This follows from the fact that the subalgebra of Proposition <ref> obtained by truncation to degrees $0$ and $1$, and subsequent replacement of $\theta$ with $-\theta$, is precisely the stated $L_\infty$-structure on $\CCA$. It is also an easy direct check using the Leibniz rules for the exterior
derivative $\dd$, the Poisson bracket on $C^\infty(U)$, and the
$\Omega^1(U)$-valued bracket of Remark <ref>.

The significance of this observation is that it offers a natural path
towards a homotopy algebraic construction of noncommutative gauge
transformations beyond the semi-classical level. Similarly to the
discussion of Remark <ref>, the
$P_\infty$-algebra of Proposition <ref> can be
quantized to an
$A_\infty$-structure on $\CCA[[\hbar]]$ over $\real[[\hbar]]$ which is
a deformation of the algebra structure on $\CCA$ and whose
semi-classical limit induces the $P_\infty$-structure
$\{\ell_n^{\,\theta}\}_{n=1}^\infty$ through the
alternatization $\{\ell_n^{\,\star}\}_{n=1}^\infty$ of the structure
maps of this $A_\infty$-algebra: $\ell_n^{\,\theta} =
\frac1\hbar\,\ell_n^{\,\star}\big|_{\hbar=0}$. Then the
Poisson gauge algebra is a semi-classical limit of a noncommutative
gauge algebra which is organized by the “quantum”
$L_\infty$-structure $\{\ell_n^{\,\star}\}_{n=1}^\infty$, making the discussion at the beginning of
Section <ref> somewhat more precise. While this is an
interesting approach to the construction of noncommutative gauge
symmetries, it also lies beyond the scope of the present paper and we
leave it for future investigation.

Almost Poisson gauge algebras.

The situation
is much more complicated in the case of a general almost Poisson
bivector $\theta$. Now, because of the presence of the non-vanishing brackets $\rho_{n+2}^{\,\theta}$ in Proposition <ref> involving two functions and forms of higher degree, the corresponding truncation does not determine a subalgebra, as this would violate the homotopy Jacobi identities.
The difference between the $L_\infty$-structure maps of Propositions <ref> and <ref> (aside from numerical factors) lies entirely in the terms involving the Lagrangian multipliers of the almost Poisson gauge algebroid, whose inclusion restores the homotopy Jacobi identities. As we will now show, these terms violate the derivation properties of the original $P_\infty$-algebra from Proposition <ref>.

For this,
let us study in detail the derivation properties of the brackets
$\ell^{\,\theta}_{n+2}(f,g,A^{\otimes n})$ and $\ell^{\,\theta}_{n+1}(f,A^{\otimes n})$ from
Proposition <ref> with the same graded
commutative algebra $\CCA$ as in
Proposition <ref>. First of all, for the
multiplication of gauge parameters it is clear that the derivation

\begin{align*}
\ell^{\,\theta}_{n+2}(f \cdot h,g,A^{\otimes n})&=f\cdot
\ell^{\,\theta}_{n+2}(h,g,A^{\otimes
n}) + \ell^{\,\theta}_{n+2}(f,g,A^{\otimes
n})\cdot h \
, \\[4pt]
\ell^{\,\theta}_{n+1}(f \cdot h,A^{\otimes n})&=f\cdot
\ell^{\,\theta}_{n+2}(h,A^{\otimes
n}) + \ell^{\,\theta}_{n+2}(f,A^{\otimes n})\cdot h
\end{align*}

For the multiplication of gauge fields by gauge parameters, consider
first the brackets involving two gauge parameters. Since brackets
with three gauge parameters vanish for degree reasons, the desired derivation
property is just $C^\infty(U)$-linearity

\begin{equation}\label{ph2}
\ell^{\,\theta}_{n+2}(f ,g, h\cdot A,A^{\otimes n-1})=h\cdot \ell^{\,\theta}_{n+2}(f
,g,A^{\otimes n}) \ .
\end{equation}

This essentially means that the field dependent gauge parameter
$[\![f,g]\!]_\theta$ from Proposition <ref>, when evaluated on the
argument $h\cdot A$, should not depend on the derivatives of $h$. The
potentially problematic terms are the ones involving the Lagrangian
multiplier vector fields. However, the functions $\Lambdam^{ij}(A)$
constructed in Corollary <ref> satisfy

\begin{equation}
\Lambdam^{ij}(A)\,A_i=0 \ , \label{tr1}
\end{equation}

and together with Proposition <ref> it now easily follows that

\begin{equation*}
L^{ij}\big(A,\partial(h\,A)\big)=L^{ij}(A,h\,\partial A) \ .
\end{equation*}

Moreover, from (<ref>) the identity (<ref>) also implies

\begin{equation*}
L^{ij}(A,\partial A)\,A_i=0 \ ,
\end{equation*}

and as a consequence the terms with derivatives of $h$ disappear
from $s_{h\,A}^*\{\Phi_{h\,A}(L_f),\Phi_{h\,A}(L_g)\}_{\omega^{-1}}$. It follows that
all terms involving derivatives of $h$ also disappear from
$[\![f,g]\!]_\theta (h\,A)$ and this implies (<ref>). In other
words, the
map $A\mapsto\ell^{\,\theta}_{n+2}(f ,g, \alpha_1,\dots,\alpha_{n-1},A)$
is a derivation, for all $n\geq1$ and $\alpha_1,\dots,\alpha_{n-1}\in\Omega^1(U)$.

For the brackets with a single gauge parameter the desired derivation
property reads

\begin{equation}\label{tr3}
\ell^{\,\theta}_{n+1}(f,h\cdot A, A^{\otimes
n-1})=h\cdot\ell^{\,\theta}_{n+1}(f,A^{\otimes n}) +(-1)^{n-1}
\ell^{\,\theta}_{n+1}(f,h,A^{\otimes n-1})\cdot A \ .
\end{equation}

This relation is more complicated since it involves brackets of
different nature and different graded symmetry. In particular, since
the bracket with two gauge parameters is skew-symmetric in $f$ and
$h$, the relation (<ref>) implies the consistency condition

\begin{equation}\label{tr4}
\ell^{\,\theta}_{n+1}(f,h\cdot A, A^{\otimes
n-1})+\ell^{\,\theta}_{n+1}(h,f\cdot A, A^{\otimes
n})+f\cdot\ell^{\,\theta}_{n+1}(h,A^{\otimes n}) \ .
\end{equation}

For $n=1$ it is easy to
see that the bracket $\ell_2^{\,\theta}$ from
Proposition <ref> satisfies (<ref>), while
for $n=2$ the relevant brackets are

\begin{align*}
\ell^{\,\theta}_3(f,h,A)&=\Pim(\dd f,\dd h,A) \ , \\[4pt]
\ell^{\,\theta}_3(f,A,B)&=-\gamma^{(2)}(\dd f,A,B) + \Pim^\otimes(\dd
f,A,\DD B) + \Pim^\otimes(\dd f,B,\DD A) \\
& \quad \, +
\tfrac12\,\dd A\big(\Pim(\dd
f,B,\,\cdot\,),\,\cdot\,\big) +
\tfrac12\,\dd B\big(\Pim(\dd
f,A,\,\cdot\,),\,\cdot\,\big) \ ,
\end{align*}

for $f,h\in C^\infty(U)$ and $A,B\in\Omega^1(U)$.
One then calculates explicitly

\begin{align*}
\ell^{\,\theta}_3(f,h\cdot
A,A) &= -h\cdot\gamma^{(2)}(\dd f,A,A) + h\cdot\Pim^\otimes(\dd
f,A,\DD A) + \Pim^\otimes(\dd f,A, \dd h\otimes A + h\,\DD A) \\
& \quad \, +\tfrac12\,(\dd h\wedge A)\big(\Pim(\dd
f,A,\,\cdot\,),\,\cdot\,\big) + \tfrac12\, h\cdot\dd A\big(\Pim(\dd
f,A,\,\cdot\,),\,\cdot\,\big) \\
& \quad \, + \tfrac12\, h\cdot\dd A\big(\Pim(\dd
f,A,\,\cdot\,),\,\cdot\,\big) \\[4pt]
&= h\cdot\big(-\gamma^{(2)}(\dd f,A,A) + 2\,\Pim^\otimes(\dd f,A,\DD
A)+\dd A(\Pim(\dd f,A,\,\cdot\,),\,\cdot\,) \big) \\
& \quad \, +\Pim(\dd f,A,\dd h)\,\cdot A +\tfrac12\,\Pim(\dd f, A,\dd
h)\cdot A -
\tfrac12\,\dd h\cdot\Pim(\dd f,A,A) \\[4pt]
A \ ,
\end{align*}

showing that already for $n=2$ the derivation property (<ref>) is
violated. Starting from the next order $n=3$, even the consistency
condition (<ref>) is violated. In the case of a Poisson structure
the higher brackets all satisfy (<ref>) because of their
$\CCA$-linearity, but for generic almost Poisson structures the
derivation property imposes severe restrictions on the
$L_\infty$-algebra. We summarise the present discussion in

Let $(M,\theta)$ be an almost Poisson manifold with non-zero Jacobiator
$\Pim$, and let $U\subseteq M$ be an open
subset. Then the $L_\infty$-structure of
Proposition <ref> and the graded commutative
algebra structure of Proposition <ref> do not
combine into a compatible $P_\infty$-structure on

We do not solve the problem of finding a $P_\infty$-structure on the
almost Poisson gauge algebroid in this paper, but let us briefly mention
some possible ways that one may proceed.

As we have shown above, the
derivation properties are violated by precisely the terms involving
the Lagrangian multipliers $L_f$, which were introduced to cancel
the second term in the commutator of two gauge transformations in
(<ref>). Instead of including $L_f$, one could try to cure the
problem by identifying this term as the contribution from a non-zero
homogeneous subspace $V_2$ in degree $2$. This would lead to a
$3$-term $L_\infty$-algebra, which is likely a $P_\infty$-algebra
under a suitable extension of the truncated exterior product on
$C^\infty(U)\oplus \Omega^1(U)$. However, the gauge theory
interpretation of the space $V_2$ is not clear, and this appears to be
a general feature of physical systems based on almost Poisson algebras:
to maintain all desirable features one inevitably needs to introduce
some auxiliary (unphysical) degrees of freedom, such that the
elimination of these auxiliary variables comes at the price of losing
some of the desired properties (see [46] for an example of
this). In the present situation, we lose the Leibniz rule for the
$L_\infty$-structure, but still retain a well-defined gauge algebra.

Another possibility would be to work instead with a curved
$L_\infty$-algebra. A curving of an $L_\infty$-structure
$\{\ell_n\}_{n=1}^\infty$ on a graded vector space $V$ is an
additional map $\ell_0$ of degree $2$ from the ground field into $V$,
which intertwines with the higher brackets through the homotopy Jacobi
identities extended to $\{\ell_n\}_{n=0}^\infty$; then $\ell_1$ is no
longer a differential and one loses the underlying cochain complex. In
our situation, it
may be possible to redefine the brackets $\ell_n^{\,\theta}$ (via an $L_\infty$-quasi-isomorphism) to absorb
the violation of the Leibniz rule, which may then violate the standard
$L_\infty$-relations, but may instead form a curved
$L_\infty$-algebra. However, a curved $L_\infty$-algebra also necessarily
contains a non-zero homogeneous subspace $V_2$ in degree $2$, whose
meaning at the purely kinematic level of gauge symmetries is not

Finally, one could work with a weaker notion of $P_\infty$-structure,
where the graded commutative product is also replaced by a sequence of
higher products such that the commutativity of the product and the
Leibniz rule hold only up to homotopy. It would be interesting to
explore all of these modifications of the $L_\infty$-structure
exhibited in Proposition <ref> in order
to fully elucidate the structure of the almost Poisson gauge algebroid,
and its extension to noncommutative and nonassociative
gauge transformations.

Our constructions of $P_\infty$-structures above differ in several ways from the treatments of <cit.> and <cit.>, where an $L_\infty$-structure on the de Rham complex, truncated at degree $2$, was constructed for generic almost Poisson structures.

Firstly, the $L_\infty$-structure of Proposition <ref> differs for brackets involving forms of degree higher than one: for example, the $2$-bracket of $E\in\Omega^2(U)$ and $f\in C^\infty(U)$ is simply the almost Poisson  bracket in [11, 44], while in our case the $2$-bracket is

\begin{align*}
\varrho^{\,\theta}_2(f,E) = \{f,E\}_\theta - 2\,\gamma^{(1)\otimes}(\dd f,E) \ .
\end{align*}

On the one hand the brackets of [11, 44] do not define a $P_\infty$-structure, while on the other hand the brackets of Proposition <ref> are not designed to close an arbitrary gauge algebra. For a Poisson bivector $\theta$, the $P_\infty$-algebra of Proposition <ref> contains the Poisson gauge algebra of Proposition <ref>, and following the standard $L_\infty$-algebra formulation of field theories [34, 35], it also contains the higher degree spaces needed to formulate the dynamics of a particular Poisson gauge theory. For an almost Poisson structure, one can use the $P_\infty$-algebra in this way to formulate a gauge algebra without the need of Lagrangian multipliers, at the price of obtaining a closure condition that involves the field equations in addition to the field dependent gauge transformations [34, 35].
For example, in $d=3$ dimensions the brackets appear to define an almost Poisson Chern-Simons theory which is different from that of [11, 44]; it would be interesting to further develop this field theory, which in our symplectic embedding approach can be written down concisely and explicitly to all orders using the brackets (<ref>).

Secondly, an $A_\infty$-structure on $\CCA[[\hbar]]$  is
sketched in <cit.>, where the first few
multiplication maps are deduced up to order $\hbar^2$. However, this
structure is different from what is proposed above, as in their case
the classical limit reduces the $A_\infty$-algebra to the differential
graded commutative algebra $(\CCA,\dd)$, whereas here we propose that
the classical limit should simply be the algebra $\CCA$ without
further structure. In other words, even the differential of the
$A_\infty$-structure should be considered as part of the deformation
quantization of the graded commutative algebra $\CCA$.

§.§ Comparison with the $L_\infty$-bootstrap

According to the prescription of the $L_\infty$-bootstrap approach to
constructing almost Poisson gauge algebroids [11], one
starts with the natural structure maps

\begin{align*}
\ell_1(f)=\dd f \qquad \mbox{and} \qquad \ell_2(f,g)=-\{f,g\}_\theta \ ,
\end{align*}

and attempts to construct the rest of the $L_\infty$-structure by
consistently solving the homotopy Jacobi identities order by order.
Let us briefly comment on the benefits of our approach to deformations of
gauge transformations, based on symplectic
embeddings, over the approach based on the $L_\infty$-bootstrap, which
was used in [43] to propose recursion relations
for the construction of the gauge $L_\infty$-algebras:

* In the $L_\infty$-bootstrap approach, one can define
contributions to the gauge transformations order by order in the
formal deformation parameter $t$. Symplectic embeddings are more
appropriate for computing explicit all orders expressions, which are
sometimes asymptotic expansions of analytic functions known in
closed form. We will
illustrate this in several examples in Section <ref> below.

* Following the method proposed in [43], one
can recursively construct the brackets of the form $\ell_{n+1}(f,\ell_1(g),
\ell_1(h),\dots)$ from previously defined brackets at lower
orders. In the case of Poisson deformations of gauge theories, there is no problem
in restoring $\ell_{n+1}(f,A^{\otimes n})$ from the given brackets
$\ell_{n+1}(f,\ell_1(g), \ell_1(h),\dots)$. However, in the case of
almost Poisson deformations the situation is much more complicated,
as the passage from the brackets $\ell_{n+1}(f,\ell_1(g), \ell_1(h),\dots)$ to
$\ell_{n+1}(f,A^{\otimes n})$ is extremely non-trivial. This problem
is circumvented when working with symplectic embeddings.

* From purely technical and calculational standpoints, the approach of this paper
based on symplectic embeddings is significantly simpler then the
bootstrap approach proposed in [43].

* The $L_\infty$-bootstrap approach describes a linearization of
the complete $A_\infty$-structure of noncommutative gauge
variations, which loses part of the information required to
completely determine the semi-classical limit of the full
noncommutative gauge transformations. The missing information is a
derivation property, which requires a $P_\infty$-algebra to
describe the semi-classical limit, and this is naturally captured
by our symplectic embedding construction.

In previous sections we already looked at the two simplest examples of
Poisson bivector fields: the case of a manifold $M$ with the trivial
Poisson structure $\theta_0=0$, for which the symplectic embedding is
given by the associated symplectic groupoid $(T^*M,\omega_0)$ for $M$,
and $M=\real^d$ with a constant Poisson structure $\theta(x)=\theta$,
whose symplectic embedding is given by the strict deformation of the
cotangent bundle $T^*\real^d$ with symplectic form
$\omega=\omega_0+t\,\theta^*$; the corresponding Poisson
gauge transformations were described in Example <ref> and the
associated $L_\infty$-structure on the gauge algebroid in
Example <ref>. The purpose of this final section is to extend these
basic examples to some more complicated examples, and in particular to
consider an example of an almost Poisson structure. These cases
will generally involve symplectic embeddings of $(M,\theta)$ given by a
formal deformation of the cotangent bundle of $M$, while the
(almost) Poisson gauge algebroid is correspondingly described by an
$L_\infty$-algebra which in general is no longer simply a differential graded Lie
algebra and involves infinitely many brackets.

§.§ Linear Poisson structures

We consider first a large class of Poisson structures whereby the symplectic
embedding can be described as a strict deformation of
$(T^*M,\omega_0)$. Let $\mfg$ be a Lie algebra of dimension $d$ with structure constants in a given basis
denoted by $f^{ij}_k$. On $M=\real^d$ we can define the linear Poisson
bivector field [1]

\begin{equation}\label{e1}
\theta^{ij}(x)=f^{ij}_k\,x^k \ ,
\end{equation}

which, by regarding $\real^d$ as the dual of the Lie algebra $\mfg$, is the
Kirillov-Kostant Poisson structure on $\mfg^*$. In this case the
function $\Sigma(p,p',x)$ in (<ref>) is a
generating function for the Dynkin series for the Baker-Campbell-Hausdorff formula for the
Lie algebra $\mfg=(\real^d)^*$; if $G$ is any Lie group whose Lie
algebra is $\mfg$, then an integrating symplectic groupoid is
$T^*G\simeq G \ltimes\mfg^*$, regarded as the action groupoid with
respect to the coadjoint action, see e.g. [16]. Using the
polydifferential representation constructed
in [29, 26, 49] one finds, in the notation
of Section <ref>, that a local
symplectic embedding of the Poisson structure (<ref>) can be
written as

\begin{equation}\label{e3}
\gamma^i_j(p)= \sum_{n=1}^\infty \, \frac{t^{n-1}\,B_n}{n!} \,
p_{j_1}\cdots p_{j_n} \ ,
\end{equation}

where $B_n$ are the Bernoulli numbers
($B_n=-\frac12,\frac16,0,-\frac1{30},0,\dots$ for $n=1,2,3,4,5,\dots$).

Suppose that the structure constants of $\mfg$ are chosen such that
the $d{\times}d$
matrix $\sf M$, with elements ${\sf M}^{i}_l(p):={f}^{ij_1}_k\, {f}^{kj_2}_l\,p_{j_1}\,p_{j_2}$,
is diagonalizable for all transverse coordinates $p$. Following [49], one may then
construct an analytic function $\gamma^i_j(p)$ whose Taylor expansion
around $t=0$ coincides with the asymptotic series (<ref>). For
this, we observe that (<ref>) can be rewritten as

\begin{equation}\label{e4}
\gamma^i_j(p)=-\tfrac12\,f^{il}_j\,p_{l}+\tfrac1t\,\mathcal{X}\big(-t^2\,{\sf
M}/{2}\big)^i_j \ ,
\end{equation}

where $\mathcal{X}({\sf M})^i_j$ is the matrix-valued function with

\begin{equation*}
\mathcal{X}(u)=\sqrt{\tfrac{u}{2}}\cot\sqrt{\tfrac{u}{2}}-1=\sum_{n=1}^\infty\,
\frac{(-2)^n\,B_{2n}\,u^{n}}{(2n)!} \ ,
\end{equation*}

and the power series converges for $u\in\complex$ with $|u|<\frac12$.
Since ${\sf M}$ is diagonalizable, there exists a non-degenerate
$d{\times}d$ matrix $\sf S$ such that

\begin{equation*}
{\sf M}={\sf S}\, {\sf D}\, {\sf S}^{-1} \ ,
\end{equation*}

where $\sf D$ is the diagonal matrix whose entries are the eigenvalues
$\lambda_1(p),\dots,\lambda_d(p)$ of $\sf M$ on the diagonal. Thus (<ref>) becomes

\begin{equation*}
\gamma^i_j(p)=-\tfrac12\,f^{il}_j\,p_{l}+\tfrac1t\,\big[{\sf S}\,
\mathcal{X}\big(-t^2\,{\sf D}/{2}\big)\, {\sf S}^{-1}\big]^{i}_j \ ,
\end{equation*}

\begin{equation*}
\mathcal{X}({\sf D})=\begin{pmatrix}
\mathcal{X}(\lambda_1) & & \\ & \ddots & \\ & &
\mathcal{X}(\lambda_d) \end{pmatrix}
\ .
\end{equation*}

Let us now consider two particular examples.

Let $\mfg=\mathfrak{su}(2)$ with the Lie-Poisson structure

\begin{align*}
\theta^{ij}(x) = 2\,\varepsilon^{ij}{}_k\,x^k
\end{align*}

on $\mathfrak{su}(2)^*=\real^3$, where
$\varepsilon^{ijk}$ is the Levi-Civita symbol in three dimensions, and
we used the standard Euclidean inner product on $\real^3$ to raise and lower indices: $\varepsilon^{ij}{}_k:=\varepsilon^{ijl}\,\delta_{lk}$;
the factor of $2$ is just for convenience. In this case the generalized Bopp shift (<ref>) is given by [49]

\begin{align*}
\pi_\theta(x,p)^i = x^i - t\,\varepsilon^{ij}{}_k\,p_j\,x^k + t^2\,\chi\big(t^2\,|p|^2\big)\,\big(x^i\,|p|^2-p^i\,p_j\,x^j\big) \ ,
\end{align*}

from which one calculates

\begin{equation*}
\gamma^i_j(p)= -
\varepsilon_j{}^{ik}\,p_k + t\, \chi\big(t^2\,|p|^2\big) \,\big( |p|^2\,
\delta^i_j- p_j\,p^i\big) \ ,
\end{equation*}

\begin{equation*}
\chi(u)=\tfrac1u\,\big(\sqrt{u}\cot\sqrt{u}-1\big) \qquad \mbox{with}
\quad \chi(0) = -\tfrac13 \ ,
\end{equation*}

and $|p|^2:=\delta^{ij}\,p_i\,p_j$.

According to (<ref>), the corresponding deformation of abelian
gauge transformations yields the Poisson gauge transformations [43]

\begin{equation*}
\delta_{f}^\theta A=\dd f+t\, x\cdot (\nabla A\times\nabla f)+t\,A\times\nabla
f+t^2\, \chi\big(t^2\,|A|^2\big)\,\big(|A|^2\,\dd f- (A\cdot\nabla
f)\,A\big) \ ,
\end{equation*}

where $A\times\nabla f:=\ast(A\wedge\dd f)$, with $\ast$ the Hodge duality operator on
the Euclidean vector space $\real^3$, while
$|A|^2:=\delta^{ij}\,A_i\,A_j$ and $A\cdot\nabla
f:=\delta^{ij}\,A_i\,\partial_j f$; we also abbreviated
$\{A,f\}_\theta=: x\cdot (\nabla A\times\nabla f)$ at $x\in\real^3$. This may be verified directly to
close the Poisson gauge algebra (<ref>) and to have the
correct deformation property (<ref>). It encodes the gauge symmetry
of rotationally invariant Poisson gauge theories, which by
Proposition <ref> is generated
by the action of the $P_\infty$-algebra with infinitely many non-vanishing brackets in coincident
gauge field entries given by

\begin{align*}
\ell^{\,\theta}_1(f) &= \dd f \ , \\[4pt]
\ell^{\,\theta}_2(f,g) &= -x\cdot(\nabla f \times \nabla g) \ , \\[4pt]
\ell^{\,\theta}_2(f,A) &= x\cdot (\nabla A\times\nabla f)+ A\times\nabla f \ ,
\\[4pt]
\ell^{\,\theta}_{2n+1}\big(f,A^{\otimes 2n}\big) &= B_{2n}\,\big(|A|^2\big)^{n-1}
\, \big(|A|^2\,\dd f - (A\cdot\nabla f)\,
A\big) \ ,
\end{align*}

for $n\geq1$.

Let $\mfg$ be the $d$-dimensional $\kappa$-Minkowski algebra which
yields the Kirillov-Kostant structure

\begin{equation*}
\theta_a^{ij}(x)=2\,\big(a^i\,x^j-a^j\,x^i\big) \ ,
\end{equation*}

parameterized by a fixed constant vector $(a^i)\in\real^d$. For this Poisson structure one finds [50]

\begin{equation*}
\gamma^i_j(p)= \big(\tfrac1t\,\sqrt{1+t^2\,\langle a, p\rangle^2}-\tfrac1t+\langle a,
p\rangle \big) \,
\delta^i_j -a^i\,p_j \ ,
\end{equation*}

where $\langle a, p\rangle :=a^i\,p_i$ is the usual pairing between
$\real^d=\mfg^*$ and $\mfg$.
The corresponding deformation of abelian gauge transformations
becomes the Poisson gauge transformations

\begin{equation*}
\delta_{f}^{\theta_a} A=\big(\sqrt{1+t^2\,(\iota_aA)^2}+t\,\iota_a A\big) \, \dd
f + t\,\{A,f\}_{\theta_a} - t\,(\iota_a\dd f)\, A \ ,
\end{equation*}

where $\iota_a$ denotes interior multiplication with the constant
vector field $a^i\,\partial_i$ on $\real^d$. By expanding the square root
in its Taylor series around $t=0$ using the binomial series

\begin{align*}
\sqrt{1+u^2} = 1 - \sum_{n=1}^\infty \, \frac2n \, \binom{2n-2}{n-1}
\, \Big(-\frac{u^2}4\Big)^n \ ,
\end{align*}

we can calculate the non-zero coincident gauge field
brackets of the corresponding $P_\infty$-algebra from
Proposition <ref> to get

\begin{align*}
\ell^{\,\theta_a}_1(f) &= \dd f \ , \\[4pt]
\ell^{\,\theta_a}_2(f,g) &= -\{f,g\}_{\theta_a} \ , \\[4pt]
\ell^{\,\theta_a}_2(f,A) &= \{A,f\}_{\theta_a} + \iota_a(A\wedge\dd
f) \ , \\[4pt]
\ell^{\,\theta_a}_{2n+1}\big(f,A^{\otimes 2n}\big) &= -\frac1{2^{2n-1}} \,
\frac{(2n-2)!\,(2n)!}{(n-1)!\,n!}
\ (\iota_aA)^{2n} \,
\dd f \ ,
\end{align*}

for $n\geq1$.

§.§ Poisson families

We will now show that a rather large class of
non-linear Poisson structures
on $M=\real^2$ admit symplectic embeddings that yield Poisson gauge
algebras whose homotopy algebraic structures are given by differential graded Poisson algebras,
though in a different form than what we have encountered thus far in
this paper. The basic idea is to start from the outset with smooth
families of bivectors $\theta_t\in\mfX^2(\real^2)$,
parameterized smoothly by $t\in[0,1]$. Any such bivector defines a family of
Poisson structures on $\real^2$ by dimensional
reasons and can be written as

\begin{align}\label{e24}
\theta_t^{ij}(x) = \vartheta_t(x)\,\varepsilon^{ij} \ ,
\end{align}

where $\varepsilon^{ij}$ is the Levi-Civita symbol in two
dimensions. We assume that the smooth functions $\vartheta_t(x)$
satisfy two properties:

(a) $\vartheta_t(x)\neq0$ for all $x\in\real^2$ and $t\in[0,1]$; and

(b) $\vartheta_0(x)=1$ for all $x\in\real^2$.

From this we can construct a symplectic embedding of
$(\real^2,\theta_t)$ by finding a one-form $b=b_i(x)\,\dd x^i$ on
$\real^2$ whose components solve the first order differential equation

\begin{equation}\label{e25}
\ ,
\end{equation}

together with the Jacobian equation

\begin{equation}\label{e26}
\det(\partial_{i}b_{j}) = 0 \ .
\end{equation}

For given $\vartheta_t(x)$ the solution of (<ref>) and (<ref>)
was constructed in [28], where a first order classical mechanics on
the cotangent bundle of $\real^2$ was proposed whose canonical
quantization gives a quantization of Poisson algebras
with non-constant bivectors. The action functional of this one-dimensional
topological sigma-model is given by

\begin{align*}
{\mathcal S}(X,P) = \int_\real \, \langle P,\dd X\rangle + \frac
t2\,(P+X^*b)\wedge\dd(P+X^*b) \ ,
\end{align*}

where $(X,P):\real\to T^*\real^2=\real^2\times(\real^2)^*$ are
(suitably supported) smooth maps, with $\langle\,\cdot\,,\,\cdot\,\rangle$ the
pairing between $(\real^2)^*$ and $\real^2$. Hamiltonian reduction of the
phase space of this sigma-model by its rank $2$ second class constraints
defines Dirac brackets which, in our language, yield a symplectic embedding of
the family (<ref>) with cosymplectic structure

\begin{align}\label{e29}
\omega_t^{-1} =
\tfrac t2\,\vartheta_t(x)\, \varepsilon^{ij}\, \partial_i\wedge\partial_j +
\tfrac12\,\gamma_t{}^i_j(x)\,
\big(\partial_i\wedge\tilde\partial{}^j+\tilde\partial{}^j\wedge\partial_i\big)
\ ,
\end{align}

\begin{align}\label{eq:gammaijx}
\gamma_t{}^i_j(x) =
\vartheta_t(x)\,\big(\delta_j^i-t\,\varepsilon^{ik}\,
\partial_kb_j(x)\big) \ ,
\end{align}

and the Jacobian condition (<ref>) ensures the Lagrangian section
By condition (a) above this is indeed non-degenerate for all
$t\in[0,1]$, and by condition (b) it coincides with the canonical
cosymplectic structure on $T^*\real^2$ at $t=0$. In particular, it
defines a strict deformation of the cotangent symplectic
groupoid $(T^*\real^2,\omega_0)$ for $\real^2$.

The important new feature here, compared to our previous formulation
in (<ref>), is that the matrix (<ref>) does not depend
on the transverse coordinates $p$. Regarding it as a $(1,1)$-tensor field on
$\real^2$, according to (<ref>) the deformation of abelian gauge
transformations by the Poisson family (<ref>) is given by the Poisson
gauge transformations

\begin{align}\label{e30}
\delta_f^{\theta_t}A = \gamma_t(\dd f,\,\cdot\,) +
t\,\{A,f\}_{\theta_t} \ .
\end{align}

The $L_\infty$-algebra formulation of these gauge symmetries has the
remarkable property given by

The Poisson gauge transformations (<ref>) are generated by the
action of the
family of differential graded Poisson algebras on
$C^\infty(\real^2)\oplus\Omega^1(\real^2)$ with non-vanishing brackets

\begin{align*}
\ell_1^{\,\theta_t}(f) = \gamma_t(\dd f,\,\cdot\,) \ , \quad
\ell_2^{\,\theta_t}(f,g) = -\{f,g\}_{\theta_t} \qquad \mbox{and}
\qquad \ell_2^{\,\theta_t}(f,A) = \{A,f\}_{\theta_t} \ ,
\end{align*}

for $f,g\in C^\infty(\real^2)$ and $A\in\Omega^1(\real^2)$.

This is a simple consequence of the fact that the components of the
Poisson bivector (<ref>) do not depend on $p$, the Jacobi
identities for the Poisson brackets, and the Lagrangian zero section condition
$\{p_i,p_j\}_{\omega_t^{-1}}=0$. In coordinates

\begin{align*}
\ell^{\,\theta_t}_1(f) = \{f,p_i\}_{\omega_t^{-1}}\,\dd x^i \ ,
\end{align*}

and the differential condition $\big(\ell_1^{\,\theta_t}\big)^2=0$ follows
from the Jacobi identity for the Poisson bracket
$\{\,\cdot\,,\,\cdot\,\}_{\omega_t^{-1}}$ along with
$\{p_i,p_j\}_{\omega_t^{-1}}=0$. Similarly, the derivation property

\begin{align*}
\ell_1^{\,\theta_t}\{f,g\}_{\theta_t} =
\big\{\ell_1^{\,\theta_t}(f),g\big\}_{\theta_t} +
\big\{f,\ell^{\,\theta_t}_1(g)\big\}_{\theta_t}
\end{align*}

holds as a consequence of the Jacobi identity for the bracket
$\{\,\cdot\,,\,\cdot\,\}_{\omega_t^{-1}}$. The homotopy Jacobi
identity $\CJ_3=0$ is simply the graded Jacobi identity for the Lie bracket
$\{\,\cdot\,,\,\cdot\,\}_{\theta_t}$, and the derivation properties of the brackets are clear.

Proposition <ref> implies that there is no need to introduce higher
brackets in the $L_\infty$-structure on the gauge algebra in this
case, provided that one “twists” the differential, which in our
previous treatments always coincided with the exterior derivative
$\dd$, in such a way that the new differential is a derivation of the
Poisson algebra corresponding to (<ref>). This generalizes
Example <ref> for constant Poisson structures, recovered here in the case that $\vartheta_t(x)=1$ for all
$x\in\real^2$ and $t\in[0,1]$, for which $b=0$. The “twisting” here is
captured automatically by the symplectic embedding approach to the
deformation of gauge transformations that we developed in the present

Consider the family of rotationally symmetric Poisson structures
(<ref>) with

\begin{align*}
\vartheta_t(x) = \frac1{1+t\,|x|^2} \ ,
\end{align*}

where $|\cdot|$ is the standard Euclidean norm. In this case, one

\begin{align*}
b_i(x) = -\tfrac14\,|x|^2\,\varepsilon_{ij}\,x^j
\end{align*}

as a solution to (<ref>) and (<ref>). The twisted differential
from Proposition <ref> then has components

\begin{align*}
\ell_1^{\,\theta_t}(f)_i = \frac1{1+t\,|x|^2} \, \Big( \big(1+\tfrac t4\,|x|^2\big)\,\partial_if +
\tfrac
t2\,\varepsilon_{ik}\,x^k\,x^l\,\varepsilon_l{}^j\,\partial_jf \Big)
\ ,
\end{align*}

for $i=1,2$.

§.§ Magnetic Poisson structures

Let $M=T^*Q$ be the cotangent bundle of a $d$-dimensional manifold
$Q$, with bundle projection $\varpi:M\to Q$. The manifold $M$ is a symplectic manifold with
canonical symplectic two-form which we denote by $\sigma_0$. We write local
coordinates on $M$ as $(x^i)=(q^a,q^*_a)$, with $a=1,\dots,d$ and
$i=1,\dots,2d$, where $(q^a)$ are local coordinates on $Q$ and
$(q^*_a)$ are canonically conjugate coordinates in the normal
directions to the zero section $Q\subset M$. We denote the
corresponding derivatives by $(\partial_i)=(\partial_a,\partial_*^a)$,
where $\partial_a=\partial/\partial q^a$ and
$\partial_*^a=\partial/\partial q^*_a$.

Let $B\in\Omega^2(Q)$ be an arbitrary two-form on the base manifold
$Q$. Its pullback to $M$ deforms the symplectic structure $\sigma_0$ to an almost
symplectic form

\begin{align*}
\sigma_B = \sigma_0-\varpi^*B
\end{align*}

which is closed if and only if $B$ is a closed two-form on $Q$. The
inverse $\theta_B=\sigma_B^{-1}$ is an $H$-twisted Poisson structure
on $M$, with twisting three-form $H\in\Omega^3(M)$ given by

\begin{align*}
H = \varpi^*\dd B \ .
\end{align*}

The bivector $\theta_B$ defines a magnetic Poisson structure on
$M$; the terminology comes from monopole physics where the
two-form $B$ plays the role of a magnetic field.
In local coordinates, where $B=\frac12\,B_{ab}(q) \, \dd q^a\wedge\dd
q^b$ and $H=\frac1{3!}\,H_{abc}(q)\,\dd q^a\wedge\dd q^b\wedge\dd
q^c$, the bivector $\theta_B$ reads

\begin{align*}
\theta_B = \tfrac12\,(\partial_a\wedge
\partial_*^a+\partial_*^a\wedge\partial_a) + \tfrac12\,B_{ab}(q)\,
\partial_*^a\wedge\partial_*^b \ ,
\end{align*}

and the Jacobiator

\begin{align*}
\Pim_B=[\theta_B,\theta_B]=\tfrac1{3!}\,
\end{align*}

has non-vanishing components
only in the normal directions to the zero section $Q\subset M$.

Deformation quantization of such twisted Poisson manifolds was originally considered
in [55] using Kontsevich's formalism. Their higher
geometric quantization was developed in [13] by regarding the
three-form $H$ as the curvature of a trivial gerbe on $M$, and the
extension to non-trivial gerbes is discussed in [14];
see [67] for a review of the different perspectives to
quantization of magnetic Poisson structures. In the language of
Remark <ref>, the almost cosymplectic structure
$\sigma_B^{-1}$ is equivalent to the canonical cosymplectic structure
$\sigma_0^{-1}$ by means of a $B$-transformation, which leads to many simplifications in its symplectic
embedding construction: many components of the tensors $\gamma^{(n)}$ and
$\theta^{(n)}$ of the symplectic embedding vanish as various
combinations of derivatives acting on the components of $\theta_B$ and
$\Pim_B$, which are functions on the base manifold $Q$, are identically zero; in particular
$\Pim_B^{ijk}\,\partial_k\theta_B^{lm}=0$, see e.g. [45].

Linear magnetic Poisson structures.

We specialise now to
$Q=\real^d$ with the linear magnetic Poisson structure defined by the two-form

\begin{align*}
B_{ab}(q) = \tfrac12\,H_{abc}\,q^a \ ,
\end{align*}

where $H$ is a constant three-form on $\real^d$. This is
the twisted Poisson analog of a constant Poisson structure: In this
case the components of the Jacobiator $\Pim_B$ are constant and the generalized Bopp shift (<ref>) reads

\begin{align*}
\pi_{\theta_B}(x,p)^i = x^i-\tfrac t2\,\theta_B^{ij}(x)\,p_j \ ,
\end{align*}

so $\gamma^{(n)}=0$ and $\theta^{(n)}=0$ for all
$n\geq2$ [46]. We denote the fibre coordinates of $T^*M$ by
$(p_i)=(\xi_a,\xi_*^a)$. In the notation of Sections <ref>
and <ref>, the non-zero components of the symplectic
embedding are then given by

\begin{align*}
\gamma_{ab}(\xi_*) = -\tfrac12\,H_{abc}\,\xi_*^c \ , \quad
\underline{\theta}\,^a_b=-\underline{\theta}\,_{a}^b = \delta^a_b
\qquad \mbox{and} \qquad \underline{\theta}_{\,ab}(q,\xi_*) =
H_{abc}\,(q^c-t\,\xi_*^c) \ .
\end{align*}

For the corresponding deformation of abelian gauge transformations, we
note that the tensors $\Upm^{(n)}$ determined from Corollary <ref> also vanish for all
$n\geq2$, so that in this case $\Lambdam^{ij}(A) =
-\tfrac12\,\Pim_B^{ijk}\,A_k$. This depends only on the transverse
components of the gauge fields $A\in\Omega^1(M)$ to the zero section
$Q\subset M$, so we decompose gauge fields as

\begin{align}\label{eq:Aqdecomp}
A = A_i(x)\, \dd x^i = \alpha_a(q,q^*)\, \dd q^a + \alpha^a_*(q,q^*)\, \dd q_a^*
\end{align}

and obtain explicitly

\begin{align*}
\Lambdam_{ab}(\alpha_*) = -\tfrac12\,H_{abc}\,\alpha_*^c
\end{align*}

as the only non-zero components of $\Lambdam^{ij}(A)$.
The Lagrangian multipliers $L_f$ are constructed from
Proposition <ref> as formal power series
which are now given explicitly to all orders by the constant Jacobiator
$\Pim_B$ on $M$. We can write them in terms of an analytic
function on the jet space $J^1T^*M$ whose Taylor series around
$t=0$ coincides with the asymptotic series (<ref>). For this, we
introduce the $2d{\times}2d$ matrix ${\sf M}$ with elements

\begin{align*}
{\sf M}^{a}_b(\alpha_*,\partial_* \alpha_*)=-H_{bce}\,\alpha_*^c\,\partial_*^a\alpha_*^e
\end{align*}

and observe that the non-vanishing components in (<ref>) can be rewritten as

\begin{align}\label{eq:Labsum}
L_{ab}(\alpha_*,\partial_* \alpha_*) = -\frac12\,H_{ace}\,\alpha_*^e \
\sum_{n=0}^\infty \, \Big(\frac{t^2}2\Big)^n \, \big({\sf
M}^n\big)^{c}_{b} = -\frac12\,H_{ace}\,\alpha_*^e\,\big[\big(\one -
\tfrac{t^2}2 \,{\sf M}\big)^{-1}\big]_{b}^{c} \ .
\end{align}

The Lagrangian multipliers

\begin{align*}
L_f={L_f}_a\,\partial_*^a \qquad \mbox{with} \quad {L_f}_a = t^2\, L_{ab}(\alpha_*,\partial_*\alpha_*) \, \partial_*^bf
\end{align*}

are then transverse to $Q\subset M$, depend only on the transverse
components of the gauge fields and their normal derivatives, and are
determined entirely by the normal derivatives of gauge parameters
$f\in C^\infty(M)$.

From Remark <ref>, the corresponding deformation
of abelian gauge transformations is given by the almost Poisson gauge

\begin{align*}
\delta_f^{\theta_B} A = \delta_f^{\theta_B}\alpha_a \, \dd q^a +
\delta_f^{\theta_B}\alpha^a_* \, \dd q^*_a \ ,
\end{align*}

\begin{align}
\delta_f^{\theta_B}\alpha_a &=
\partial_af+t\,\{\alpha_a,f\}_{\theta_B}
+ \tfrac t2\,
H_{abc}\,\alpha_*^c\,\partial_*^bf +
\nonumber \\
& \quad \, \hspace{1cm} +\tfrac{t^2}2\, H_{bln}\,\alpha_*^n\,\partial_*^kf\, \big[\big(\one -
\tfrac{t^2}2 \,{\sf M}\big)^{-1}\big]_{k}^{l} \,
\big(\partial_a\alpha_*^b-\partial_*^b\alpha_a + \tfrac t2\,
+t\,\{\alpha_a,\alpha_*^b\}_{\theta_B} \nonumber \\
& \quad \, \hspace{7cm} + t^2\,
\big) \ , \nonumber \\[4pt]
\delta_f^{\theta_B}\alpha_*^a &= \partial_*^af +
t\,\{\alpha_*^a,f\}_{\theta_B} +
\nonumber \\
& \quad \, \hspace{1cm}+\tfrac{t^2}2\, H_{bln}\,\alpha_*^n\,\partial_*^kf\, \big[\big(\one -
\tfrac{t^2}2 \,{\sf M}\big)^{-1}\big]_{k}^{l} \,
\big(\partial_*^a\alpha_*^b-\partial_*^b\alpha_*^a
+t\,\{\alpha_*^a,\alpha_*^b\}_{\theta_B} \nonumber\\
& \quad \, \hspace{7cm} +
\big) \ . \label{eq:deltaalpha}
\end{align}

These satisfy the almost Poisson gauge algebra (<ref>)
with the field dependent gauge parameter

\begin{align}
[\![f,g]\!]_{\theta_B}(\alpha_*) &= \{f,g\}_{\theta_B} -
\nonumber \\
& \quad \, -\tfrac{t^3}4\,H_{bsn}\,H_{crm}\,\alpha_*^n\,\alpha_*^m\,
\partial_*^kf\,\partial_*^pg\, \big[\big(\one -
\tfrac{t^2}2 \,{\sf M}\big)^{-1}\big]_{k}^{s} \, \big[\big(\one -
\tfrac{t^2}2 \,{\sf M}\big)^{-1}\big]_{p}^{r} \label{eq:fgthetaBA}
\\
& \quad \, \hspace{4cm} \times
\big(\partial_*^b\alpha_*^c-\partial_*^c\alpha_*^b + t\,
\{\alpha_*^b,\alpha_*^c\}_{\theta_B}
\big) \ . \nonumber
\end{align}

Notice that, apart from the terms involving the almost Poisson brackets
$\{\,\cdot\,,\,\cdot\,\}_{\theta_B}$, the gauge variations of the
transverse components $\alpha_*^a$ in (<ref>) as well as
the brackets (<ref>) are
determined completely by normal components to the embedding $Q\subset

The $L_\infty$-algebra of these almost Poisson gauge symmetries is
given by Proposition <ref>, with infinitely
many non-vanishing brackets which in
this example greatly simplify due to the vanishing of most tensors in
the symplectic embedding and in the Lagrangian multipliers.
They can be read
off directly from (<ref>) and (<ref>) by reinstating
the formal power series expansion (<ref>). For the coincident
gauge field brackets involving a single gauge parameter, we use the
decomposition (<ref>) to similarly write
$\ell_{n+1}^{\,\theta_B}\big(f,A^{\otimes n}\big)\in\Omega^1(M)$ in
component form

\begin{align*}
\ell_{n+1}^{\,\theta_B}\big(f,A^{\otimes n}\big) =
\lambda^{\theta_B}_{n+1}\big(f,A^{\otimes n}\big)_a \, \dd q^a +
\lambda^{\theta_B}_{n+1}\big(f,\alpha_*^{\otimes n}\big)^a_* \, \dd q^*_a \ ,
\end{align*}

\begin{align*}
\lambda_1^{\theta_B}(f)_a &= \partial_af \ , \\[4pt]
\lambda_2^{\theta_B}(f,A)_a &= \{\alpha_a,f\}_{\theta_B} +
\tfrac12\,H_{abc}\,\alpha_*^c\,\partial_*^bf
\ , \\[4pt]
\lambda_3^{\theta_B}\big(f,A^{\otimes2}\big)_a &=
\,
\big(\partial_a\alpha_*^b-3\,\partial_*^b\alpha_a\big)
\ , \\[4pt]
\lambda_4^{\theta_B}\big(f,A^{\otimes3}\big)_a &= 3\,
\,
\big(\{\alpha_a,\alpha_*^b\}_{\theta_B}
\tfrac12\,H_{ace}\,\alpha_*^e\,\partial_*^c\alpha_*^b\big)
\ , \\[4pt]
\lambda_{2n-1}^{\theta_B}\big(f,A^{\otimes2n-2}\big)_a &=
\,
H_{blm}\, H_{kb_1c_1}\,H_{a_1b_2c_2}\cdots
H_{a_{n-3}b_{n-2}c_{n-2}}\,\alpha_*^m\, \alpha_*^{b_1}\cdots\alpha_*^{b_{n-2}}\,\partial_*^kf
\\
& \quad \, \hspace{1cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots
\partial_*^{a_{n-3}}\alpha_*^{c_{n-3}} \,
\partial_*^l\alpha_*^{c_{n-2}}\,\big(\partial_a\alpha_*^b-3\,\partial_*^b\alpha_a\big)
\ , \\[4pt]
\lambda_{2n}^{\theta_B}\big(f,A^{\otimes2n-1}\big)_a &=
\,
H_{blm}\, H_{kb_1c_1}\,H_{a_1b_2c_2}\cdots
H_{a_{n-3}b_{n-2}c_{n-2}}\,\alpha_*^m\, \alpha_*^{b_1}\cdots\alpha_*^{b_{n-2}}\,\partial_*^kf\,
\\
& \quad \, \hspace{1cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots\partial_*^{a_{n-3}}\alpha_*^{c_{n-3}}\,\partial_*^l\alpha_*^{c_{n-2}}
\,
\big(\{\alpha_a,\alpha_*^b\}_{\theta_B} + \tfrac12\,H_{ace}\,\alpha_*^e\,\partial_*^c\alpha_*^b
\big) \ ,
\end{align*}

\begin{align}
\lambda_1^{\theta_B}(f)_*^a &= \partial_*^af \ , \nonumber \\[4pt]
\lambda_2^{\theta_B}(f,\alpha_*)_*^a &= \{\alpha_*^a,f\}_{\theta_B} \ ,
\nonumber \\[4pt]
\lambda_3^{\theta_B}\big(f,\alpha_*^{\otimes2}\big)_*^a &=
\,
\big(\partial_*^a\alpha_*^b-3\,\partial_*^b\alpha_*^a\big)
\ ,
\nonumber \\[4pt]
\lambda_4^{\theta_B}\big(f,\alpha_*^{\otimes3}\big)_*^a &=
\,
\{\alpha_*^a,\alpha_*^b\}_{\theta_B}
\ , \label{eq:lambdamagnetic}\\[4pt]
\lambda_{2n-1}^{\theta_B}\big(f,\alpha_*^{\otimes2n-2}\big)_*^a &=
\,
H_{blm}\, H_{kb_1c_1}\,H_{a_1b_2c_2}\cdots
H_{a_{n-3}b_{n-2}c_{n-2}}\,\alpha_*^m\, \alpha_*^{b_1}\cdots\alpha_*^{b_{n-2}}\,\partial_*^kf
\nonumber \\
& \quad \, \hspace{1cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots
\partial_*^{a_{n-3}}\alpha_*^{c_{n-3}} \,
\partial_*^l\alpha_*^{c_{n-2}}\,\big(\partial_*^a\alpha_*^b-3\,\partial_*^b\alpha_*^a\big)
\ , \nonumber \\[4pt]
\lambda_{2n}^{\theta_B}\big(f,\alpha_*^{\otimes2n-1}\big)_*^a &=
\,
H_{blm}\, H_{kb_1c_1}\,H_{a_1b_2c_2}\cdots
H_{a_{n-3}b_{n-2}c_{n-2}}\,\alpha_*^m\, \alpha_*^{b_1}\cdots\alpha_*^{b_{n-2}}\,\partial_*^kf\,
\nonumber \\
& \quad \, \hspace{1cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots\partial_*^{a_{n-3}}\alpha_*^{c_{n-3}}\,\partial_*^l\alpha_*^{c_{n-2}}
\,
\{\alpha_*^a,\alpha_*^b\}_{\theta_B} \ , \nonumber
\end{align}

for $n\geq3$. For the coincident gauge field brackets involving two
gauge parameters, after some further tedious calculation and simplification we obtain for the first few brackets

\begin{align}
\ell_2^{\,\theta_B}(f,g) &= -\{f,g\}_{\theta_B} \ , \nonumber \\[4pt]
\ell_3^{\,\theta_B}(f,g,\alpha_*) &=
\ , \nonumber \\[4pt]
\ell_4^{\,\theta_B}\big(f,g,\alpha_*^{\otimes2}\big) &= 0 \ ,
\nonumber \\[4pt]
\ell_5^{\,\theta_B}\big(f,g,\alpha_*^{\otimes 3}\big) &= -\tfrac32\, H_{bke}\,H_{cpm}\,\alpha_*^e\,\alpha_*^m\,
\partial_*^kf\,\partial_*^pg\,
\big(\partial_*^b\alpha_*^c-\partial_*^c\alpha_*^b\big) \ ,
\nonumber \\[4pt]
\ell_6^{\,\theta_B}\big(f,g,\alpha_*^{\otimes 4}\big) &= -6\,H_{bke}\,H_{cpm}\,\alpha_*^e\,\alpha_*^m\,
\partial_*^kf\,\partial_*^pg\,
\{\alpha_*^b,\alpha_*^c\}_{\theta_B}
\ , \nonumber \\[4pt]
\ell_7^{\,\theta_B}\big(f,g,\alpha_*^{\otimes5}\big) &=
\nonumber \\
& \quad \, \qquad \times \,
\big(\partial_*^a\alpha_*^b\,\partial_*^r\alpha_*^c + \tfrac12\,
\partial_*^b\alpha_*^a\,\partial_*^r\alpha_*^c + \tfrac12\,
\partial_*^a\alpha_*^b\,\partial_*^c\alpha_*^r\big) \ , \label{eq:ellfgmagnetic1}
\end{align}

together with the higher order brackets

\begin{align}
\ell_{2n}^{\,\theta_B}\big(f,g,\alpha_*^{\otimes2n-2}\big) &=
\frac{(2n-2)!}{2^{n-1}}
\,
\cdots
\alpha_*^{b_{n-3}}\,\partial_*^kf\,\partial_*^pg\,\{\alpha_*^b,\alpha_*^c\}_{\theta_B}
\nonumber \\
& \quad \, \times \,\Big(H_{a_1b_2c_2}\cdots
\nonumber \\
& \quad \, \hspace{2cm} \times \, \big(\delta_k^s
\, H_{pb_1c_1}\,\partial_*^r\alpha_*^{c_{n-3}} + \delta_p^r\,
H_{kb_1c_1}\,\partial_*^s\alpha_*^{c_{n-3}}\big) \nonumber \\
& \quad \, \qquad + \sum_{l=1}^{n-4} \,
H_{a_{l-1}b_lc_l} \nonumber \\
& \quad \, \qquad \hspace{1.5cm} \times \, H_{pb_{l+1}c_{l+1}}\,H_{a_{l+1}b_{l+2}c_{l+2}}\cdots
H_{a_{n-4}b_{n-3}c_{n-3}} \nonumber \\
& \quad \, \qquad \hspace{2cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots\partial_*^{a_{l-1}}\alpha_*^{c_{l-1}}\,\partial_*^s\alpha_{*}^{c_l}
\nonumber \\
& \quad \, \qquad \hspace{2.5cm} \times \, \partial_*^{a_{l+1}}\alpha_*^{c_{l+1}}\cdots\partial_*^{a_{n-4}}\alpha_*^{c_{n-4}}\,\partial_*^r\alpha_*^{c_{n-3}}
\Big) \ , \label{eq:ellfgmagnetic2}
\end{align}

\begin{align}
\ell_{2n+1}^{\,\theta_B}\big(f,g,\alpha_*^{\otimes2n-1}\big) &=
\,
\nonumber \\
& \quad \, \times \, \bigg(H_{a_1b_2c_2}\cdots
\nonumber \\
& \quad \, \qquad \times \, \Big(\frac12\,
- \partial_*^c\alpha_*^b\big) \nonumber \\
& \quad \qquad \hspace{2cm} \times \, \big(\delta_k^s\,
H_{pb_1c_1}\,\partial_*^r\alpha_*^{c_{n-2}} + \delta_p^r\,
H_{kb_1c_1}\,\partial_*^s\alpha_*^{c_{n-2}}\big) \nonumber \\
& \quad \, \qquad \qquad +
H_{pb_1c_1}\,\partial_*^r\alpha_*^{c_{n-3}} + \delta_p^r\,
H_{kb_1c_1}\,\partial_*^s\alpha_*^{c_{n-3}}\big) \Big) \nonumber \\
& \quad \, \qquad +
\Big(\frac12\,H_{a_{n-3}b_{n-2}c_{n-2}}\,\partial_*^{a_{n-3}}\alpha_*^{c_{n-3}}\,\partial_*^r\alpha_*^{c_{n-2}}\,\big(\partial_*^b\alpha_*^c
- \partial_*^c\alpha_*^b\big) \nonumber \\
& \quad \, \qquad \hspace{2cm} +
\nonumber \\
& \quad \, \qquad \qquad \times \, \sum_{l=1}^{n-4} \,
H_{a_{l-1}b_lc_l} \nonumber \\
& \quad \qquad \qquad \hspace{1.5cm} \times \, H_{pb_{l+1}c_{l+1}}\,H_{a_{l+1}b_{l+2}c_{l+2}}\cdots
H_{a_{n-4}b_{n-3}c_{n-3}} \nonumber \\
& \quad \qquad \qquad \hspace{1.5cm} \times
\partial_*^{a_1}\alpha_*^{c_1}\cdots\partial_*^{a_{l-1}}\alpha_*^{c_{l-1}}\,\partial_*^s\alpha_*^{c_l}\,\partial_*^{a_{l+1}}\alpha_*^{c_{l+1}}\cdots\partial_*^{a_{n-4}}\alpha_*^{c_{n-4}}
\nonumber \\
& \quad \, \qquad + \frac12\,H_{kb_1c_1}\,H_{a_1b_2c_2}\cdots
H_{a_{n-4}b_{n-3}c_{n-3}}\,H_{pb_{n-2}c_{n-2}} \nonumber \\
& \quad \, \qquad \hspace{2cm} \times \,
\partial_*^{a_1}\alpha_*^{c_1}\cdots\partial_*^{a_{n-4}}\alpha_*^{a_{n-4}c_{n-4}}\,\partial_*^s\alpha_*^{c_{n-3}}\,\partial_*^r\alpha_*^{c_{n-2}}\bigg)
\ , \label{eq:ellfgmagnetic3}
\end{align}

for $n\geq4$.

The forms of the transverse gauge transformations in (<ref>) and
the brackets (<ref>), as well as their corresponding
$L_\infty$-structures (<ref>) and
(<ref>)–(<ref>), suggest a natural truncation of
almost Poisson gauge transformations in this case to gauge parameters and gauge
fields along the normal directions to the zero section $Q\subset
M$. Let $Q^*$ be
the submanifold defined by the equations $q_a=0$ in $M$. Given $f,g\in
C^\infty(Q^*)$ and $\alpha_*\in\Omega^1(Q^*)$, the pullbacks
$\delta_f^{\theta_B}\alpha_*\big|_{Q^*}$ and
$[\![f,g]\!]_{\theta_B}(\alpha_*)\big|_{Q^*}$ eliminate only the terms
involving almost Poisson brackets $\{\,\cdot\,,\,\cdot\,\}_{\theta_B}$,
leaving a non-trivial
dependence on the three-form $H$ which deforms the standard abelian
gauge transformations on the manifold
$Q^*$. In [3] it was shown that this pullback
operation turns the nonassociative star-product on
$C^\infty(M)[[\hbar]]$, which quantizes the
twisted Poisson structure $\theta_B$, into a sequence of `triproducts' on
$C^\infty(Q^*)[[\hbar]]$, which quantizes the
trivector $\Pim_B\in\mfX^3(Q^*)$ that
for $d=3$ defines a Nambu-Poisson structure (of degree $3$) on
$Q^*=\real^3$. This pullback operation can thus be thought of as
defining a `Nambu-Poisson gauge symmetry' which ought to be related to
a 3-Lie algebra (or equivalently a related Lie 2-algebra) action on
$\Omega^1(Q^*)$. Indeed, the pullback of the $L_\infty$-structure to
$Q^*$ does not involve any $2$-brackets, as
$\ell_2^{\,\theta_B}(f,\alpha_*)\big|_{Q^*}=0$ and
$\ell_2^{\,\theta_B}(f,g)\big|_{Q^*}=0$, and should be regarded as the
homotopy algebra action underlying this higher Lie algebra gauge
symmetry. It would be interesting to work out the details and
understand the underlying structures better.

Cosymplectic brackets on the constraint locus

Let $(T^*M,\omega)$ be a local symplectic embedding of an almost Poisson
manifold $(M,\theta)$, let $U\subseteq M$ be an open subset, and let
$\lambda_0=p_i\,\dd x^i$ be the Liouville one-form on $T^*U$. Let
$f,g,h\in C^\infty(U)$ and $A\in\Omega^1(U)$. In this appendix we
derive some useful identities for the cosymplectic brackets on the
constraint locus ${\sf im}(s_A)\subset T^*U$, where
$\Phi_A=(p_i-A_i(x))\,\dd x^i=0$, which we use in the main text.

We will need the relation

\begin{align}
& s_A^*\{\pi^*f,\{\pi^*g,\lambda_0\}_{\omega^{-1}}\}_{\omega^{-1}} -
\nonumber \\[4pt]
& \hspace{5cm} = t\,s_A^*\{\pi^*f,\gamma(\pi^*\dd
g,\,\cdot\,)\}_{\omega^{-1}} - t\,
g,\,\cdot\,)\}_{\omega^{-1}} \nonumber \\[4pt]
& \hspace{5cm} =
\nonumber \\[4pt]
& \hspace{5cm} =
\, s_A^*\{\pi^*f,(\Phi_A)_i\}_{\omega^{-1}} \ .
\label{r1}\end{align}

We also need

\begin{align}
& s_A^*\{\pi^*f,\{\pi^*g,\pi^*h\}_{\omega^{-1}}\}_{\omega^{-1}} -
\notag \\[4pt]
& \hspace{5cm} =
- t\,
\notag \\[4pt]
& \hspace{5cm} =
\, \big(s_A^*\{\pi^*f,p_i\}_{\omega^{-1}}
\nonumber \\[4pt]
& \hspace{5cm} =
\, s_A^*\{\pi^*f,(\Phi_A)_i\}_{\omega^{-1}} \ .
\label{r2}\end{align}

Now the combination of (<ref>) and (<ref>) implies

\begin{align}
& s_A^*\{\pi^*f,\{\pi^*g,\Phi_A\}_{\omega^{-1}}\}_{\omega^{-1}} -
\notag \\[4pt]
& \hspace{5cm}               =
s_A^*\big(\tilde\partial^i\{\pi^*g,\Phi_A\}_{\omega^{-1}}\big) \,
s_A^*\{\pi^*f,(\Phi_A)_i\}_{\omega^{-1}} \ .
\label{r3}\end{align}

Next we calculate

\begin{align}
& s_A^*\{\{\pi^*g,\pi^*f\}_{\omega^{-1}},\Phi_A\}_{\omega^{-1}} -
\notag \\[4pt]
& \hspace{1.5cm} =
\,
\big(-s_A^*\{\pi^*A_i,\lambda_0\}_{\omega^{-1}}
- s_A^*\{p_i,\pi^*A\}_{\omega^{-1}} +
\notag \\[4pt]
& \hspace{1.5cm} =
\,
\ .
\label{r4}\end{align}

One may also check

\begin{align}
& s_A^*\{\{\pi^*f,(\Phi_A)_j\}_{\omega^{-1}},\Phi_A\}_{\omega^{-1}} -
\notag \\[4pt]
& \hspace{5cm} =
\, s_A^*\{(\Phi_A)_i,\Phi_A\}_{\omega^{-1}} \ ,
\label{r5}\end{align}

\begin{align}
\notag \\[4pt]
& \hspace{5cm} =
\,
\ .
\label{r6}\end{align}

Closure of almost Poisson gauge transformations

In this appendix we prove that the gauge transformations defined in
Proposition <ref> close the almost Poisson gauge algebra
(<ref>). For this, we use (<ref>) to calculate the left-hand
side of (<ref>), and after rearranging the terms we find

\begin{align}
\delta^\theta_f\big(s_A^*\{\pi^*g,\Phi_A\}_{\omega^{-1}}+L_g^j\,s_A^*\{(\Phi_A)_j,\Phi_A\}_{\omega^{-1}}\big)-\delta^\theta_g\big(s_A^*\{\pi^*f,\Phi_A\}_{\omega^{-1}}+L_f^i\,s_A^*\{(\Phi_A)_i,\Phi_A\}_{\omega^{-1}}\big)
\notag \\[4pt]
& =
\\
& \hspace{0.5cm}
\notag
\\
& \hspace{1cm}
\notag \\
& \hspace{1.5cm}
& \hspace{2cm}
\notag
\\
& \hspace{2.5cm}
+L_g^j\,s_A^*\big(\tilde\partial{}^k\{(\Phi_A)_j,\Phi_A\}_{\omega^{-1}}\big)\,\big(s_A^*\{\pi^*f,(\Phi_A)_k\}_{\omega^{-1}}+L_f^i\,s_A^*\{(\Phi_A)_i,(\Phi_A)_k\}_{\omega^{-1}}\big)  \notag\\
& \hspace{3cm} -L_g^j\,s_A^*\{\pi^*s_A^*\{\pi^*f,(\Phi_A)_j\}_{\omega^{-1}} +
\notag
\\
& \hspace{3.5cm} +L_f^i\,s_A^*\{\pi^*s_A^*\{\pi^*g,(\Phi_A)_i\}_{\omega^{-1}} +
\notag \\
& \hspace{4cm} +\big(\delta_f^\theta L_g^i - \delta_g^\theta
L_f^i\big)\,s_A^*\{(\Phi_A)_i,\Phi_A\}_{\omega^{-1}} \ .
\label{c6}\end{align}

By explicit calculation and using the formulas (<ref>), (<ref>)
and (<ref>) from Appendix <ref> one may check

\begin{align*}
& s_A^*\{\pi^*f + L_f^i\,(\Phi_A)_i,\{\pi^*g +
L_g^j\,(\Phi_A)_j,\Phi_A\}_{\omega^{-1}}\}_{\omega^{-1}} \\
& \hspace{6cm} - s_A^*\{\pi^*f + L_f^i\,(\Phi_A)_i,\pi^*s_A^*\{\pi^*g +
L_g^j\,(\Phi_A)_j,\Phi_A\}_{\omega^{-1}}\}_{\omega^{-1}} \\[4pt]
& \hspace{1cm} =
\big(s_A^*(\tilde\partial{}^k\{\pi^*g,\Phi_A\}_{\omega^{-1}})
\big)
\\
& \hspace{5cm} \times \
\big(s_A^*\{\pi^*f,(\Phi_A)_k\}_{\omega^{-1}}+L_f^i\,s_A^*\{(\Phi_A)_i,(\Phi_A)_k\}_{\omega^{-1}}\big)
\ .
\end{align*}

Using this expression we rewrite (<ref>) as

\begin{align*}
\\
& \hspace{6cm}
\\
& \hspace{1cm} + s_A^*\{L_f^i,\Phi_A\}_{\omega^{-1}} \,
s_A^*\{\pi^*g,(\Phi_A)_i\}_{\omega^{-1}} -
s_A^*\{L_g^i,\Phi_A\}_{\omega^{-1}} \,
s_A^*\{\pi^*f,(\Phi_A)_i\}_{\omega^{-1}} \\
& \hspace{1cm} +
\\
& \hspace{1cm} + 2\, \big( L_f^i\,s_A^*\{L_g^j,\Phi_A\}_{\omega^{-1}}
- L_g^i\,s_A^*\{L_f^j,\Phi_A\}_{\omega^{-1}} \big) \,
s_A^*\{(\Phi_A)_j,(\Phi_A)_i\}_{\omega^{-1}} \\
& \hspace{1cm} + 2\,
+\big(\delta_f^\theta L_g^i - \delta_g^\theta
\ .
\end{align*}

Using the Jacobi identity in the first two lines and simplifying the
remaining lines, we rewrite this expression as

\begin{align*}
& s_A^*\{\{\pi^*f+L_f^i\,(\Phi_A)_i,\pi^*g+L_g^j\,(\Phi_A)_j\}_{\omega^{-1}},\Phi_A\}_{\omega^{-1}} +\big(\delta_f^\theta L_g^i - \delta_g^\theta
\\
& \hspace{2cm} - s_A^*\{L_g^j\,\pi^*s_A^*\{\pi^*f,(\Phi_A)_j\}_{\omega^{-1}} +
L_f^i\,\pi^*s_A^*\{(\Phi_A)_i,\pi^*g\}_{\omega^{-1}},\Phi_A\}_{\omega^{-1}} \\
& \hspace{4cm} +
\ .
\end{align*}

Using $\Phi_A=0$ on the constraint locus ${\sf im}(s_A)$, we rewrite the first line so
that this expression becomes

\begin{align*}
& s_A^*\{\{\pi^*f,\pi^*g\}_{\omega^{-1}} {+}
L_g^j\,\{\pi^*f,(\Phi_A)_j\}_{\omega^{-1}} {+}
L_f^i\,\{(\Phi_A)_i,\pi^*g\}_{\omega^{-1}} {+}
\\
& \quad +\big(s_A^*\{L_f^k,\pi^*g\}_{\omega^{-1}} +
s_A^*\{\pi^*f,L_g^k\}_{\omega^{-1}} +
L_f^i\,s_A^*\{(\Phi_A)_i,L_g^k\}_{\omega^{-1}} \\
& \hspace{2cm} +
L_g^j\,s_A^*\{L_f^k,(\Phi_A)_j\}_{\omega^{-1}}\big) \,
s_A^*\{(\Phi_A)_k,\Phi_A\}_{\omega^{-1}} +\big(\delta_f^\theta L_g^i - \delta_g^\theta
\\
& \hspace{4cm} - s_A^*\{L_g^j\,\pi^*s_A^*\{\pi^*f,(\Phi_A)_j\}_{\omega^{-1}} +
L_f^i\,\pi^*s_A^*\{(\Phi_A)_i,\pi^*g\}_{\omega^{-1}},\Phi_A\}_{\omega^{-1}} \\
& \hspace{6cm} +
\ .
\end{align*}

Applying again the formulas (<ref>)–(<ref>) from
Appendix <ref> in the first line of this
expression, after simplification we end up with

\begin{align*}
& s_A^*\{\pi^*s_A^*\{\pi^*f,\pi^*g\}_{\omega^{-1}} -
\\
& \quad + \Big(  \delta_f^\theta L_g^k - \delta_g^\theta
L_f^k +
s_A^*\big(\tilde\partial^k\{\pi^*f,\pi^*g\}_{\omega^{-1}}\big) +
\\
& \hspace{1cm} +
+ s_A^*\{L_f^k,\pi^*g\}_{\omega^{-1}} \\
& \hspace{1.5cm} +
s_A^*\{\pi^*f,L_g^k\}_{\omega^{-1}} + L_f^i\,s_A^*\{(\Phi_A)_i,L_g^k\}_{\omega^{-1}} +
\Big) \,
\ .
\end{align*}

By (<ref>) and (<ref>) this is equal to
$s_A^*\{t\,[\![f,g]\!]_\theta(A) + L^k_{t\,
which proves the closure formula (<ref>).

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