ $\displaystyle m$ $\displaystyle\in$ $\displaystyle\mathbb{N}.$

Using the general rule (84), we get the following nontrivial $\ell$-brackets:

$\displaystyle\ell_{2m+1}\left(A^{\otimes(2m+1)}\right)$ $\displaystyle=$
$\displaystyle
4^{m}\,\left[\partial_{a}(A_{b}Z^{m})+\frac{1}{\lambda}\,\frac{(2m+1)}{8\,m}\,\varepsilon_{ab}^{~{}~{}\,c}\,\\{Z^{m},A_{c}\\}\right]\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b},$
$\displaystyle\ell_{2m}\left(A^{\otimes(2m)}\right)$ $\displaystyle=$
$\displaystyle
4^{m-1}\left[\lambda\varepsilon_{abp}\partial_{c}(A^{p}A^{c}Z^{m-1})-\frac{1}{\lambda}\varepsilon_{ab}^{~{}~{}\,c}\partial_{c}(Z^{m})-\\{A_{a}Z^{m-1},A_{b}\\}\right]\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b},$
$\displaystyle m$ $\displaystyle\in$ $\displaystyle\mathbb{N},$ (111)

which in addition to the brackets $\ell_{2}(f,g)$ and $\ell_{n+1}(f,A^{\otimes
n})$ constructed in [11, Section 6.1] define the L${}_{\infty}^{full}$ algebra
corresponding to the $\mathfrak{su}(2)$-like Poisson gauge theory. The results
of this section can be generalized to the four-dimensional case (with a
commutative fourth coordinate) and/or to the angular non-commutativity [21]
using the explicit expressions for $\mathcal{F}$, presented in [22].

##  5 Comments on P∞

The Poisson infinity or P∞ structures appeared in the context of quantization
of co-isotropic sub-manifolds of Poisson manifolds [23, 24]. In fact P∞
algebra is an L∞ algebra $(\ell_{n})_{n=1}^{\infty}$ on $V$ in which, in
addition, it is introduced an operation of the associative multiplication
between the elements. To each pare of elements, $\forall u,v\in V$, it is
assigned the third element $u\otimes v\to u\cdot v\in V$ in such a way that
$(u\cdot v)\cdot w=u\cdot(v\cdot w)$. And the corresponding $\ell$-brackets
should satisfy the Leibniz rule with respect to this product,

$\ell_{n}(v_{1},\dots,v_{n-1},u\cdot v)=\ell_{n}(v_{1},\dots,v_{n-1},u)\cdot
v+u\cdot\ell_{n}(v_{1},\dots,v_{n-1},v)\,.$ (112)

Following [11, Proposition 5.23] we define the multiplication by truncating
the product on the exterior algebra at degree $2$, that is,

$\displaystyle f\cdot g$ $\displaystyle=f\,g\ ,\quad f\cdot A=f\,A\,,\qquad
f\cdot E=f\,E\,,$ (113) $\displaystyle A\cdot B$ $\displaystyle=A\wedge
B\,,\qquad A\cdot E=0\,,\qquad E\cdot F=0\,.$

for all zero-forms $f$ and $g$, one-forms $A$ and $B$ and two-forms $E$ and
$F$. We remind that the spaces of higher forms are empty by construction.

The same [11, Proposition 5.23] shows that the subalgebra of
L${}_{\infty}^{full}$ called L${}_{\infty}^{gauge}$ which is concentrated in
the degrees $0$ and $1$ and describes the action of the gauge symmetries on
the gauge fields defines the P∞ algebra, i.e., the derivation properties of
the corresponding brackets are satisfied,

$\displaystyle\ell_{2}(f,g\cdot h)$ $\displaystyle=$
$\displaystyle\ell_{2}(f,g)\cdot h+g\cdot\ell_{2}(f,h)\,,$ (114)
$\displaystyle\ell_{n}\left(A^{\otimes n-1},f\cdot g\right)$ $\displaystyle=$
$\displaystyle\ell_{n}\left(A^{\otimes n-1},f\right)\cdot
g+f\cdot\ell_{n}\left(A^{\otimes n-1},g\right)\,,$
$\displaystyle\ell_{n}\left(f,A^{\otimes n-2},g\cdot A\right)$
$\displaystyle=$ $\displaystyle\ell_{n}\left(f,A^{\otimes n-2},g\right)\cdot
A+g\cdot\ell_{n}\left(f,A^{\otimes n-1}\right)\,.$

To check whether L${}_{\infty}^{full}$ algebra can be made into the P∞
structure we have to check the derivation properties (112) on the
$\ell$-brackets which belongs to the degree two, i.e., $\ell_{2}(f,E)$ and
$\ell_{n}\left(A^{\otimes n}\right)$. The check involving $\ell_{2}(f,E)$ is
straightforward. The Leibniz rule for the bracket $\ell_{n}\left(A^{\otimes
n}\right)$ reads,

$\ell_{n}\left(A^{\otimes n-1},f\cdot A\right)=f\cdot\ell_{n}\left(A^{\otimes
n}\right)+\ell_{n}\left(A^{\otimes n-1},f\right)\cdot A\,,\qquad n>1\,.$ (115)

Using the explicit form of the bracket $\ell_{n}\left(A^{\otimes n}\right)$
given by (58) the left hand side of (115) can be represented as,

$\displaystyle\ell_{n}\left(A^{\otimes n-1},f\cdot A\right)$ $\displaystyle=$
$\displaystyle f\cdot\ell_{n}\left(A^{\otimes n}\right)+$
$\displaystyle\frac{n!}{2}\,(-1)^{\frac{n(n-1)}{2}}\left(P_{ab}{}^{cd(n-1)}\,A_{d}\,\partial_{c}f+R_{ab}{}^{cd(n-2)}\,A_{d}\left\\{A_{c},f\right\\}\right){\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.$

According to the same formulas (58) and (113) one finds,

$\displaystyle\ell_{2}(A,f)\cdot A$ $\displaystyle=$
$\displaystyle-\left(\gamma^{c(1)}_{a}\,\partial_{c}f+\\{A_{a},f\\}\right)A_{b}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,$
(117) $\displaystyle\ell_{n}\left(A^{\otimes n-1},f\right)\cdot A$
$\displaystyle=$
$\displaystyle(n-1)!\,(-1)^{\frac{n(n-1)}{2}}\,\gamma_{a}^{c(n-1)}\partial_{c}f\,A_{b}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,,\qquad
n>2\,.$

In order (115) to hold the second term on the right of (5) should be equal to
the right hand side of (117). For the constant NC parameter $\Theta^{ij}$,
$\gamma_{a}^{b}(A)=\rho_{a}^{b}(A)=\delta_{a}^{b}$. In this case the non-
vanishing brackets are $\ell_{2}(A,f)=-\\{A_{a},f\\}\,{\mathsf{d}}x^{a}$ and
$\ell_{2}(A,A)=-\\{A_{a},A_{b}\\}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}$,
while all higher brackets vanish. So, for the canonical non-commutativity the
second term in the r.h.s. of (5) is precisely equal to (117) and the
L${}_{\infty}^{full}$ algebra turns into P∞ structure.

For non-constant $\Theta^{ij}(x)$ the situation is more complicated. The are
non-vanishing higher brackets and the contribution to (5) containing the
Poisson bracket $\\{A_{c},f\\}$ will not be compensated by (117), since it
does not have any term with the Poisson brackets. But the violation of (115)
can be seen already for $n=2$, since

$\displaystyle
P_{ab}{}^{cd(1)}\,A_{d}\,\partial_{c}f\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}$
$\displaystyle=$
$\displaystyle-2\,\partial_{a}\Theta^{cd}A_{d}\,A_{b}\,\partial_{c}f\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}$
$\displaystyle\neq$
$\displaystyle-\frac{1}{2}\,\partial_{a}\Theta^{cd}A_{d}\,A_{b}\,\partial_{c}f\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}=\gamma^{c(1)}_{a}\,\partial_{c}f\,A_{b}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.$

We conclude that the Leibniz rule in L${}_{\infty}^{full}$ algebra holds true
only in case of the canonical non-commutativity. Let us note also that
L${}_{\infty}^{gauge}$ algebra has has the structure of P∞ only for Poisson
gauge transformations, for almost-Poisson case when the bi-vector
$\Theta^{ij}(x)$ does not satisfy the Jacobi identity not even
L${}_{\infty}^{gauge}$ enjoys the derivation properties [11]. At least so far
we were not able to link the P∞ algebras to deformations of gauge theories.

##  6 Towards full non-commutative gauge theory

As it has been already mentioned in the introduction the L∞-bootstrap approach
is a powerful tool for the construction of order by order in $\Theta$ and
$\hbar$ corrections of given undeformed gauge theory. Suppose we have a full
non-commutative gauge theory with a corresponding L∞ algebra defined by the
brackets $\ell^{NC}_{n}$ satisfying the relations (3.1) schematically denoted
by,

${\cal
J}_{n}:=\sum^{n}_{i=1}\,(-1)^{i\,(n-i)}\ell^{NC}_{n-i+1}\ell^{NC}_{i}=0\,.$
(119)

According to the prescription of the L∞ bootstrap (1) and (2) the initial
setup reads,

$\ell^{NC}_{1}(f)={\mathsf{d}}f\,,\qquad\mbox{and}\qquad\ell^{NC}_{2}(f,g)=i[f,g]_{\star}\,.$
(120)

Let us set, $\ell_{1}(f)=\ell^{NC}_{1}(f)$, and

$\ell_{n}=\lim_{\hbar\to 0}\,\frac{1}{\hbar^{n-1}}\,\ell^{NC}_{n}\,,\qquad
n>1\,.$ (121)

In particular,

$\ell_{2}(f,g)=\lim_{\hbar\to
0}\,\frac{1}{\hbar}\,[f,g]_{\star}=-\\{f,g\\}\,.$ (122)

If the brackets $\ell^{NC}_{n}$ satisfy the L∞ relations (119), then the
brackets $\ell_{n}$ also do, and the brackets $\ell_{1}(f)$ and
$\ell_{2}(f,g)$ are exactly the starting point for the L∞ algebra of Poisson
gauge theory. In this sense we may call $\\{\ell_{n}\\}$ as a quasi-classical
limit of $\\{\ell^{NC}_{n}\\}$ and the Poisson gauge theory is a quasi-
classical limit of the full non-commutative gauge theory as it was mentioned
in the introduction. The brackets $\ell_{n}$ given by the Proposition 57
provide the leading in $\hbar$ order of the brackets $\ell_{n}^{NC}$,

$\ell^{NC}_{n}=\hbar^{n-1}\ell_{n}+{\cal O}(\hbar^{n})\,.$ (123)

The question is how to construct the higher order in $\hbar$ contributions to
the brackets $\ell^{NC}_{n}$. The aim of this section is to outline the
solution to this problem within the L∞-bootstrap approach.

For the future convenience we start introducing the condensed indices,

$(i)^{p}=(i_{1}\dots
i_{p})\,,\qquad\left(\partial_{i}\right)^{p}=\partial_{i_{1}}\dots\partial_{i_{p}}\,,$
(124)

where the round bracket means symmetrization. In particular,

$K^{(i)^{p}j}\left(\partial_{i}\right)^{p}A_{j}=K^{i_{1}\dots
i_{p}j}\partial_{i_{1}}\dots\partial_{i_{p}}A_{j}\,.$

The non-commutative deformation of the space-time is introduced through the
given Hermitean, $(f\star g)^{\ast}=g^{\ast}\star f^{\ast}$, unital, $f\star
1=1\star f=f$, star product defined by the expression,

$f\star
g=f\,g+\sum_{n=1}^{\infty}\left(\frac{i\hbar}{2}\right)^{n}B_{n}(f,g)\,,\qquad
B_{1}=\\{f,g\\}\,.$ (125)

The bi-differential operators

$B_{n}(f,g)=\sum_{k,l=1}^{n}B_{n}^{(i)^{k}|(j)^{l}}(\partial_{i})^{k}f\,(\partial_{j})^{l}g\,,$
(126)

are determined to satisfy the associativity condition, $f\star(g\star
h)=(f\star g)\star h$. The coefficient functions $B_{n}^{(i)^{k}|(j)^{l}}$ can
be constructed following the prescription of the Formality Theorem [25], or
also using the poly-differential approach proposed in [26]. The Hermiticity of
the star product implies that, $B_{2n}(f,g)=B_{2n}(g,f)$, and
$B_{2n+1}(f,g)=-B_{2n+1}(g,f)$. Thus the expression for the bracket
$\ell_{2}^{NC}(f,g)$ becomes,

$\ell^{NC}_{2}(f,g)=2i\sum_{n=0}^{\infty}\left(\frac{i\hbar}{2}\right)^{2n+1}B_{2n+1}(f,g)=-\hbar\,\\{f,g\\}+{\cal
O}(\hbar^{3})\,.$ (127)

To determine the bracket $\ell_{2}^{NC}(f,A)$ one has to solve the homotopy
relation, ${\cal J}_{2}(f,g)=0$, written as,

$\ell^{NC}_{2}(\ell^{NC}_{1}(f),g)+\ell_{2}^{NC}(f,\ell^{NC}_{1}(g))=\ell^{NC}_{1}(\ell_{2}^{NC}(f,g))\,.$
(128)

Remind that the element $\ell^{NC}_{1}(f)$ belongs to the space of the gauge
fields, that is, we have a combination of yet undetermined brackets of the
form $\ell_{2}(f,A)$ on the left expressed in terms of the combination of
given brackets on the right. One may check that setting,

$\displaystyle\ell_{2}^{NC}(f,A)=\left\\{i[f,A_{a}]_{\star}+\sum_{k,l=1}^{\infty}\tilde{B}_{a}^{(i)^{k}|(j)^{l}}(\partial_{i})^{k}f\,(\partial_{j})^{l-1}A_{j_{l}}\right\\}{\mathsf{d}}x_{a}\,,$
(129)

where,

$\tilde{B}_{a}^{(i)^{k}|(j)^{l}}=i\sum_{n=0}^{\infty}\left(\frac{i\hbar}{2}\right)^{2n+1}\partial_{a}B_{2n+1}^{(i)^{k}|(j)^{l}}\,,$
(130)

and defining according to the graded symmetry,
$\ell_{2}^{NC}(A,f):=-\ell_{2}^{NC}(f,A)$, one solves (128).

The non-trivial technical issue appears on the following steps of the
bootstrap procedure. Let us discuss the definition of the bracket
$\ell_{2}^{NC}(A,B)$ entering the decomposition of the non-commutative field
strength (6). It should be found from the identity, ${\cal J}(f,A)=0$, written
in the form,

$\displaystyle\ell^{NC}_{2}(\ell^{NC}_{1}(f),A)+\ell^{NC}_{2}(f,\ell^{NC}_{1}(A))=\ell^{NC}_{1}(\ell^{NC}_{2}(f,A))\,.$
(131)

The bracket, $\ell_{2}^{NC}(f,A)$ on the right is given by (129), while on the
left we have a combination of two yet undetermined brackets of the form
$\ell_{2}^{NC}(A,B)$ and $\ell^{NC}_{2}(f,\ell^{NC}_{1}(A))$ reminding that
the element $\ell^{NC}_{1}(A)$ belongs to the space space of gauge covariant
objects $V_{2}$. Following the reasoning of the Proposition 57 for the element
${\cal F}\in V_{2}$ we just set,

$\displaystyle\ell^{NC}_{2}(f,{\cal F})=i[f,{\cal F}]_{\star}\,.$ (132)

In this case, the associativity of the star product implies the Jacobi
identity for the bracket (132), meaning that the homotopy relation, ${\cal
J}_{3}(f,g,{\cal F})=0$, will be automatically satisfied and there is no need
for the higher brackets $\ell_{n+2}^{NC}(f,{\cal F},A^{n})$. The gauge
covariance condition thus becomes, $\delta_{f}^{NC}{\cal F}=i[f,{\cal
F}]_{\star}$. Now we rewrite the eq. (131) as,

$\displaystyle\ell^{NC}_{2}(\ell^{NC}_{1}(f),A)={\cal J}_{2}^{R}(f,A)\,,$
(133) $\displaystyle{\cal
J}_{2}^{R}(f,A):=\ell^{NC}_{1}\left(\ell_{2}^{NC}(f,A)\right)-i[f,\ell^{NC}_{1}(A)]_{\star}\,.$
(134)

Again, yet undetermined bracket on the left is expressed as a combination of
the previously determined brackets on the right. One may schematically
represent the r.h.s. as,

$\displaystyle{\cal
J}_{2}^{R}(f,A)=\sum_{p,q=1}^{\infty}Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}(\partial_{i})^{p}f\,(\partial_{j})^{q-1}A_{j_{q}}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}$
(135)

where the coefficient functions $Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}$ are
constructed from the Poisson bi-vector $\Theta^{ij}(x)$ and its derivatives
according to the prescription (134). The bracket $\ell_{2}^{NC}(A,B)$ can be
also schematically represented as a sum,

$\ell_{2}^{NC}(A,B)=\sum_{p,q=1}^{\infty}R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}(\partial_{i})^{p-1}A_{i_{p}}\,(\partial_{j})^{q-1}B_{j_{q}}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.$
(136)

The graded symmetry of the bracket, $\ell_{2}^{NC}(A,B)=\ell_{2}^{NC}(B,A)$,
implies the symmetry relation of the corresponding coefficients,

$R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}=R_{ab}{}^{(j)^{q-1}j_{q}|(i)^{p-1}i_{p}}\,.$
(137)

To sum up, the coefficient functions $Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}$ are
given while $R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}$ should be determined
from the equation (133) in such a way that (137) holds true.

The bracket (136) will be constructed following [10]. First we observe that
the graded symmetry of the bracket,
$\ell_{2}^{NC}(\ell^{NC}_{1}(f),\ell^{NC}_{1}(g))=\ell_{2}^{NC}(\ell^{NC}_{1}(g),\ell^{NC}_{1}(f)),$
implies the consistency condition on the right hand side of the equation
(133),

$\displaystyle{\cal J}_{2}^{R}(f,\ell^{NC}_{1}(g))-{\cal
J}_{2}^{R}(g,\ell^{NC}_{1}(f))=\,0\,.$ (138)

The later however is automatically satisfied due to the previously solved
homotopy relation, ${\cal J}_{2}(f,g)=0$, given by (128). Indeed, one
calculates,

$\displaystyle\ell^{NC}_{1}\left(\ell_{2}^{NC}(f,\ell^{NC}_{1}(g))\right)-\ell^{NC}_{1}\left(\ell_{2}^{NC}(g,\ell^{NC}_{1}(f))\right)=\ell^{NC}_{1}\left(\left[\ell^{NC}_{1}\left(\ell_{2}^{NC}(f,g)\right)-{\cal
J}_{2}(f,g)\right]\right)\equiv\,0\,.$ (139)

The relation (138) implies that,

$\displaystyle\sum_{p,q=1}^{\infty}Q_{ab}{}^{(i)^{p}|(j)^{q}}(\partial_{i})^{p}f\,(\partial_{j})^{q}g\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}=\sum_{p,q=1}^{\infty}Q_{ab}{}^{(i)^{p}|(j)^{q}}(\partial_{i})^{p}g\,(\partial_{j})^{q}f\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,,$
(140)

which in turn results in the symmetry of the corresponding coefficient
functions under the permutation of the symmetriezed groups of indices,

$Q_{ab}{}^{(i)^{p}|(j)^{q}}=Q_{ab}{}^{(j)^{q}|(i)^{p}}\,.$ (141)

This relation is essential for the construction of the coefficients
$R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}$ with required symmetry properties,
and it follows from the bootstrap procedure. Note that the coefficient
$Q_{ab}{}^{(i)^{p}|(j)^{q}}$ is symmetric over all $j$-indices, while the
coefficient $Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}$ is only symmetric with respect
to first $(q-1)$ $j$-indices. So we introduce the notation for the “not
completely symmetric” part,

$Q_{ab}{}^{(i)^{p}|[j]^{q}}:=Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}-Q_{ab}{}^{(i)^{p}|(j)^{q}}\,,\qquad\mbox{with}\qquad
Q_{ab}{}^{(i)^{p}|[j]^{q}}(\partial_{j})^{q}f=0\,.$ (142)

It is also convenient to decompose $R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}$
into maximally symmetric and partially antisymmetric parts,

$\displaystyle R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}$ $\displaystyle=$
$\displaystyle R_{ab}{}^{(i)^{p}|(j)^{q}}+R_{ab}{}^{[i]^{p}|(j)^{q}}+$
$\displaystyle R_{ab}{}^{(i)^{p}|[j]^{q}}+R_{ab}{}^{[i]^{p}|[j]^{q}}.$

The symmetry relations (137) imply that,

$\displaystyle R_{ab}{}^{(i)^{p}|(j)^{q}}$ $\displaystyle=$ $\displaystyle
R_{ab}{}^{(j)^{q}|(i)^{p}}\,,$ (144) $\displaystyle
R_{ab}{}^{[i]^{p}|(j)^{q}}$ $\displaystyle=$ $\displaystyle
R_{ab}{}^{(j)^{q}|[i]^{p}}\,,$ $\displaystyle R_{ab}{}^{[i]^{p}|[j]^{q}}$
$\displaystyle=$ $\displaystyle R_{ab}{}^{[j]^{q}|[i]^{p}}\,.$

Now consider the following contribution to the right hand side of the equation
(133),

$\displaystyle\left[Q_{ab}{}^{(i)^{p}|(j)^{q-1}j_{q}}(\partial_{i})^{p}f\,(\partial_{j})^{q-1}A_{j_{q}}+Q_{ab}{}^{(j)^{q}|(i)^{p-1}i_{p}}(\partial_{j})^{q}f\,(\partial_{i})^{p-1}A_{i_{p}}\right]{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.$
(145)

It should be compensated by the corresponding contribution to the bracket
$\ell_{2}^{NC}(B,A)$ on the left,

$\displaystyle\left[R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}(\partial_{i})^{p-1}B_{i_{p}}\,(\partial_{j})^{q-1}A_{j_{q}}+R_{ab}{}^{(j)^{q-1}j_{q}|(i)^{p-1}i_{p}}(\partial_{j})^{q-1}B_{j_{q}}(\partial_{i})^{p-1}A_{i_{p}}\right]{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b},$
(146)

with $B_{i_{p}}=\partial_{i_{p}}f$. Substituting decompositions (142) and (6)
in the eqs. (145) and (146) correspondingly and comparing the coefficients we
conclude that,

$\displaystyle R_{ab}{}^{(i)^{p}|(j)^{q}}$ $\displaystyle=$ $\displaystyle
Q_{ab}{}^{(i)^{p}|(j)^{q}}\,,$ (147) $\displaystyle
R_{ab}{}^{(i)^{p}|[j]^{q}}$ $\displaystyle=$ $\displaystyle
Q_{ab}{}^{(i)^{p}|[j]^{q}}\,,$ $\displaystyle R_{ab}{}^{[i]^{p}|(j)^{q}}$
$\displaystyle=$ $\displaystyle Q_{ab}{}^{(j)^{q}|[i]^{p}}\,,$

and since the coefficient $R_{ab}{}^{[i]^{p}|[j]^{q}}$ does not contribute to
(146) we just set it to zero. We end up with,

$R_{ab}{}^{(i)^{p-1}i_{p}|(j)^{q-1}j_{q}}=Q_{ab}{}^{(i)^{p}|(j)^{q}}+Q_{ab}{}^{(i)^{p}|[j]^{q}}+Q_{ab}{}^{(j)^{q}|[i]^{p}}\,.$
(148)

One may easily see that it satisfies the symmetry relation (137) because of
(141). And thus, we have proved the following,

###### Proposition 149

The solution to the equation (133) satisfying the required symmetry property
is given by,

$\displaystyle\ell_{2}^{NC}(A,B)=\sum_{p,q=1}^{\infty}\left[Q_{ab}{}^{(i)^{p}|(j)^{q}}+Q_{ab}{}^{(i)^{p}|[j]^{q}}+Q_{ab}{}^{(j)^{q}|[i]^{p}}\right](\partial_{i})^{p-1}A_{i_{p}}\,(\partial_{j})^{q-1}B_{j_{q}}\,{\mathsf{d}}x^{a}\wedge{\mathsf{d}}x^{b}\,.\,\,\,\,\Box$

To define the higher brackets one has to solve the higher homotopy relations
given explicitly by the Proposition 32. Just like it happened in case of the
Poisson gauge algebra the L∞ relation with three gauge parameters, ${\cal
J}_{3}(f,g,h)=0$, is satisfied automatically because of the Jacobi identity
for the bracket $\ell^{NC}_{2}(f,g)$ given by (120). The latter means that
there is no need in the higher brackets with two gauge parameters and we just
set them to zero, $\ell^{NC}_{n+2}(f,g,A^{\otimes n})=0$. To determine the
bracket with one gauge parameter and two gauge fields $\ell_{3}^{NC}(f,A,B)$
which enters the definition of the gauge variation $\delta_{f}^{NC}A$ one has
to solve the L∞ relation, ${\cal J}_{3}(f,g,A)=0$. The latter can be written
as,

$\displaystyle\ell_{3}^{NC}(\ell_{1}^{NC}(f),g,A)+\ell_{3}^{NC}(f,\ell_{1}^{NC}(g),A)={\cal
J}^{R}_{3}(f,g,A)\,,$ (150) $\displaystyle{\cal J}^{R}_{3}(f,g,A):=$ (151)
$\displaystyle-\ell_{1}^{NC}(\ell_{3}^{NC}(f,g,A))-\ell_{2}^{NC}(\ell_{2}^{NC}(f,g),A)-\ell_{2}^{NC}(\ell_{2}^{NC}(A,f),g)-\ell_{2}^{NC}(\ell_{2}^{NC}(g,A),f)\,,$

where yet undetermined brackets of the form $\ell_{3}^{NC}(f,A,B)$ on the left
are expressed as a combination of the previously determined brackets ${\cal
J}^{R}_{3}(f,g,A)$ on the right. The graded symmetry,

$\ell_{3}^{NC}\left(f,\ell_{1}^{NC}(g),\ell_{1}^{NC}(h)\right)=\ell_{3}^{NC}\left(f,\ell_{1}^{NC}(h),\ell_{1}^{NC}(g)\right)\,,$
(152)

implies the consistency condition on the right hand side of (150),

${\cal J}^{R}_{3}\left(f,g,\ell_{1}^{NC}(h)\right)+{\cal
J}^{R}_{3}\left(g,h,\ell_{1}^{NC}(f)\right)+{\cal
J}^{R}_{3}\left(h,f,\ell_{1}^{NC}(g)\right)=0\,.$ (153)

In [9] it was demonstrated that this consistency condition holds true as a
consequence of the previously solved homotopy relations, ${\cal
J}_{2}(f,g)=0$, and ${\cal J}_{3}(f,g,h)=0$. An explicit form of the bracket
$\ell_{3}^{NC}(f,A,B)$ can be constructed following the logic of the
Proposition 149 and the relation (153). The precise recurrence relations for
the construction of full non-commutative (non-associative) gauge algebra were
recently obtained in [27].

For some particular choices of the Poisson structure $\Theta^{ij}(x)$, like
the $\mathfrak{su}(2)$-structure [28]-[32] or the $\kappa$-Minkowski [33]-[37]
the expressions for the star product and the star commutator are known in all
orders in $\hbar$. In this situation one may also look for the explicit
expressions for the gauge transformations $\delta_{f}^{NC}$ and the
corresponding dynamical objects ${\cal F}^{NC}$ and ${\cal D}^{NC}(\varphi)$.
We leave it for the future investigation.

## Acknowledgements

The results of section 5 were obtained under support of the Tomsk State
University Development Programe (Priority2030). O.A. thanks Universidade
Federal do ABC (UFABC) and CAPES for support. V.G.K. was supported in parts by
the São Paulo Research Foundation (FAPESP), grant 2021/09313-8 and by the CNPq
grant 304130/2021-4.

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