,\end{split}$ (5.20)

where
$\eta=\frac{1}{2}\,\eta_{\mu\nu}\,\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}$
in coordinates $x=(x^{\mu})\in\mathbbm{R}^{1,d-1}$; as previously we use
Einstein summation conventions throughout.

Denoting the basis vector fields of $\mathbbm{R}^{1,d-1}$ by
$\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$, the action of these
generators on fields may be realised by the vector fields implementing
translations and Lorentz transformations

$\displaystyle\mathsf{P}_{\mu}=\partial_{\mu}\qquad\mbox{and}\qquad\mathsf{M}_{\mu\nu}=x_{\mu}\,\partial_{\nu}-x_{\nu}\,\partial_{\mu}\
.$ (5.21)

Let
$\mathcal{F}\in\mathsf{U}\mathbbm{R}^{1,d-1}[[\hbar]]\otimes\mathsf{U}\mathbbm{R}^{1,d-1}[[\hbar]]$
be the Moyal–Weyl twist (see Example 4.23)

$\displaystyle\mathcal{F}=\exp\big{(}-\tfrac{\mathrm{i}\,\hbar}{2}\,\theta^{\mu\nu}\,\mathsf{P}_{\mu}\otimes\mathsf{P}_{\nu}\big{)}\
,$ (5.22)

where $(\theta^{\mu\nu})$ is a constant antisymmetric real $d{\times}d$
matrix. As is well-known (see e.g. [85]), the noncommutative scalar field
theory is then translationally invariant, but it breaks Lorentz invariance in
the classical sense.

The full relativistic invariance of the noncommutative scalar field theory on
$(\mathbbm{R}^{1,d-1},\eta)$ is restored by twisting the action of the Lorentz
vector fields $\mathsf{M}_{\mu\nu}$ to a braided representation. A quick check
reveals that the twist $\mathcal{F}$ in this example acts trivially on the Lie
bracket, $[-,-]_{\mathfrak{k}}^{\star}=[-,-]_{\mathfrak{k}}$, so the
underlying braided Lie algebra of Killing vector fields coincides with the
classical Poincaré algebra. A generic element of the Poincaré algebra

$\displaystyle\xi=\varepsilon^{\mu}\,\mathsf{P}_{\mu}+\tfrac{1}{2}\,\omega^{\mu\nu}\,\mathsf{M}_{\mu\nu}$
(5.23)

is specified by a set of real translation parameters $\varepsilon^{\mu}$ and
real Lorentz transformation parameters $\omega^{\mu\nu}=-\omega^{\nu\mu}$. Its
braided action on a scalar field $\phi$ is given by

$\displaystyle\delta_{\xi}^{{\textrm{\tiny${\mathcal{R}}$}}}\phi=\mathrm{L}_{\xi}^{\star}\,\phi=\varepsilon^{\mu}\,\partial_{\mu}\phi+\omega^{\mu\nu}\,x_{\nu}\star\partial_{\mu}\phi=\mathrm{L}_{\xi}\phi-\tfrac{\mathrm{i}\,\hbar}{2}\,\omega^{\mu\nu}\,\theta_{\nu\lambda}\,\partial^{\lambda}\partial_{\mu}\phi\
.$ (5.24)

As expected, only Lorentz transformations of $\phi$ are modified from their
classical rules; note that the transformations of the coordinate functions
$(x\mapsto x^{\mu}):\mathbbm{R}^{1,d-1}\to\mathbbm{R}$ themselves are
unchanged, as $\mathrm{L}_{\xi}^{\star}x^{\mu}=\mathrm{L}_{\xi}x^{\mu}$. These
braided Poincaré transformations close the _classical_ Poincaré algebra.

Nevertheless, our braided Lorentz transformations are _not_ the same as the
‘twisted Lorentz transformations’ of [36, 90]. Those act as the classical
Lorentz symmetry on a single field, and do not follow a braided derivation
rule on star-products of fields but rather a deformed Leibniz rule defined by
the Drinfel’d twist of the coproduct $\Delta_{\textrm{\tiny$\mathcal{F}$}}$ on
the enveloping Hopf algebra of $\mathfrak{k}$. It would be interesting to
explore whether the standard Wigner classification of particles holds for
braided representations of the Lorentz algebra, as it does for the twisted
actions of [36].

##  6 Braided $BF$ theory

Let us now return to examples of field theories with gauge symmetries. In this
section we look at an example of a topological field theory which is a natural
extension of Chern–Simons theory, considered as our prototypical guiding
example of gauge theories in the (braided) $L_{\infty}$-algebra formalism in
Sections 2 and 4. The story spelled out for Chern–Simons theory is easily
adapted for arbitrary $BF$ theories, whose classical $L_{\infty}$-algebra
formulation was worked out in detail by [43]. These examples serve to nicely
illustrate generic features of field theories with higher (or reducible) gauge
symmetries in a simple framework, and provide new examples of ensuing
novelties with the corresponding higher braided gauge symmetries which also
drastically deviate from their classical and star-gauge counterparts.

###  6.1 $BF$ theory in the $L_{\infty}$-algebra formalism

Let $M$ be a closed oriented $d$-dimensional manifold. Let
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ be a Lie algebra, and let $W$ be a
$\mathfrak{g}$-module with a nondegenerate pairing
$\mathrm{Tr}_{{\textrm{\tiny$W$}}}:W\otimes\mathfrak{g}\rightarrow\mathbbm{R}$
which is invariant under the $\mathfrak{g}$-action:

$\displaystyle\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}(X\cdot w)\otimes
Y+w\otimes[X,Y]_{\mathfrak{g}}\big{)}=0\ ,$ (6.1)

for $w\in W$ and $X,Y\in\mathfrak{g}$. The standard example is when
$\mathfrak{g}$ is a quadratic Lie algebra, where $W=\mathfrak{g}$ is a
$\mathfrak{g}$-module under the adjoint action of $\mathfrak{g}$ on itself, as
in Section 2.1.

The $BF$ action functional is given by

$\displaystyle S(B,A)=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(B\wedge
F_{A})\ ,$ (6.2)

where $F_{A}\in\Omega^{2}(M,\mathfrak{g})$ is the curvature of a connection
$1$-form $A\in\Omega^{1}(M,\mathfrak{g})$ and $B\in\Omega^{d-2}(M,W)$. The
field equations of $BF$ theory are solved by pairs of a flat
$\mathfrak{g}$-connection on $M$ and a covariantly constant $(d{-}2)$-form
valued in the representation $W$ of $\mathfrak{g}$:

$\displaystyle\begin{split}\mathfrak{F}_{B}&:=F_{A}=\mathrm{d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}=0\
\in\ \Omega^{2}(M,\mathfrak{g})\ ,\\\\[4.0pt]
\mathfrak{F}_{A}&:=\mathrm{d}^{A}B=\mathrm{d}B+A\wedge B=0\ \in\
\Omega^{d-1}(M,W)\ ,\end{split}$ (6.3)

where again $A\wedge B$ computes the exterior product of the form components
while pairing the components in $\mathfrak{g}$ and $W$ via the
$\mathfrak{g}$-action.

The action functional (6.2) is invariant under standard gauge transformations
$\lambda\in\Omega^{0}(M,\mathfrak{g})$ acting as

$\displaystyle\delta_{\lambda}(B,A)=\big{(}-\lambda\cdot
B\,,\,\mathrm{d}\lambda+[A,\lambda]_{\mathfrak{g}}\big{)}\ .$ (6.4)

Compared to Chern–Simons theory, however, for $d\geq 3$ there is an extra
‘shift’ symmetry generated by $(d{-}3)$-forms $\tau\in\Omega^{d-3}(M,W)$
valued in $W$, which act as

$\displaystyle\delta_{\tau}(B,A):=(\mathrm{d}\tau+A\wedge\tau,0)\ .$ (6.5)

The corresponding Noether identities coincide with the usual Bianchi
identities

$\displaystyle\begin{split}{\mathsf{d}}_{(B,A)}(\mathfrak{F}_{B},\mathfrak{F}_{A}):\\!&=\big{(}(-1)^{d-3}\,\mathrm{d}^{A}\mathfrak{F}_{B}\,,\,\mathrm{d}^{A}\mathfrak{F}_{A}-\mathfrak{F}_{B}\wedge
B\big{)}\\\\[4.0pt]
&=\big{(}(-1)^{d-3}\,\mathrm{d}^{A}F_{A}\,,\,(\mathrm{d}^{A})^{2}B-F_{A}\wedge
B\big{)}=(0,0)\ \in\ \Omega^{3}(M,\mathfrak{g})\times\Omega^{d}(M,W)\
.\end{split}$ (6.6)

Here the first slot is the second Bianchi identity which in this example
corresponds to the shift symmetry, in contrast to Chern–Simons theory where it
corresponds to the usual gauge transformations generated by
$\Omega^{0}(M,\mathfrak{g})$. The latter symmetries correspond to the first
Bianchi identity in $BF$ theory, which is the second slot.

The cyclic $L_{\infty}$-algebra of $BF$ theory in $d$ dimensions is given by
the underlying graded vector space $V:=V^{0}\oplus V^{1}\oplus V^{2}\oplus
V^{3}$, where

$\displaystyle\begin{split}V^{0}=\Omega^{d-3}(M,W)\times\Omega^{0}\big{(}M,\mathfrak{g}\big{)}\qquad&,\qquad
V^{1}=\Omega^{d-2}(M,W)\times\Omega^{1}\big{(}M,\mathfrak{g}\big{)}\
,\\\\[4.0pt]
V^{2}=\Omega^{2}\big{(}M,\mathfrak{g}\big{)}\times\Omega^{d-1}(M,W)\qquad&,\qquad
V^{3}=\Omega^{3}(M,\mathfrak{g})\times\Omega^{d}(M,W)\ .\end{split}$ (6.7)

We denote gauge parameters by $(\tau,\lambda)\in V^{0}$, dynamical fields by
$(B,A)\in V^{1}$, field equations by $(\mathcal{B},\mathcal{A})\in V^{2}$, and
Noether identities by $(\mathcal{T},\varLambda)\in V^{3}$. The non-trivial
brackets then comprise the differential

$\displaystyle\ell_{1}(\tau,\lambda)=(\mathrm{d}\tau,\mathrm{d}\lambda)\
,\quad\ell_{1}(B,A)=(\mathrm{d}A,\mathrm{d}B)\qquad\mbox{and}\qquad\ell_{1}(\mathcal{B},\mathcal{A})=(\mathrm{d}\mathcal{B},\mathrm{d}\mathcal{A})\
,$ (6.8)

along with the $2$-brackets

$\displaystyle\begin{split}\ell_{2}\big{(}(\tau_{1},\lambda_{1})\,,\,(\tau_{2},\lambda_{2})\big{)}&=\big{(}-\lambda_{1}\cdot\tau_{2}+\lambda_{2}\cdot\tau_{1}\,,\,-[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}\big{)}\
,\\\\[4.0pt]
\ell_{2}\big{(}(\tau,\lambda)\,,\,(B,A)\big{)}&=\big{(}-\lambda\cdot
B+A\wedge\tau\,,\,-[\lambda,A]_{\mathfrak{g}}\big{)}\ ,\\\\[4.0pt]
\ell_{2}\big{(}(\tau,\lambda)\,,\,(\mathcal{B},\mathcal{A})\big{)}&=\big{(}-[\lambda,\mathcal{B}]_{\mathfrak{g}}\,,\,-\lambda\cdot\mathcal{A}+\mathcal{B}\wedge\tau\big{)}\
,\\\\[4.0pt]
\ell_{2}\big{(}(\tau,\lambda)\,,\,(\mathcal{T},\varLambda)\big{)}&=\big{(}-[\lambda,\mathcal{T}]_{\mathfrak{g}}\,,\,-\lambda\cdot\varLambda+(-1)^{d-3}\,\mathcal{T}\wedge\tau\big{)}\
,\\\\[4.0pt]
\ell_{2}\big{(}(B_{1},A_{1})\,,\,(B_{2},A_{2})\big{)}&=-\big{(}[A_{1},A_{2}]_{\mathfrak{g}}\,,\,A_{1}\wedge
B_{2}+A_{2}\wedge B_{1}\big{)}\ ,\\\\[4.0pt]
\ell_{2}\big{(}(B,A)\,,\,(\mathcal{B},\mathcal{A})\big{)}&=-\big{(}[A,\mathcal{B}]_{\mathfrak{g}}\,,\,A\wedge\mathcal{A}-\mathcal{B}\wedge
B\big{)}\ .\end{split}$ (6.9)

Thus the dynamics of $BF$ theory in any dimension $d$ is also organised by a
differential graded Lie algebra. The cyclic inner product is given naturally
as

$\displaystyle\begin{split}\langle(B,A)\,,\,(\mathcal{B},\mathcal{A})\rangle&:=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(B\wedge\mathcal{B})+\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\mathcal{A}\wedge
A)\ ,\\\\[4.0pt]
\langle(\tau,\lambda)\,,\,(\mathcal{T},\mathcal{P})\rangle&:=\int_{M}\,(-1)^{d-3}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\tau\wedge\mathcal{T})+\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\varLambda\wedge\lambda)\
.\end{split}$ (6.10)

###  6.2 Noncommutative $BF$ theory in the braided $L_{\infty}$-algebra
formalism

The differential graded Lie algebra of $d$-dimensional $BF$ theory from
Section 6.1 lives in the category of $\mathsf{U}\mathfrak{v}$-modules, as a
consequence of diffeomorphism invariance of the topological field theory.
Applying the general formalism of braided field theory from Section 4.6, we
twist deform it to the 4-term differential graded braided Lie algebra
$\big{(}V[[\hbar]],\ell_{1}^{\star},\ell_{2}^{\star}\big{)}$ with $1$-bracket
$\ell_{1}^{\star}=\ell_{1}$ and the non-trivial $2$-brackets

$\displaystyle\begin{split}\ell_{2}^{\star}\big{(}(\tau_{1},\lambda_{1})\,,\,(\tau_{2},\lambda_{2})\big{)}&=\big{(}-\lambda_{1}\star\tau_{2}+\mathsf{R}_{k}(\lambda_{2})\star\mathsf{R}^{k}(\tau_{1})\,,\,-[\lambda_{1},\lambda_{2}]^{\star}_{\mathfrak{g}}\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,(B,A)\big{)}&=\big{(}-\lambda\star
B+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(\tau)\,,\,-[\lambda,A]_{\mathfrak{g}}^{\star}\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,(\mathcal{B},\mathcal{A})\big{)}&=\big{(}-[\lambda,\mathcal{B}]^{\star}_{\mathfrak{g}}\,,\,-\lambda\star\mathcal{A}+\mathsf{R}_{k}(\mathcal{B})\wedge_{\star}\mathsf{R}^{k}(\tau)\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,(\mathcal{T},\varLambda)\big{)}&=\big{(}-[\lambda,\mathcal{T}]^{\star}_{\mathfrak{g}}\,,\,-\lambda\star\varLambda+(-1)^{d-3}\,\mathsf{R}_{k}(\mathcal{T})\wedge_{\star}\mathsf{R}^{k}(\tau)\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(B_{1},A_{1})\,,\,(B_{2},A_{2})\big{)}&=-\big{(}[A_{1},A_{2}]^{\star}_{\mathfrak{g}}\,,\,A_{1}\wedge_{\star}B_{2}+\mathsf{R}_{k}(A_{2})\wedge_{\star}\mathsf{R}^{k}(B_{1})\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(B,A)\,,\,(\mathcal{B},\mathcal{A})\big{)}&=-\big{(}[A,\mathcal{B}]^{\star}_{\mathfrak{g}}\,,\,A\wedge_{\star}\mathcal{A}-\mathsf{R}_{k}(\mathcal{B})\wedge_{\star}\mathsf{R}^{k}(B)\big{)}\
.\end{split}$ (6.11)

The notation $\lambda\star B$ again stands for the left braided representation
of $\lambda\in\Omega^{0}(M,\mathfrak{g})$ on $B\in\Omega^{d-2}(M,W)[[\hbar]]$
as described in Section 4.3, and similarly for $A\wedge_{\star}B$.

A pair of gauge parameters
$(\tau,\lambda)\in\Omega^{d-3}(M,W)\times\Omega^{0}(M,\mathfrak{g})$ defines
the (left) braided gauge transformations of the fields
$(B,A)\in\Omega^{d-2}(M,W)\times\Omega^{1}(M,\mathfrak{g})$ by

$\displaystyle\begin{split}\delta_{(\tau,\lambda)}^{{\textrm{\tiny${\mathcal{R}}$}}}(B,A):\\!&=\ell_{1}^{\star}(\tau,\lambda)+\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,(B,A)\big{)}\\\\[4.0pt]
&=\big{(}\mathrm{d}\tau+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(\tau)-\lambda\star
B\,,\,\mathrm{d}\lambda-[\lambda,A]^{\star}_{\mathfrak{g}}\big{)}\
,\end{split}$ (6.12)

encoding the expected deformation of the symmetries. Notice that in the
braided framework, the shift symmetry is twisted naturally irrespective of the
dimension $d$, gauge algebra $\mathfrak{g}$ and representation $W$. The
closure as a braided Lie algebra

$\displaystyle\big{[}\delta_{(\tau_{1},\lambda_{1})}^{{\textrm{\tiny${\mathcal{R}}$}}},\delta_{(\tau_{2},\lambda_{2})}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{]}_{\circ}^{\star}=\delta_{-\ell_{2}^{\star}((\tau_{1},\lambda_{1}),(\tau_{2},\lambda_{2}))}^{{\textrm{\tiny${\mathcal{R}}$}}}=\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{(\lambda_{1}\star\tau_{2}-\mathsf{R}_{k}(\lambda_{2})\star\mathsf{R}^{k}(\tau_{1}),[\lambda_{1},\lambda_{2}]^{\star}_{\mathfrak{g}})}$
(6.13)

follows naturally as in the commutative case, giving a braided deformation of
the semi-direct product

$\displaystyle\Omega^{d-3}(M,W)[[\hbar]]\rtimes_{\star}\Omega^{0}(M,\mathfrak{g})[[\hbar]]\
,$ (6.14)

where we regard the vector space $W$ (and hence $\Omega^{d-3}(M,W)[[\hbar]]$)
as an abelian Lie algebra.

#### Braided $BF$ equations

Following the prescription from Section 4.6, the field equations
$\mathfrak{F}_{(B,A)}^{\star}=0$ valued in
$\Omega^{2}(M,\mathfrak{g})[[\hbar]]\times\Omega^{d-1}(M,W)[[\hbar]]$ are
encoded by

$\displaystyle\begin{split}\mathfrak{F}_{(B,A)}^{\star}=(\mathfrak{F}_{B}^{\star},\mathfrak{F}_{A}^{\star}):\\!&=\ell_{1}^{\star}(B,A)-\tfrac{1}{2}\,\ell_{2}^{\star}\big{(}(B,A)\,,\,(B,A)\big{)}\\\\[4.0pt]
&=\big{(}\mathrm{d}A+\tfrac{1}{2}\,[A,A]^{\star}_{\mathfrak{g}}\,,\,\mathrm{d}B+A\wedge_{\star}B+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(B)\big{)}\\\\[4.0pt]
&=\big{(}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\,,\,\tfrac{1}{2}\,(\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}B+\mathrm{d}_{\star{\textrm{\tiny R}}}^{A}B)\big{)}\ ,\end{split}$
(6.15)

where in the third line we have identified the left and right braided
covariant derivatives from Section 4.3. The field equations state that the
braided $\mathfrak{g}$-connection $A$ is flat while the field $B$ obeys a
symmetrized braided covariant constancy condition. Such symmetrized
combinations are expected since they preserve reality of the fields (cf.
Example 4.106), and hence are a generic feature of braided field theories.

According to the general braided $L_{\infty}$-algebra formulation, we obtain
the braided gauge covariance

$\displaystyle\begin{split}\delta_{(\tau,\lambda)}^{{\textrm{\tiny${\mathcal{R}}$}}}\mathfrak{F}_{(B,A)}^{\star}&=\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,(\mathfrak{F}_{B}^{\star},\mathfrak{F}_{A}^{\star})\big{)}=\big{(}-[\lambda,\mathfrak{F}_{B}^{\star}]_{\mathfrak{g}}^{\star}\,,\,-\lambda\star\mathfrak{F}_{A}^{\star}+\mathsf{R}_{k}(\mathfrak{F}_{B}^{\star})\wedge_{\star}\mathsf{R}^{k}(\tau)\big{)}\
,\end{split}$ (6.16)

which may also be easily verified explicitly. This extends the statement from
Section 4.3 that the braided curvature, left and right covariant derivatives
are covariant under gauge transformations from $\Omega^{0}(M,\mathfrak{g})$.

#### Braided Noether identities

From the general prescription of Section 4.6, the braided Noether identities
which are valued in
$\Omega^{3}(M,\mathfrak{g})[[\hbar]]\times\Omega^{d}(M,W)[[\hbar]]$ are given
by

$\displaystyle\begin{split}{\mathsf{d}}_{(B,A)}^{\star}\mathfrak{F}_{(B,A)}^{\star}:\\!&=\ell_{1}^{\star}(\mathfrak{F}_{B}^{\star},\mathfrak{F}_{A}^{\star})-\tfrac{1}{2}\,\Big{(}\ell_{2}^{\star}\big{(}(B,A)\,,\,(\mathfrak{F}_{B}^{\star},\mathfrak{F}_{A}^{\star})\big{)}-\ell_{2}^{\star}\big{(}(\mathfrak{F}_{B}^{\star},\mathfrak{F}_{A}^{\star})\,,\,(B,A)\big{)}\Big{)}\\\
&\quad\,+\tfrac{1}{4}\,\ell_{2}^{\star}\big{(}\mathsf{R}_{k}(B,A)\,,\,\ell_{2}^{\star}(\mathsf{R}^{k}(B,A)\,,\,(B,A))\big{)}\
=\ (0,0)\ ,\end{split}$ (6.17)

where
$\mathsf{R}_{k}(B,A):=\big{(}\mathsf{R}_{k}(B),\mathsf{R}_{k}(A)\big{)}$.
These translate to a pair of differential identities among the field equations
(6.15). Expanding and collecting terms, the first slot of (6.17) is the
Noether identity corresponding to the braided shift symmetry:

$\displaystyle\tfrac{1}{2}\,\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\mathfrak{F}_{B}^{\star}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\mathfrak{F}_{B}^{\star}\big{)}+\tfrac{1}{4}\,\big{[}\mathsf{R}_{k}(A),[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\big{]}_{\mathfrak{g}}^{\star}\
=\ 0\ .$ (6.18)

The second slot is the Noether identity corresponding to the usual braided
gauge transformations generated by $\Omega^{0}(M,\mathfrak{g})$:

$\displaystyle\begin{split}\tfrac{1}{2}\,&\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\mathfrak{F}_{A}^{\star}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\mathfrak{F}_{A}^{\star}-\mathsf{R}_{k}(\mathfrak{F}_{B}^{\star})\wedge_{\star}\mathsf{R}^{k}(B)-\mathfrak{F}_{B}^{\star}\wedge_{\star}B\big{)}\\\
&\hskip
56.9055pt+\tfrac{1}{4}\,\Big{(}-[A,\mathsf{R}_{k}(A)]^{\star}_{\mathfrak{g}}\wedge_{\star}\mathsf{R}^{k}(B)+\mathsf{R}_{k}(A)\wedge_{\star}\big{(}\mathsf{R}^{k}(A)\wedge_{\star}B\big{)}\\\
&{\phantom{\hskip
56.9055pt+\tfrac{1}{4}\,\big{(}-[A,\mathsf{R}_{k}(A)]^{\star}_{\mathfrak{g}}\wedge_{\star}\mathsf{R}^{k}(B)+}}+{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(1)}}}(A)\wedge_{\star}\big{(}{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(2)}}}(A)\wedge_{\star}\mathsf{R}^{k}(B)\big{)}\Big{)}\
=\ 0\ ,\end{split}$ (6.19)

where we used the Sweedler notation
$\Delta_{\textrm{\tiny$\mathcal{F}$}}(\mathsf{R}_{k})=:{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(1)}}}\otimes{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(2)}}}$
(with summations understood). This identity may also be derived directly by
summing over the braided deformations of the first Bianchi identity in twisted
noncommutative differential geometry:

$\displaystyle\begin{split}(\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A})^{2}B&=F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}B+\tfrac{1}{2}\,A\wedge_{\star}(A\wedge_{\star}B)-\tfrac{1}{2}\,\mathsf{R}_{k}(A)\wedge_{\star}\big{(}\mathsf{R}^{k}(A)\wedge_{\star}B\big{)}\
,\\\\[4.0pt] (\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A})^{2}B&=\mathsf{R}_{k}(F_{A}^{\textrm{\tiny${\mathcal{R}}$}})\wedge_{\star}\mathsf{R}^{k}(B)\\\
&\quad\,+\tfrac{1}{2}\,\mathsf{R}_{k}\,\mathsf{R}_{l}(A)\wedge_{\star}\big{(}\mathsf{R}_{m}\,\mathsf{R}^{l}(A)\wedge_{\star}\mathsf{R}^{m}\,\mathsf{R}^{k}(B)\big{)}-\tfrac{1}{2}\,{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(1)}}}(A)\wedge_{\star}\big{(}{\mathsf{R}_{k}}_{\bar{\textrm{\tiny(2)}}}(A)\wedge_{\star}\mathsf{R}^{k}(B)\big{)}\
,\end{split}$ (6.20)

and the mixed derivative expressions. These follow by simply expanding the
braided covariant derivatives. The last two terms cancel each other in the
classical limit, giving the usual identity $(\mathrm{d}^{A})^{2}B=F\wedge B$.

#### Braided $BF$ functional

When $M$ is closed and oriented, the inner product (6.10) twists to a
(braided) cyclic pairing given by

$\displaystyle\begin{split}\langle(B,A)\,,\,(\mathcal{B},\mathcal{A})\rangle_{\star}&:=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(B\wedge_{\star}\mathcal{B})+\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\mathcal{A}\wedge_{\star}A)\
,\\\\[4.0pt]
\langle(\tau,\lambda)\,,\,(\mathcal{T},\varLambda)\rangle_{\star}&:=\int_{M}\,(-1)^{d-3}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\tau\wedge_{\star}\mathcal{T})+\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\varLambda\wedge_{\star}\lambda)\
.\end{split}$ (6.21)

We shall write out the braided action functional in two forms: First by using
a braided cyclic pairing (or forgetting the $\wedge_{\star}$-cyclicity in the
case of compatible twists), and then by employing a compatible twist and its
resulting $\wedge_{\star}$-cyclicity under the integral. This is instructive
for considerations of more complicated braided field theories.

For a braided cyclic pairing, the action functional is given by the
prescription of Section 4.6, which in the example at hand reads

$\displaystyle\begin{split}S_{\star}(B,A):\\!&=\frac{1}{2}\,\big{\langle}(B,A)\,,\,\ell_{1}^{\star}(B,A)\big{\rangle}_{\star}-\frac{1}{6}\,\big{\langle}(B,A)\,,\,\ell_{2}^{\star}\big{(}(B,A),(B,A)\big{)}\big{\rangle}_{\star}\\\\[4.0pt]
&=\frac{1}{2}\,\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}\mathrm{d}A\big{)}+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}A\wedge_{\star}\mathrm{d}B\big{)}\\\
&\quad\,+\frac{1}{6}\,\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}[A,A]^{\star}_{\mathfrak{g}}\big{)}+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\Big{(}A\wedge_{\star}\big{(}A\wedge_{\star}B+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(B)\big{)}\Big{)}\\\\[4.0pt]
&=\int_{M}\,\frac{1}{2}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}\mathrm{d}A\big{)}+\frac{1}{2}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathrm{d}A\wedge_{\star}B\big{)}\\\
&\qquad\qquad\,+\frac{1}{6}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}[A,A]_{\mathfrak{g}}^{\star}\big{)}+\frac{1}{6}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}\mathsf{R}^{k}([A,A]_{\mathfrak{g}}^{\star})\big{)}\\\
&\qquad\qquad\,+\frac{1}{6}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\big{)}\
,\end{split}$ (6.22)

where in the last equality we integrated by parts and used invariance of
$\mathrm{Tr}_{{\textrm{\tiny$W$}}}(-\wedge_{\star}-)$ under the left braided
$\mathfrak{g}$-action, which follows from the classical invariance (6.1) order
by order in the deformation parameter $\hbar$. Identifying the braided
curvature $2$-form
$F_{A}^{\textrm{\tiny${\mathcal{R}}$}}=\mathrm{d}A+\frac{1}{2}\,[A,A]_{\mathfrak{g}}^{\star}$,
we arrive at

$\displaystyle\begin{split}S_{\star}(B,A)&=\int_{M}\,\frac{1}{2}\,\Big{(}\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}B\big{)}\Big{)}\\\
&\qquad\qquad\,-\frac{1}{12}\,\Big{(}\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}[A,A]_{\mathfrak{g}}^{\star}\big{)}+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}\mathsf{R}^{k}([A,A]_{\mathfrak{g}}^{\star})\big{)}\Big{)}\\\
&\qquad\qquad\,+\frac{1}{6}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\big{)}\
.\end{split}$ (6.23)

The first two terms of (6.23) are “naive” deformations which give the
classical action functional (6.2) in the classical limit. The last three terms
cancel each other in the classical limit, but are individually non-covariant.
However, one may easily check that together they form a braided covariant
combination. The possible non-covariance arises from terms proportional to
$\mathrm{d}\lambda$ in a gauge transformation of $A$, while the covariant
terms vanish by invariance of $\mathrm{Tr}_{{\textrm{\tiny$W$}}}$. Indeed one
finds

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}[A,A]_{\mathfrak{g}}^{\star}\big{)}&=\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\big{)}\\\
&\quad\,+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}_{l}(A),\mathsf{R}^{l}\,\mathsf{R}^{k}(\mathrm{d}\lambda)]_{\mathfrak{g}}^{\star}\big{)}\\\\[4.0pt]
&=2\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]^{\star}_{\mathfrak{g}}\big{)}\
,\end{split}$ (6.24)

where we used the braided symmetry of the bracket in the second equality.
Similarly, we find

$\displaystyle\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}\mathsf{R}^{k}([A,A]_{\mathfrak{g}}^{\star})\big{)}=2\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}\mathsf{R}^{k}([\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star})\big{)}$
(6.25)

and

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\big{)}&=\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\big{)}\\\
&\quad\,+\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}\mathsf{R}_{k}(B)\wedge_{\star}\mathsf{R}^{k}([\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star})\big{)}\
.\end{split}$ (6.26)

Adding the three variations with the prefactors in (6.23) shows that they
cancel, and the action functional is braided gauge invariant as expected by
the general discussion of Section 4.6.

Restricting to compatible Drinfel’d twists, the pairing becomes a strictly
cyclic structure on the braided $L_{\infty}$-algebra
$\big{(}V[[\hbar]],\ell_{1}^{\star},\ell_{2}^{\star}\big{)}$. In this case,
the action functional (6.23) further simplifies to

$\displaystyle
S_{\star}(B,A)=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}\big{(}B\wedge_{\star}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}\
,$ (6.27)

where the first two terms of (6.23) combine by $\wedge_{\star}$-cyclicity,
while the last three terms cancel for the same reason together with an
$\mathcal{R}$-matrix identity (4.19) applied to the final term. Thus the
braided noncommutative $BF$ functional is the expected “naive” deformation. By
varying it with respect to each of fields $B$ and $A$, it is easy to
explicitly derive the field equations (6.15) using $\wedge_{\star}$-cyclicity
under the integral, invariance of $\mathrm{Tr}_{{\textrm{\tiny$W$}}}$ and
$\mathcal{R}$-matrix identities (4.19). Similarly, braided gauge invariance
under (6.12) and the corresponding Noether identities (6.17) are verified
explicitly. For Hermitian twists and real fields the action functional is
real-valued as discussed in Section 4.6.

###  6.3 Higher braided gauge symmetry

Another new feature of $BF$ theories, compared to Chern–Simons gauge theory,
is that they generally possess additional higher shift symmetries: The shift
symmetry (6.5) is on-shell reducible in dimensions $d\geq 4$. This means that,
strictly speaking, we should also include in (6.7) the negatively-graded
vector spaces $V^{-k}=\Omega^{d-3-k}(M,W)$ for $k=1,\dots,d-3$, which
parameterize ‘higher gauge transformations’, together with their duals
$V^{k+3}=\Omega^{k+3}(M,\mathfrak{g})$ and the obvious brackets in (6.8) and
(6.9). We spell out this out explicitly for $BF$ theories in the simplest case
of dimension $d=4$.

Any element $\tau\in\Omega^{1}(M,W)$ generates the shift symmetry

$\displaystyle\delta_{\tau}(B,A)=\big{(}\mathrm{d}^{A}\tau,0\big{)}\ \in\
\Omega^{2}(M,W)\times\Omega^{1}(M,\mathfrak{g})\ .$ (6.28)

Now consider
$\tau^{\prime}:=\tau+\mathrm{d}^{A}\varepsilon\,\in\Omega^{1}(M,W)$ for any
$\varepsilon\in\Omega^{0}(M,W)$. This generates the shift symmetry

$\displaystyle\begin{split}\delta_{\tau^{\prime}}(B,A)&=\big{(}\mathrm{d}^{A}\tau^{\prime},0\big{)}=\big{(}\mathrm{d}^{A}\tau+(\mathrm{d}^{A})^{2}\varepsilon,0\big{)}=\big{(}\delta_{\tau}B+F_{A}\cdot\varepsilon,0\big{)}=\big{(}\delta_{\tau}B+\mathfrak{F}_{B}\cdot\varepsilon,0\big{)}\
,\end{split}$ (6.29)

and so the two transformations differ by a term proportional to a field
equation, that is, $\tau$ and
$\tau^{\prime}=:\tau+\delta_{(\varepsilon,A)}\tau$ generate the same symmetry
on-shell. This leads to a redundancy in the subspace of $V^{0}$ generating the
action of shift symmetries on $V^{1}$; the redundancy lives in the subspace of
covariantly exact $1$-forms valued in $W$. We may parameterize this by
$V^{-1}:=\Omega^{0}(M,W)$; for $d=4$ this is enough and no further gauge
redundancy in the description exists, while in higher dimensions $d>4$ the
form degrees change and similarly higher-to-higher gauge transformations are
required, and so on.

Following [43], let us now describe the complete extended $L_{\infty}$-algebra
of $BF$ theory in $d=4$ dimensions. We extend the cochain complex by
introducing

$\displaystyle V^{-1}:=\Omega^{0}(M,W)\qquad\text{and}\qquad
V^{4}:=\Omega^{4}(M,\mathfrak{g})\ ,$ (6.30)

and denote the corresponding elements by $\varepsilon\in V^{-1}$ and
$\mathcal{E}\in V^{4}$. The brackets (6.8) and (6.9) are extended as

$\displaystyle\ell_{1}(\varepsilon)=(\mathrm{d}\varepsilon,0)\qquad\text{and}\qquad\ell_{1}(\mathcal{T},\varLambda)=\mathrm{d}\mathcal{T}\
,$ (6.31)

together with

$\displaystyle\begin{split}\ell_{2}\big{(}\varepsilon\,,\,(\tau,\lambda)\big{)}=\lambda\cdot\varepsilon\quad,\quad\ell_{2}\big{(}\varepsilon\,,\,(B,A)\big{)}&=-(A\cdot\varepsilon,0)\quad,\quad\ell_{2}\big{(}\varepsilon\,,\,(\mathcal{B},\mathcal{A})\big{)}=(\mathcal{B}\cdot\varepsilon,0)\
,\\\\[4.0pt]
\ell_{2}\big{(}\varepsilon\,,\,(\mathcal{T},\varLambda)\big{)}=-(0,\mathcal{T}\cdot\varepsilon)\quad,\quad\ell_{2}(\varepsilon,\mathcal{E})&=(0,\mathcal{E}\cdot\varepsilon)\quad,\quad\ell_{2}\big{(}(\tau,\lambda)\,,\,\mathcal{E}\big{)}=-[\lambda,\mathcal{E}]_{\mathfrak{g}}\
,\\\\[4.0pt]
\ell_{2}\big{(}(B,A)\,,\,(\mathcal{T},\varLambda)\big{)}=-[A,\mathcal{T}]_{\mathfrak{g}}\quad&,\quad\ell_{2}\big{(}(\mathcal{B}_{1},\mathcal{A}_{1})\,,\,(\mathcal{B}_{2},\mathcal{A}_{2})\big{)}=-[\mathcal{B}_{1},\mathcal{B}_{2}]_{\mathfrak{g}}\
.\end{split}$ (6.32)

The cyclic inner product extends naturally as

$\displaystyle\langle\varepsilon,\mathcal{E}\rangle=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\varepsilon\wedge\mathcal{E})\
.$ (6.33)

The higher gauge transformation is now encoded in the expected way via (2.58)
as

$\displaystyle\delta_{(\varepsilon,(B,A))}(\tau,\lambda)=\ell_{1}(\varepsilon)-\ell_{2}\big{(}\varepsilon,(B,A)\big{)}=\big{(}\mathrm{d}^{A}\varepsilon,0\big{)}\
.$ (6.34)

Its adjoint defines the higher Noether operator

$\displaystyle{\mathsf{d}}_{(B,A)}^{{\textrm{\tiny(1)}}}(\mathcal{T},\varLambda)=\ell_{1}(\mathcal{T},\varLambda)-\ell_{2}\big{(}(B,A),(\mathcal{T},\varLambda)\big{)}=\mathrm{d}^{A}\mathcal{T}\
,$ (6.35)

which encodes the level 1 Noether identity via (2.64) as

$\displaystyle{\mathsf{d}}_{(B,A)}^{{\textrm{\tiny(1)}}}\circ{\mathsf{d}}_{(B,A)}(\mathcal{B},\mathcal{A})=-\ell_{2}\big{(}(\mathfrak{F}_{B},\mathfrak{F}_{A}),(\mathcal{B},\mathcal{A})\big{)}=[\mathfrak{F}_{B},\mathcal{B}]_{\mathfrak{g}}\
.$ (6.36)

The extension of the $L_{\infty}$-algebra carries through the usual
deformation to produce a braided $L_{\infty}$-algebra without any issues.
However, we currently do not have a good physical interpretation of these
‘higher’ braided gauge symmetries. The issue is related to the problem of
defining the braided moduli space of classical solutions. We first briefly
describe the problem without explicitly writing out the detailed extended
braided $L_{\infty}$-algebra, and then show how it is a general consequence
within the braided $L_{\infty}$-algebra framework.

For $d=4$ the braided shift symmetry generated by $\tau\in\Omega^{1}(M,W)$
acts as
$\delta_{\tau}^{{\textrm{\tiny${\mathcal{R}}$}}}(B,A)=(\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\tau,0)$. For a higher gauge parameter $\varepsilon\in\Omega^{0}(M,W)$, we
would expect the deformation of the action on gauge parameters to be
$\delta_{(\varepsilon,A)}^{{\textrm{\tiny${\mathcal{R}}$}}}\tau=\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\varepsilon\,\in\Omega^{1}(M,W)[[\hbar]]$. This, however, does not seem to
generate an equivalent (in the classical sense) braided gauge transformation
via $\tau^{\prime}:=\tau+\mathrm{d}^{A}_{\star{\textrm{\tiny R}}}\varepsilon$,
since

$\displaystyle\begin{split}\delta_{\tau^{\prime}}^{{\textrm{\tiny${\mathcal{R}}$}}}(B,A)&=\big{(}\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\tau^{\prime},0\big{)}\\\\[4.0pt]
&=\big{(}\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\tau+(\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}})^{2}\varepsilon,0\big{)}\\\\[4.0pt]
&=\big{(}\delta_{\tau}^{{\textrm{\tiny${\mathcal{R}}$}}}B+\mathsf{R}_{k}(\mathfrak{F}^{\star}_{B})\star\mathsf{R}^{k}(\varepsilon)-\tfrac{1}{2}\,\mathsf{R}_{k}(A)\wedge_{\star}\big{(}\mathsf{R}_{l}(A)\star\mathsf{R}^{l}\,\mathsf{R}^{k}(\varepsilon)\big{)}\\\
&\hskip
142.26378pt+\tfrac{1}{2}\,\mathsf{R}_{k}\,\mathsf{R}_{l}(A)\wedge_{\star}\big{(}\mathsf{R}_{m}\,\mathsf{R}^{l}(A)\star\mathsf{R}^{m}\,\mathsf{R}^{k}(\varepsilon)\big{)}\,,\,0\big{)}\
.\end{split}$ (6.37)

That is, they do not generate the same symmetry on-shell. The last two
polynomial terms spoil this symmetry, which is a consequence of the breaking
of the first Bianchi identity (6.20) used in the third equality. We now
explain that this is a generic feature of any higher braided gauge theory, by
using its braided $L_{\infty}$-algebra incarnation.

For this, we extend the braided brackets (6.11) by twisting the corresponding
classical extended brackets, with $\ell^{\star}_{1}=\ell_{1}$ as usual and the
braided 2-brackets

$\displaystyle\begin{split}\ell^{\star}_{2}\big{(}\varepsilon\,,\,(\tau,\lambda)\big{)}=\lambda\star\varepsilon\qquad&,\qquad\ell_{2}^{\star}\big{(}\varepsilon\,,\,(B,A)\big{)}=-\big{(}\mathsf{R}_{k}(A)\star\mathsf{R}^{k}(\varepsilon),0\big{)}\
,\\\\[4.0pt]
\ell^{\star}_{2}\big{(}\varepsilon\,,\,(\mathcal{B},\mathcal{A})\big{)}=\big{(}\mathsf{R}_{k}(\mathcal{B})\star\mathsf{R}^{k}(\varepsilon),0\big{)}\qquad&,\qquad\ell_{2}^{\star}\big{(}\varepsilon\,,\,(\mathcal{T},\varLambda)\big{)}=-\big{(}0,\mathsf{R}_{k}(\mathcal{T})\star\mathsf{R}^{k}(\varepsilon)\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}(\varepsilon,\mathcal{E})=\big{(}0,\mathsf{R}_{k}(\mathcal{E})\star\mathsf{R}^{k}(\varepsilon)\big{)}\qquad&,\qquad\ell_{2}^{\star}\big{(}(\tau,\lambda)\,,\,\mathcal{E}\big{)}=-[\lambda,\mathcal{E}]_{\mathfrak{g}}^{\star}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(B,A)\,,\,(\mathcal{T},\varLambda)\big{)}=-[A,\mathcal{T}]_{\mathfrak{g}}^{\star}\qquad&,\qquad\ell_{2}^{\star}\big{(}(\mathcal{B}_{1},\mathcal{A}_{1})\,,\,(\mathcal{B}_{2},\mathcal{A}_{2})\big{)}=-[\mathcal{B}_{1},\mathcal{B}_{2}]_{\mathfrak{g}}^{\star}\
.\end{split}$ (6.38)

The cyclic inner product extends as

$\displaystyle\langle\varepsilon,\mathcal{E}\rangle_{\star}=\int_{M}\,\mathrm{Tr}_{{\textrm{\tiny$W$}}}(\varepsilon\wedge_{\star}\mathcal{E})\
.$ (6.39)

Using these brackets, we see that the (left) higher braided gauge
transformation is expressed exactly as in the classical case (2.58), that is

$\displaystyle\delta_{(\varepsilon,(B,A))}^{{\textrm{\tiny${\mathcal{R}}$}}}(\tau,\lambda)=\big{(}\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\varepsilon,0\big{)}=\ell^{\star}_{1}(\varepsilon)-\ell_{2}^{\star}\big{(}\varepsilon\,,\,(B,A)\big{)}\
.$ (6.40)

However, using the braided graded Jacobi identities one easily verifies that
the classical closure property (2.59) no longer holds, as

$\displaystyle\begin{split}\delta_{(\varepsilon,(B,A))}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{(}\delta_{\tau}^{{\textrm{\tiny${\mathcal{R}}$}}}(B,A)\big{)}&=\ell_{2}^{\star}\big{(}\varepsilon,\mathfrak{F}^{\star}_{(B,A)}\big{)}+\ell_{2}^{\star}\big{(}\ell_{2}^{\star}\big{(}\varepsilon,\mathsf{R}_{k}(B,A)\big{)}\,,\,\mathsf{R}^{k}(B,A)\big{)}\\\
&\hskip
113.81102pt-\tfrac{1}{2}\,\ell_{2}^{\star}\big{(}\varepsilon,\ell_{2}^{\star}\big{(}(B,A)\,,\,(B,A)\big{)}\big{)}\
,\end{split}$ (6.41)

whereby expanding the brackets explicitly recovers the formula (6.37), as
expected. Since this is a feature of any differential graded braided Lie
algebra, this will be a generic trait of any higher gauge theory based on a
braided $L_{\infty}$-algebra, where higher brackets will generally contribute
higher tensor powers of the fields to this expression.

One may also use cyclicity of the extended pairing to define the adjoint
${\mathsf{d}}^{{\textrm{\tiny(1)}}{\textrm{\tiny${\mathcal{R}}$}}}_{(B,A)}:V^{3}[[\hbar]]\to
V^{4}[[\hbar]]$ of the operator
$\delta_{(\varepsilon,(B,A))}^{{\textrm{\tiny${\mathcal{R}}$}}}$, exactly as
in the classical case (2.62). However, the meaning of this operator is
currently unclear, due to both the breakdown of the on-shell closure propery
of its adjoint operation and also the peculiar definition of the level 0
Noether operator (4.96) in the braided field theory. It would be interesting
to further explore these issues and develop a proper formulation of higher
braided gauge theories. As before, the braided BV formalism of Section 4.8
circumvents these issues, while an appropriate higher category generalization
of the description of Section 4.9 ought to make sense of higher braided gauge
symmetries in the setting of the braided Maurer–Cartan moduli spaces discussed
there.353535This would require the $\infty$-category setting mentioned in
Footnote 29.

##  7 Braided noncommutative Yang–Mills theory

In this final section we look at our first example of a noncommutative field
theory with gauge symmetry based on a braided $L_{\infty}$-algebra which is
not a differential graded braided Lie algebra. We twist deform the
$L_{\infty}$-algebra formulation of Yang–Mills theory discussed in Section 2.5
to uncover a new noncommutative deformation of Yang–Mills theory with novel
and surprising features. This is in marked contrast to scalar field theory,
whose braided deformation gives the standard noncommutative field theory, and
the braided Chern–Simons and $BF$ theories, which are based on differential
graded braided Lie algebras, and whose field equations and variational
principles are also given by the naive replacements of products of fields with
star-products.363636Of course, this does not preclude the different
kinematical sectors of these theories due to the braided Lie brackets, as well
as the novel general nature of classical solutions in braided field theory and
the corresponding modified forms of gauge redundancies through the braided
Noether identities.

###  7.1 Noncommutative Yang–Mills theory in the braided $L_{\infty}$-algebra
formalism

In this section we consider the braided noncommutative deformation of the
Yang–Mills gauge theory described in Section 2.5. As in the example of
noncommutative scalar field theory, we suppose now that the background
spacetime $(M,g)$ has a non-trivial Killing Lie subalgebra of vector fields
$\mathfrak{k}\subset\mathfrak{v}={\sf\Gamma}(TM)$. For the same reasons as
spelled out in Section 5.2, the Yang–Mills cyclic $L_{\infty}$-algebra
$(V,\ell_{1},\ell_{2},\ell_{3},\langle-,-\rangle)$ lives in the category of
$\mathsf{U}\mathfrak{k}$-modules.

We now apply the general formalism of braided field theory from Section 4.6 to
the more elaborate example of Yang–Mills theory. A choice of a Killing
Drinfel’d twist373737In $d=4$ dimensions, Yang–Mills theory is further
invariant under conformal transformations. In this dimension, one may also use
Drinfel’d twists generated by the conformal Killing Lie algebra of $(M,g)$. An
analogous statement holds in $d=2$ dimensions, wherein Yang–Mills theory is
invariant under all area-preserving diffeomorphisms.
$\mathcal{F}\in\mathsf{U}\mathfrak{k}[[\hbar]]\otimes\mathsf{U}\mathfrak{k}[[\hbar]]$
deforms the underlying $L_{\infty}$-algebra to a 4-term braided
$L_{\infty}$-algebra
$\big{(}V[[\hbar]],\ell_{1}^{\star},\ell_{2}^{\star},\ell_{3}^{\star}\big{)}$
with 1-bracket $\ell_{1}^{\star}=\ell_{1}$, the 2-brackets

$\displaystyle\begin{split}\ell_{2}^{\star}(\lambda_{1},\lambda_{2})=-[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}\qquad&,\qquad\ell_{2}^{\star}(\lambda,A)=-[\lambda,A]_{\mathfrak{g}}^{\star}\
,\\\\[4.0pt]
\ell_{2}^{\star}(\lambda,\mathcal{A})=-[\lambda,\mathcal{A}]_{\mathfrak{g}}^{\star}\qquad,\qquad\ell_{2}^{\star}(\lambda,\varLambda)&=-[\lambda,\varLambda]_{\mathfrak{g}}^{\star}\qquad,\qquad\ell_{2}^{\star}(A,\mathcal{A})=-[A,\mathcal{A}]_{\mathfrak{g}}^{\star}\
,\\\\[4.0pt] \ell_{2}^{\star}(A_{1},A_{2})=-\mathrm{d}\ast_{\textrm{\tiny
H}}[A_{1},A_{2}]_{\mathfrak{g}}^{\star}-&[A_{1},\ast_{\textrm{\tiny
H}}\mathrm{d}A_{2}]_{\mathfrak{g}}^{\star}+(-1)^{d}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}A_{1},A_{2}]_{\mathfrak{g}}^{\star}\ ,\end{split}$ (7.1)

and finally the non-zero 3-bracket

$\displaystyle\begin{split}\ell_{3}^{\star}(A_{1}&,A_{2},A_{3})\\\\[4.0pt]
&=-\big{[}A_{1},\,\ast_{\textrm{\tiny
H}}[A_{2},A_{3}]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\big{[}\mathsf{R}_{k}(A_{2}),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A_{1}),A_{3}]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+(-1)^{d}\,\big{[}\ast_{\textrm{\tiny
H}}[A_{1},A_{2}]_{\mathfrak{g}}^{\star}\,,A_{3}\big{]}_{\mathfrak{g}}^{\star}\
,\end{split}$ (7.2)

where
$[-,-]_{\mathfrak{g}}^{\star}:=[-,-]_{\mathfrak{g}}\circ\mathcal{F}^{-1}$.

The (left) braided gauge transformation of a gauge field
$A\in\Omega^{1}(M,\mathfrak{g})$ by a gauge parameter
$\lambda\in\Omega^{0}(M,\mathfrak{g})$ follows the standard rule

$\displaystyle\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}A:=\ell_{1}^{\star}(\lambda)+\ell_{2}^{\star}(\lambda,A)=\mathrm{d}\lambda-[\lambda,A]_{\mathfrak{g}}^{\star}\
,$ (7.3)

closing as a braided Lie algebra:

$\displaystyle\big{[}\delta_{\lambda_{1}}^{{\textrm{\tiny${\mathcal{R}}$}}},\delta_{\lambda_{2}}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{]}_{\circ}^{\star}=\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{-\ell_{2}^{\star}(\lambda_{1},\lambda_{2})}=\delta_{[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}}^{{\textrm{\tiny${\mathcal{R}}$}}}\
,$ (7.4)

for all $\lambda_{1},\lambda_{2}\in\Omega^{0}(M,\mathfrak{g})$.

#### Braided Yang–Mills equations

The braided Yang–Mills field equations turn out to have a rather surprising
form. Following the prescription from Section 4.6, the field equations
$\mathfrak{F}^{\star}_{A}=0$ valued in $\Omega^{d-1}(M,\mathfrak{g})[[\hbar]]$
are given by

$\displaystyle\begin{split}\begin{tabular}[]{|l|}\hline\cr\\\
$\displaystyle\mathfrak{F}_{A}^{\star}=\tfrac{1}{2}\,\big{(}\mathrm{d}^{A}_{\star{\textrm{\tiny
L}}}\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}+\mathrm{d}^{A}_{\star{\textrm{\tiny
R}}}\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}$\\\\[5.69054pt]
$\displaystyle\quad\,\hskip
28.45274pt+\,\tfrac{1}{6}\,\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\tfrac{1}{12}\,\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{(-1)^{d}}{12}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}$\\\ \\\
\hline\cr\end{tabular}\end{split}$ (7.5)

where
$F_{A}^{\textrm{\tiny${\mathcal{R}}$}}=\mathrm{d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}^{\star}\in\Omega^{2}(M,\mathfrak{g})[[\hbar]]$
is the braided curvature 2-form. Indeed, some simple algebra gives

$\displaystyle\begin{split}\mathfrak{F}^{\star}_{A}&=\ell_{1}^{\star}(A)-\tfrac{1}{2}\,\ell_{2}^{\star}(A,A)-\tfrac{1}{3!}\,\ell_{3}^{\star}(A,A,A)\\\\[4.0pt]
&=\mathrm{d}\ast_{\textrm{\tiny
H}}\mathrm{d}A+\tfrac{1}{2}\,\mathrm{d}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}+\tfrac{1}{2}\,[A,\ast_{\textrm{\tiny
H}}\mathrm{d}A]_{\mathfrak{g}}^{\star}-\tfrac{(-1)^{d}}{2}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}A,A]_{\mathfrak{g}}^{\star}\\\
&\quad\,+\tfrac{1}{6}\,\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{1}{6}\,\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\tfrac{(-1)^{d}}{6}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\\\\[4.0pt]
&=\mathrm{d}\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}+\tfrac{1}{2}\,\big{[}A,\ast_{\textrm{\tiny
H}}\mathrm{d}A+\tfrac{1}{2}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\tfrac{(-1)^{d}}{2}\,\big{[}\ast_{\textrm{\tiny
H}}\mathrm{d}A+\tfrac{1}{2}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,\,A\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,-\tfrac{1}{12}\,\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{1}{6}\,\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{(-1)^{d}}{12}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\
,\end{split}$ (7.6)

and the result follows by identifying the curvature and left/right covariant
derivatives.

The first term of the field equations (7.5) is the expected symmetrized
braided covariant derivative of $\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}$, which reduces to the covariant
constancy condition in the classical limit, similarly to the example of
braided $BF$ theory from Section 6.2. However, the appearance of the last
three terms is completely unexpected, since in the classical limit
${\mathcal{R}}=1\otimes 1$ and the terms conspire to cancel. Hence these terms
could not have been guessed or even motivated by naively deforming the
classical field equations. The braided $L_{\infty}$-algebra perspective
necessitates their presence. Similar features occur in four-dimensional
braided gravity [45], and we expect such genuinely new interaction terms to be
a generic feature of gauge theories whose underlying $L_{\infty}$-algebra is
not simply a differential graded Lie algebra.

The field equations (7.5) differ drastically from those in the usual
approaches to noncommutative Yang–Mills theory based on star-gauge symmetry
(cf. Section 3.1), wherein only the unbraided version of the first term
appears (with left and right covariant derivatives coinciding). This either
imposes restrictions on the gauge algebra $\mathfrak{g}$ or the introduction
of (infinitely many) new degrees of freedom. Our braided field theory instead
makes sense for any Lie algebra $\mathfrak{g}$ without introducing any
spurious degrees of freedom. The braided field equations (7.5) are furthermore
real when restricted to real gauge fields, as follows from the general
discussion of Section 4.6.

The general braided $L_{\infty}$-algebra formulation guarantees the gauge
covariance of the field equations (7.5) under the braided gauge
transformations (7.3). Gauge covariance in the present example reads in the
expected way as

$\displaystyle\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathfrak{F}_{A}^{\star}=\ell_{2}^{\star}(\lambda,\mathfrak{F}_{A}^{\star})=-[\lambda,\mathfrak{F}_{A}^{\star}]_{\mathfrak{g}}^{\star}\
.$ (7.7)

Given the explicit form of (7.5), this is a non-trivial statement. The first
term is manifestly braided covariant, as it involves covariant derivatives. On
the other hand, the last three terms are not separately braided covariant
objects. However, it is straightforward to show that together they do form a
braided covariant combination, as guaranteed by the braided homotopy relations
of Section 4.6. We will demonstrate how this works explicitly in Section 7.2
below.

#### Braided Noether identities

From the general prescription of Section 4.6, the gauge redundancies of
braided Yang–Mills theory are encoded by the braided Noether identities

$\displaystyle\begin{split}{\mathsf{d}}_{A}^{\star}\mathfrak{F}^{\star}_{A}&:=\ell_{1}^{\star}(\mathfrak{F}_{A}^{\star})-\tfrac{1}{2}\,\big{(}\ell_{2}^{\star}(A,\mathfrak{F}_{A}^{\star})-\ell_{2}^{\star}(\mathfrak{F}_{A}^{\star},A)\big{)}\\\
&\quad\,-\tfrac{1}{12}\,\ell_{1}^{\star}\big{(}\ell_{3}^{\star}(A,A,A)\big{)}+\tfrac{1}{4}\,\ell_{2}^{\star}\big{(}\mathsf{R}_{k}(A),\,\ell_{2}^{\star}(\mathsf{R}^{k}(A),A)\big{)}\\\
&\quad\,+\tfrac{1}{12}\,\Big{(}\ell_{2}^{\star}\big{(}\ell_{3}^{\star}(A,A,A),A\big{)}-\ell_{2}^{\star}\big{(}A,\ell_{3}^{\star}(A,A,A)\big{)}\Big{)}\
=\ 0\ ,\end{split}$ (7.8)

where we used $\ell^{\star}_{3}\big{(}\mathcal{A},A_{1},A_{2}\big{)}=0$.
Expanding and collecting terms, this translates to the differential identity

$\displaystyle\begin{split}\begin{tabular}[]{|l|}\hline\cr\\\
$\displaystyle\tfrac{1}{2}\,\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\mathfrak{F}_{A}^{\star}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\mathfrak{F}_{A}^{\star}\big{)}+\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}Z_{A}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}Z_{A}-3\,\mathrm{d}Z_{A}+\tfrac{1}{4}\,\big{(}\mathrm{d}\tilde{Z}_{A}-[A,\tilde{Z}_{A}]_{\mathfrak{g}}^{\star}-[\tilde{Z}_{A},A]_{\mathfrak{g}}^{\star}\big{)}$\\\\[5.69054pt]
$\hskip
42.67912pt+\,\tfrac{1}{4}\,\big{[}\mathsf{R}_{k}(A)\,,\,\mathrm{d}\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}+[\mathsf{R}^{k}(A),\ast_{\textrm{\tiny
H}}\mathrm{d}A]_{\mathfrak{g}}^{\star}-(-1)^{d}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\
=\ 0$\\\ \\\ \hline\cr\end{tabular}\end{split}$ (7.9)

where we define $Z_{A}$ to be the new covariant combination from the field
equations (7.5):

$\displaystyle\begin{split}Z_{A}:=\tfrac{1}{6}\,\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\tfrac{1}{12}\,\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{(-1)^{d}}{12}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\
,\end{split}$ (7.10)

and $\tilde{Z}_{A}$ to be the first term of this expression:

$\displaystyle\tilde{Z}_{A}:=\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\
.$ (7.11)

In the classical limit $Z_{A}$ vanishes, while the rest of the inhomogeneous
terms cancel upon application of the classical Jacobi identity for the graded
Lie algebra
$\big{(}\Omega^{\bullet}(M,\mathfrak{g}),[-,-]_{\mathfrak{g}}\big{)}$. Thus we
obtain the classical Noether identity $\mathrm{d}^{A}\mathfrak{F}_{A}=0$, as
expected.

#### Braided Yang–Mills functional

By twisting the classical Yang–Mills inner product (2.74), we obtain the
(braided) cyclic structure

$\displaystyle\langle
A,\mathcal{A}\rangle_{\star}:=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(A\wedge_{\star}\mathcal{A})\qquad\mbox{and}\qquad\langle\lambda,\mit\Lambda\rangle_{\star}:=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(\lambda\wedge_{\star}\mit\Lambda)\
.$ (7.12)

We shall use this to write out and simplify the braided Yang–Mills action
functional in two forms, as we did with the braided $BF$ theory of Section
6.2. First we consider an arbitrary Drinfel’d twist $\mathcal{F}$, and hence a
braided cyclic pairing (or for a compatible twist we simply forget the strict
cyclicity). We define the action functional via the prescription of Section
4.6:

$\displaystyle\begin{split}S_{\star}(A):\\!&=\frac{1}{2}\,\langle
A,\ell^{\star}_{1}(A)\rangle_{\star}-\frac{1}{3!}\,\langle
A,\ell^{\star}_{2}(A,A)\rangle_{\star}-\frac{1}{4!}\,\langle
A,\ell^{\star}_{3}(A,A,A)\rangle_{\star}\\\\[4.0pt]
&=\int_{M}\,\frac{1}{2}\,\mathrm{Tr}_{\mathfrak{g}}(A\wedge_{\star}\mathrm{d}\ast_{\textrm{\tiny
H}}\mathrm{d}A)\\\
&\qquad\qquad-\frac{1}{6}\,\mathrm{Tr}_{\mathfrak{g}}\Big{(}A\wedge_{\star}\big{(}-\mathrm{d}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}-[A,\ast_{\textrm{\tiny
H}}\mathrm{d}A]_{\mathfrak{g}}^{\star}+(-1)^{d}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}A,A]_{\mathfrak{g}}^{\star}\big{)}\,\Big{)}\\\
&\qquad\qquad-\frac{1}{24}\,\mathrm{Tr}_{\mathfrak{g}}\Big{(}A\wedge_{\star}\big{(}-\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\\\
&\hskip 227.62204pt+(-1)^{d}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\big{)}\,\Big{)}\
.\end{split}$ (7.13)

We simplify (7.13) by making the term
$\mathrm{Tr}_{\mathfrak{g}}(F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}})$ appear, and hence view the
remaining terms as a deformation of the “naive” noncommutative Yang–Mills
action functional. This requires the use of the $\mathcal{R}$-matrix
identities (4.19), as well as the invariance of
$\mathrm{Tr}_{\mathfrak{g}}(-\wedge_{\star}-)$ under the left and right
braided adjoint actions, which follows from the classical invariance order by
order in the deformation parameter $\hbar$. For instance, we may derive the
useful identity

$\displaystyle\begin{split}\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}A\wedge_{\star}[A,\ast_{\textrm{\tiny
H}}\mathrm{d}A]_{\mathfrak{g}}^{\star}\,\big{)}&=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathsf{R}_{k}(A),\mathsf{R}^{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\ast_{\textrm{\tiny
H}}\mathrm{d}A\big{)}\\\\[4.0pt]
&=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,A]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\ast_{\textrm{\tiny
H}}\mathrm{d}A\big{)}\\\\[4.0pt]
&=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\mathrm{d}A\big{)}\
,\end{split}$ (7.14)

where in the first line we use the braided ${\rm ad}(\mathfrak{g})$-invariance
of $\mathrm{Tr}_{\mathfrak{g}}(-\wedge_{\star}-)$, followed by braided
symmetry of $[-,-]_{\mathfrak{g}}^{\star}$ on 1-forms and strict symmetry of
the Hodge duality operator $\ast_{\textrm{\tiny H}}$ with respect to the
cyclic inner product. In a completely analogous way, we find

$\displaystyle\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}A\wedge_{\star}\,\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\,\big{)}=-(-1)^{d}\,\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}\ .$ (7.15)

By treating the rest of the terms in a similar fashion, after a long but
straightforward calculation we reach a simplified and more familiar form of
the braided noncommutative Yang–Mills functional:

$\displaystyle\begin{split}S_{\star}(A)&=\int_{M}\,\frac{1}{2}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}+\frac{1}{6}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\ast_{\textrm{\tiny
H}}\mathsf{R}^{k}(F_{A}^{\textrm{\tiny${\mathcal{R}}$}})\big{)}\\\
&\quad\qquad\,-\frac{1}{12}\,\Big{(}\mathrm{Tr}_{\mathfrak{g}}\big{(}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\wedge_{\star}\,F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}\Big{)}\\\
&\quad\qquad\,+\frac{1}{24}\,\Big{(}\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}\\\
&\qquad\qquad\qquad\qquad\,-\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast_{\textrm{\tiny
H}}\mathsf{R}^{k}([A,A]_{\mathfrak{g}}^{\star}\,)\big{)}\Big{)}\ .\end{split}$
(7.16)

The first term is the “naive” deformation of the classical Yang–Mills action
functional (2.66), similarly to that discussed in Section 3.1. It is invariant
under braided gauge transformations, due to braided ${\rm
ad}(\mathfrak{g})$-invariance of $\mathrm{Tr}_{\mathfrak{g}}$ and covariance
of the braided curvature $F_{A}^{\textrm{\tiny${\mathcal{R}}$}}$. The
remaining terms are unexpected new interactions that represent the non-trivial
deformation we uncover in the braided $L_{\infty}$-algebra framework. They
vanish in the classical limit where $\mathcal{R}=1\otimes 1$, recovering the
classical action functional. This is completely in line with the extra terms
we find in the braided field equations (7.5). Given that the (braided)
$L_{\infty}$-algebra formulation of field theories is a broad generalization
of Chern–Simons theory, it seems natural that Chern–Simons type terms,
involving the gauge field $A$ on its own, turn up in our definition of braided
field theories. Although these terms are individually non-covariant, together
they form braided gauge covariant combinations. We discuss this point in more
detail in Section 7.2 below.

Restricting to compatible Drinfel’d twists, the pairing becomes a strictly
cyclic structure for the braided $L_{\infty}$-algebra
$(V[[\hbar]],\ell_{1}^{\star},\ell_{2}^{\star},\ell_{3}^{\star})$. In this
case, the action functional (7.16) simplifies further to

$\displaystyle\begin{split}\begin{tabular}[]{|l|}\hline\cr\\\ $\displaystyle
S_{\star}(A)=\frac{1}{2}\,\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}\big{)}+\frac{1}{24}\,\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}$\\\\[11.38109pt]
$\displaystyle\quad\,\hskip
28.45274pt-\frac{1}{24}\,\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,A]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}$\\\ \\\
\hline\cr\end{tabular}\end{split}$ (7.17)

where the first three unexpected terms cancel each other by
$\wedge_{\star}$-cyclicity under the integral, and the last term similarly
simplifies. One may derive the braided field equations (7.5) by varying this
explicit action functional via similar manipulations as above. Of course, this
is simply a confirmation of the general arguments from Section 4.6. For
Hermitian twists and real gauge fields, it is real-valued.

###  7.2 Braided gauge symmetry

It is instructive to see how gauge invariance of braided Yang–Mills theory is
realised explicitly. We start with the field equations (7.5). Consider a gauge
transformation of the last term:

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{[}\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}&=\big{[}\ast_{\textrm{\tiny
H}}[\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}+\big{[}\ast_{\textrm{\tiny
H}}[\mathsf{R}_{k}(A),\delta_{\mathsf{R}^{k}(\lambda)}^{{\textrm{\tiny${\mathcal{R}}$}}}A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,+\big{[}\ast_{\textrm{\tiny
H}}\mathsf{R}_{k}([A,A]_{\mathfrak{g}}^{\star})\,,\delta_{\mathsf{R}^{k}(\lambda)}^{{\textrm{\tiny${\mathcal{R}}$}}}A\big{]}_{\mathfrak{g}}^{\star}\\\\[4.0pt]
&=\big{[}\ast_{\textrm{\tiny
H}}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}-\big{[}\ast_{\textrm{\tiny
H}}[[\lambda,A]_{\mathfrak{g}}^{\star},A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,+\big{[}\ast_{\textrm{\tiny
H}}[\mathsf{R}_{k}(A),\mathsf{R}^{k}(\mathrm{d}\lambda)]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}-\big{[}\ast_{\textrm{\tiny
H}}[\mathsf{R}_{k}(A),[\mathsf{R}^{k}(\lambda),A]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,+\big{[}\ast_{\textrm{\tiny
H}}\mathsf{R}_{k}([A,A]_{\mathfrak{g}}^{\star})\,,\mathsf{R}^{k}(\mathrm{d}\lambda)\big{]}_{\mathfrak{g}}^{\star}-\big{[}\ast_{\textrm{\tiny
H}}\mathsf{R}_{k}([A,A]_{\mathfrak{g}}^{\star})\,,[\mathsf{R}^{k}(\lambda),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\\\\[4.0pt]
&=2\,\big{[}\ast_{\textrm{\tiny
H}}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}-(-1)^{d}\,\big{[}\mathrm{d}\lambda,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,-\big{[}\lambda,[\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\
,\end{split}$ (7.18)

where in the last equality we used the linearity of the Hodge duality operator
and the braided Jacobi identity to combine the terms cubic in $A$ into the
sought covariant form. The breaking of covariance arises from the terms
involving $\mathrm{d}\lambda$.

Similarly, one may check that

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{[}A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}&=2\,\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\big{[}\mathrm{d}\lambda,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,-\big{[}\lambda,[A,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\
,\end{split}$ (7.19)

and

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}[\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,]_{\mathfrak{g}}^{\star}&=\big{[}\mathrm{d}\lambda,\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\big{[}\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\\\
&\quad\,-(-1)^{d}\,\big{[}\ast_{\textrm{\tiny
H}}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}-\big{[}\lambda,[\mathsf{R}_{k}(A),\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}\
.\end{split}$ (7.20)

Taking the weighted sum of these three braided variations specified by (7.5),
we see that the non-covariant terms cancel. Hence the combination in the field
equations (7.5) is indeed braided gauge covariant.

Let us now turn to the general form (7.16) of the action functional. The
second, third and fourth terms form a covariant combination by the same
calculation for the braided $BF$ functional from Section 6.2. The last two
terms also give a surprising covariant combination. The potential non-
covariance arises from the terms proportional to $\mathrm{d}\lambda$ in the
gauge transformation, and dropping the covariant components which vanish by
braided ${\rm ad}(\mathfrak{g})$-invariance of $\mathrm{Tr}_{\mathfrak{g}}$,
we find

$\displaystyle\begin{split}&\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\big{)}\\\\[4.0pt]
&\qquad\quad=\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathsf{R}_{l}(A),\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[\mathsf{R}^{l}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star})\big{)}\\\
&\qquad\quad\quad\,+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathsf{R}_{l}(A),\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[\mathsf{R}_{m}(A),\mathsf{R}^{m}\,\mathsf{R}^{l}(\mathrm{d}\lambda)]_{\mathfrak{g}}^{\star})\big{)}\\\
&\qquad\quad\quad\,+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathsf{R}_{l}(A),\mathsf{R}_{k}\,\mathsf{R}^{m}\,\mathsf{R}^{l}(\mathrm{d}\lambda)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}\,\mathsf{R}_{m}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\big{)}\\\\[4.0pt]
&\qquad\quad=\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{k}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{)}\\\
&\qquad\quad\quad\,+\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{m}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[\mathsf{R}^{l}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,\mathsf{R}^{m}([\mathsf{R}_{l}(A),\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star})\big{)}\\\
&\qquad\quad\quad\,+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathsf{R}_{l}(A),\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\mathsf{R}^{k}(\ast_{\textrm{\tiny
H}}[\mathsf{R}^{l}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star})\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}\\\\[4.0pt]
&\qquad\quad=\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{k}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}\\\
&\qquad\quad\quad\,+2\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{l}(\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,[\mathsf{R}^{l}\,\mathsf{R}_{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}\\\\[4.0pt]
&\qquad\quad=\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{k}(\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}\\\
&\qquad\quad\quad\,+2\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\ast_{\textrm{\tiny
H}}[\mathrm{d}\lambda,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}\
,\end{split}$ (7.21)

where we repeatedly used braided symmetry of
$\mathrm{Tr}_{\mathfrak{g}}(-\wedge_{\star}-)$ and of
$[-,-]_{\mathfrak{g}}^{\star}$ on 1-forms along with the $\mathcal{R}$-matrix
identities (4.19). In a completely analogous way, one may check that

$\displaystyle\begin{split}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\,\mathrm{Tr}_{\mathfrak{g}}&\big{(}[A,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}\\\\[4.0pt]
&=2\,\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,\mathsf{R}_{k}(A)]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}+\mathrm{Tr}_{\mathfrak{g}}\big{(}[\mathrm{d}\lambda,A]_{\mathfrak{g}}^{\star}\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[A,A]_{\mathfrak{g}}^{\star}\,\big{)}\\\
&\quad\,+\mathrm{Tr}_{\mathfrak{g}}\big{(}\mathsf{R}_{k}([A,A]_{\mathfrak{g}}^{\star})\wedge_{\star}\,\ast_{\textrm{\tiny
H}}[\mathsf{R}^{k}(\mathrm{d}\lambda),A]_{\mathfrak{g}}^{\star}\,\big{)}\
.\end{split}$ (7.22)

Thus the non-covariant terms cancel by strict symmetry of the Hodge duality
operator, ensuring gauge invariance of the action functional as expected from
the general theory of Section 4.

###  7.3 Global braided symmetry

Similarly to the example of noncommutative scalar theory from Section 5.3,
braided Yang–Mills theory is invariant under the finite-dimensional braided
Lie algebra of twisted Killing symmetries
$(\mathfrak{k},[-,-]_{\mathfrak{k}}^{\star})$, acting via the braided Lie
derivative. These are incorporated into the braided $L_{\infty}$-algebra
framework of Section 7.1 by extending the degree $0$ and $3$ homogeneous
subspaces of the underlying cochain complex of Section 2.5 to get383838The
caveat discussed in Footnote 33 should also be borne in mind here.

$\displaystyle V^{0}=\mathfrak{k}\times\Omega^{0}(M,\mathfrak{g})\ ,\quad
V^{1}=\Omega^{1}(M,\mathfrak{g})\ ,\quad V^{2}=\Omega^{d-1}(M,\mathfrak{g})\
,\quad V^{3}=\mathfrak{k}^{*}\times\Omega^{d}(M,\mathfrak{g})\ ,$ (7.23)

with the differential extended as

$\displaystyle\ell_{1}^{\star}(\xi,\lambda)=\mathrm{d}\lambda\
,\quad\ell_{1}^{\star}(A)=\mathrm{d}\ast_{\textrm{\tiny
H}}\mathrm{d}A\qquad\mbox{and}\qquad\ell_{1}^{\star}(\mathcal{A})=(0,\mathrm{d}\mathcal{A})\
,$ (7.24)

and by modifying the $2$-brackets of (7.1) as

$\displaystyle\begin{split}\ell_{2}^{\star}\big{(}(\xi_{1},\lambda_{1})\,,\,(\xi_{2},\lambda_{2})\big{)}=\big{(}-[\xi_{1},\xi_{2}]_{\mathfrak{k}}^{\star}&\
,\,-\mathrm{L}_{\xi_{1}}^{\star}\lambda_{2}+\mathrm{L}^{\star}_{\mathsf{R}_{k}(\xi_{2})}\mathsf{R}^{k}(\lambda_{1})-[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}\big{)}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(\xi,\lambda)\,,\,A\big{)}=\mathrm{L}_{\xi}^{\star}A-[\lambda,A]_{\mathfrak{g}}^{\star}\qquad&,\qquad\ell_{2}^{\star}\big{(}(\xi,\lambda)\,,\,\mathcal{A}\big{)}=\mathrm{L}_{\xi}^{\star}\mathcal{A}-[\lambda,\mathcal{A}]_{\mathfrak{g}}^{\star}\
,\\\\[4.0pt]
\ell_{2}^{\star}\big{(}(\xi,\lambda)\,,\,(\xi^{\textrm{\tiny$\vee$}},\varLambda)\big{)}=\big{(}\xi\star\xi^{\textrm{\tiny$\vee$}}\,,\,\mathrm{L}_{\xi}^{\star}\varLambda-[&\lambda,\varLambda]_{\mathfrak{g}}^{\star}\big{)}\qquad,\qquad\ell_{2}^{\star}(A,\mathcal{A})=\big{(}0,-[A,\mathcal{A}]_{\mathfrak{g}}^{\star}\big{)}\
,\end{split}$ (7.25)

while the brackets $\ell_{2}^{\star}(A_{1},A_{2})$ and
$\ell_{3}^{\star}(A_{1},A_{2},A_{3})$ are unchanged. The cyclic structure
(7.12) is naturally extended as

$\displaystyle\langle
A,\mathcal{A}\rangle_{\star}=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(A\wedge_{\star}\mathcal{A})\qquad\mbox{and}\qquad\langle(\xi,\lambda)\,,\,(\xi^{\textrm{\tiny$\vee$}},\varLambda)\rangle_{\star}=\langle\xi,\xi^{\textrm{\tiny$\vee$}}\rangle_{\star}+\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(\lambda\wedge_{\star}\varLambda)\
.$ (7.26)

The extended braided transformations of the gauge fields

$\displaystyle\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{(\xi,\lambda)}A=\ell_{1}^{\star}(\xi,\lambda)+\ell_{2}^{\star}\big{(}(\xi,\lambda)\,,\,A\big{)}=\mathrm{L}_{\xi}^{\star}A+\mathrm{d}\lambda-[\lambda,A]_{\mathfrak{g}}^{\star}$
(7.27)

satisfy the gauge algebra

$\displaystyle\big{[}\delta_{(\xi_{1},\lambda_{1})}^{{\textrm{\tiny${\mathcal{R}}$}}},\delta_{(\xi_{2},\lambda_{2})}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{]}_{\circ}^{\star}=\delta_{-\ell_{2}^{\star}((\xi_{1},\lambda_{1}),(\xi_{2},\lambda_{2}))}^{{\textrm{\tiny${\mathcal{R}}$}}}=\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{([\xi_{1},\xi_{2}]_{\mathfrak{k}}^{\star},\mathrm{L}_{\xi_{1}}^{\star}\lambda_{2}-\mathrm{L}^{\star}_{\mathsf{R}_{k}(\xi_{2})}\mathsf{R}^{k}(\lambda_{1})+[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star})}\
,$ (7.28)

with bracket of the braided semi-direct product Lie algebra

$\displaystyle\mathfrak{k}[[\hbar]]\ltimes_{\star}\Omega^{0}(M,\mathfrak{g})[[\hbar]]\
.$ (7.29)

The braided noncommutative Yang–Mills equations (7.5) are correspondingly
covariant,

$\displaystyle\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{(\xi,\lambda)}\mathfrak{F}_{A}^{\star}=\ell_{2}^{\star}\big{(}(\xi,\lambda)\,,\,\mathfrak{F}_{A}^{\star}\big{)}=\mathrm{L}_{\xi}^{\star}\mathfrak{F}_{A}^{\star}-[\lambda,\mathfrak{F}_{A}^{\star}]_{\mathfrak{g}}^{\star}\
,$ (7.30)

and likewise the action functional (7.16) is invariant,
$\delta_{(\xi,\lambda)}^{{\textrm{\tiny${\mathcal{R}}$}}}S_{\star}(A)=0$.
These extended properties also follow directly from the closure and derivation
properties of the braided Lie derivative $\mathrm{L}_{\xi}^{\star}$.

## References

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