$\displaystyle\mathcal{F}_{12}\,(\Delta\otimes\mathbbm{1})\mathcal{F}=\mathcal{F}_{23}\,(\mathbbm{1}\otimes\Delta)\mathcal{F}\
,$ (4.12)

where $\mathcal{F}_{12}=\mathcal{F}\otimes 1$ and
$\mathcal{F}_{23}=1\otimes\mathcal{F}$, together with the normalization
condition $\mathcal{F}=1\otimes 1+O(\hbar)$.

We write the power series expansion of the twist as $\mathcal{F}=:{\sf
f}^{k}\otimes{\sf
f}_{k}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$,
with the sum over $k$ understood, and likewise for the inverse twist
$\mathcal{F}^{-1}=:\bar{{\sf f}}^{k}\otimes\bar{{\sf
f}}_{k}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
which satisfies a condition similar to (4.12) that can be expressed as

$\displaystyle\bar{\sf f}^{k}\otimes\Delta(\bar{\sf f}_{k})\,(\bar{\sf
f}^{l}\otimes\bar{\sf f}_{l})=\Delta(\bar{\sf f}^{k})\,(\bar{\sf
f}^{l}\otimes\bar{\sf f}_{l})\otimes\bar{\sf f}_{k}\ .$ (4.13)

The normalization condition ensures that the twist deformations induced by
$\mathcal{F}$ below have good classical limits at $\hbar=0$.

A Drinfel’d twist $\mathcal{F}$ defines a new Hopf algebra structure on the
universal enveloping algebra $\mathsf{U}\mathfrak{v}[[\hbar]]$, which we
denote by $\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$. As algebras
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}=\mathsf{U}\mathfrak{v}[[\hbar]]$,
while the new coproduct $\Delta_{\textrm{\tiny$\mathcal{F}$}}$ is given by

$\displaystyle\Delta_{{\textrm{\tiny$\mathcal{F}$}}}(X):=\mathcal{F}\,\Delta(X)\,\mathcal{F}^{-1}\
,$ (4.14)

for all $X\in\mathsf{U}\mathfrak{v}[[\hbar]]$. This new Hopf algebra is not
cocommutative in general, but the cocommutativity is controlled up to a
_braiding_ given by the invertible _universal ${\mathcal{R}}$-matrix_
${\mathcal{R}}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
induced by the twist as

$\displaystyle{\mathcal{R}}=\mathcal{F}_{21}\,\mathcal{F}^{-1}=:\mathsf{R}^{k}\otimes\mathsf{R}_{k}\
,$ (4.15)

where $\mathcal{F}_{21}=\tau(\mathcal{F})={\sf f}_{k}\otimes{\sf f}^{k}$ is
the twist with its legs swapped. Explicitly

$\displaystyle\tau\circ\Delta_{{\textrm{\tiny$\mathcal{F}$}}}(X)={\mathcal{R}}\,\Delta_{{\textrm{\tiny$\mathcal{F}$}}}(X)\,{\mathcal{R}}^{-1}\
.$ (4.16)

It is easy to see that the ${\mathcal{R}}$-matrix is triangular, that is

$\displaystyle{\mathcal{R}}_{21}={\mathcal{R}}^{-1}=\mathsf{R}_{k}\otimes\mathsf{R}^{k}\
,$ (4.17)

and moreover that

$\displaystyle(\Delta_{{\textrm{\tiny$\mathcal{F}$}}}\otimes\mathbbm{1}){\mathcal{R}}={\mathcal{R}}_{13}\,{\mathcal{R}}_{23}\qquad\mbox{and}\qquad(\mathbbm{1}\otimes\Delta_{{\textrm{\tiny$\mathcal{F}$}}}){\mathcal{R}}={\mathcal{R}}_{13}\,{\mathcal{R}}_{12}\
,$ (4.18)

where ${\mathcal{R}}_{13}=\mathsf{R}^{k}\otimes 1\otimes\mathsf{R}_{k}$, or
equivalently

$\displaystyle\Delta_{\textrm{\tiny$\mathcal{F}$}}(\mathsf{R}^{k})\otimes\mathsf{R}_{k}=\mathsf{R}^{l}\otimes\mathsf{R}^{k}\otimes\mathsf{R}_{l}\,\mathsf{R}_{k}\qquad\mbox{and}\qquad\mathsf{R}^{k}\otimes\Delta_{\textrm{\tiny$\mathcal{F}$}}({\mathsf{R}_{k}})=\mathsf{R}^{l}\,\mathsf{R}^{k}\otimes\mathsf{R}_{k}\otimes\mathsf{R}_{l}\
.$ (4.19)

These identities tell us that passing an element “at once” over a pair of
elements in a triple tensor product
$\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
is the same as passing successively over “each one” individually.

Drinfel’d twist deformation quantization consists in twisting the enveloping
Hopf algebra $\mathsf{U}\mathfrak{v}$ to a non-cocommutative Hopf algebra
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$, while simultaneously
twisting all of its modules [17]. A Drinfel’d twist $\mathcal{F}$ defines a
symmetric monoidal category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ of left
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-modules and equivariant
maps, that is, linear maps which commute with the action of ${\sf\Gamma}(TM)$
via the twisted coproduct $\Delta_{\textrm{\tiny$\mathcal{F}$}}$. The category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ is
functorially equivalent to ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ [20]. The
braiding isomorphism $\tau_{{\textrm{\tiny${\mathcal{R}}$}}}$ of
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ is now
non-trivial and given by composing the transposition $\tau$ with the action of
the inverse ${\mathcal{R}}^{-1}$ of the universal ${\mathcal{R}}$-matrix.
Since ${\mathcal{R}}$ is triangular, ${\mathcal{R}}_{21}={\mathcal{R}}^{-1}$,
the braiding is symmetric, i.e. the braiding isomorphism
$\tau_{{\textrm{\tiny${\mathcal{R}}$}}}$ squares to the identity morphism.

If $(\mathcal{A},\mu)$ is a (left) $\mathsf{U}\mathfrak{v}$-module algebra,
then we can deform the product $\mu$ on $\mathcal{A}$ by precomposing it with
the inverse of the twist $\mathcal{F}$ to get a new product

$\displaystyle\mu_{\star}(a\otimes b)=\mu\circ\mathcal{F}^{-1}(a\otimes
b)=\mu\big{(}\bar{\sf f}^{k}(a)\otimes\bar{\sf f}_{k}(b)\big{)}\ ,$ (4.20)

for $a,b\in\mathcal{A}$, where on the right-hand side we extend $\mu$
(linearly over $\mathbbm{C}$) to
$\mathcal{A}[[\hbar]]\otimes\mathcal{A}[[\hbar]]\simeq(\mathcal{A}\otimes\mathcal{A})[[\hbar]]$
by applying it term by term to the coefficients of a formal power series. The
cocycle condition (4.12) on $\mathcal{F}$ guarantees that this produces an
associative star-product $\mu_{\star}$ on $\mathcal{A}[[\hbar]]$, and it
generally defines a noncommutative
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module algebra
$(\mathcal{A}[[\hbar]],\mu_{\star})$, that is, an algebra in the category of
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-modules, because it
carries a representation of the twisted Hopf algebra
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$:

$\displaystyle\begin{split}X\big{(}\mu_{\star}(a\otimes
b)\big{)}&=X\big{(}\mu\circ\mathcal{F}^{-1}(a\otimes b)\big{)}\\\\[4.0pt]
&=\mu\circ\Delta(X)\big{(}\mathcal{F}^{-1}(a\otimes b)\big{)}\\\\[4.0pt]
&=\mu\circ\mathcal{F}^{-1}\big{(}\Delta_{\textrm{\tiny$\mathcal{F}$}}(X)(a\otimes
b)\big{)}=\mu_{\star}\big{(}\Delta_{\textrm{\tiny$\mathcal{F}$}}(X)(a\otimes
b)\big{)}\ ,\end{split}$ (4.21)

for all $X\in\mathsf{U}\mathfrak{v}$ and $a,b\in\mathcal{A}$.

If the algebra $(\mathcal{A},\mu)$ is commutative, i.e. $\mu\circ\tau=\mu$,
then $(\mathcal{A}[[\hbar]],\mu_{\star})$ is _braided commutative_ , i.e.
$\mu_{\star}\circ\tau_{{\textrm{\tiny${\mathcal{R}}$}}}=\mu_{\star}$. The
noncommutativity in this case is controlled by the triangular
${\mathcal{R}}$-matrix as

$\displaystyle\mu_{\star}(a\otimes
b)=\mu_{\star}\big{(}\mathsf{R}_{k}(b)\otimes\mathsf{R}^{k}(a)\big{)}\ ,$
(4.22)

which is easily seen by recalling that
${\mathcal{R}}=\mathcal{F}_{21}\,\mathcal{F}^{-1}$.

Let us now illustrate the formalism above by giving some concrete examples of
Drinfel’d twists which commonly appear in noncommutative field theory.

######  Example 4.23.

The _Moyal–Weyl twist_ on $M=\mathbbm{R}^{1,d-1}$ is given by

$\displaystyle\begin{split}\mathcal{F}&=\exp\big{(}-\tfrac{\mathrm{i}\,\hbar}{2}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}\\\\[4.0pt]
&=1\otimes
1+\sum_{n=1}^{\infty}\,\Big{(}-\frac{\mathrm{i}\,\hbar}{2}\Big{)}^{n}\,\frac{1}{n!}\,\theta^{\mu_{1}\nu_{1}}\cdots\theta^{\mu_{n}\nu_{n}}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\,\otimes\,\partial_{\nu_{1}}\cdots\partial_{\nu_{n}}=:{\sf
f}^{k}\otimes{\sf f}_{k}\ ,\end{split}$ (4.24)

where $(\theta^{\mu\nu})$ is a constant $d{\times}d$ antisymmetric real-valued
matrix. This twist is built on the enveloping Hopf algebra
$\mathsf{U}\mathbbm{R}^{1,d-1}\subset\mathsf{U}{\sf\Gamma}(T\mathbbm{R}^{1,d-1})$
of the abelian Lie algebra of infinitesimal spacetime translations. In this
case $\mathcal{F}_{21}=\mathcal{F}^{-1}$ so that the universal
${\mathcal{R}}$-matrix is given by

$\displaystyle{\mathcal{R}}=\mathcal{F}^{-2}=\exp\big{(}\,\mathrm{i}\,\hbar\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}=:\mathsf{R}^{k}\otimes\mathsf{R}_{k}\
.$ (4.25)

The twist deformation
$\big{(}C^{\infty}(\mathbbm{R}^{1,d-1})[[\hbar]],\mu_{\star}\big{)}$ of the
algebra of functions on $\mathbbm{R}^{1,d-1}$ gives precisely the Moyal–Weyl
star-product (1.1). More generally, the twist deformation
$\big{(}\Omega^{\bullet}(\mathbbm{R}^{1,d-1})[[\hbar]],\wedge_{\star}\big{)}$
of the exterior algebra of differential forms on $\mathbbm{R}^{1,d-1}$ gives
the noncommutative differential calculus discussed in Section 3.1; for the
holonomic coframe $\mathrm{L}_{\partial_{\nu}}\mathrm{d}x^{\mu}=0$, so that
indeed
$\mathrm{d}x^{\mu}\wedge_{\star}\mathrm{d}x^{\nu}=\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$
and
$\lambda\star\mathrm{d}x^{\mu}=\mathrm{d}x^{\mu}\star\lambda=\lambda\cdot\mathrm{d}x^{\mu}$
for $\lambda\in\Omega^{0}(\mathbbm{R}^{1,d-1})$.222222For these and other
considerations in this paper, the Cartan formula
$\mathrm{L}_{\xi}=\mathrm{d}\circ\imath_{\xi}+\imath_{\xi}\circ\mathrm{d}$ for
the action of the Lie derivative on forms is very useful, where $\imath_{\xi}$
is the interior multiplication with the vector field $\xi\in\mathfrak{v}$.
Similarly, for the holonomic frame
$\mathrm{L}_{\partial_{\mu}}\partial_{\nu}=[\partial_{\mu},\partial_{\nu}]_{\mathfrak{v}}=0$,
so that indeed $\lambda\star\partial_{\mu}=\lambda\cdot\partial_{\mu}$.

The Moyal–Weyl twist (4.24) has a natural extension from $\mathbbm{R}^{1,d-1}$
to any manifold $M$ with a non-constant Poisson bivector $\theta$ whose
components in a local holonomic frame satisfy

$\displaystyle\theta^{\mu\rho}\,\partial_{\rho}\theta^{\nu\lambda}=0\ ,$
(4.26)

see (3.52). An important class of examples arises in the noncommutative gauge
theories on D-branes in non-geometric T-duals to string theory
compactifications with duality twists, see e.g. [57, 12].

######  Example 4.27.

The _angular twist_ on $M=\mathbbm{R}^{1,d-1}$ is given by [46]

$\displaystyle\begin{split}\mathcal{F}&=\exp\big{(}-\tfrac{\mathrm{i}\,\hbar}{2}\,(\partial_{0}\otimes\mathsf{M}_{\theta}-\mathsf{M}_{\theta}\otimes\partial_{0})\big{)}\\\\[4.0pt]
&=1\otimes
1+\sum_{n=1}^{\infty}\,\Big{(}-\frac{\mathrm{i}\,\hbar}{2}\Big{)}^{n}\
\sum_{l=0}^{n}\,\frac{(-1)^{l}}{l!\,(n-l)!}\,(\partial_{0})^{n-l}\,(\mathsf{M}_{\theta})^{l}\,\otimes\,(\partial_{0})^{l}\,(\mathsf{M}_{\theta})^{n-l}=:{\sf
f}^{k}\otimes{\sf f}_{k}\ ,\end{split}$ (4.28)

where

$\displaystyle\mathsf{M}_{\theta}:=\theta^{ij}\,\mathsf{M}_{ij}\qquad\mbox{with}\quad\mathsf{M}_{ij}=x_{i}\,\partial_{j}-x_{j}\,\partial_{i}$
(4.29)

for a constant $(d{-}1){\times}(d{-}1)$ antisymmetric real matrix
$(\theta^{ij})$ and $i,j=1,\dots,d-1$. Here we denote coordinates on
$\mathbbm{R}^{1,d-1}$ by
$(x^{\mu})=\big{(}x^{0},(x^{i})\big{)}=(x^{0},x^{1},\dots,x^{d-1})$ with
$x^{0}$ the time coordinate. This twist is built on the enveloping Hopf
subalgebra of $\mathsf{U}{\sf\Gamma}(T\mathbbm{R}^{1,d-1})$ generated by the
direct product Lie algebra $\mathbbm{R}\times\mathfrak{so}(d-1)$ of
infinitesimal time translations and spatial rotations. The corresponding
triangular ${\mathcal{R}}$-matrix is again given by
${\mathcal{R}}=\mathcal{F}^{-2}$.

For the corresponding twist deformation
$\big{(}\Omega^{\bullet}(\mathbbm{R}^{1,d-1})[[\hbar]],\wedge_{\star}\big{)}$
of the exterior algebra, the holonomic coframe again has undeformed exterior
products
$\mathrm{d}x^{\mu}\wedge_{\star}\mathrm{d}x^{\nu}=\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$.
To describe the deformation of the $C^{\infty}(\mathbbm{R}^{1,d-1})$-bimodule
structure it is convenient to perform a spatial rotation of the coordinates
which brings the matrix $(\theta^{ij})$ into its Jordan canonical form with
non-zero skew-eigenvalues $\theta^{a}$ for $a=1,\dots,r$, where $2r$ is the
rank of $(\theta^{ij})$. Then

$\displaystyle\begin{split}\mathrm{d}x^{2a-1}\star\lambda&=\cos(\mathrm{i}\,\hbar\,\theta^{a}\,\partial_{0})\lambda\star\mathrm{d}x^{2a-1}+\sin(\mathrm{i}\,\hbar\,\theta^{a}\,\partial_{0})\lambda\star\mathrm{d}x^{2a}\
,\\\\[4.0pt]
\mathrm{d}x^{2a}\star\lambda&=-\sin(\mathrm{i}\,\hbar\,\theta^{a}\,\partial_{0})\lambda\star\mathrm{d}x^{2a-1}+\cos(\mathrm{i}\,\hbar\,\theta^{a}\,\partial_{0})\lambda\star\mathrm{d}x^{2a}\
,\end{split}$ (4.30)

for $a=1,\dots,r$ and $\lambda\in\Omega^{0}(\mathbbm{R}^{1,d-1})$, whereas
$\mathrm{d}x^{0}\star\lambda=\lambda\star\mathrm{d}x^{0}$ and
$\mathrm{d}x^{i}\star\lambda=\lambda\star\mathrm{d}x^{i}$ for
$i=2r+1,\dots,d-1$.

######  Example 4.31.

Both the Moyal–Weyl twist of Example 4.23 and the angular twist of Example
4.27 are special instances of a more general class of twists called _abelian
twists_ [16]. These can be defined on _any_ manifold $M$ and are given by

$\displaystyle\begin{split}\mathcal{F}&=\exp\big{(}-\tfrac{\mathrm{i}\,\hbar}{2}\,\theta^{ab}\,X_{a}\otimes
X_{b}\big{)}\\\\[4.0pt] &=1\otimes
1+\sum_{n=1}^{\infty}\,\Big{(}-\frac{\mathrm{i}\,\hbar}{2}\Big{)}^{n}\,\frac{1}{n!}\,\theta^{a_{1}b_{1}}\cdots\theta^{a_{n}b_{n}}\,X_{a_{1}}\cdots
X_{a_{n}}\,\otimes\,X_{b_{1}}\cdots X_{b_{n}}=:{\sf f}^{k}\otimes{\sf f}_{k}\
,\end{split}$ (4.32)

where $(\theta^{ab})$ is a constant antisymmetric real-valued matrix and
$\\{X_{a}\\}\subset{\sf\Gamma}(TM)$ is a set of mutually commuting vector
fields, $[X_{a},X_{b}]_{\mathfrak{v}}=0$. These twists are built from an
abelian subalgebra of the Lie algebra $\mathfrak{v}$ of vector fields on $M$.
For such twists the universal ${\mathcal{R}}$-matrix is always given as
${\mathcal{R}}=\mathcal{F}^{-2}$. They are a common choice in noncommutative
geometry and field theory because they share many of the special simplifying
properties enjoyed by the Moyal–Weyl twist. In particular, in an open
neighbourhood of every point of $M$ outside a set of measure zero, an abelian
twist is equivalent to a Moyal–Weyl twist via a suitable change of local frame
[11]. This is useful for practical considerations, as it means that the legs
of the twist $\mathcal{F}$ commute with the local basis vector fields in
almost every open set.

######  Example 4.33.

Twists need not only be associated to commuting vector fields. As an example
of a Drinfel’d twist which is _not_ an abelian twist, consider the _Jordanian
twist_ on $M=\mathbbm{R}^{1,d-1}$ given by [13]

$\displaystyle\begin{split}\mathcal{F}=\exp\big{(}-\mathsf{D}\otimes\log(1-\mathrm{i}\,\hbar\,\partial_{0})\big{)}=1\otimes
1+\sum_{n=1}^{\infty}\,\frac{(\mathrm{i}\,\hbar)^{n}}{n!}\,\mathsf{D}^{\underline{n}}\,\otimes\,(\partial_{0})^{n}=:{\sf
f}^{k}\otimes{\sf f}_{k}\ ,\end{split}$ (4.34)

where

$\displaystyle\mathsf{D}:=x^{\mu}\,\partial_{\mu}\ ,$ (4.35)

and the lower factorial $X^{\underline{n}}\in\mathsf{U}\mathfrak{v}$ of a
vector field $X\in\mathfrak{v}$ for $n\geq 1$ is defined by

$\displaystyle X^{\underline{n}}=X\,(X-1\big{)}\cdots\big{(}X-(n-1)\big{)}\ .$
(4.36)

This is built on the enveloping Hopf algebra of the semi-direct product Lie
algebra $\mathbbm{R}\ltimes\mathbbm{R}$ of infinitesimal dilatations and time
translations. The corresponding universal ${\mathcal{R}}$-matrix is given by

$\displaystyle\begin{split}{\mathcal{R}}&=\mathcal{F}_{21}\,\mathcal{F}^{-1}=\exp\big{(}-\log(1-\mathrm{i}\,\hbar\,\partial_{0})\otimes\mathsf{D}\big{)}\exp\big{(}\mathsf{D}\otimes\log(1-\mathrm{i}\,\hbar\,\partial_{0})\big{)}\\\\[4.0pt]
&=1\otimes 1+\sum_{n=1}^{\infty}\,(-\mathrm{i}\,\hbar)^{n}\
\sum_{l=0}^{n}\,\frac{(-1)^{l}}{l!\,(n-l)!}\,(\partial_{0})^{n-l}\,\mathsf{D}^{\underline{l}}\,\otimes\,(\partial_{0})^{l}\,\mathsf{D}^{\underline{n-l}}=:\mathsf{R}^{k}\otimes\mathsf{R}_{k}\
.\end{split}$ (4.37)

The holonomic coframe $\mathrm{d}x^{\mu}$ of the deformed exterior algebra
$\big{(}\Omega^{\bullet}(\mathbbm{R}^{1,d-1})[[\hbar]],\wedge_{\star}\big{)}$
once again has trivial star-exterior products
$\mathrm{d}x^{\mu}\wedge_{\star}\mathrm{d}x^{\nu}=\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$.
The $C^{\infty}(\mathbbm{R}^{1,d-1})$-bimodule structure is non-trivial and
given by

$\displaystyle\lambda\star\mathrm{d}x^{\mu}=\mathrm{d}x^{\mu}\star(1-\mathrm{i}\,\hbar\,\partial_{0})^{-1}\lambda\
,$ (4.38)

for all $\lambda\in\Omega^{0}(\mathbbm{R}^{1,d-1})$.

These twists appear in $\kappa$-deformed field theories on Minkowski
spacetimes which are covariant under the action of a $\kappa$-Poincaré quantum
group [13]. Jordanian twists can be more generally associated to the Borel
subalgebra of any Lie algebra
$\mathfrak{g}\subset\mathfrak{v}={\sf\Gamma}(TM)$ [16]. By further combining
these with the abelian twists of Example 4.31, we obtain more general
Drinfel’d twists which provide realizations of the extended Jordanian twists
of [64]. Such twists arise in the holographic dual gauge theories to
backgrounds obtained via Yang–Baxter deformations of string theory in
$\mathbbm{AdS}_{5}{\times}\mathbbm{S}^{5}$ [89].

###  4.3 Braided gauge symmetry and noncommutative kinematics

Let us now spell out the kinematical ingredients of braided noncommutative
gauge symmetry, incorporating dynamics later on. Let
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
be any Drinfel’d twist on a manifold $M$, and let $\mathfrak{g}$ be any Lie
algebra. As in Section 2.1, let
$\Omega^{\bullet}(M,\mathfrak{g})=\Omega^{\bullet}(M)\otimes\mathfrak{g}$ be
the exterior algebra of $\mathfrak{g}$-valued differential forms on $M$ with
graded Lie bracket $[-,-]_{\mathfrak{g}}$ given by the tensor product of
exterior multiplication with the Lie bracket of $\mathfrak{g}$. The graded Lie
algebra $\big{(}\Omega^{\bullet}(M,\mathfrak{g}),[-,-]_{\mathfrak{g}}\big{)}$
is naturally a $\mathsf{U}\mathfrak{v}$-module with $\mathsf{U}\mathfrak{v}$
acting trivially on $\mathfrak{g}$ and via the Lie derivative on
$\Omega^{\bullet}(M)$ (see Example 4.8), that is, it is a graded Lie algebra
in the category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$.

Drinfel’d twist deformation then produces a
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module
$\big{(}\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]],[-,-]_{\mathfrak{g}}^{\star}\big{)}$
with the star-bracket

$\displaystyle[\alpha,\beta]_{\mathfrak{g}}^{\star}:=[-,-]_{\mathfrak{g}}\circ\mathcal{F}^{-1}(\alpha\otimes\beta)=[\bar{\sf
f}^{k}(\alpha),\bar{\sf f}_{k}(\beta)]_{\mathfrak{g}}\ .$ (4.39)

This bracket makes $\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]]$ into a
_braided_ (or _quantum_) _graded Lie algebra_ in the sense of [91, 73], that
is, a graded Lie algebra in the category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ [21].
It is braided graded antisymmetric:

$\displaystyle[\alpha_{1},\alpha_{2}]_{\mathfrak{g}}^{\star}=-(-1)^{|\alpha_{1}|\,|\alpha_{2}|}\,[\mathsf{R}_{k}(\alpha_{2}),\mathsf{R}^{k}(\alpha_{1})]_{\mathfrak{g}}^{\star}\
,$ (4.40)

and satisfies the braided graded Jacobi identity:

$\displaystyle[\alpha_{1},[\alpha_{2},\alpha_{3}]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}=[[\alpha_{1},\alpha_{2}]_{\mathfrak{g}}^{\star},\alpha_{3}]_{\mathfrak{g}}^{\star}+(-1)^{|\alpha_{1}|\,|\alpha_{2}|}\,[\mathsf{R}_{k}(\alpha_{2}),[\mathsf{R}^{k}(\alpha_{1}),\alpha_{3}]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\
,$ (4.41)

for all homogeneous
$\alpha_{1},\alpha_{2},\alpha_{3}\in\Omega^{\bullet}(M,\mathfrak{g})$. This
braided graded Lie algebra has a natural braided Lie subalgebra
$\big{(}\Omega^{0}(M,\mathfrak{g})[[\hbar]],[-,-]_{\mathfrak{g}}^{\star}\big{)}$
which will play an important role below.

It is important to note that this definition makes sense for _any_ Lie algebra
$\mathfrak{g}$. In the special instance that $\mathfrak{g}$ is a matrix Lie
algebra, the braided commutator does _not_ coincide with the graded star-
commutator (4.1):

$\displaystyle[\alpha,\beta]_{\mathfrak{g}}^{\star}=\alpha\wedge_{\star}\beta-(-1)^{|\alpha|\,|\beta|}\,\mathsf{R}_{k}(\beta)\wedge_{\star}\mathsf{R}^{k}(\alpha)\
\neq\
[\alpha\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\beta]_{\mathfrak{g}}\
,$ (4.42)

where we used
$\mathcal{F}_{21}^{-1}=\mathcal{F}^{-1}\,{\mathcal{R}}^{-1}=\mathcal{F}^{-1}\,{\mathcal{R}}_{21}$.
In particular, if $\mathfrak{g}$ is an abelian Lie algebra, then the braided
commutator (4.42) always vanishes on $\Omega^{0}(M,\mathfrak{g})[[\hbar]]$,
while the star-commutator (4.1) does not.

Recall that the de Rham differential
$\mathrm{d}:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ makes
$\big{(}\Omega^{\bullet}(M,\mathfrak{g}),[-,-]_{\mathfrak{g}}\big{)}$ into a
differential graded Lie algebra, as a consequence of the Leibniz rule on the
underlying exterior algebra. By the Cartan structure equations,
$\mathrm{d}\circ\mathrm{L}_{\xi}=\mathrm{L}_{\xi}\circ\mathrm{d}$ for all
$\xi\in\mathfrak{v}$, so the exterior derivative commutes with the Lie
derivatives that enter into the definition of the star-exterior product via
the inverse twist $\mathcal{F}^{-1}$. It follows that the standard undeformed
de Rham differential is also an ordinary graded derivation of the underlying
star-exterior algebra
$\big{(}\Omega^{\bullet}(M)[[\hbar]],\wedge_{\star}\big{)}$:

$\displaystyle\mathrm{d}(\alpha\wedge_{\star}\beta)=\mathrm{d}\alpha\wedge_{\star}\beta+(-1)^{|\alpha|}\,\alpha\wedge_{\star}\mathrm{d}\beta\
.$ (4.43)

This makes
$\big{(}\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]],[-,-]_{\mathfrak{g}}^{\star},\mathrm{d}\big{)}$
into a differential graded braided Lie algebra, and provides a natural
noncommutative differential calculus for the construction of field theories on
$M$.

We are now ready to define notions of braided gauge fields and matter fields.
The idea is to deform representations of the Lie algebra $\mathfrak{g}$ in
which fields transform to braided representations of $\mathfrak{g}$, of which
there are two types, regarded as acting from the ‘left’ and from the ‘right’.
The _left_ and _right braided gauge transformations_ of a gauge field
$A\in\Omega^{1}(M,\mathfrak{g})$ by a gauge parameter
$\lambda\in\Omega^{0}(M,\mathfrak{g})$ are given respectively by twisting the
action of a classical gauge variation $\delta_{\lambda}$ as

$\displaystyle\begin{split}{}^{\textrm{\tiny
L}}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}A&:=\delta_{\bar{\sf
f}^{k}(\lambda)}\bar{\sf
f}_{k}(A)=\mathrm{d}\lambda-[\lambda,A]_{\mathfrak{g}}^{\star}\ ,\\\\[4.0pt]
{}^{\textrm{\tiny
R}}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}A&:={}^{\textrm{\tiny
L}}\delta_{\mathsf{R}_{k}(\lambda)}^{\textrm{\tiny${\mathcal{R}}$}}\mathsf{R}^{k}(A)=\delta_{\bar{\sf
f}_{k}(\lambda)}\bar{\sf
f}^{k}(A)=\mathrm{d}\lambda+[A,\lambda]_{\mathfrak{g}}^{\star}\ ,\end{split}$
(4.44)

where in the second line we again used
$\mathcal{F}^{-1}\,{\mathcal{R}}_{21}=\mathcal{F}_{21}^{-1}$. Since
$[A,\lambda]_{\mathfrak{g}}^{\star}=-[\mathsf{R}_{k}(\lambda),\mathsf{R}^{k}(A)]_{\mathfrak{g}}^{\star}$,
these are different transformations. In contrast, star-gauge transformations
do not see this left/right distinction, as
$[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda]_{\mathfrak{g}}=-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]_{\mathfrak{g}}$.

Similarly, if $\phi\in\Omega^{p}(M,W)$ is a $p$-form matter field in a linear
representation $W$ of $\mathfrak{g}$, then its left and right gauge
transformations are defined respectively by

$\displaystyle{}^{\textrm{\tiny
L}}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\phi=-\lambda\star\phi\qquad\mbox{and}\qquad{}^{\textrm{\tiny
R}}\delta_{\lambda}^{{\textrm{\tiny${\mathcal{R}}$}}}\phi=-\mathsf{R}_{k}(\lambda)\star\mathsf{R}^{k}(\phi)\
,$ (4.45)

where here $\lambda\star\phi:=\bar{\sf f}^{k}(\lambda)\cdot\bar{\sf
f}_{k}(\phi)$ is given by the tensor product of multiplication of forms by
functions and the $\mathfrak{g}$-action on $W$, and
$\mathsf{R}_{k}(\lambda)\star\mathsf{R}^{k}(\phi)=\bar{\sf
f}_{k}(\lambda)\cdot\bar{\sf f}^{k}(\phi)$. As left and right braided gauge
symmetries behave in a completely analogous way [45], we will consider only
left variations from now on and drop the superscript ${}^{\textrm{\tiny L}}$
from the notation for simplicity; we will usually also omit the adjective
‘left’.

By setting
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\big{|}_{\Omega^{0}(M,\mathfrak{g})}=0$
and
$\mathcal{V}:=\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]]\oplus\Omega^{\bullet}(M,W)[[\hbar]]$,
a braided gauge transformation is extended to a map
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}:\mathcal{V}\to\mathcal{V}$
of bidegree $(0,0)$ by commuting with arbitrary maps
$\mathcal{V}\to\mathcal{V}$ and as a braided derivation of operations
$\mu_{\star}$ defined on $\mathcal{V}\otimes\mathcal{V}$ in the category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$, that
is, it obeys the _braided Leibniz rule_

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\mu_{\star}(\upsilon_{1}\otimes\upsilon_{2})=\mu_{\star}\big{(}\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\upsilon_{1}\otimes\upsilon_{2}\big{)}+\mu_{\star}\big{(}\mathsf{R}_{k}(\upsilon_{1})\otimes\delta^{\textrm{\tiny${\mathcal{R}}$}}_{\mathsf{R}^{k}(\lambda)}\upsilon_{2}\big{)}$
(4.46)

for all $\upsilon_{1},\upsilon_{2}\in\mathcal{V}$. Generally, only braided
(not ordinary) derivations are compatible with braided symmetry properties of
such maps $\mu_{\star}$ [45]. This rule also defines the tensor products of
braided representations. They furthermore close under the _braided commutator_

$\displaystyle\big{[}\delta_{\lambda_{1}}^{\textrm{\tiny${\mathcal{R}}$}},\delta_{\lambda_{2}}^{\textrm{\tiny${\mathcal{R}}$}}\big{]}_{\circ}^{\star}:=\delta_{\lambda_{1}}^{\textrm{\tiny${\mathcal{R}}$}}\circ\delta_{\lambda_{2}}^{\textrm{\tiny${\mathcal{R}}$}}-\delta_{\mathsf{R}_{k}(\lambda_{2})}^{\textrm{\tiny${\mathcal{R}}$}}\circ\delta^{\textrm{\tiny${\mathcal{R}}$}}_{\mathsf{R}^{k}(\lambda_{1})}=\delta^{\textrm{\tiny${\mathcal{R}}$}}_{[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}}\
.$ (4.47)

The braided commutator generally makes the braided derivations of a
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module algebra into a
braided Lie algebra [21]. That (4.44) and (4.45) give the representation
(4.47) of the braided Lie algebra
$\big{(}\Omega^{0}(M,\mathfrak{g})[[\hbar]],[-,-]_{\mathfrak{g}}^{\star}\big{)}$
on $\mathcal{V}$ is proven in [45].

Braided noncommutative kinematics is now defined by the _braided left and
right covariant derivatives_

$\displaystyle\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\phi:=\mathrm{d}\phi+A\wedge_{\star}\phi\qquad\mbox{and}\qquad\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\phi:=\mathrm{d}\phi+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(\phi)$
(4.48)

in $\Omega^{p+1}(M,W)[[\hbar]]$, where $A\wedge_{\star}\phi=\bar{\sf
f}^{k}(A)\wedge\bar{\sf f}_{k}(\phi)$ is given by the tensor product of
exterior multiplication of forms with the action of $\mathfrak{g}$ on the
representation $W$, and
$\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(\phi)=\bar{\sf
f}_{k}(A)\wedge\bar{\sf f}^{k}(\phi)$. They are both _braided covariant_ :

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\phi\big{)}=-\lambda\star\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\phi\big{)}\qquad\mbox{and}\qquad\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\big{(}\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\phi\big{)}=-\lambda\star\big{(}\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\phi\big{)}\ ,$ (4.49)

as proven in [45].

Similarly, one defines the _braided curvature_

$\displaystyle
F_{A}^{\textrm{\tiny${\mathcal{R}}$}}:=\mathrm{d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}^{\star}\
\in\ \Omega^{2}(M,\mathfrak{g})[[\hbar]]\ ,$ (4.50)

where now there is no left/right distinction due to braided symmetry of the
bracket $[-,-]_{\mathfrak{g}}^{\star}$ on $1$-forms, i.e.
$[A,A]_{\mathfrak{g}}^{\star}=[\mathsf{R}_{k}(A),\mathsf{R}^{k}(A)]_{\mathfrak{g}}^{\star}$.
The braided curvature is also braided covariant:

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}=-[\lambda,F_{A}^{\textrm{\tiny${\mathcal{R}}$}}]_{\mathfrak{g}}^{\star}\
.$ (4.51)

Note that for a matrix Lie algebra $\mathfrak{g}$, one can write

$\displaystyle
F^{\textrm{\tiny${\mathcal{R}}$}}_{A}=\mathrm{d}A+\tfrac{1}{2}\,\big{(}A\wedge_{\star}A+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(A)\big{)}\
\neq\ F_{A}^{\star}\ ,$ (4.52)

which differs from the usual field strength (4.4) in conventional
noncommutative gauge theories (see e.g. (3.5) and (3.34)).

#### Braided diffeomorphisms

The discussion above carries through _verbatum_ to spacetime symmetries of
field theories, such as the diffeomorphism symmetry of gravity, which is in
fact where the formalism was originally developed (see e.g. [16, 17, 14]) and
which inspired the present treatment of gauge symmetries. This is generated
infinitesimally by vector fields $\xi\in\mathfrak{v}={\sf\Gamma}(TM)$ acting
via the Lie derivative $\mathrm{L}_{\xi}$ on generic tensor fields $T$:
$\delta_{\xi}T=\mathrm{L}_{\xi}\,T$ (see Example 4.8). The Cartan structure
equations tell us that this gives a representation of the Lie algebra
$(\mathfrak{v},[-,-]_{\mathfrak{v}})$ of vector fields on tensor fields:
$[\mathrm{L}_{\xi_{1}},\mathrm{L}_{\xi_{2}}]_{\circ}=\mathrm{L}_{[\xi_{1},\xi_{2}]_{\mathfrak{v}}}$.

This is obviously a $\mathsf{U}\mathfrak{v}$-module, and following the
twisting procedure above we define braided diffeomorphisms
$\delta_{\xi}^{\textrm{\tiny${\mathcal{R}}$}}T=\mathrm{L}_{\xi}^{\star}\,T$
through the _braided Lie derivative_ $\mathrm{L}_{\xi}^{\star}$ given by

$\displaystyle\mathrm{L}_{\xi}^{\star}\,T:=\mathrm{L}_{\bar{\sf
f}^{k}(\xi)}\bar{\sf f}_{k}(T)\ .$ (4.53)

The braided Lie derivative is a braided derivation of the star-tensor product
$\otimes_{\star}=\otimes\circ\mathcal{F}^{-1}$ which closes the braided Lie
algebra $\big{(}\mathfrak{v}[[\hbar]],[-,-]_{\mathfrak{v}}^{\star}\big{)}$:

$\displaystyle\big{[}\mathrm{L}_{\xi_{1}}^{\star},\mathrm{L}_{\xi_{2}}^{\star}\big{]}_{\circ}^{\star}=\mathrm{L}_{[\xi_{1},\xi_{2}]_{\mathfrak{v}}^{\star}}^{\star}\
.$ (4.54)

Note that the braided Lie bracket of vector fields may be represented as

$\displaystyle[\xi_{1},\xi_{2}]_{\mathfrak{v}}^{\star}=\xi_{1}\star\xi_{2}-\mathsf{R}_{k}(\xi_{2})\star\mathsf{R}^{k}(\xi_{1})\
\neq\
[\xi_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\xi_{2}]_{\mathfrak{v}}$
(4.55)

in the twist deformation of the universal enveloping algebra
$\mathsf{U}\mathfrak{v}$, which by construction is again a vector field in
$\mathfrak{v}[[\hbar]]$, in contrast to the star-bracket
$[\xi_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\xi_{2}]_{\mathfrak{v}}$.

It is worth stressing once more that these braided diffeomorphisms are _not_
the same as the ‘twisted diffeomorphisms’ of [19]. Twisted diffeomorphisms act
through the twisted enveloping Hopf algebra
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$, whose underlying
algebra is the same as that of the classical Hopf algebra
$\mathsf{U}\mathfrak{v}$. In particular, an infinitesimal twisted
diffeomorphism of a single field simply follows the classical rule with the
usual Lie derivative $\mathrm{L}_{\xi}$ so that they also close on the
classical Lie algebra $(\mathfrak{v},[-,-]_{\mathfrak{v}})$ of
diffeomorphisms, while on star-products of fields they follow a deformed
Leibniz rule defined by the twisted coproduct
$\Delta_{\textrm{\tiny$\mathcal{F}$}}(\mathrm{L}_{\xi})$ from (4.14).

###  4.4 Example: Braided Chern–Simons theory

As a simple illustration of a field theory with braided gauge symmetry where
we can explore some of the novelties that arise, let us go back to the
beginning of our story with the simplest prototypical example, namely the
Chern–Simons gauge theory from Section 2.1, and consider its braided
noncommutative deformation. We define the braided noncommutative version of
the Chern–Simons functional (2.2) in the obvious way as

$\displaystyle
S_{\star}(A)=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\Big{(}\frac{1}{2}\,A\wedge_{\star}\mathrm{d}A+\frac{1}{3!}\,A\wedge_{\star}[A,A]^{\star}_{\mathfrak{g}}\Big{)}\
,$ (4.56)

for a gauge field $A\in\Omega^{1}(M,\mathfrak{g})$.

In order to be able to use this in a standard variational principle, we need a
technical restriction on the Drinfel’d twist $\mathcal{F}$: we require that
the usual integral over $M$ is _graded cyclic_ with respect to the deformed
exterior product $\wedge_{\star}$, that is,

$\displaystyle\int_{M}\,\alpha\wedge_{\star}\beta=(-1)^{|\alpha|\,|\beta|}\,\int_{M}\,\beta\wedge_{\star}\alpha\
,$ (4.57)

for all homogeneous forms $\alpha,\beta\in\Omega^{\bullet}(M)[[\hbar]]$ in
complementary degrees. For example, this holds for abelian twists (see Example
4.31), whereby ${\mathcal{R}}^{-1}=\mathcal{F}^{2}$ and (4.57) follows from
integration by parts with respective to the Lie derivative.

Using this graded cyclicity repeatedly, along with invariance of the quadratic
form $\mathrm{Tr}_{\mathfrak{g}}$ on $\mathfrak{g}$ and the
${\mathcal{R}}$-matrix identities (4.19) as well as the braided Leibniz rule
(4.46), one checks explicitly that the action functional (4.56) is invariant
under the (left) braided gauge transformations

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A=\mathrm{d}\lambda-[\lambda,A]_{\mathfrak{g}}^{\star}\
,$ (4.58)

for an arbitrary gauge parameter $\lambda\in\Omega^{0}(M,\mathfrak{g})$, that
is, $\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=0$. This
holds for arbitrary quadratic Lie algebras $\mathfrak{g}$ without the
introduction of extra degrees of freedom for the gauge fields $A$. On the
other hand, by the ordinary Leibniz rule its variation is given by

$\displaystyle\delta
S_{\star}(A)=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\Big{(}\delta
A\wedge_{\star}\mathrm{d}A+\frac{1}{2}\,\delta
A\wedge_{\star}[A,A]_{\mathfrak{g}}^{\star}\Big{)}$ (4.59)

for arbitrary variations $\delta A\in\Omega^{1}(M,\mathfrak{g})$ of the gauge
fields, as in the classical case. This leads to the equations of motion

$\displaystyle\mathfrak{F}_{A}^{\star}=F_{A}^{\textrm{\tiny${\mathcal{R}}$}}=0$
(4.60)

describing a flat braided $\mathfrak{g}$-connection, which are braided
covariant by virtue of (4.51):

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}_{A}^{\star}=-[\lambda,\mathfrak{F}_{A}^{\star}]_{\mathfrak{g}}^{\star}\
.$ (4.61)

So far everything we have said is just the obvious braided deformation of what
was said in the classical case from Section 2.1, as in the conventional
noncommutative field theories of Section 3. At this point, however, the
braided field theory deviates from the naive expectations. Despite the
covariance of the field equations, braided gauge symmetries _do not_ produce
new solutions: An infinitesimal gauge variation produces the field equation

$\displaystyle\mathfrak{F}^{\star}_{A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A}=\mathrm{d}(A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A)+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}^{\star}+\tfrac{1}{2}\,\big{(}[\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A,A]_{\mathfrak{g}}^{\star}+[A,\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A]_{\mathfrak{g}}^{\star}\big{)}+O(\lambda^{2})\
,$ (4.62)

and due to the braided Leibniz rule (4.46) this cannot be combined to the form
$\mathfrak{F}^{\star}_{A}+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}_{A}^{\star}+O(\lambda^{2})$.
That is,

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}_{A}^{\star}\
\neq\
\mathfrak{F}^{\star}_{A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A}-\mathfrak{F}_{A}^{\star}+O(\lambda^{2})\
,$ (4.63)

and hence a braided gauge transformation of a solution $A$ to
$\mathfrak{F}_{A}^{\star}=0$ may produce a field configuration with
$\mathfrak{F}^{\star}_{A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A}\neq
0$. In other words, there is generally no moduli space of flat
$\mathfrak{g}$-connections modulo braided gauge transformations in this sense,
and the space of physical states cannot be described as in the commutative
case.232323The trivial (and uninteresting) exception is of course when
$\mathfrak{g}$ is an abelian Lie algebra and the resulting Chern–Simons theory
is a non-interacting field theory, whose classical and braided versions are
the same. We will say more about this unusual yet consistent feature of the
“braided moduli space” from a more technical standpoint in Section 4.9.

How then should we interpret, or even justify, the meaning of the term
‘braided gauge symmetry’? The key point is our alternative interpretation of
gauge symmetries via Noether’s second theorem, which for the standard
Chern–Simons theories from Sections 2.1 and 3.1 coincides with the Bianchi
identity for the curvature 2-form. The latter is known to be generally
violated in braided noncommutative differential geometry (see e.g. [22, 14,
9]): The braided version of the Bianchi identity is readily calculated by
taking the exterior derivative of the expression (4.50) and rewriting it in
terms of $F_{A}^{\textrm{\tiny${\mathcal{R}}$}}$ to get the differential
identity

$\displaystyle\mathrm{d}F_{A}^{\textrm{\tiny${\mathcal{R}}$}}-\tfrac{1}{2}\,[F_{A}^{\textrm{\tiny${\mathcal{R}}$}},A]_{\mathfrak{g}}^{\star}+\tfrac{1}{2}\,[A,F_{A}^{\textrm{\tiny${\mathcal{R}}$}}]_{\mathfrak{g}}^{\star}+\tfrac{1}{4}\,\big{(}[[A,A]_{\mathfrak{g}}^{\star},A]_{\mathfrak{g}}^{\star}-[A,[A,A]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\big{)}=0\
.$ (4.64)

The last term vanishes in the classical case with trivial
${\mathcal{R}}$-matrix ${\mathcal{R}}=1\otimes 1$ by symmetry of the Lie
bracket $[-,-]_{\mathfrak{g}}$ on $1$-forms $\Omega^{1}(M,\mathfrak{g})$, but
it is non-vanishing in general. We can rewrite the modified Bianchi identity
in terms of the left and right covariant derivatives (4.48) as

$\displaystyle\tfrac{1}{2}\,\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\mathfrak{F}_{A}^{\star}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\mathfrak{F}_{A}^{\star}\big{)}=-\tfrac{1}{4}\,[\mathsf{R}_{k}(A),[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\
,$ (4.65)

where we used the braided (odd) Jacobi identity (4.41) on 1-forms with
$\alpha_{1}=\alpha_{2}=\alpha_{3}=A$. Again this identity holds off-shell, but
it differs from its classical form by inhomogeneous terms involving the gauge
field $A$ itself.

In analogy to the classical case, it is natural to interpret this as the
‘braided Noether identity’ that should follow via braided gauge invariance of
the action functional. This is indeed the case, but one can no longer use the
formula (4.59) as in the classical case, because braided gauge transformation
obey a braided Leibniz rule and so are no longer special directions of a
general field variation, which obeys the standard Leibniz rule; that is,

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)\
\neq\
\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A\wedge_{\star}\mathfrak{F}_{A}^{\star}\big{)}\
.$ (4.66)

Nonetheless, the braided Noether identity (4.65) follows from
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=0$ by using
cyclicity and integration by parts, along with ${\rm
ad}(\mathfrak{g})$-invariance of the inner product
$\mathrm{Tr}_{\mathfrak{g}}$ and the braided Jacobi identity, to isolate the
gauge parameter $\lambda$ while making the Euler–Lagrange derivative
$\mathfrak{F}_{A}^{\star}$ appear; that is,

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=-\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\bigg{(}\lambda\wedge_{\star}\Big{(}\frac{1}{2}\,\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\mathfrak{F}_{A}^{\star}+\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\mathfrak{F}_{A}^{\star}\big{)}+\frac{1}{4}\,[\mathsf{R}_{k}(A),[\mathsf{R}^{k}(A),A]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\Big{)}\bigg{)}\
.$ (4.67)

This provides a braided version of Noether’s second theorem: braided gauge
symmetries still induce interdependence among the equations of motion, and
hence gauge redundancies. This justifies the interpretation of local braided
symmetries as “gauge”. Note that there is only one braided Noether identity,
corresponding to the fact that left and right braided gauge transformations
are not independent (see (4.44)).

We now notice that, analogously to Section 2.1, the formulation of braided
Chern–Simons gauge theory can be written entirely in terms of the ‘bracket’
operations

$\displaystyle\ell_{1}^{\star}=\mathrm{d}\qquad\mbox{and}\qquad\ell_{2}^{\star}=-[-,-]_{\mathfrak{g}}^{\star}$
(4.68)

which define a differential graded braided Lie algebra structure on the graded
vector space $V:=\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]]$, as well as the
(strictly) cyclic inner product

$\displaystyle\langle\alpha,\beta\rangle_{\star}:=\langle-,-\rangle\circ\mathcal{F}^{-1}(\alpha\otimes\beta)=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(\alpha\wedge_{\star}\beta)$
(4.69)

on $V$, where strict cyclicity follows using the classical properties together
with the graded commutativity (4.57) and the ${\mathcal{R}}$-matrix identities
(4.19). That is, the braided gauge transformations of the fields, the
equations of motion and the gauge redundancy via the braided Noether
identities are encoded respectively by

$\displaystyle\begin{split}\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A=\ell_{1}^{\star}(\lambda)+\ell_{2}^{\star}(\lambda,A)&\qquad,\qquad\mathfrak{F}_{A}^{\star}=\ell^{\star}_{1}(A)-\tfrac{1}{2}\,\ell_{2}^{\star}(A,A)\
,\\\\[4.0pt] {\mathsf{d}}^{\star}_{A}\mathfrak{F}_{A}^{\star}:=\
\ell_{1}^{\star}(\mathfrak{F}_{A}^{\star})+\tfrac{1}{2}\,\big{(}\ell^{\star}_{2}(\mathfrak{F}_{A}^{\star},A)-&\,\ell_{2}^{\star}(A,\mathfrak{F}_{A}^{\star})\big{)}+\tfrac{1}{4}\,\ell_{2}^{\star}\big{(}\mathsf{R}_{k}(A),\ell_{2}^{\star}(\mathsf{R}^{k}(A),A)\big{)}\
=\ 0\ ,\end{split}$ (4.70)

with gauge closure and braided covariance represented by

$\displaystyle\big{[}\delta_{\lambda_{1}}^{\textrm{\tiny${\mathcal{R}}$}},\delta_{\lambda_{2}}^{\textrm{\tiny${\mathcal{R}}$}}\big{]}^{\star}_{\circ}A=\delta^{\textrm{\tiny${\mathcal{R}}$}}_{-\ell_{2}^{\star}(\lambda_{1},\lambda_{2})}A\qquad\mbox{and}\qquad\delta^{\textrm{\tiny${\mathcal{R}}$}}_{\lambda}\,\mathfrak{F}_{A}^{\star}=\ell_{2}^{\star}(\lambda,\mathfrak{F}_{A}^{\star})\
.$ (4.71)

The braided Chern–Simons functional is encoded as

$\displaystyle S_{\star}(A)=\tfrac{1}{2}\,\langle
A,\ell_{1}^{\star}(A)\rangle_{\star}-\tfrac{1}{3!}\,\langle
A,\ell_{2}^{\star}(A,A)\rangle_{\star}\ ,$ (4.72)

from which the equations of motion $\mathfrak{F}_{A}^{\star}=0$ are derived by
varying and using cyclicity of the inner product.

The structure maps in this formulation of the braided gauge theory are all
evidently simply twist deformations, via the procedure of Section 4.2, of
those from Section 2.1. This motivates an extension of the notion of an
$L_{\infty}$-algebra to the braided setting, and the construction of more
general noncommutative field theories with braided gauge symmetry by twisting
the formalism from Section 2.4. This extension will occupy the remainder of
this section.

###  4.5 Braided $L_{\infty}$-algebras

In this section we shall work with $L_{\infty}$-algebras as presented in
Section 2 in terms of brackets and relations, and define a braided
generalization in these terms, deferring a discussion of their incarnations as
braided versions of coalgebras and Chevalley–Eilenberg algebras until Section
4.8. The key observation is that the definition of a classical
$L_{\infty}$-algebra takes place in the category of vector spaces and linear
maps, which is a symmetric monoidal category with braiding given by the
trivial transposition isomorphism $\tau$. This definition can be made in _any_
symmetric monoidal category with a non-trivial braiding, just like the
definition of a Lie algebra. In this paper we will only need the definition in
the category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ of left
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-modules, and we
restrict to this setting for definiteness.

######  Definition 4.73.

A _braided $L_{\infty}$-algebra_ is a $\mathbbm{Z}$-graded
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module242424We regard
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$ itself as a
$\mathbbm{Z}$-graded
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module sitting in
degree $0$. $V=\bigoplus_{k\in\mathbbm{Z}}\,V^{k}$ with a collection
$\ell_{n}:V^{\otimes n}\to V$, $n\geq 1$ of equivariant graded braided
antisymmetric multilinear maps of degree $2-n$ that satisfy the _braided
homotopy Jacobi identities_

$\displaystyle\sum_{i=1}^{n}\,(-1)^{i\,(n-i)}\,\ell_{n-i+1}\circ\big{(}\ell_{i}\otimes\mathbbm{1}^{\otimes
n-i}\big{)}\ \circ\ \sum_{\sigma\in{\rm Sh}(i;n-i)}\,{\rm
sgn}(\sigma)\,\tau_{{\textrm{\tiny${\mathcal{R}}$}}}^{\sigma}=0$ (4.74)

for each $n\geq 1$, where
$\tau_{{\textrm{\tiny${\mathcal{R}}$}}}^{\sigma}:V^{\otimes n}\to V^{\otimes
n}$ denotes the action of the permutation $\sigma$ via the symmetric braiding
$\tau_{{\textrm{\tiny${\mathcal{R}}$}}}$ of the category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ times
the Koszul sign multiplication which interchanges factors in a tensor product
of graded $\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-modules.

Let us unravel this definition in a bit more detail. Graded braided
antisymmetry is the property

$\displaystyle\ell_{n}(v_{1},\dots,v_{n})=-(-1)^{|v_{i}|\,|v_{i+1}|}\
\ell_{n}\big{(}v_{1},\dots,v_{i-1},\mathsf{R}_{k}(v_{i+1}),\mathsf{R}^{k}(v_{i}),v_{i+2},\dots,v_{n}\big{)}$
(4.75)

for $i=1,\dots,n-1$ and homogeneous elements $v_{1},\dots,v_{n}\in V$. The
permutation action $\tau_{{\textrm{\tiny${\mathcal{R}}$}}}^{\sigma}$ includes,
in addition to the usual Koszul signs, appropriate insertions of the
${\mathcal{R}}$-matrix as in (4.75). The braided homotopy Jacobi identities
(4.74) are unchanged from their classical counterparts for $n=1,2$, that is,
(2.26) and (2.27) continue to hold. Thus, similarly to the classical case,
every braided $L_{\infty}$-algebra has an underlying cochain complex
$(V,\ell_{1})$ in the category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ and
$\ell_{2}:V\otimes V\to V$ is a cochain map in
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$. The
first homotopy relation which differs from the classical case is for $n=3$,
which is a modification of (2.28) in accordance with the non-trivial braiding
(4.75):

$\displaystyle\begin{split}&\ell_{2}\big{(}\ell_{2}(v_{1},v_{2}),v_{3}\big{)}-(-1)^{|v_{2}|\,|v_{3}|}\,\ell_{2}\big{(}\ell_{2}(v_{1},\mathsf{R}_{k}(v_{3})),\mathsf{R}^{k}(v_{2})\big{)}\\\
&\hskip
113.81102pt+(-1)^{(|v_{2}|+|v_{3}|)\,|v_{1}|}\,\ell_{2}\big{(}\ell_{2}(\mathsf{R}_{k}(v_{2}),\mathsf{R}_{l}(v_{3})),\mathsf{R}^{l}\,\mathsf{R}^{k}(v_{1})\big{)}\\\\[4.0pt]
&\hskip
28.45274pt=-\ell_{3}\big{(}\ell_{1}(v_{1}),v_{2},v_{3}\big{)}-(-1)^{|v_{1}|}\,\ell_{3}\big{(}v_{1},\ell_{1}(v_{2}),v_{3}\big{)}-(-1)^{|v_{1}|+|v_{2}|}\,\ell_{3}\big{(}v_{1},v_{2},\ell_{1}(v_{3})\big{)}\\\
&\hskip 113.81102pt-\ell_{1}\big{(}\ell_{3}(v_{1},v_{2},v_{3})\big{)}\
.\end{split}$ (4.76)

When the only non-zero bracket is $\ell_{2}$, a braided $L_{\infty}$-algebra
is a braided Lie algebra in the sense of [73] and it coincides with the notion
of Lie algebra in
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$
discussed by [21]. Thus braided $L_{\infty}$-algebras are homotopy coherent
generalizations of braided Lie algebras. In particular, the cohomology
$H^{\bullet}(V,\ell_{1})$ of a braided $L_{\infty}$-algebra is a
$\mathbbm{Z}$-graded braided Lie algebra.

In the following we regard the one-dimensional vector space
$\mathbbm{C}[[\hbar]]$ as a trivial graded left
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-module, sitting in
degree $0$.

######  Definition 4.77.

A _cyclic braided $L_{\infty}$-algebra_ is a braided $L_{\infty}$-algebra
$(V,\\{\ell_{n}\\})$ together with a non-degenerate graded braided symmetric
cochain map $\langle-,-\rangle:V\otimes V\to\mathbbm{C}[[\hbar]]$ that
satisfies the _braided cyclicity condition_

$\displaystyle\langle-,-\rangle\,\circ\,(\mathbbm{1}\otimes\ell_{n})={\rm
sgn}(\sigma)\,\langle-,-\rangle\,\circ\,(\mathbbm{1}\otimes\ell_{n})\,\circ\,\tau_{{\textrm{\tiny${\mathcal{R}}$}}}^{\sigma}\
,$ (4.78)

for all $n\geq 1$ and for all cyclic permutations $\sigma\in C_{n+1}$.

Evaluated on elements $v_{0},v_{1},\dots,v_{n}\in V$, cyclicity in the
category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ reads
as

$\displaystyle\big{\langle}v_{0},\ell_{n}(v_{1},v_{2},\dots,v_{n})\big{\rangle}=\pm\,\big{\langle}\mathsf{R}_{k_{0}}\,\mathsf{R}_{k_{1}}\cdots\mathsf{R}_{k_{n-1}}(v_{n}),\ell_{n}\big{(}\mathsf{R}^{k_{0}}(v_{0}),\mathsf{R}^{k_{1}}(v_{1}),\dots,\mathsf{R}^{k_{n-1}}(v_{n-1})\big{)}\big{\rangle}\
.$ (4.79)

In this paper we are predominantly interested in braided $L_{\infty}$-algebras
that result from twist deformations of classical $L_{\infty}$-algebras, via
the techniques described in Section 4.2. This can be done starting from an
$L_{\infty}$-algebra $(V,\\{\ell_{n}\\})$ in the category
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ of left
$\mathsf{U}\mathfrak{v}$-modules. This means that
$V=\bigoplus_{k\in\mathbbm{Z}}\,V_{k}$ is a $\mathbbm{Z}$-graded
$\mathsf{U}\mathfrak{v}$-module and the $n$-brackets $\ell_{n}:V^{\otimes
n}\to V$ are equivariant maps. Given any Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$,
we can deform the brackets $\ell_{n}$ to new brackets
$\ell_{n}^{\star}:V[[\hbar]]^{\otimes n}\to V[[\hbar]]$ which are morphisms in
the twisted representation category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ of
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-modules. Following the
standard prescription (4.20), we set $\ell_{1}^{\star}:=\ell_{1}$ and

$\displaystyle\ell_{n}^{\star}(v_{1}\otimes\cdots\otimes
v_{n}):=\ell_{n}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n})$ (4.80)

for $n\geq 2$, where $v\otimes_{\star}v^{\prime}:=\mathcal{F}^{-1}(v\otimes
v^{\prime})=\bar{\sf f}^{k}(v)\otimes\bar{\sf f}_{k}(v^{\prime})$ for
$v,v^{\prime}\in V$. The functorial equivalence between the symmetric monoidal
categories ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ and
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$ then
implies the following central result, whose proof is found in [45, Proposition
4.8].

######  Proposition 4.81.

If $(V,\\{\ell_{n}\\})$ is an $L_{\infty}$-algebra in the category
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$, then
$(V[[\hbar]],\\{\ell_{n}^{\star}\\})$ is a braided $L_{\infty}$-algebra.

This result is a far reaching generalization of what we did in Section 4.4,
where the differential graded Lie algebra
$\big{(}\Omega^{\bullet}(M,\mathfrak{g}),\ell_{1},\ell_{2}\big{)}$ underlying
Chern–Simons gauge theory on a three-manifold $M$ was twist deformed to the
differential graded braided Lie algebra
$\big{(}\Omega^{\bullet}(M,\mathfrak{g})[[\hbar]],\ell_{1}^{\star},\ell_{2}^{\star}\big{)}$
of braided Chern–Simons theory on $M$. We will discuss how this procedure
extends to more general field theories organised by generic
$L_{\infty}$-algebras in Section 4.6.

Similarly, we can twist deform cyclic structures on $L_{\infty}$-algebras to
obtain cyclic braided $L_{\infty}$-algebras. Let
$(V,\\{\ell_{n}\\},\langle-,-\rangle)$ be a cyclic $L_{\infty}$-algebra in the
category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ of left
$\mathsf{U}\mathfrak{v}$-modules, with $\mathbbm{R}$ regarded as a trivial
graded $\mathsf{U}\mathfrak{v}$-module sitting in degree $0$. This means that
$(V,\\{\ell_{n}\\})$ is an $L_{\infty}$-algebra in
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ and the cyclic inner product
$\langle-,-\rangle:V\otimes V\to\mathbbm{R}$ is
$\mathsf{U}\mathfrak{v}$-invariant:

$\displaystyle X\langle v_{1}\otimes
v_{2}\rangle=\langle\Delta(X)(v_{1}\otimes v_{2})\rangle=0\ ,$ (4.82)

for all $X\in\mathsf{U}\mathfrak{v}$ and $v_{1},v_{2}\in V$. Following the
standard prescription (4.20), we can twist deform the inner product
$\langle-,-\rangle$ to a new pairing $\langle-,-\rangle_{\star}$ defined by

$\displaystyle\langle
v_{1},v_{2}\rangle_{\star}:=\langle-,-\rangle\circ\mathcal{F}^{-1}(v_{1}\otimes
v_{2})=\langle\bar{\sf f}^{k}(v_{1}),\bar{\sf f}_{k}(v_{2})\rangle\ .$ (4.83)

######  Proposition 4.84.

If $(V,\\{\ell_{n}\\},\langle-,-\rangle)$ is a cyclic $L_{\infty}$-algebra in
the category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$, then
$(V[[\hbar]],\\{\ell_{n}^{\star}\\},\langle-,-\rangle_{\star})$ is a cyclic
braided $L_{\infty}$-algebra.

###### Proof.

Non-degeneracy of the pairing $\langle-,-\rangle_{\star}:V[[\hbar]]\otimes
V[[\hbar]]\to\mathbbm{C}[[\hbar]]$ follows from the untwisted non-degeneracy
and an order by order argument in the formal deformation parameter $\hbar$.
Graded symmetry of the cyclic inner product $\langle-,-\rangle$ implies that
the twisted inner product $\langle-,-\rangle_{\star}$ is naturally graded
braided symmetric.

To check braided cyclicity, we compute

$\displaystyle\begin{split}\big{\langle}v_{0},\ell^{\star}_{n}(v_{1},\dots,v_{n})\big{\rangle}_{\star}&=\big{\langle}\bar{\sf
f}^{k}(v_{0})\otimes\bar{\sf f}_{k}\big{(}\ell_{n}(\bar{\sf
f}^{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1})\otimes\bar{\sf
f}_{l}(v_{n}))\big{)}\big{\rangle}\\\\[4.0pt]
&=\big{\langle}(\mathbbm{1}\otimes\ell_{n})\big{(}\bar{\sf
f}^{k}(v_{0})\otimes\Delta(\bar{\sf f}_{k})\big{[}\bar{\sf
f}^{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1})\otimes\bar{\sf
f}_{l}(v_{n})\big{]}\big{)}\big{\rangle}\\\\[4.0pt]
&=\big{\langle}(\mathbbm{1}\otimes\ell_{n})\big{(}\Delta(\bar{\sf
f}^{k})\big{[}\bar{\sf f}^{l}(v_{0})\otimes\bar{\sf
f}_{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1})\big{]}\otimes\bar{\sf
f}_{k}(v_{n})\big{)}\big{\rangle}\\\\[4.0pt] &=\pm\,\big{\langle}\bar{\sf
f}_{k}(v_{n})\otimes\ell_{n}\big{(}\Delta\big{(}\bar{\sf
f}^{k})\big{[}\bar{\sf f}^{l}(v_{0})\otimes\bar{\sf
f}_{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1})\big{]}\big{)}\big{\rangle}\\\\[4.0pt]
&=\pm\,\big{\langle}\bar{\sf f}_{k}(v_{n})\otimes\bar{\sf
f}^{k}\big{(}\ell_{n}(\bar{\sf f}^{l}(v_{0})\otimes\bar{\sf
f}_{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1}))\big{)}\big{\rangle}\\\\[4.0pt]
&=\pm\,\big{\langle}\bar{\sf f}^{k}\,\mathsf{R}_{m}(v_{n})\otimes\bar{\sf
f}_{k}\,\mathsf{R}^{m}\big{(}\ell_{n}(\bar{\sf f}^{l}(v_{0})\otimes\bar{\sf
f}_{l}(v_{1}\otimes_{\star}\cdots\otimes_{\star}v_{n-1}))\big{)}\big{\rangle}\\\\[4.0pt]
&=\pm\,\big{\langle}\mathsf{R}_{m}(v_{n})\otimes\mathsf{R}^{m}\big{(}\ell_{n}^{\star}(v_{0}\otimes
v_{1}\otimes\cdots\otimes v_{n-1})\big{)}\big{\rangle}_{\star}\\\\[4.0pt]
&=\pm\,\big{\langle}(\mathbbm{1}\otimes\ell_{n}^{\star})\big{(}\mathsf{R}_{m}(v_{n})\otimes\Delta_{\textrm{\tiny$\mathcal{F}$}}(\mathsf{R}^{m})\big{[}v_{0}\otimes(v_{1}\otimes\cdots\otimes
v_{n-1})\big{]}\big{)}\big{\rangle}_{\star}\\\\[4.0pt]
&=\pm\,\big{\langle}\mathsf{R}_{k_{0}}\,\mathsf{R}_{k_{1}}\cdots\mathsf{R}_{k_{n-1}}(v_{n}),\ell_{n}^{\star}\big{(}\mathsf{R}^{k_{0}}(v_{0}),\mathsf{R}^{k_{1}}(v_{1}),\dots,\mathsf{R}^{k_{n-1}}(v_{n-1})\big{)}\big{\rangle}_{\star}\
.\end{split}$ (4.85)

In the first and seventh lines we use the definitions of the star-pairing and
the star-brackets, in the second and fifth lines we use commutativity of
$\ell_{n}$ with the $\mathsf{U}\mathfrak{v}$-action, in the third line we use
the cocycle property (4.13) for the inverse of the Drinfel’d twist along with
$\mathsf{U}\mathfrak{v}$-invariance (4.82) of the classical pairing, in the
fourth line we use classical cyclicity, in the sixth line we use the
definition of the ${\mathcal{R}}$-matrix, in the eighth line we use
commutativity of $\ell_{n}^{\star}$ with the
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-action, and in the
ninth line we use the ${\mathcal{R}}$-matrix identities (4.19). ∎

However, this level of generality is not suitable for applications to field
theory, as the loss of strict graded symmetry and cyclicity would lead to
problems with the variational principle for the corresponding action
functionals. Following the concrete example of braided Chern–Simons theory
from Section 4.4, we need to restrict the types of Drinfel’d twists by
abstracting the graded cyclicity property (4.57) for integration of twisted
differential forms.

######  Definition 4.86.

A Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
is _compatible_ with a cyclic structure $\langle-,-\rangle:V\otimes
V\to\mathbbm{R}$ on an $L_{\infty}$-algebra in
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ if

$\displaystyle\langle
v_{1},v_{2}\rangle_{\star}=\langle\mathsf{R}_{k}(v_{1}),\mathsf{R}^{k}(v_{2})\rangle_{\star}=(-1)^{|v_{1}|\,|v_{2}|}\,\langle
v_{2},v_{1}\rangle_{\star}$ (4.87)

for all homogeneous $v_{1},v_{2}\in V$.

In other words, a compatible Drinfel’d twist turns braided graded symmetry of
the inner product into _strict_ graded symmetry. It also turns the braided
cyclicity condition into the strict cyclicity condition (see (2.31)), as
inferred by

######  Corollary 4.88.

Let $(V,\\{\ell_{n}\\},\langle-,-\rangle)$ be a cyclic $L_{\infty}$-algebra in
the category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$, and let $\mathcal{F}$
be a compatible Drinfel’d twist. Then
$(V[[\hbar]],\\{\ell_{n}^{\star}\\},\langle-,-\rangle_{\star})$ is a strictly
cyclic braided $L_{\infty}$-algebra.

###### Proof.

The seventh line of the computation (4.85) shows

$\displaystyle\begin{split}\big{\langle}v_{0},\ell^{\star}_{n}(v_{1},v_{2},\dots,v_{n})\big{\rangle}_{\star}&=\pm\,\big{\langle}\mathsf{R}_{k}(v_{n}),\mathsf{R}^{k}\big{(}\ell_{n}^{\star}(v_{0},v_{1},\dots,v_{n-1})\big{)}\big{\rangle}_{\star}\\\\[4.0pt]
&=\pm\,\big{\langle}v_{n},\ell_{n}^{\star}(v_{0},v_{1},\dots,v_{n-1})\big{\rangle}_{\star}\
,\end{split}$ (4.89)

where in the second line we used (4.87). ∎

###  4.6 Braided $L_{\infty}$-algebras of noncommutative field theories

We shall now define noncommutative field theories in the braided
$L_{\infty}$-algebra formalism by employing a twist deformation of the
approach discussed in Section 2.4, whereby a classical field theory is defined
by its 4-term cyclic $L_{\infty}$-algebra as the initial input.252525Here we
treat only theories with irreducible gauge symmetries. We shall discuss an
example with higher gauge symmetries in Section 6. We call a field theory
obtained in this way a _braided field theory_. That is, a braided field theory
is defined by a 4-term cyclic braided $L_{\infty}$-algebra as the initial
input.

Starting from a classical field theory on a manifold $M$, described
algebraically by a 4-term cyclic $L_{\infty}$-algebra
$(V,\\{\ell_{n}\\},\langle-,-\rangle)$ (over $\mathbbm{R}$) with underlying
graded vector space

$\displaystyle V=V^{0}\oplus V^{1}\oplus V^{2}\oplus V^{3}$ (4.90)

in the representation category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$,
following the prescription of Section 4.5 we construct the corresponding
4-term cyclic braided $L_{\infty}$-algebra
$(V[[\hbar]],\\{\ell_{n}^{\star}\\},\langle-,-\rangle_{\star})$ (over
$\mathbbm{C}[[\hbar]]$) in the twisted representation category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$. Since
$\ell_{1}^{\star}=\ell_{1}$, this braided $L_{\infty}$-algebra has the same
underlying cochain complex (2.45). In other words the noncommutative field
theories obtained in this way have the same underlying free field theory as
their classical versions, as in the case of the star-gauge theories of Section
3.1. Only the interactions are different and they are encoded in the higher
braided brackets $\ell_{n}^{\star}$ with $n\geq 2$.

The (left) braided gauge transformation of a field $A\in V^{1}$ by a gauge
parameter $\lambda\in V^{0}$ is defined by

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A:=\ell_{1}^{\star}(\lambda)+\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell_{n+1}^{\star}(\lambda,A^{\otimes n})\
,$ (4.91)

as in the classical case (2.46), where now the positioning of $\lambda$ inside
the slots of the brackets $\ell_{n+1}^{\star}$ matters because of braided
symmetry (4.75). The braided gauge variations are extended to maps
$\delta^{\textrm{\tiny${\mathcal{R}}$}}_{\lambda}:V\to V$ of degree $0$ in the
usual way as braided derivations. For the field theories we work with in this
paper, whereby the algebra of gauge variations closes off-shell with field-
independent gauge parameters, they close a braided Lie algebra under the
braided commutator:

$\displaystyle\big{[}\delta_{\lambda_{1}}^{{\textrm{\tiny${\mathcal{R}}$}}},\delta_{\lambda_{2}}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{]}_{\circ}^{\star}=\delta_{-\ell^{\star}_{2}(\lambda_{1},\lambda_{2})}^{{\textrm{\tiny${\mathcal{R}}$}}}\
.$ (4.92)

The braided closure property (4.92) is shown in [45] using the braided
homotopy Jacobi identity (4.74) evaluated on
$(\lambda_{1},\lambda_{2},A^{\otimes n})$, order by order in the expansion in
$A^{\otimes n}$. The calculations are considerably more involved than in the
classical case [60], due to the braided Leibniz rule (4.46) obeyed by the
braided gauge variations, and because the braided symmetry properties (4.75)
prevent one from combining various sums and simplifying the homotopy
relations. The identities (4.19) are instrumental in manipulating the
proliferation of ${\mathcal{R}}$-matrix factors which ensue. We take the
validity of this result, and the others which follow below, as a highly non-
trivial vindication that braided gauge symmetries and covariant dynamics are
controlled by the tightly knit algebraic structure of a braided
$L_{\infty}$-algebra.

The equations of motion are dictated by a braided version of the Maurer–Cartan
equations $\mathfrak{F}_{A}^{\star}=0$ in $V^{2}[[\hbar]]$ for a field $A\in
V^{1}$, where

$\displaystyle\mathfrak{F}^{\star}_{A}:=\ell_{1}^{\star}(A)+\sum_{n\geq
2}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell^{\star}_{n}(A^{\otimes n})\ ,$ (4.93)

exactly as in the classical case (2.48). The dynamics are braided covariant,
that is, the expression (4.93) is covariant under the braided gauge variations
(4.91):

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}^{\star}_{A}=\ell_{2}^{\star}(\lambda,\mathfrak{F}_{A}^{\star})+\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{(n+1)!}\
\sum_{i=0}^{n}\,(-1)^{i}\,\ell^{\star}_{n+2}\big{(}\lambda,A^{\otimes
i},\mathfrak{F}^{\star}_{A},A^{\otimes n-i}\big{)}\ ,$ (4.94)

for all gauge parameters $\lambda\in V^{0}$, as shown in [45] order by order
using the braided homotopy Jacobi identity (4.74) evaluated on
$(\lambda,A^{\otimes n})$. In the classical case, where
${\mathcal{R}}=1\otimes 1$, the expression (4.94) agrees with (2.49).

The absence of a (naive) moduli space of solutions to
$\mathfrak{F}_{A}^{\star}=0$ that we observed in the case of braided
Chern–Simons theory can be seen in this formalism to be a generic feature of
noncommutative field theories with braided gauge symmetries: Although (4.94)
implies that
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\mathfrak{F}_{A}^{\star}=0$
on solutions $A\in V^{1}$ of the field equations $\mathfrak{F}_{A}^{\star}=0$,
computing the first order variation in any infinitesimal gauge parameter
$\lambda\in V^{0}$ gives

$\displaystyle\mathfrak{F}^{\star}_{A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A}=\ell_{1}^{\star}\big{(}A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A\big{)}+\sum_{n\geq
2}\,\frac{(-1)^{n\choose 2}}{n!}\,\Big{(}\ell_{n}^{\star}(A^{\otimes
n})+\sum_{i=0}^{n-1}\,\ell_{n}^{\star}\big{(}A^{\otimes
i},\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A,A^{\otimes
n-1-i}\big{)}\Big{)}+O(\lambda^{2})\ .$ (4.95)

This cannot be brought to the form
$\mathfrak{F}^{\star}_{A}+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}_{A}^{\star}+O(\lambda^{2})$
because of the braided Leibniz rule (4.46), and it will generally differ from
the result of (4.94) by ${\mathcal{R}}$-matrix factors in the sum over $i$.
Hence
$\mathfrak{F}^{\star}_{A+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A}\neq\mathfrak{F}^{\star}_{A}+\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\,\mathfrak{F}^{\star}_{A}+O(\lambda^{2})$
in general, and it does not make sense to take the (naive) quotient of the
subspace in $V^{1}$ of classical solutions by braided gauge transformations.
We will return to this point in Section 4.9 where we explain how a braided
version of the Maurer–Cartan moduli space
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(V[[\hbar]],\\{\ell^{\star}_{n}\\})$
can still be defined from a more abstract perspective.

From a physical perspective, generalized braided gauge symmetries are well-
defined through a braided version of the Noether identities which hold off-
shell and encode gauge redundancies, analogously to the classical case. These
identities follow from the braided $L_{\infty}$-algebra formalism, which also
naturally controls kinematical violations through the braided homotopy Jacobi
identities, such as the well-known failure of the Bianchi identity in braided
geometry that we discussed in Section 4.4. Analogously to the classical case,
the braided Noether identities follow from a weighted sum over $n\geq 1$ of
the braided homotopy Jacobi identity (4.74) evaluated on $A^{\otimes n}$.
However, because of the braided symmetry properties (4.75), we do not have
available the simplification (2.54) which took place in the classical case and
the weighted sum is very cumbersome to work with.

For the field theories we treat in this paper, where the gauge transformations
are at most linear in the field $A$, the braided Noether identities are
encoded by [45]

$\displaystyle\begin{split}{\mathsf{d}}^{\star}_{A}\mathfrak{F}^{\star}_{A}&:=\ell_{1}^{\star}(\mathfrak{F}^{\star}_{A})+\frac{1}{2}\,\big{(}\ell_{2}^{\star}(\mathfrak{F}^{\star}_{A},A)-\ell^{\star}_{2}(A,\mathfrak{F}^{\star}_{A})\big{)}-\sum_{n\geq
3}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{1}^{\star}\big{(}\ell^{\star}_{n}(A^{\otimes n})\big{)}\\\
&\quad\,-\sum_{n\geq 2}\,\frac{(-1)^{n\choose
2}}{2\,n!}\,\Big{(}\ell^{\star}_{2}\big{(}\ell^{\star}_{n}(A^{\otimes
n}),A\big{)}-\ell_{2}^{\star}\big{(}A,\ell_{n}^{\star}(A^{\otimes
n})\big{)}\Big{)}\ =\ 0\ .\end{split}$ (4.96)

This uses the simplification provided by the braided homotopy Jacobi identity
(4.74) evaluated on $A^{\otimes n+1}$ which in this case reads

$\displaystyle\begin{split}\sum_{i=1}^{n-1}\,\ell_{2}^{\star}\big{(}\ell_{n}^{\star}(A^{\otimes
n-i},\mathsf{R}_{k}(A^{\otimes
i})),\mathsf{R}^{k}(A)\big{)}&=\ell_{2}^{\star}\big{(}A,\ell_{n}^{\star}(A^{\otimes
n})\big{)}-\ell_{2}^{\star}\big{(}\ell_{n}^{\star}(A^{\otimes n}),A\big{)}\\\
&\quad\,-(-1)^{n}\,\ell_{1}^{\star}\big{(}\ell_{n+1}^{\star}(A^{\otimes
n+1})\big{)}\ ,\end{split}$ (4.97)

for all $n\geq 2$. Unlike the classical case (2.53), the braided Noether
identity (4.96) is no longer linear in the field equations
$\mathfrak{F}_{A}^{\star}$, while the inhomogeneous terms involving solely the
field $A$ cancel in the classical limit ${\mathcal{R}}=1\otimes 1$ after using
(4.97). This generalizes the braided Chern–Simons Noether identity in the
second line of (4.70), which is recovered after applying (4.97).

While the braided Noether identities evidently exhibit gauge redundancies
through inhomogeneous differential identities among the field equations, we
can understand their precise correspondence to braided gauge symmetries by
appealing to a variational principle for the braided field theory. Using the
cyclic inner product $\langle-,-\rangle_{\star}$ on our braided
$L_{\infty}$-algebra we define a braided version of the Maurer–Cartan action
functional by

$\displaystyle S_{\star}(A)=\frac{1}{2}\,\langle
A,\ell_{1}^{\star}(A)\rangle_{\star}+\sum_{n\geq 2}\,\frac{(-1)^{n\choose
2}}{(n+1)!}\,\langle A,\ell^{\star}_{n}(A^{\otimes n})\rangle_{\star}\ ,$
(4.98)

for all fields $A\in V^{1}$, exactly as in the classical case (2.56).

If $\mathcal{F}$ is a compatible Drinfel’d twist, in the sense of Definition
4.86, then using the ordinary Leibniz rule for general field variations
$\delta A$ on the tangent space $TV^{1}$ to the space of fields we can easily
compute the variation of (4.98) exactly as in the classical case to get

$\displaystyle\begin{split}\delta S_{\star}(A)&=\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{(n+1)!}\,\Big{(}\langle\delta
A,\ell_{n}^{\star}(A^{\otimes n})\rangle_{\star}+\sum_{i=0}^{n-1}\,\langle
A,\ell_{n}^{\star}(A^{\otimes i},\delta A,A^{\otimes
n-i-1})\rangle_{\star}\Big{)}\\\\[4.0pt] &=\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{n!}\,\langle\delta A,\ell^{\star}_{n}(A^{\otimes
n})\rangle_{\star}=\langle\delta A,\mathfrak{F}^{\star}_{A}\rangle_{\star}\
,\end{split}$ (4.99)

where in the second equality we used strict cyclicity for compatible twists
(see Corollary 4.88) to set all terms in the sum over $i$ equal to the first
term. Hence by non-degeneracy of the pairing, the extrema of the action
functional (4.98) are precisely the solutions $A\in V^{1}$ of the equations of
motion $\mathfrak{F}^{\star}_{A}=0$.

Braided gauge invariance of the action functional (4.98) is a bit more subtle,
because the braided derivation property of
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}$ is not generally compatible
with the strict symmetry (4.87) of the inner product. Nevertheless, the gauge
variation $\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)$ of
the action functional (4.98) _is_ well-defined, because the strict cyclicity
condition (4.89) ensures compatibility with strict symmetry of the pairing
order by order in tensor powers of the fields $A^{\otimes n}$ for $n\geq 1$
[45], that is,

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\big{\langle}A,\ell^{\star}_{n}\big{(}A^{\otimes
n}\big{)}\big{\rangle}_{\star}=\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}\big{\langle}\ell^{\star}_{n}\big{(}A^{\otimes
n}\big{)},A\big{\rangle}_{\star}\ .$ (4.100)

However, in contrast to the classical case, a braided gauge variation
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A$ of a field $A\in V^{1}$
along a gauge parameter $\lambda\in V^{0}$ is no longer a special direction of
a general variation $\delta A$ of the field in the tangent space $TV^{1}$,
because the gauge variations are now defined as _braided_ derivations, in
contrast to the general variations. In other words, in the braided case the
first equality in (2.57) simply no longer holds in general:

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)\
\neq\
\langle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A,\mathfrak{F}^{\star}_{A}\rangle_{\star}\
,$ (4.101)

which can be seen explicitly by direct calculation.

Braided gauge invariance of the action functional, that is,
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=0$ for all
$\lambda\in V^{0}$, follows from an order by order calculation using the
braided homotopy Jacobi identity (4.74) applied to $A^{\otimes n+1}$ [45];
this holds irrespectively of the strict or braided cyclicity of the inner
product, that is, it is true for generic twists $\mathcal{F}$. Despite
(4.101), the gauge variation of the action functional (4.98) can moreover be
written in the form

$\displaystyle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=-\langle\lambda,{\mathsf{d}}_{A}^{\star}\mathfrak{F}^{\star}_{A}\rangle_{\star}\
,$ (4.102)

exactly as in the classical case (2.57), by using cyclicity together with the
${\mathcal{R}}$-matrix identities (4.19) to isolate the gauge parameter
$\lambda$. In this sense the operation ${\sf d}^{\star}_{A}$ is the ‘braided
adjoint’ to $-\delta^{\textrm{\tiny${\mathcal{R}}$}}_{\lambda}$ with respect
to the cyclic inner product, and the braided Noether identities
${\mathsf{d}}^{\star}_{A}\mathfrak{F}^{\star}_{A}=0$ then follow from braided
gauge invariance
$\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}S_{\star}(A)=0$ of the action
functional (4.98). We stress, however, that ${\sf d}^{\star}_{A}$ is not what
would normally be called the adjoint operator with respect to a non-degenerate
pairing, as for example in (2.62), as this notion would take as starting point
the right-hand side
$\langle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A,\mathfrak{F}^{\star}_{A}\rangle_{\star}$
of the non-equality (4.101). Since
$\langle\delta_{\lambda}^{\textrm{\tiny${\mathcal{R}}$}}A,\mathfrak{F}^{\star}_{A}\rangle_{\star}\neq
0$, the usual adjoint operator does not compose to zero with the braided
Maurer–Cartan operator, as is reflected directly in the braided Noether
identities (4.96).

The procedure described here yields systematic constructions of _new_ examples
of noncommutative field theories which are braided deformations of classical
field theories. These braided theories contain no new degrees of freedom
compared with their classical versions, while still possessing good classical
limits at $\hbar=0$. Unlike our motivating example of Section 4.4, in some
cases the braided field theories exhibit some surprising features that are not
simply the naive deformations of those found in the classical cases, in
startling contrast to the conventional noncommutative field theories discussed
in Section 3. In addition to the braided Chern–Simons theory discussed in
Section 4.4, the examples of braided Einstein–Cartan–Palatini theories of
gravity in three and four dimensions are worked out in detail in [45].

In the subsequent sections we present three new examples: the standard
noncommutative scalar field theory (regarded as a braided field theory), a
braided version of $BF$ theory in arbitrary dimensionality, and a braided
version of noncommutative Yang–Mills theory for arbitrary gauge algebras. In
the remainder of the present section we address some further technical points
surrounding the constructions above.

###  4.7 Reality conditions

We have not yet addressed the reality conditions that are needed to make
physical sense of the field equations as well as of the action functional of a
braided field theory. For this, we need to restrict to special twists.

######  Definition 4.103.

A Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
is _Hermitian_ if it satisfies the reality condition
$\mathcal{F}^{*}=\mathcal{F}_{21}$, where ∗ denotes complex conjugation.

For the inverse twist $\mathcal{F}^{-1}=\bar{\sf f}^{k}\otimes\bar{\sf f}_{k}$
the condition of Definition 4.103 reads

$\displaystyle\bar{\sf f}^{k}{}^{*}\otimes\bar{\sf f}_{k}{}^{*}=\bar{\sf
f}_{k}\otimes\bar{\sf f}^{k}\ .$ (4.104)

In particular, this turns the conjugation isomorphism into an anti-involution
of the braided exterior algebra, that is,

$\displaystyle(\alpha\wedge_{\star}\beta)^{*}=(-1)^{|\alpha|\,|\beta|}\,\beta^{*}\wedge_{\star}\alpha^{*}$
(4.105)

for homogeneous forms $\alpha,\beta\in\Omega^{\bullet}(M)[[\hbar]]$, and in
this case we say that $\wedge_{\star}$ is a _Hermitian star-product_. A large
class of examples is provided by abelian twists (see Example 4.31), which are
Hermitian when we use the convention $\hbar^{\ast}=\hbar$ for the formal
deformation parameter.

######  Example 4.106.

In the setting of Section 4.3, let $A\in\Omega^{1}(M,\mathfrak{g})[[\hbar]]$
be a braided gauge field and $\phi\in\Omega^{p}(M,W)[[\hbar]]$ a braided
matter field in a braided representation $W$ of a gauge algebra
$\mathfrak{g}$, which are both real: $A^{*}=A$ and $\phi^{*}=\phi$. If the
twist $\mathcal{F}$ is Hermitian, then the complex conjugate of the left
braided covariant derivative of $\phi$ from (4.48) is given by

$\displaystyle\begin{split}\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}\phi\big{)}^{*}&=\big{(}\mathrm{d}\phi+A\wedge_{\star}\phi\big{)}^{*}\\\\[4.0pt]
&=\mathrm{d}\phi^{*}+\big{(}\bar{\sf f}^{k}(A)\wedge\bar{\sf
f}_{k}(\phi)\big{)}^{\ast}\\\\[4.0pt] &=\mathrm{d}\phi^{*}+\bar{\sf
f}_{k}(A^{*})\wedge\bar{\sf f}^{k}(\phi^{*})\\\\[4.0pt]
&=\mathrm{d}\phi+\mathsf{R}_{k}(A)\wedge_{\star}\mathsf{R}^{k}(\phi)=\mathrm{d}_{\star{\textrm{\tiny
R}}}^{A}\phi\ ,\end{split}$ (4.107)

where in the last line we used
$\mathcal{F}_{21}^{-1}=\mathcal{F}^{-1}\,{\mathcal{R}}^{-1}$. Thus neither the
left nor right braided covariant derivative preserves the reality of the
matter field $\phi$ in general. However, their symmetrized combination

$\displaystyle\tfrac{1}{2}\,\big{(}\mathrm{d}_{\star{\textrm{\tiny
L}}}^{A}+\mathrm{d}_{\star{\textrm{\tiny R}}}^{A}\big{)}$ (4.108)

does preserve reality, and in examples this is the combination that will
naturally appear, as guaranteed by the general formalism below (cf. Section
4.4).

Let $(V,\\{\ell_{n}\\},\langle-,-\rangle)$ be a cyclic 4-term
$L_{\infty}$-algebra over $\mathbbm{R}$ encoding a classical Lagrangian field
theory on a manifold $M$. This theory describes real dynamical fields, that
is, they belong to a real vector space $V^{1}$ which is a module over
$C^{\infty}(M,\mathbbm{R})$. Similarly, the field equations are real and the
action functional is real-valued. Twist deformation via a compatible Hermitian
Drinfel’d twist $\mathcal{F}$ yields a 4-term strictly cyclic braided
$L_{\infty}$-algebra
$(V[[\hbar]],\\{\ell_{n}^{\star}\\},\langle-,-\rangle_{\star})$ over
$\mathbbm{C}[[\hbar]]$. We will now show that the reality of the classical
field theory survives this deformation.

Firstly, the braided field equations $\mathfrak{F}_{A}^{\star}\in
V^{2}[[\hbar]]$ are real when restricted to real fields $A\in V^{1}[[\hbar]]$.
Conjugating the brackets $\ell_{n}^{\star}(A^{\otimes n})\in V^{2}[[\hbar]]$
yields brackets in the complex conjugate vector space $V^{2}[[\hbar]]^{*}$
given by

$\displaystyle\begin{split}\ell_{n}^{\star}(A^{\otimes
n})^{*}&=\ell_{n}\big{(}\bar{\sf f}^{k_{1}}(A)\otimes\bar{\sf
f}_{k_{1}}(\bar{\sf f}^{k_{2}}(A)\otimes\bar{\sf f}_{k_{2}}(\cdots\bar{\sf
f}^{k_{n-1}}(A)\otimes\bar{\sf f}_{k_{n-1}}(A))\cdots))\big{)}^{*}\\\\[4.0pt]
&=\ell_{n}\big{(}\bar{\sf f}_{k_{1}}(A^{*})\otimes\bar{\sf f}^{k_{1}}(\bar{\sf
f}_{k_{2}}(A^{*})\otimes\bar{\sf f}^{k_{2}}(\cdots\bar{\sf
f}_{k_{n-1}}(A^{*})\otimes\bar{\sf
f}^{k_{n-1}}(A^{*}))\cdots))\big{)}\\\\[4.0pt]
&=\ell_{n}\big{(}\mathsf{R}_{k_{1}}(A^{*})\otimes_{\star}\mathsf{R}^{k_{1}}(\mathsf{R}_{k_{2}}(A^{*})\otimes_{\star}\mathsf{R}^{k_{2}}(\cdots\mathsf{R}_{k_{n-1}}(A^{*})\otimes_{\star}\mathsf{R}^{k_{n-1}}(A^{*}))\cdots))\big{)}\\\\[4.0pt]
&=\ell_{n}^{\star}\big{(}\mathsf{R}_{k_{1}}(A^{*})\otimes\mathsf{R}^{k_{1}}(\mathsf{R}_{k_{2}}(A^{*})\otimes\mathsf{R}^{k_{2}}(\cdots\mathsf{R}_{k_{n-1}}(A^{*})\otimes\mathsf{R}^{k_{n-1}}(A^{*}))\cdots))\big{)}\\\\[4.0pt]
&=\ell_{n}^{\star}(A^{*}{}^{\otimes n})\ ,\end{split}$ (4.109)

where $A^{*}\in V^{1}[[\hbar]]^{*}$. In the second line we repeatedly used
Hermiticity of the twist and compatibility of the classical brackets with
complex conjugation, ${}^{*}\circ\ell_{n}=\ell_{n}\circ{}^{*}$ (as they are
$\mathbbm{R}$-multilinear maps extended linearly over the complexification of
the underlying vector space), and in the third line we used the definition of
the $\mathcal{R}$-matrix $\mathcal{R}=\mathcal{F}_{21}\,\mathcal{F}^{-1}$. In
the last two lines we used the definition of the braided brackets along with
their braided graded antisymmetry. This shows

$\displaystyle(\mathfrak{F}_{A}^{\star})^{*}=\mathfrak{F}_{A^{*}}^{\star}\ ,$
(4.110)

and in particular the field equations are real on the subspace of fields where
the conjugation isomorphism $A\mapsto A^{*}$ is the identity.

Secondly, the braided action functional
$S_{\star}:V^{1}[[\hbar]]\to\mathbbm{C}[[\hbar]]$ is real-valued when
restricted to real fields. This follows similarly to the computation above:

$\displaystyle\begin{split}\big{[}\langle A,\ell_{n}^{\star}(A^{\otimes
n})\rangle_{\star}\big{]}^{*}&=\big{[}\big{\langle}\bar{\sf f}^{k}(A),\bar{\sf
f}_{k}\big{(}\ell_{n}^{\star}(A^{\otimes
n})\big{)}\big{\rangle}\big{]}^{*}\\\\[4.0pt] &=\big{\langle}\bar{\sf
f}_{k}(A^{*}),\bar{\sf f}^{k}\big{(}\ell_{n}^{\star}(A^{\otimes
n})^{*}\big{)}\big{\rangle}\\\\[4.0pt]
&=\big{\langle}\mathsf{R}_{k}(A^{*}),\mathsf{R}^{k}\big{(}\ell_{n}^{\star}(A^{*}{}^{\otimes
n})\big{)}\big{\rangle}_{\star}=\langle
A^{*},\ell_{n}^{\star}(A^{*}{}^{\otimes n})\rangle_{\star}\ ,\end{split}$
(4.111)

where in the last equality we additionally used the compatibility of the
twist. This shows that $S_{\star}(A)^{*}=S_{\star}(A^{*})$, and the required
reality follows.

###  4.8 Braided BV formalism

Let us now briefly explain how the notion of cyclic braided
$L_{\infty}$-algebra
$(V[[\hbar]],\\{\ell_{n}^{\star}\\},\langle-,-\rangle_{\star})$ in terms of
star-brackets and braided homotopy relations is related to braided versions of
the algebraic and coalgebraic formulations from Section 2.2. The Drinfel’d
twist deformation prescription defines the braided symmetric algebra with
deformed symmetric product $\odot_{\star}:=\odot\circ\mathcal{F}^{-1}$, which
we denote by $\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])$. On
generators $v,v^{\prime}\in V[1]$ this product is braided graded commutative:

$\displaystyle
v\odot_{\star}v^{\prime}=(-1)^{|v|\,|v^{\prime}|}\,\mathsf{R}_{k}(v^{\prime})\odot_{\star}\mathsf{R}^{k}(v)\
.$ (4.112)

Following [77], it is straightforward to develop the Chevalley–Eilenberg
algebra of a braided $L_{\infty}$-algebra (as a braided noncommutative
deformation of Definition 2.32), that is, a braided version of the BV
formalism presented in Section 2.3. For this, we extend the braided
$L_{\infty}$-algebra structure on $V[[\hbar]]$ to
$\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])\otimes V[[\hbar]]$ via the
brackets

$\displaystyle\begin{split}\ell_{n}^{\star\,{\rm ext}}(\zeta_{1}\otimes
v_{1},\dots,\zeta_{n}\otimes v_{n}):&\\!=\ell_{n}^{\,\rm
ext}\big{(}(\zeta_{1}\otimes
v_{1})\otimes_{\star}\cdots\otimes_{\star}(\zeta_{n}\otimes
v_{n})\big{)}\\\\[4.0pt]
&=\pm\,\big{(}\zeta_{1}\odot_{\star}\mathsf{R}_{k_{1}^{\textrm{\tiny
1}}}(\zeta_{2})\odot_{\star}\cdots\odot_{\star}\mathsf{R}_{k_{n-1}^{1}}\cdots\mathsf{R}_{k_{1}^{n-1}}(\zeta_{n})\big{)}\\\
&\hskip
56.9055pt\,\otimes\,\ell_{n}^{\star}\big{(}\mathsf{R}^{k_{n-1}^{1}}\cdots\mathsf{R}^{k_{1}^{1}}(v_{1}),\dots,\mathsf{R}^{k_{1}^{n-1}}(v_{n-1}),v_{n}\big{)}\
,\end{split}$ (4.113)

for all $n\geq 1$,
$\zeta_{1},\dots,\zeta_{n}\in\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])$
and $v_{1},\dots,v_{n}\in V$. Similarly, we extend the cyclic structure via
the pairing

$\displaystyle\begin{split}\langle\zeta_{1}\otimes v_{1},\zeta_{2}\otimes
v_{2}\rangle_{\star}^{\rm ext}:=\langle(\zeta_{1}\otimes
v_{1})\otimes_{\star}(\zeta_{2}\otimes v_{2})\rangle^{\rm
ext}=\pm\,\big{(}\zeta_{1}\odot_{\star}\mathsf{R}_{k}(\zeta_{2})\big{)}\,\langle\mathsf{R}^{k}(v_{1}),v_{2}\rangle_{\star}\
.\end{split}$ (4.114)

We can now define a braided version of the BV differential $Q_{\textrm{\tiny
BV}}^{\star}:\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])^{*}\to\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])^{*}$
by

$\displaystyle Q_{\textrm{\tiny BV}}^{\star}\xi=-\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell_{n}^{\star\,\rm ext}(\xi^{\otimes n})\
,$ (4.115)

where the contracted coordinate functions

$\displaystyle\xi:=\tau^{\alpha}\otimes\tau_{\alpha}\ \in\
\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])\otimes V[[\hbar]]$ (4.116)

of degree $1$ are defined by a choice of basis $\\{\tau_{\alpha}\\}\subset V$
with dual basis $\\{\tau^{\alpha}\\}\subset V^{*}\simeq V[3]$. Similarly, we
define a braided version of the BV symplectic $2$-form

$\displaystyle\omega_{\textrm{\tiny
BV}}^{\star}:=-\tfrac{1}{2}\,\langle\delta\xi,\delta\xi\rangle_{\star}^{\rm
ext}\ \in\ \Omega^{2}(V[1])[[\hbar]]\ ,$ (4.117)

whose inverse is a braided graded Poisson bracket $\\{-,-\\}_{\textrm{\tiny
BV}}^{\star}$, that is, a braided graded Lie bracket which is a braided graded
derivation on $\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])$ in each of
its slots, and which moreover is compatible with the differential
$Q_{\textrm{\tiny BV}}^{\star}$. The action of the braided BV differential can
then be represented in terms of this braided Poisson bracket as

$\displaystyle Q_{\textrm{\tiny BV}}^{\star}=\\{S^{\star}_{\textrm{\tiny
BV}},-\\}^{\star}_{\textrm{\tiny BV}}\ ,$ (4.118)

where the braided BV action functional

$\displaystyle S^{\star}_{\textrm{\tiny BV}}=\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{(n+1)!}\,\langle\xi,\ell_{n}^{\star\,\rm
ext}(\xi^{\otimes n})\rangle_{\star}^{\rm ext}\ \in\
\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])$ (4.119)

of degree $0$ satisfies the classical master equation

$\displaystyle\\{S^{\star}_{\textrm{\tiny BV}},S^{\star}_{\textrm{\tiny
BV}}\\}^{\star}_{\textrm{\tiny BV}}=0\ .$ (4.120)

A crucial point here is that the classical master equation, or equivalently
nilpotency $(Q_{\textrm{\tiny BV}}^{\star})^{2}=0$, follows from precisely the
same calculation as in the ordinary case [60, Section 4.3]. This is because
the contracted coordinate functions $\xi$ are
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant elements of
$\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[2])\otimes
V[[\hbar]]$,262626Invariance of $\xi$ follows from either a direct calculation
or by the $\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-isomorphism
$V\otimes V^{*}\simeq{\rm
Hom}_{{}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}}(V,V)$,
under which $\xi$ corresponds to the identity morphism on $V$. and hence all
appearances of ${\mathcal{R}}$-matrices in the properties of the extended
brackets $\ell_{n}^{\star\,\rm ext}$ and of the extended pairing
$\langle-,-\rangle_{\star}^{\rm ext}$ disappear when evaluated on tensor
powers of the $\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant
element $\xi$. Another important feature is that this differential graded
algebra perspective circumvents the issues discussed in Section 4.5 with
defining a moduli space
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(V[[\hbar]],\\{\ell^{\star}_{n}\\})$
of classical solutions in the braided setting. This is because it
characterises this space by its differential graded algebra
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{A}_{\textrm{\tiny
MC}}$ of equivariant functions, and so the (coarse) moduli space may be
defined through the degree $0$ cohomology of the differential graded braided
commutative algebra
$\big{(}\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])^{*},Q_{\textrm{\tiny
BV}}^{\star}\big{)}$. These two observations will be combined together in
Section 4.9 below to sketch a precise description of the moduli space
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(V[[\hbar]],\\{\ell^{\star}_{n}\\})$
which resolves the technical issues mentioned in Section 4.5.

In a similar vein, one can derive a duality with differential graded braided
cocommutative coalgebras (as a braided noncommutative deformation of
Definition 2.19). The braided symmetric algebra
$\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])$ has a natural free
noncocommutative coalgebra structure with coproduct
$\Delta_{\textrm{\tiny$V$}}^{\star}:\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])\to\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])\otimes\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])$
obtained by Drinfel’d twist deformation of the cocommutative coproduct (2.18)
in the usual way, replacing permutation actions $\tau^{\sigma}:V[1]^{\otimes
n}\to V[1]^{\otimes n}$ with $\tau_{\textrm{\tiny${\mathcal{R}}$}}^{\sigma}$
for $\sigma\in S_{n}$. The braided graded symmetric multilinear maps

$\displaystyle b_{n}^{\star}:=s\circ\ell_{n}^{\star}\circ(s^{-1})^{\otimes n}$
(4.121)

of degree $1$ define a map $\sum_{n\geq
1}\,b_{n}^{\star}:\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])\to(V[1])[[\hbar]]$,
which uniquely extends to a coderivation

$\displaystyle
D_{\star}:\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])\longrightarrow\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1])$
(4.122)

of degree $1$. This is a differential, $(D_{\star})^{2}=0$, as a consequence
of the braided homotopy Jacobi identities, and it too provides a natural
setting for the moduli space of classical solutions
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(V[[\hbar]],\\{\ell^{\star}_{n}\\})$
as the degree $0$ cohomology of the corresponding differential graded braided
cocommutative coalgebra
$(\mathrm{Sym}_{\textrm{\tiny${\mathcal{R}}$}}(V[1]),D_{\star})$.

The coalgebra formulation also provides a natural interpretation of a braided
$L_{\infty}$-algebra as a deformation of an $L_{\infty}$-algebra in the
category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ [52]: The classical
differential graded coalgebra $(\mathrm{Sym}(V[1]),D)$ naturally inherits the
structure of a cocommutative Hopf algebra compatible with the differential
$D$,272727More precisely, here one needs to work with the unreduced symmetric
algebra and curved $L_{\infty}$-algebras. which can be twisted to a new
noncocommutative Hopf algebra by the techniques of Section 4.2 with a
compatible differential $D_{\textrm{\tiny$\mathcal{F}$}}$. The braided
$L_{\infty}$-algebra $(V[[\hbar]],D_{\star})$ may then be interpreted as an
$L_{\infty}$-module for the resulting twisted $L_{\infty}$-algebra.

###  4.9 Moduli spaces of classical solutions

We conclude this section by briefly sketching how to potentially make sense of
the moduli space of classical solutions
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(V[[\hbar]],\\{\ell^{\star}_{n}\\})$
in braided field theory, which we refer to as the ‘braided Maurer–Cartan
moduli space’. We shall be intentionally brief and gloss over many technical
details in order to illustrate the main ideas;282828Nevertheless, this is the
most technical part of the paper. The reader not interested in the formal
descriptions of moduli spaces may safely skip this section, which is not
essential to the rest of the paper. a more thorough and rigorous construction
will be presented elsewhere. We start by recalling the description of the
classical Maurer–Cartan moduli space
$\mathscr{M}\mathscr{C}(V,\\{\ell_{n}\\})$ of an $L_{\infty}$-algebra
$(V,\\{\ell_{n}\\})$, whose construction is originally due to Getzler [51],
see also [80] for a more explicit alternative construction and [75] for the
special case when $(V,\ell_{1},\ell_{2})$ is a differential graded Lie algebra
(as in the example of Chern–Simons gauge theory).

We begin with some preliminary notation. Let ${\sf dAlg}$ denote the category
of differential graded commutative algebras over $\mathbbm{R}$, whose
morphisms are algebra homomorphisms of degree $0$ which intertwine the
differentials.292929 The appropriate homotopical analogue generalizing the
spectrum of a commutative algebra to differential graded commutative algebras
takes place in the $\infty$-category of ‘simplicial’ sets, but this is not
needed in our presentation below and we shall work solely with the category
${\sf Set}$ of sets. Similarly, to define appropriate notions of
representability of functors we should work with the $\infty$-category of
‘Artin’ differential graded commutative algebras in non-positive degrees, but
we do not indicate this detail and stick to the full source category ${\sf
dAlg}$ below. Let $\mathcal{L}=(V,\\{\ell_{n}\\})$ be a classical
$L_{\infty}$-algebra (in the category of real vector spaces). Given a
differential graded commutative algebra
$\mathsf{A}=(\mathcal{A},{\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}})$ (with
the product on $\mathcal{A}$ denoted by juxtaposition), there is a tensor
product $L_{\infty}$-algebra
$\mathsf{A}\otimes\mathcal{L}=\big{(}\mathcal{A}\otimes
V,\big{\\{}\ell_{n}^{\textrm{\tiny$\mathsf{A}$}}\big{\\}}\big{)}$ with
homogeneous subspaces $(\mathcal{A}\otimes
V)^{k}=\bigoplus_{l\in\mathbbm{Z}}\,\mathcal{A}^{-l}\otimes V^{l+k}$ for
$k\in\mathbbm{Z}$ and brackets given by

$\displaystyle\begin{split}\ell_{1}^{\textrm{\tiny$\mathsf{A}$}}(a_{1}\otimes
v_{1})=({\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}+\ell_{1})(a_{1}\otimes
v_{1}):&\\!={\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}a_{1}\otimes
v_{1}+(-1)^{|a_{1}|}\,a_{1}\otimes\ell_{1}(v_{1})\ ,\\\\[4.0pt]
\ell_{n}^{\textrm{\tiny$\mathsf{A}$}}(a_{1}\otimes v_{1},\dots,a_{n}\otimes
v_{n})&=\pm\,(a_{1}\cdots a_{n})\otimes\ell_{n}(v_{1},\dots,v_{n})\
,\end{split}$ (4.123)

for all $n\geq 2$, $a_{1},\dots,a_{n}\in\mathcal{A}$ and $v_{1},\dots,v_{n}\in
V$. This generalizes the extended $L_{\infty}$-algebra construction of Section
2.3.

Following [80], the (coarse) Maurer–Cartan moduli space
$\mathscr{M}\mathscr{C}(\mathcal{L})$ of the $L_{\infty}$-algebra
$\mathcal{L}=(V,\\{\ell_{n}\\})$ is the set of solutions $A\in V^{1}$ of the
Maurer–Cartan equation $\mathfrak{F}_{A}=0$, modulo the gauge equivalence
relation $A\sim A^{\prime}$ if there exists a gauge parameter $\lambda\in
V^{0}$ and a path of fields $A(t)\in V^{1}$ for $t\in[0,1]$ with $A(0)=A$,
$A(1)=A^{\prime}$ which satisfies the differential equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}A(t)=\delta_{\lambda}A(t)$ (4.124)

for all $t\in[0,1]$. The infinitesimal version of (4.124) (its evaluation at
$t=0$) gives the usual infinitesimal gauge transformation $\delta_{\lambda}A$
of $A$. With the given initial condition and an appropriate completeness
condition on the $L_{\infty}$-algebra $\mathcal{L}$, the differential equation
(4.124) always admits a solution $A(t)$ which satisfies the Maurer–Cartan
equation for all $t\in[0,1]$, that is, $\mathfrak{F}_{A(t)}=0$. In the case
that $\mathcal{L}$ is a differential graded Lie algebra, this definition is
equivalent to the equivalence relation arising from the (integrated) group
action of $V^{0}$ on $V^{1}$ (see e.g. [75]).

The moduli space $\mathscr{M}\mathscr{C}(\mathcal{L})$ is a (formal derived)
stack that is described by its ‘functor of points’
$\mathscr{F}_{\mathcal{L}}:{\sf dAlg}\to{\sf Set}$, as we now explain. This
defines a presheaf on the category of differential graded affine schemes over
$\mathbbm{R}$, and so from this perspective the moduli space is described not
as a set of objects but rather as a space which parametrizes those objects.

######  Definition 4.125.

Let $\mathcal{L}=(V,\\{\ell_{n}\\})$ be an $L_{\infty}$-algebra and
$\mathsf{A}=(\mathcal{A},{\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}})$ a
differential graded commutative algebra.

  1. (a)

A _family of classical solutions_ in $\mathcal{L}$ parametrized by
$\mathsf{A}$ is an element $\alpha=a^{l}\otimes v_{l}$, with
$a^{l}\in\mathcal{A}^{-l}$ and $v_{l}\in V^{l+1}$ for $l\in\mathbbm{Z}$, which
solves the Maurer–Cartan equation

$\displaystyle({\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}+\ell_{1})\alpha+\sum_{n\geq
2}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n}^{\textrm{\tiny$\mathsf{A}$}}(\alpha^{\otimes n})=0$ (4.126)

in the $L_{\infty}$-algebra $\mathsf{A}\otimes\mathcal{L}$.

  2. (b)

Two families $\alpha$ and $\alpha^{\prime}$ of classical solutions in
$\mathcal{L}$ are _gauge equivalent_ if there exists $\lambda=b^{l}\otimes
w_{l}$, with $b^{l}\in\mathcal{A}^{-k}$ and $w_{l}\in V^{k}$ for
$l\in\mathbbm{Z}$, and a path $\alpha(t)=a^{l}(t)\otimes v_{l}(t)$ in the
$L_{\infty}$-algebra $\mathsf{A}\otimes\mathcal{L}$ which satisfies the
differential equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\alpha(t)=({\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}+\ell_{1})\lambda+\sum_{n\geq
1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n+1}^{\textrm{\tiny$\mathsf{A}$}}\big{(}\lambda,\alpha(t)^{\otimes
n}\big{)}\ ,$ (4.127)

such that $\alpha=\alpha(0)$ and $\alpha^{\prime}=\alpha(1)$.

For a family parametrized by a one-point space, that is, for
$\mathsf{A}=\mathbbm{R}_{0}:=(\mathbbm{R},0)$ with $\mathbbm{R}$ sitting in
degree $0$, this definition reduces to the notion of gauge equivalence of
classical solutions discussed above.

######  Definition 4.128.

Let $\mathcal{L}$ be an $L_{\infty}$-algebra.

  1. (a)

The _moduli functor_ is the covariant functor $\mathscr{F}_{\mathcal{L}}:{\sf
dAlg}\to{\sf Set}$ which associates to every differential graded commutative
algebra $\mathsf{A}$ the set $\mathscr{F}_{\mathcal{L}}(\mathsf{A})$ of gauge
equivalence classes of families of classical solutions in $\mathcal{L}$
parametrized by $\mathsf{A}$, and to every morphism
$f:\mathsf{A}\to\mathsf{B}$ of differential graded commutative algebras it
associates the pushforward $\mathscr{F}_{\mathcal{L}}(f)$ sending families
parametrized by $\mathsf{A}$ to families parametrized by $\mathsf{B}$.

  2. (b)

A (_fine_) _Maurer–Cartan moduli space_ $\mathscr{M}\mathscr{C}(\mathcal{L})$
is a representation of the moduli functor, that is, a pair
$(\hat{\mathsf{A}},\mathsf{\Phi})$ where $\hat{\mathsf{A}}$ is an object of
the category ${\sf dAlg}$ and ${\mathsf{\Phi}}:{\rm Hom}_{\sf
dAlg}(\hat{\mathsf{A}},-)\to\mathscr{F}_{\mathcal{L}}$ is a natural
transformation of functors.

The idea behind this definition is that the functor ${\rm Hom}_{\sf
dAlg}(\hat{\mathsf{A}},-):{\sf dAlg}\to{\sf Set}$ describes all possible ways
that any other differential graded affine scheme may be mapped to the moduli
space, which completely captures the (derived) geometric structure of the
moduli scheme given by the spectrum of the differential graded commutative
algebra $\hat{\mathsf{A}}$. The distinguished element
$\hat{\alpha}={\sf\Phi}(\mathbbm{1}_{\hat{A}})\in\mathscr{F}_{\mathcal{L}}(\hat{\mathsf{A}})$
is called the _universal classical solution_ : For every pair
$(\mathsf{A},\alpha)$ with $\mathsf{A}$ an object of ${\sf dAlg}$ and
$\alpha\in\mathscr{F}_{\mathcal{L}}(\mathsf{A})$, there is a unique morphism
${\sf\Lambda}\in{\rm Hom}_{\sf dAlg}(\hat{\mathsf{A}},\mathsf{A})$ such that
$(\mathscr{F}_{\mathcal{L}}\circ{\sf\Lambda})(\hat{\alpha})=\alpha$. When it
exists, a universal object $(\hat{\mathsf{A}},\hat{\alpha})$ is unique up to a
unique isomorphism, and in this sense it is the space of _all_ classical
solutions modulo gauge transformations. For a more accurate and explicit
description of the Maurer–Cartan moduli space using simplicial methods, see
[80].

This description of the moduli space of classical solutions can be adapted to
the braided noncommutative setting in the following way, which uses a
description of functors of points similar to those studied in [23] in the
context of toric noncommutative geometry. For this, let now
$\mathcal{L}=(V,\\{\ell_{n}\\})$ be an $L_{\infty}$-algebra in the category
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ of left
$\mathsf{U}\mathfrak{v}$-modules. Similarly, we work with differential graded
commutative algebras in the category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$.
These form the category ${}^{\mathsf{U}\mathfrak{v}}{\sf dAlg}$ of
differential graded commutative left $\mathsf{U}\mathfrak{v}$-module algebras,
whose morphisms are equivariant algebra homomorphisms of degree $0$
intertwining the differentials. The description of the Maurer–Cartan moduli
space $\mathscr{M}\mathscr{C}(\mathcal{L})$ proceeds exactly as above, by
restricting the moduli functor $\mathscr{F}_{\mathcal{L}}:{\sf dAlg}\to{\sf
Set}$ to the subcategory ${}^{\mathsf{U}\mathfrak{v}}{\sf dAlg}$.

Given a Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$,
by Proposition 4.81 we obtain a braided $L_{\infty}$-algebra
$\mathcal{L}^{\star}=(V[[\hbar]],\\{\ell_{n}^{\star}\\})$. Similarly, by the
usual constructions of Section 4.2 we obtain a (functorially equivalent)
category ${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf dAlg}$
of differential graded braided commutative algebras. An object
$\mathsf{A}^{\star}=(\mathcal{A},{\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}})$
of this category consists of a graded braided commutative algebra
$\mathcal{A}$ (with the product on $\mathcal{A}$ denoted by $\star$), that is,

$\displaystyle a\star
a^{\prime}=(-1)^{|a|\,|a^{\prime}|}\,\mathsf{R}_{k}(a^{\prime})\star\mathsf{R}^{k}(a)\
,$ (4.129)

for all homogeneous $a,a^{\prime}\in\mathcal{A}$, together with a differential
${\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}:\mathcal{A}\to\mathcal{A}$ of
degree $1$ which is a graded derivation, that is,
${\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}(a\star
a^{\prime})={\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}a\star
a^{\prime}+(-1)^{|a|}\,a\star{\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}a^{\prime}$.
There is a braided tensor product $L_{\infty}$-algebra
$\mathsf{A}^{\star}\otimes\mathcal{L}^{\star}=\big{(}\mathcal{A}\otimes
V[[\hbar]],\big{\\{}\ell_{n}^{\star{\textrm{\tiny$\mathsf{A}$}}}\big{\\}}\big{)}$
whose differential is unchanged from the classical case (4.123), while the
higher brackets of (4.123) are modified to

$\displaystyle\begin{split}\ell_{n}^{\star{\textrm{\tiny$\mathsf{A}$}}}(a_{1}\otimes
v_{1},\dots,a_{n}\otimes
v_{n})&=\pm\,\big{(}a_{1}\star\mathsf{R}_{k_{1}^{\textrm{\tiny
1}}}(a_{2})\star\cdots\star\mathsf{R}_{k_{n-1}^{1}}\cdots\mathsf{R}_{k_{1}^{n-1}}(a_{n})\big{)}\\\
&\hskip
56.9055pt\,\otimes\,\ell_{n}^{\star}\big{(}\mathsf{R}^{k_{n-1}^{1}}\cdots\mathsf{R}^{k_{1}^{1}}(v_{1}),\dots,\mathsf{R}^{k_{1}^{n-1}}(v_{n-1}),v_{n}\big{)}\
,\end{split}$ (4.130)

for all $n\geq 2$, $a_{1},\dots,a_{n}\in\mathcal{A}$ and $v_{1},\dots,v_{n}\in
V$. This generalizes the extended braided $L_{\infty}$-algebra construction of
Section 4.8.

######  Definition 4.131.

Let $\mathcal{L}^{\star}=(V[[\hbar]],\\{\ell^{\star}_{n}\\})$ be a braided
$L_{\infty}$-algebra and
$\mathsf{A}^{\star}=(\mathcal{A},{\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}})$
a differential graded braided commutative algebra.

  1. (a)

A _family of classical solutions_ in $\mathcal{L}^{\star}$ parametrized by
$\mathsf{A}^{\star}$ is a
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant element
$\alpha=a^{l}\otimes v_{l}$, with $a^{l}\in\mathcal{A}^{-l}$ and $v_{l}\in
V^{l+1}[[\hbar]]$ for $l\in\mathbbm{Z}$, which solves the braided
Maurer–Cartan equation

$\displaystyle({\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}+\ell^{\star}_{1})\alpha+\sum_{n\geq
2}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n}^{\star{\textrm{\tiny$\mathsf{A}$}}}(\alpha^{\otimes n})=0$
(4.132)

in the braided $L_{\infty}$-algebra
$\mathsf{A}^{\star}\otimes\mathcal{L}^{\star}$.

  2. (b)

Two families $\alpha$ and $\alpha^{\prime}$ of classical solutions in
$\mathcal{L}^{\star}$ are _braided gauge equivalent_ if there exists a
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant element
$\lambda=b^{l}\otimes w_{l}$, with $b^{l}\in\mathcal{A}^{-k}$ and $w_{l}\in
V^{k}[[\hbar]]$ for $l\in\mathbbm{Z}$, and a
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant path
$\alpha(t)=a^{l}(t)\otimes v_{l}(t)$ in the braided $L_{\infty}$-algebra
$\mathsf{A}^{\star}\otimes\mathcal{L}^{\star}$ which satisfies the
differential equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\alpha(t)=({\mathsf{d}}_{\textrm{\tiny$\mathcal{A}$}}+\ell_{1}^{\star})\lambda+\sum_{n\geq
1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n+1}^{\star{\textrm{\tiny$\mathsf{A}$}}}\big{(}\lambda,\alpha(t)^{\otimes
n}\big{)}\ ,$ (4.133)

such that $\alpha=\alpha(0)$ and $\alpha^{\prime}=\alpha(1)$.

This is our key definition: part (b) of the definition now makes sense non-
trivially, because all ${\mathcal{R}}$-matrix insertions disappear when
evaluated on the
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant elements of
the set of tensor product solutions $\alpha$ to the braided Maurer–Cartan
equations $\mathfrak{F}_{\alpha}^{\star}=0$ in
$\mathsf{A}^{\star}\otimes\mathcal{L}^{\star}$, and so braided gauge
transformations act as in the classical setting of Definition 4.125. Note that
this does not preclude noncommutative deformations of the solutions compared
to Definition 4.125, as both $\mathsf{A}^{\star}$ and $\mathcal{L}^{\star}$
are objects in the symmetric monoidal category
${}_{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}$, and so
each individually involve braided operations: only in their braided tensor
product $\mathsf{A}^{\star}\otimes\mathcal{L}^{\star}$ do we look for
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant elements.
This also clarifies what goes wrong in the setting of Section 4.6: to recover
that situation we would set $\mathsf{A}^{\star}=\mathbbm{R}_{0}$ in this
definition, which means that the classical solutions should be
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant fields. Then
all issues concerning the action of braided gauge transformations, the braided
Noether identities and related matters disappear. Of course we are mostly
interested in non-invariant fields, but to detect those one has to consider
differential graded braided commutative algebras $\mathsf{A}^{\star}$ which
carry a non-trivial
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-action. In this sense,
the braided moduli space does not contain any (interesting) ‘points’, as we
further exemplify below.

The description of the braided Maurer–Cartan moduli space is now a simple
extension of the classical case, as the latter definitions make sense for
covariant functors from any (locally small) category to the category of sets.
The definition of moduli functor from part (a) of Definition 4.128 generalizes
immediately to give a moduli functor
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}:{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf
dAlg}\to{\sf Set}$ which sends each differential graded braided commutative
algebra $\mathsf{A}^{\star}$ to the set
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}(\mathsf{A}^{\star})$
of braided gauge equivalence classes of families of classical solutions in the
braided $L_{\infty}$-algebra $\mathcal{L}^{\star}$ parametrized by
$\mathsf{A}^{\star}$. Part (b) then generalizes easily in the following
manner.

######  Definition 4.134.

Let $\mathcal{L}^{\star}$ be a braided $L_{\infty}$-algebra. A _braided
Maurer–Cartan moduli space_
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{M}\mathscr{C}(\mathcal{L}^{\star})$
is a pair $(\hat{\mathsf{A}}^{\star},\hat{\alpha}^{\star})$, where
$\hat{\mathsf{A}}^{\star}$ is an object of the category
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf dAlg}$ and
$\hat{\alpha}^{\star}\in{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}(\hat{A}^{\star})$,
which is a universal object representing the moduli functor
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}:{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf
dAlg}\to{\sf Set}$.

This definition has the same meaning as in the classical case,303030By
Yoneda’s lemma, this definition is bijectively equivalent to the existence of
a natural transformation ${\sf\Phi}^{\star}:{\rm
Hom}_{{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf
dAlg}}(\hat{\mathsf{A}}^{\star},-)\to{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}$
with
$\hat{\alpha}^{\star}={\sf\Phi}^{\star}(\mathbbm{1}_{\hat{\mathsf{A}}^{\star}})$.
but with a crucial difference: since the moduli space is necessarily an object
in the source category of the functor
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}$,
and we allow noncommutative parameter spaces (regarded as dual to differential
graded braided commutative algebras), the braided Maurer–Cartan moduli space
is naturally a (differential graded) ‘noncommutative scheme’, which is the
tradeoff for making sense of the braided gauge equivalence classes. Of course,
the highly non-trivial aspect of this construction, which we do not address
here, is whether there actually exists a representation of the functor
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}$
which brings our arguments above to fruition, and hence provides a realization
of the moduli space of classical solutions in braided field theory.

Noncommutative moduli spaces have previously appeared in braided
noncommutative geometry as solution spaces to noncommutative instanton
equations in [32], where it is shown that the noncommutativity of the
parameter spaces is a gauge artefact: one may always recover an equivalent
description in terms of the usual notion of classical moduli space by a
suitable choice of gauge transformation and a noncommutative quotient
construction. It would be interesting to understand the braided Maurer–Cartan
moduli space (if it really exists in this sense) in a similar vein, which
would elucidate the physical meaning of the abstract arguments presented here.

This braided Maurer–Cartan moduli space may also be extracted from the braided
BV formalism of Section 4.8 and its braided noncommutative algebra of
functions
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{A}_{\textrm{\tiny
MC}}$. For this, one considers the functor of points
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}^{\textrm{\tiny$\vee$}}:{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf
dAlg}\to{\sf Set}$ defined on objects by
${}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{F}_{\mathcal{L}^{\star}}^{\textrm{\tiny$\vee$}}(\mathsf{A}^{\star})={\rm
Hom}_{{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}{\sf
dAlg}}\big{(}{}^{\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}}\mathscr{A}_{\textrm{\tiny
MC}},\mathsf{A}^{\star}\big{)}$. Taking again
$\mathsf{A}^{\star}=\mathbbm{R}_{0}$, one only detects
$\mathsf{U}_{\textrm{\tiny$\mathcal{F}$}}\mathfrak{v}$-invariant points,
whereas the interesting fields appear for non-trivial differential graded
braided commutative algebras $\mathsf{A}^{\star}$.

##  5 Noncommutative scalar field theory

The $L_{\infty}$-algebra framework also accommodates field theories without
gauge symmetries. This simply translates to trivial homogeneous subspaces
$V^{0}=V^{3}=\\{0\\}$ and corresponding zero brackets. The appearance of the
$L_{\infty}$-algebra structure is then tautological, as it is just given by
polynomial expansion of the field equations. In this section we will review
the formalism in the example of an interacting scalar field theory, and
subsequently twist deform this $L_{\infty}$-algebra to obtain the standard
noncommutative scalar field theory. This example also serves to illustrate how
to handle more general theories which are not diffeomorphism invariant, via
twist deformation quantization along the global spacetime symmetries of the
classical field theory. It moreover shows how to incorporate global symmetries
into the braided $L_{\infty}$-algebra framework, which we describe following
the classical treatment of [60].313131Of course, we are not interested in a
moduli space of globally (braided) symmetric solutions (as these are not gauge
symmetries). In principle, one is interested instead in the weakly conserved
currents corresponding to these infinitesimal symmetries, which would follow
from a braided version of Noether’s first theorem. We do not develop this
interesting point in the present paper.

###  5.1 Scalar field theory in the $L_{\infty}$-algebra formalism

Let $(M,g)$ be a $d$-dimensional oriented Lorentzian manifold.323232The
discussion is presented for Lorentzian signature in order to compare later on
with previous points of view on global symmetries. The more general pseudo-
Riemannian cases differ only by a sign in the Laplace–Beltrami operator. Let
$\mathrm{d}{\rm vol}_{g}\in\Omega^{d}(M)$ be the induced volume form and
$\ast_{\textrm{\tiny H}}:\Omega^{k}(M)\to\Omega^{d-k}(M)$ the corresponding
Hodge duality operator. Denote by ${\square}:=\ast_{\textrm{\tiny
H}}\mathrm{d}\ast_{\textrm{\tiny H}}\mathrm{d}:C^{\infty}(M,\mathbbm{R})\to
C^{\infty}(M,\mathbbm{R})$ the Laplace–Beltrami operator acting on smooth
functions.

The action functional for an interacting real scalar field $\phi\in
C^{\infty}(M,\mathbbm{R})$ is defined by

$\displaystyle
S(\phi)=\int_{M}\,\Big{(}\frac{1}{2}\,\phi\,\big{(}\square-m^{2}\big{)}\phi-\sum_{n\geq
3}\,\frac{\kappa_{n-1}}{n!}\,\phi^{n}\Big{)}\ \mathrm{d}{\rm vol}_{g}\ ,$
(5.1)

for a mass parameter $m^{2}\geq 0$ and coupling constants
$\kappa_{n-1}\in\mathbbm{R}$. The corresponding field equations are given by
$\mathfrak{F}_{\phi}=0$ where

$\displaystyle\mathfrak{F}_{\phi}=\big{(}\square-m^{2}\big{)}\phi-\sum_{n\geq
2}\,\frac{\kappa_{n}}{n!}\,\phi^{n}\ ,$ (5.2)

from which the cyclic $L_{\infty}$-algebra is immediately read off.

The underlying graded vector space is concentrated in degrees $1$ and $2$,
comprised of two copies of the space of functions on $M$:

$\displaystyle V=V^{1}\oplus V^{2}\qquad\mbox{with}\quad
V^{1}=V^{2}=C^{\infty}(M,\mathbbm{R})\ ,$ (5.3)

and we denote its homogeneous elements by $\phi\in V^{1}$ and $\varPhi\in
V^{2}$. The underlying cochain complex $(V,\ell_{1})$ has as differential the
Klein–Gordon operator, and the non-trivial brackets are

$\displaystyle\ell_{1}(\phi)=\big{(}\square-m^{2}\big{)}\phi\qquad\mbox{and}\qquad\ell_{n}(\phi_{1},\dots,\phi_{n})=-(-1)^{n\choose
2}\,\kappa_{n}\,\phi_{1}\cdots\phi_{n}\ ,$ (5.4)

where $n\geq 2$, while the cyclic pairing is

$\displaystyle\langle\phi,\varPhi\rangle:=\int_{M}\,\phi\ast_{\textrm{\tiny
H}}\varPhi=\int_{M}\,\phi\ \varPhi\ \mathrm{d}{\rm vol}_{g}\ .$ (5.5)

The homotopy Jacobi identities follow trivially for degree reasons. Cyclicity
with respect to $\ell_{n}$ for $n\geq 2$ is tautological, due to commutativity
of pointwise multiplication of functions. Cyclicity with respect to $\ell_{1}$
follows since $\square$ is a (formally) self-adjoint operator with respect to
the inner product (5.5) on $C^{\infty}(M,\mathbbm{R})$.

###  5.2 Noncommutative scalar field theory in the braided
$L_{\infty}$-algebra formalism

Suppose now that the spacetime $(M,g)$ has a non-trivial set of isometries.
That is, there exists a non-zero Lie subalgebra of Killing vector fields
$\mathfrak{k}\subset\mathfrak{v}={\sf\Gamma}(TM)$, such that
$\mathrm{L}_{\xi}\,g=0$ for all $\xi\in\mathfrak{k}$. This implies that

$\displaystyle\mathrm{L}_{\xi}\circ\ast_{\textrm{\tiny H}}=\ast_{\textrm{\tiny
H}}\circ\mathrm{L}_{\xi}$ (5.6)

for all $\xi\in\mathfrak{k}$, and so the Laplace–Beltrami operator
$\square=\ast_{\textrm{\tiny H}}\mathrm{d}\ast_{\textrm{\tiny H}}\mathrm{d}$
also commutes with the action of all Killing vector fields. It follows that
all brackets (5.4) of the scalar field $L_{\infty}$-algebra commute with the
action of $\mathfrak{k}$:

$\displaystyle\mathrm{L}_{\xi}\circ\ell_{n}=\ell_{n}\circ\mathrm{L}_{\xi}$
(5.7)

for all $\xi\in\mathfrak{k}$ and $n\geq 1$. In other words,
$(V,\\{\ell_{n}\\})$ is an $L_{\infty}$-algebra in the category of
$\mathsf{U}\mathfrak{k}$-modules.

By the construction of Section 4.5, a choice of a _Killing_ Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{k}[[\hbar]]\otimes\mathsf{U}\mathfrak{k}[[\hbar]]$
results in a 2-term braided $L_{\infty}$-algebra
$\big{(}V[[\hbar]],\\{\ell^{\star}_{n}\\}\big{)}$ with non-zero brackets

$\displaystyle\begin{split}\ell_{1}^{\star}(\phi)=\big{(}\square-m^{2}\big{)}\phi\qquad\mbox{and}\qquad\ell_{n}^{\star}(\phi_{1},\dots,\phi_{n})=-(-1)^{n\choose
2}\,\kappa_{n}\,\phi_{1}\star\cdots\star\phi_{n}\ ,\end{split}$ (5.8)

where $n\geq 2$ and $\star$ denotes the star-product of functions. By the
prescription of Section 4.6, the braided field equations landing in
$V^{2}[[\hbar]]=C^{\infty}(M,\mathbbm{R})[[\hbar]]$ are encoded by

$\displaystyle\mathfrak{F}^{\star}_{\phi}=\sum_{n\geq 1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n}^{\star}(\phi^{\otimes
n})=\big{(}\square-m^{2}\big{)}\phi-\sum_{n\geq
2}\,\frac{\kappa_{n}}{n!}\,\phi^{\star n}\ ,$ (5.9)

where $\phi^{\star n}:=\phi\star\cdots\star\phi$ ($n$ times). This is just the
usual noncommutative deformation of scalar field equations.

The cyclic inner product (5.5) is invariant under the action of
$\mathfrak{k}$, since $\mathrm{L}_{\xi}\,\mathrm{d}{\rm vol}_{g}=0$ for all
Killing vector fields $\xi\in\mathfrak{k}$. By twisting we obtain the braided
cyclic pairing

$\displaystyle\langle\phi,\varPhi\rangle_{\star}:=\int_{M}\,\phi\star\varPhi\
\mathrm{d}{\rm vol}_{g}\ ,$ (5.10)

from which we may define the braided action functional following the
prescription of Section 4.6:

$\displaystyle\begin{split}S_{\star}(\phi):=\sum_{n\geq
1}\,\frac{(-1)^{n\choose
2}}{(n+1)!}\,\langle\phi,\ell_{n}^{\star}(\phi^{\otimes
n})\rangle_{\star}=\int_{M}\,\Big{(}\frac{1}{2}\,\phi\star\big{(}\square-m^{2}\big{)}\phi-\sum_{n\geq
3}\,\frac{\kappa_{n-1}}{n!}\,\phi^{\star n}\Big{)}\ \mathrm{d}{\rm vol}_{g}\
,\end{split}$ (5.11)

which, again, is the usual action functional for noncommutative scalar field
theory; note that, for fields of suitable asymptotic decay, the kinetic term
$\phi\star\square\,\phi\ \mathrm{d}{\rm vol}_{g}$ may be integrated by parts
to $\mathrm{d}\phi\wedge_{\star}\ast_{\textrm{\tiny H}}\mathrm{d}\phi$. Of
course, in order to derive the field equations as the critical locus of this
braided action functional in the usual sense, we have to restrict to
compatible Drinfel’d twists which make the pairing strictly cyclic, as
discussed in Section 4.6.

We emphasize that even standard noncommutative scalar field theory is
naturally encoded in a _braided_ $L_{\infty}$-algebra, even though no braided
gauge symmetries are present; in other words, it is also an example of a
braided field theory. This can be compared to the standard classical
$L_{\infty}$-algebra formulation of noncommutative scalar field theory: in
order to maintain strict symmetry of the brackets, as opposed to braided
symmetry, one needs to instead symmetrize star-products of fields to write the
higher brackets as

$\displaystyle\frac{1}{n!}\,\sum_{\sigma\in
S_{n}}\,\ell_{n}^{\star}(\phi_{\sigma(1)},\dots,\phi_{\sigma(n)})=-(-1)^{n\choose
2}\,\frac{\kappa_{n}}{n!}\,\sum_{\sigma\in
S_{n}}\,\phi_{\sigma(1)}\star\cdots\star\phi_{\sigma(n)}\ ,$ (5.12)

for $n\geq 2$, where the sum runs over all permutations of degree $n$. At the
classical level this results in the same noncommutative field theory, but at
the quantum level different interaction vertices arise. In particular, in the
braided case one loses the distinction between planar and non-planar Feynman
diagrams from dropping the symmetrization [77], raising the possibility that
no UV/IR mixing occurs in braided quantum field theory, which as a consequence
would have drastically improved renormalizability features over the standard
noncommutative quantum field theory [84]. The braided $L_{\infty}$-algebra
formalism also alters the implementation of global symmetries, as we discuss
in detail in Section 5.3 below.

###  5.3 Global braided symmetry

The global symmetries of the classical scalar field theory are the isometries
of the spacetime $(M,g)$, which are generated by the Killing Lie algebra
$\mathfrak{k}$ acting on fields via the Lie derivative. In the noncommutative
scalar field theory, only the subalgebra of $\mathfrak{k}$ that commutes with
the legs ${\sf f}^{k},{\sf f}_{k}\in\mathsf{U}\mathfrak{k}$ of the Killing
twist $\mathcal{F}$ generates global symmetries in this sense. However,
irrespective of the twist, the noncommutative scalar field theory is invariant
under the _braided_ Lie algebra of twisted Killing symmetries
$(\mathfrak{k}[[\hbar]],[-,-]_{\mathfrak{k}}^{\star})$ acting on fields via
the braided Lie derivative. This is a finite-dimensional “global” or “rigid”
braided Lie algebra of symmetries of the field theory. We shall now explain
how this is also naturally understood within the braided $L_{\infty}$-algebra
framework.

To incorporate these global symmetries, we extend the 2-term
$L_{\infty}$-algebra of Section 5.1 to a 4-term $L_{\infty}$-algebra by adding
in non-zero homogeneous subspaces of degrees $0$ and $3$, with corresponding
brackets for the action of the Lie algebra $\mathfrak{k}$.333333 Strictly
speaking, in view of the derived quotient interpretation of the kinematical
part $V^{0}\oplus V^{1}$ of the original $L_{\infty}$-algebra, we should
really be specifying an external action of the Lie algebra $\mathfrak{k}$ on
$V^{0}\oplus V^{1}\oplus V^{2}\oplus V^{3}$ as we are not interested in the
(derived) quotient by global symmetries. Thus the underlying graded vector
space is now $V=V^{0}\oplus V^{1}\oplus V^{2}\oplus V^{3}$ with

$\displaystyle V^{0}=\mathfrak{k}\ ,\quad
V^{1}=V^{2}=C^{\infty}(M,\mathbbm{R})\qquad\mbox{and}\qquad
V^{3}=\mathfrak{k}^{*}\ ,$ (5.13)

where $\mathfrak{k}^{*}$ is the linear dual of $\mathfrak{k}$. We denote
elements in degree 0 by $\xi\in\mathfrak{k}$ and in degree 3 by
$\xi^{\textrm{\tiny$\vee$}}\in\mathfrak{k}^{*}$. Note that these are not
fields, but rather should be regarded as constant maps on $M$.

Abstractly, the Lie algebra $\mathfrak{k}$ naturally acts on itself by the
adjoint action, $\xi_{1}\cdot\xi_{2}:=[\xi_{1},\xi_{2}]_{\mathfrak{k}}$ for
all $\xi_{1},\xi_{2}\in\mathfrak{k}$, and on its dual by the coadjoint action
$\xi\cdot\xi^{\textrm{\tiny$\vee$}}$ defined by

$\displaystyle\big{\langle}\xi^{\prime},\xi\cdot\xi^{\textrm{\tiny$\vee$}}\big{\rangle}=-\big{\langle}\xi\cdot\xi^{\prime},\xi^{\textrm{\tiny$\vee$}}\big{\rangle}=-\big{\langle}[\xi,\xi^{\prime}]_{\mathfrak{k}},\xi^{\textrm{\tiny$\vee$}}\big{\rangle}$
(5.14)

for all $\xi,\xi^{\prime}\in\mathfrak{k}$ and
$\xi^{\textrm{\tiny$\vee$}}\in\mathfrak{k}^{*}$, where
$\langle-,-\rangle:V^{0}\otimes V^{3}\to\mathbbm{R}$ is the canonical dual
pairing between $\mathfrak{k}$ and $\mathfrak{k}^{*}$. Concretely,
$\mathfrak{k}$ can be represented by vector fields on $M$ acting on fields via
the Lie derivative, with $[-,-]_{\mathfrak{k}}$ the restriction of the Lie
bracket of vector fields to the subalgebra $\mathfrak{k}$. These
representations obviously all live in the category of
$\mathsf{U}\mathfrak{k}$-modules, and hence are amenable to twisting by
$\mathcal{F}\in\mathsf{U}\mathfrak{k}[[\hbar]]\otimes\mathsf{U}\mathfrak{k}[[\hbar]]$
via the construction of Section 4.

The braided $L_{\infty}$-algebra $(V[[\hbar]],\\{\ell_{n}^{\star}\\})$ is now
extended by including, in addition to the brackets (5.8), the non-zero
$2$-brackets

$\displaystyle\begin{split}\ell_{2}^{\star}(\xi_{1},\xi_{2})=-[\xi_{1},\xi_{2}]_{\mathfrak{k}}^{\star}\
,\quad\ell_{2}^{\star}(\xi,\phi)=\mathrm{L}_{\xi}^{\star}\,\phi\
,\quad\ell_{2}^{\star}(\xi,\varPhi)=\mathrm{L}_{\xi}^{\star}\,\varPhi\qquad\mbox{and}\qquad\ell_{2}^{\star}(\xi,\xi^{\textrm{\tiny$\vee$}})=\xi\star\xi^{\textrm{\tiny$\vee$}}\
,\end{split}$ (5.15)

where $\xi\star\xi^{\textrm{\tiny$\vee$}}$ is the left braided coadjoint
action of $\xi\in\mathfrak{k}$ on $\xi^{\vee}\in\mathfrak{k}^{*}[[\hbar]]$.
That these extra brackets are compatible with the braided homotopy Jacobi
identities is a consequence of the fact that they are a twist deformation of
the classical brackets of [60], from which the statement follows trivially
again for degree reasons and since the $\mathfrak{k}$-action commutes with the
Laplace–Beltrami operator.343434Note that [60] only considers classical
$\phi^{4}$-theory on Minkowski space, but the generalization is immediate. The
cyclic structure $\langle-,-\rangle_{\star}$ is naturally extended to the new
homogeneous components by twisting the canonical dual pairing
$\langle-,-\rangle:V^{0}\otimes V^{3}\to\mathbbm{R}$ in the usual way.

These brackets encode the global braided transformation rules for the scalar
fields:

$\displaystyle\delta_{\xi}^{{\textrm{\tiny${\mathcal{R}}$}}}\phi=\ell_{2}^{\star}(\xi,\phi)=\mathrm{L}_{\xi}^{\star}\,\phi\
.$ (5.16)

These close the braided Lie algebra

$\displaystyle\big{[}\delta_{\xi_{1}}^{{\textrm{\tiny${\mathcal{R}}$}}},\delta_{\xi_{2}}^{{\textrm{\tiny${\mathcal{R}}$}}}\big{]}_{\circ}^{\star}=\delta^{{\textrm{\tiny${\mathcal{R}}$}}}_{-\ell_{2}^{\star}(\xi_{1},\xi_{2})}=\delta_{[\xi_{1},\xi_{2}]_{\mathfrak{k}}^{\star}}^{{\textrm{\tiny${\mathcal{R}}$}}}\
,$ (5.17)

which follows directly by the closure property of the braided Lie derivative
$\mathrm{L}_{\xi}^{\star}$. The braided field equations (5.9) are covariant,

$\displaystyle\delta_{\xi}^{{\textrm{\tiny${\mathcal{R}}$}}}\mathfrak{F}_{\phi}^{\star}=\ell_{2}^{\star}(\xi,\mathfrak{F}_{\phi}^{\star})=\mathrm{L}_{\xi}^{\star}\,\mathfrak{F}_{\phi}^{\star}\
,$ (5.18)

which follows directly from the braided derivation property of
$\mathrm{L}_{\xi}^{\star}$. Finally, the noncommutative action functional
(5.11) is invariant under these global braided transformations,
$\delta_{\xi}^{{\textrm{\tiny${\mathcal{R}}$}}}S_{\star}(\phi)=0$, which may
also be checked directly. In the classical limit, these are all just the
correct global symmetry transformation properties of classical scalar field
theory.

#### Abelian twists

Consider the special case where the Drinfel’d twist
$\mathcal{F}\in\mathsf{U}\mathfrak{t}[[\hbar]]\otimes\mathsf{U}\mathfrak{t}[[\hbar]]$
is constructed on the enveloping Hopf algebra of the Cartan subalgebra
$\mathfrak{t}\subseteq\mathfrak{k}$, i.e. from a commuting set of Killing
vector fields of the spacetime $(M,g)$. Then $\mathcal{F}$ is an abelian twist
(see Example 4.31). For vector fields $\xi\in\mathfrak{t}$, the braided Lie
derivative $\mathrm{L}_{\xi}^{\star}=\mathrm{L}_{\xi}$ coincides with the
ordinary Lie derivative and the Lie bracket $[-,-]_{\mathfrak{t}}^{\star}=0$
is also unchanged by the twist deformation (in fact, all brackets involving
elements from $\mathfrak{t}$ are unchanged). That is, the braided symmetry
transformations from the Cartan subalgebra $\mathfrak{t}$ act exactly as in
the classical theory, and we recover the anticipated statement that the
noncommutative scalar field theory is invariant under the classical action of
global symmetry transformations from $\mathfrak{t}$. However, this is not
generally true for vector fields $\xi\notin\mathfrak{t}$. The novelty of our
approach is that, when the Killing Lie algebra $\mathfrak{k}$ is nonabelian,
the noncommutative scalar field theory is nevertheless invariant under the
braided action of symmetry transformations from all of $\mathfrak{k}$. We
consider in detail the traditional example in Section 5.4 below.

###  5.4 Example: Braided Lorentz invariance

Let $M=\mathbbm{R}^{1,d-1}$ with the standard flat Minkowski metric $g=\eta$.
The Killing Lie algebra of $(\mathbbm{R}^{1,d-1},\eta)$ is the Poincaré
algebra

$\displaystyle\mathfrak{k}=\mathfrak{iso}(1,d-1)=\mathbbm{R}^{1,d-1}\rtimes\mathfrak{so}(1,d-1)\
.$ (5.19)

The abelian translation algebra $\mathbbm{R}^{1,d-1}$ is generated by
$\\{\mathsf{P}_{\mu}\\}_{\mu=0,1,\dots,d-1}$ and the nonabelian Lorentz
algebra $\mathfrak{so}(1,d-1)$ by
$\\{\mathsf{M}_{\mu\nu}\\}_{\mu,\nu=0,1,\dots,d-1}$, with
$\mathsf{M}_{\mu\nu}=-\mathsf{M}_{\nu\mu}$ and the semi-direct product Lie
brackets

$\displaystyle\begin{split}[\mathsf{P}_{\mu},\mathsf{P}_{\nu}]_{\mathfrak{k}}&=0\qquad,\qquad[\mathsf{M}_{\mu\nu},\mathsf{P}_{\lambda}]_{\mathfrak{k}}=\eta_{\mu\lambda}\,\mathsf{P}_{\nu}-\eta_{\nu\lambda}\,\mathsf{P}_{\mu}\
,\\\\[4.0pt]
[\mathsf{M}_{\mu\nu},\mathsf{M}_{\lambda\rho}]_{\mathfrak{k}}&=\eta_{\mu\lambda}\,\mathsf{M}_{\nu\rho}-\eta_{\mu\rho}\,\mathsf{M}_{\nu\lambda}-\eta_{\nu\lambda}\,\mathsf{M}_{\mu\rho}+\eta_{\nu\rho}\,\mathsf{M}_{\mu\lambda}\