Richard J. Szabo

# The $L_{\infty}$-structure of noncommutative gravity

###### Abstract

We summarise recent perspectives on symmetries of noncommutative field
theories based on homotopy algebras. We show how these viewpoints naturally
lead to a new class of noncommutative field theories which possess braided
gauge symmetries, and explain in detail their uses in gravity. We review how
these considerations lead to a new theory of noncommutative gravity in four
dimensions within the Einstein-Cartan-Palatini formalism.

##  1 Introduction

In this contribution we briefly review recent developments aimed at
understanding gauge symmetries of noncommutative field theories in the
framework of homotopy algebras, and in particular $L_{\infty}$-algebras. For a
detailed exposition of the subject and its applications to field theory, see
[1]. Here we shall summarise the applications to gravity, which were developed
in a series of papers [2, 3, 4]. In fact, these investigations were motivated
in part by a decade old problem inspired by string theory, that we now briefly
recall.

In certain non-geometric flux compactifications of string theory, it is
conjectured that the low-energy effective dynamics of closed strings are
described by noncommutative or even nonassociative deformations of gravity [5,
6, 7, 8, 9] (see [10, 11] for reviews with further references). From a
technical standpoint, such effective theories are difficult to construct,
because the metric aspects of nonassociative differential geometry are only
partially developed [12, 13, 14, 15, 16, 17]. In particular, no version of the
Einstein-Hilbert action is currently known in this framework. One way around
this difficulty is to regard nonassociative gravity as a deformation of ‘gauge
theory’, that is, to use the Einstein-Cartan principal bundle formulation,
where the corresponding action is the Palatini action [15, 18].

However, one immediately runs into problems with the naive notion of gauge
symmetry. Consider a noncommutative ${U}(N)$ gauge theory on $\mathbb{R}^{d}$
with a nonassociative star-product $\star$ for which the usual de Rham
differential ${\rm d}$ is a derivation, i.e. for which the deformed exterior
algebra $(\Omega^{\bullet}(\mathbb{R}^{d}),\wedge_{\star})$ forms a
noncommutative differential calculus with ${\rm d}$. A prominent example is
the nonassociative star-product that arises via cochain twist quantization of
the basic non-geometric $R$-flux deformations of spacetime [12, 14, 17]. For a
connection one-form $A\in\Omega^{1}(\mathbb{R}^{d},\mathfrak{u}(N))$ and a
gauge parameter $\lambda\in\Omega^{0}(\mathbb{R}^{d},\mathfrak{u}(N))$, the
naive infinitesimal star-gauge transformation would be

$\displaystyle\delta_{\lambda}^{\star}A\ =\ {\rm
d}\lambda-[\lambda\stackrel{{\scriptstyle\star}}{{,}}A]\ =\ {\rm
d}\lambda-(\lambda\star A-A\star\lambda)\ .$ (1.1)

The field strength tensor
$F_{A}^{\star}\in\Omega^{2}(\mathbb{R}^{d},\mathfrak{u}(N))$ is the
noncommutative curvature of the connection $A$ defined by

$\displaystyle F_{A}^{\star}\ =\ {\rm
d}A+\tfrac{1}{2}\,[A\stackrel{{\scriptstyle\star}}{{,}}A]\ =\ {\rm
d}A+A\wedge_{\star}A\ .$ (1.2)

Applying (1.1) to the definition (1.2), the transformation law of
$F_{A}^{\star}$ is given by

$\displaystyle\begin{split}\delta_{\lambda}^{\star}F_{A}^{\star}&\ =\
-\big{(}\lambda\star{\rm d}A+(\lambda\star
A)\wedge_{\star}A\big{)}+\big{(}{\rm
d}A\star\lambda+A\wedge_{\star}(A\star\lambda)\big{)}\\\
&\qquad+(A\star\lambda)\wedge_{\star}A-A\wedge_{\star}(\lambda\star A)\
.\end{split}$ (1.3)

In the associative case, the first line gives exactly
$-[\lambda\stackrel{{\scriptstyle\star}}{{,}}F_{A}^{\star}]$, while the second
line vanishes. However, for a general nonassociative star-product this is not
the case. Thus in general the infinitesimal gauge transformation of
$F_{A}^{\star}$ does not have the exprected covariant form:

$\displaystyle\delta_{\lambda}^{\star}F_{A}^{\star}\ \neq\
-[\lambda\stackrel{{\scriptstyle\star}}{{,}}F_{A}^{\star}]\ .$ (1.4)

A related problem is that nonassociativity generally obstructs the closure of
the infinitesimal gauge transformations (1.1):

$\displaystyle(\delta^{\star}_{\lambda}\,\delta^{\star}_{\rho}-\delta^{\star}_{\rho}\,\delta^{\star}_{\lambda})A\
\neq\ \delta^{\star}_{[\lambda\stackrel{{\scriptstyle\star}}{{,}}\rho]}A\ ,$
(1.5)

and indeed the star-commutator $[-\stackrel{{\scriptstyle\star}}{{,}}-]$ is
not generally a Lie bracket on $\Omega^{0}(\mathbb{R}^{d},\mathfrak{u}(N))$ as
it violates the Jacobi identity when the star-product $\star$ is
nonassociative.

A natural setting in which to overcome these problems is the framework of
$L_{\infty}$-algebras. In the physics literature these homotopy algebraic
structures first appeared as the natural objects which organise the
“generalized” gauge symmetries of closed string field theory via an infinite
sequence of higher brackets [19]. Subsequently it was clarified that any
classical perturbative field theory with “generalized” gauge symmetries is
organised by an $L_{\infty}$-algebra with finitely-many brackets, due to the
duality with the BV–BRST formalism [20, 21]. We briefly review these ideas in
Section 2.

The $L_{\infty}$-structure of general noncommutative and nonassociative gauge
theories for star-products which quantize generic almost Poisson structures
was partially developed in [22, 23, 24, 25] as a means of building the correct
notions of gauge symmetry that overcome the problems discussed above, as well
as those related to the failure of the de Rham differential ${\rm d}$ to
generally obey the Leibniz rule. These constructions also lead to
$L_{\infty}$-algebras with infinitely-many brackets in general. The drawback
of these treatments is that so far they are only understood at the semi-
classical level, and hence do not capture the complete structure underlying
noncommutative gauge symmetries. Instead, one can reformulate some of these
theories via deformations with braided gauge symmetries [4, 1], which involve
the same finite number of brackets as their classical counterparts and are
applicable at the full quantized level. This formalism is reviewed in Section
3.

The $L_{\infty}$-algebras of Einstein-Hilbert gravity are discussed in [20,
26, 27]. They require perturbing the theory around a flat background and also
involve infinitely-many brackets. The $L_{\infty}$-structure of the Einstein-
Cartan-Palatini (ECP) theory was developed in [3] and it only requires
finitely-many brackets. We review the $L_{\infty}$-algebra formulation of ECP
gravity in Section 4.

One of the driving motivations behind our development of braided gauge
symmetries was the observation that the twisted diffeomorphism symmetry of the
usual formulations of noncommutative gravity does not fit naturally into the
standard $L_{\infty}$-algebra picture (if it does at all). We elaborate on
this point in Section 5, which further motivates a deformation of the
underlying $L_{\infty}$-structure itself in order to make it compatible with
the twisted diffeomorphisms. We then review the braided $L_{\infty}$-algebra
which organises noncommutative gravity in the example of the four-dimensional
ECP theory.

##  2 $L_{\infty}$-algebras of classical field theories

In this section we give a brief introduction to $L_{\infty}$-algebras along
with their uses in organising the gauge symmetries and dynamics of classical
field theories. From a mathematical perspective, their applicability in these
contexts comes from their duality with differential graded commutative
algebras [28, 29]. Our presentation will be mostly informal, in order to
highlight the main ideas without too much technical clutter; see [20, 21, 30,
1] for more elaborate reviews.

###  ${\bm{L_{\infty}}}$-algebras.

Let $V\ =\ \oplus_{k\in{\mathbb{Z}}}\,V_{k}$ be a ${\mathbb{Z}}$-graded vector
space, with degree shifted graded exterior algebra $\Lambda_{V}\ =\
\wedge^{\bullet}(V[1])$ viewed as a free cocommutative coalgebra. An
_$L_{\infty}$ -structure_ on $V$ is a coderivation
$L:\Lambda_{V}\longrightarrow\Lambda_{V}$ of degree $|L|\ =\ 1$ satisfying
$L^{2}\ =\ 0$. We can decompose the coderivation $L$ into ‘components’
$\\{\ell_{n}\\}_{n\geq 1}$ where $\ell_{n}:\wedge^{n}(V[1])\longrightarrow
V[1]$ are multilinear maps of degree $|\ell_{n}|\ =\ 1$, or restoring the
original grading, maps $\ell_{n}:\wedge^{n}V\longrightarrow V$ with degree
$|\ell_{n}|\ =\ 2-n$. Then the differential condition $L^{2}\ =\ 0$ leads to
an infinite sequence of _homotopy relations_ , the first three of which read,
together with their meaning, as

$\displaystyle\begin{split}\ell_{1}(\ell_{1}(v))&\ =\ 0\hskip
130.88284pt{(V,\ell_{1})\mbox{\sf\ is a cochain complex }}\\\\[4.0pt]
\ell_{1}(\ell_{2}(v,w))&\ =\
\ell_{2}(\ell_{1}(v),w)\pm\ell_{2}(v,\ell_{1}(w))\hskip
28.45274pt{\ell_{1}\mbox{\sf\ is a derivation of }\ell_{2}}\\\\[4.0pt]
\ell_{2}(v,\ell_{2}(w,u))+{\rm cyclic}&\ =\
(\ell_{1}\circ\ell_{3}\pm\ell_{3}\circ\ell_{1})(v,w,u)\hskip
28.45274pt{\mbox{\sf Jacobi identity up to homotopy}}\end{split}$ (2.1)

where the signs depend on the degrees of the elements $v,w,u\in V$. This list
continues to include “higher homotopy Jacobi identities” involving higher
brackets. Thus $L_{\infty}$-algebras are generalizations of differential
graded Lie algebras, which are the special cases where $\ell_{n}\ =\ 0$ for
all $n\geq 3$.

Dualizing this coalgebra picture gives a graded commutative algebra derivation
$Q\ =\ L^{*}:\Lambda_{V}^{*}\longrightarrow\Lambda_{V}^{*}$ of degree $|Q|\ =\
1$ satisfying $Q^{2}\ =\ 0$. Thus $L_{\infty}$-algebras are dual to
differential graded commutative algebras.

### Quasi-isomorphisms.

An _$L_{\infty}$ -morphism_ from an $L_{\infty}$-algebra $(V,L)$ to an
$L_{\infty}$-algebra $(V^{\prime},L^{\prime})$ is a degree-preserving
coalgebra homomorphism $\Psi:\Lambda_{V}\longrightarrow\Lambda_{V^{\prime}}$
which intertwines the codifferentials: $\Psi\circ L\ =\ L^{\prime}\circ\Psi$.
In the ‘components’ $\\{\psi_{n}\\}_{n\geq 1}$ of $\Psi$, where
$\psi_{n}:\wedge^{n}V\longrightarrow V^{\prime}$ are maps of degree
$|\psi_{n}|\ =\ 1-n$, this condition gives

$\displaystyle\begin{split}&\hskip 31.2982pt{\psi_{1}\mbox{\sf\ is a cochain
map: }}\ \psi_{1}\,\ell_{1}\ =\ \ell_{1}^{\prime}\,\psi_{1}\\\\[4.0pt]
&\psi_{1}(\ell_{2}(v,w))-\ell_{2}^{\prime}(\psi_{1}(v),\psi_{1}(w))\
{\mbox{\sf\ is a homotopy in }\psi_{2}}\end{split}$ (2.2)

plus cumbersome higher relations. Thus $L_{\infty}$-morphisms generalize
homomorphisms of differential graded Lie algebras.

An $L_{\infty}$-morphism is an $L_{\infty}$-quasi-isomorphism if the induced
map on cohomology $\psi_{1*}:H^{\bullet}(V,\ell_{1})\longrightarrow
H^{\bullet}(V^{\prime},\ell_{1}^{\prime})$ is an isomorphism. Quasi-
isomorphism is an equivalence relation on $L_{\infty}$-algebras, contrary to
quasi-isomorphism on the smaller category of differential graded Lie algebras.

### Cyclic ${\bm{L_{\infty}}}$-algebras.

A _cyclic structure_ on an $L_{\infty}$-algebra $(V,L)$ is a graded inner
product $\langle-,-\rangle:V\otimes V\longrightarrow\mathbb{R}$ satisfying

$\displaystyle\langle v_{0},\ell_{n}(v_{1},v_{2},\dots,v_{n})\rangle\ =\
\pm\,\langle v_{n},\ell_{n}(v_{0},v_{1},\dots,v_{n-1})\rangle$ (2.3)

for all $n\geq 1$ and $v_{i}\in V$. Thus cyclic $L_{\infty}$-algebras
generalize quadratic differential graded Lie algebras. In the dual
differential graded commutative algebra picture $(V^{*},Q)$, a cyclic
structure translates to a graded symplectic two-form
$\omega\in\Omega^{2}(V[1])$ which is $Q$-invariant.

An $L_{\infty}$-morphism $\Psi:\Lambda_{V}\longrightarrow\Lambda_{V^{\prime}}$
between two cyclic $L_{\infty}$-algebras $(V,L,\langle-,-\rangle)$ and
$(V^{\prime},L^{\prime},\langle-,-\rangle^{\prime})$ is _cyclic_ if it
preserves the inner products:

$\displaystyle\begin{split}\langle\psi_{1}(v),\psi_{1}(w)\rangle^{\prime}&\ =\
\langle v,w\rangle\ ,\\\\[4.0pt]
\sum_{i=1}^{n-1}\,\langle\psi_{i}(v_{1},\dots,v_{i}),\psi_{n-i}(v_{i+1},\dots,v_{n})\rangle^{\prime}&\
=\ 0\quad,\quad n>2\ .\end{split}$ (2.4)

###  ${\bm{L_{\infty}}}$-structure of classical field theory.

Consider a classical field theory with generalized gauge symmetries defined on
a manifold $M$. The Batalin-Vilkovisky (BV) formalism constructs a
differential graded algebra $(C_{\bullet}^{\infty}(V[1]),Q_{\textrm{\tiny
BV}})$ on the graded vector space $V$ of BV fields consisting of ghosts,
physical fields and antifields. The action of the BV differential
$Q_{\textrm{\tiny BV}}$ on $V[1]$ is a polynomial in the ghosts, fields and
antifields and their derivatives, which is dual to a sum over brackets
$\ell_{n}$ on $V$. The BV symplectic form $\omega_{\textrm{\tiny BV}}$, which
is dual to the antibracket, is of degree $-1$ on $V$ and it induces a cyclic
inner product of degree $-3$.

The homogeneous subspace $V_{0}$ contains the gauge parameters, $V_{1}$ the
fields, $V_{2}$ the field equations, and $V_{3}$ the corresponding Noether
identities. The negatively graded homogeneous subspaces $V_{-k}$ for $k\geq 1$
encode ‘higher gauge transformations’ (corresponding to ghosts-for-ghosts, and
so on) for reducible symmetries, and dually $V_{k+3}$ contain the
corresponding ‘higher Noether identities’. The brackets $\ell_{n}$ of the
$L_{\infty}$-structure on $V$ then encode all information about the gauge
symmetries and dynamics of the classical field theory in the following way.

Gauge transformations of fields $A\in V_{1}$ by gauge parameters $\lambda\in
V_{0}$ are given by

$\displaystyle\delta_{\lambda}A\ =\
\ell_{1}(\lambda)+\ell_{2}(\lambda,A)-\tfrac{1}{2}\,\ell_{3}(\lambda,A,A)+\cdots\
,$ (2.5)

where the ellipses denote higher order brackets involving higher orders of the
fields $A$. If the gauge symmetries close off-shell, then the closure of gauge
algebra is encoded by the field-dependent gauge transformations

$\displaystyle[\delta_{\lambda_{1}},\delta_{\lambda_{2}}]A\ =\
\delta_{-C(\lambda_{1},\lambda_{2};A)}A\ ,$ (2.6)

where

$\displaystyle C(\lambda_{1},\lambda_{2};A)\ =\
\ell_{2}(\lambda_{1},\lambda_{2})+\ell_{3}(\lambda_{1},\lambda_{2},A)+\cdots\
.$ (2.7)

This follows from the homotopy relations of the $L_{\infty}$-algebra.

The field equations $\mathcal{F}_{A}\ =\ 0$ are given by

$\displaystyle\mathcal{F}_{A}\ =\
\ell_{1}(A)-\tfrac{1}{2}\,\ell_{2}(A,A)-\tfrac{1}{3!}\,\ell_{3}(A,A,A)+\cdots\
.$ (2.8)

From the homotopy relations it follows that they are covariant:

$\displaystyle\delta_{\lambda}\mathcal{F}_{A}\ =\
\ell_{2}(\lambda,\mathcal{F}_{A})+\ell_{3}(\lambda,\mathcal{F}_{A},A)+\cdots\
.$ (2.9)

The moduli space of classical physical states is the quotient of the space of
solutions to the field equations by the gauge transformations. Quasi-
isomorphic $L_{\infty}$-algebras lead to isomorphic moduli spaces, and so give
physically equivalent field theories in this sense. The corresponding Noether
identities are

$\displaystyle\mathcal{I}_{\lambda}\ :=\
\ell_{1}(\mathcal{F}_{A})+\ell_{2}(\mathcal{F}_{A},A)-\tfrac{1}{2}\,\ell_{3}(\mathcal{F}_{A},A,A)+\cdots\
\equiv\ 0\ ,$ (2.10)

which are identically zero _off-shell_ as a consequence of the homotopy
relations.

The cyclic structure of the $L_{\infty}$-algebra induces an action

$\displaystyle S\ =\ \tfrac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\tfrac{1}{3!}\,\langle
A,\ell_{2}(A,A)\rangle-\tfrac{1}{4!}\,\langle A,\ell_{3}(A,A,A)\rangle+\cdots\
.$ (2.11)

Cyclicity implies that the Euler-Lagrange equations of this action are
precisely $\mathcal{F}_{A}\ =\ 0$, and that gauge invariance of $S$ is
equivalent to the Noether identities $\mathcal{I}_{\lambda}\ =\ 0$.

### Example: Chern-Simons theory.

The prototypical example is provided by Chern-Simons gauge theory. Let $M$ be
a three-dimensional oriented manifold, and let $\mathfrak{g}$ be a quadratic
Lie algebra invariant pairing $\>{\rm Tr}_{\mathfrak{g}}$. The underlying
cochain complex of the $L_{\infty}$-structure is de Rham complex of
$\mathfrak{g}$-valued differential forms $V\ =\
\Omega^{\bullet}(M,\mathfrak{g})$ with differential $\ell_{1}\ =\ {\rm d}$.
The only other non-zero brackets are given by $\ell_{2}\ =\
-[-,-]_{\mathfrak{g}}$, the Lie bracket of $\mathfrak{g}$ composed with
exterior multiplication of differential forms. The cyclic structure of degree
$-3$ is defined by the inner product

$\displaystyle\displaystyle{\langle\alpha,\beta\rangle\ =\ \int_{M}\>{\rm
Tr}_{\mathfrak{g}}(\alpha\wedge\beta)}\ ,$ (2.12)

for forms $\alpha,\beta\in\Omega^{\bullet}(M,\mathfrak{g})$ in complementary
degrees.

The gauge transformation of a field $A\in V_{1}\ =\
\Omega^{1}(M,\mathfrak{g})$ by a gauge parameter $\lambda\in V_{0}\ =\
\Omega^{0}(M,\mathfrak{g})$ is then the standard one:

$\displaystyle\delta_{\lambda}A\ =\ {\rm d}\lambda-[\lambda,A]_{\mathfrak{g}}\
,$ (2.13)

which closes in the Lie algebra
$(\Omega^{0}(M,\mathfrak{g}),[-,-]_{\mathfrak{g}})$. The field equations
$\mathcal{F}_{A}\ =\ 0$ are given by

$\displaystyle\mathcal{F}_{A}\ =\ F_{A}\ :=\ {\rm
d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}\qquad\mbox{with}\quad\delta_{\lambda}F_{A}\
=\ -[\lambda,F_{A}]_{\mathfrak{g}}\ ,$ (2.14)

and the corresponding moduli space of classical physical states is the moduli
space of flat connections on $M$. The Noether identity is simply the Bianchi
identity

$\displaystyle\mathcal{I}_{\lambda}\ =\ {\rm d}F_{A}-[F_{A},A]_{\mathfrak{g}}\
=\ 0\ .$ (2.15)

The action reproduces the Chern-Simons functional

$\displaystyle\displaystyle{S\ =\ \int_{M}\>{\rm
Tr}_{\mathfrak{g}}\Big{(}\frac{1}{2}\,A\wedge{\rm
d}A+\frac{1}{3!}\,A\wedge[A,A]_{\mathfrak{g}}\Big{)}}\ .$ (2.16)

Thus Chern-Simons gauge theory is organised by a differential graded Lie
algebra.

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

The standard noncommutative field theories with star-gauge symmetry have
$L_{\infty}$-structures with finitely-many brackets which simply follow from
the analogue structures of classical non-abelian gauge theories [1]. In this
section we will instead define noncommutative field theories with braided
gauge symmetries. We will discuss how to formulate them through a general
notion of braided ${L_{\infty}}$-algebras, and illustrate how to use this to
systematically construct new examples of noncommutative gauge theories.

### Braided noncommutative deformation.

Let $U\Gamma(TM)$ be the universal enveloping algebra of the Lie algebra
$\Gamma(TM)$ of vector fields on a manifold $M$, endowed with its canonical
cocommutative Hopf algebra structure. Let $\mathscr{F}\ =\ {\rm
f}^{\alpha}\otimes{\rm f}_{\alpha}\in U\Gamma(TM)\otimes U\Gamma(TM)$ be a
_Drinfel’d twist_ , with inverse $\mathscr{F}^{-1}\ =\ \bar{\rm
f}^{\alpha}\otimes\bar{\rm f}_{\alpha}$ and with $\mathscr{F}_{21}\ =\ {\rm
f}_{\alpha}\otimes{\rm f}^{\alpha}$ the twist with its legs in $U\Gamma(TM)$
swapped. We do not review the theory of Drinfel’d twists here, and for
concreteness the reader may wish to simply consider the standard Moyal-Weyl
twist

$\displaystyle\mathscr{F}\ =\ \exp\big{(}-\tfrac{{\,{\rm
i}\,}}{2}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}\ ,$
(3.1)

which provides the simplest example of the general formalism (see e.g. [1] for
a more detailed account with many further examples).

Physical fields are described by functions, forms and tensors on $M$, and so
are elements of an algebra ${\mathcal{A}}$ on which $\Gamma(TM)$ acts via the
Lie derivative and the Leibniz rule; that is, ${\mathcal{A}}$ is a
$U\Gamma(TM)$-module algebra. Using the twist we deform the product
$\,\cdot\,$ on ${\mathcal{A}}$ into an associative star-product

$\displaystyle a\star b\ =\ \,\bm{\cdot}\ \mathscr{F}^{-1}(a\otimes b)\ =\
\bar{\rm f}^{\alpha}(a)\cdot\bar{\rm f}_{\alpha}(b)\ .$ (3.2)

This defines a noncommutative algebra ${\mathcal{A}}_{\star}$ carrying a
representation of a twisted Hopf algebra $U_{\mathscr{F}}\Gamma(TM)$. If
${\mathcal{A}}$ is a commutative algebra, then ${\mathcal{A}}_{\star}$ is
_braided_ -commutative:

$\displaystyle a\star b\ =\ \bar{\rm R}^{\alpha}(b)\star\bar{\rm
R}_{\alpha}(a)\ ,$ (3.3)

where $\mathscr{R}\ =\ \mathscr{F}_{21}\,\mathscr{F}^{-1}\ =\ {\rm
R}^{\alpha}\otimes{\rm R}_{\alpha}$ is the triangular $R$-matrix with inverse
$\mathscr{R}^{-1}\ =\ \bar{\rm R}^{\alpha}\otimes\bar{\rm R}_{\alpha}$.

### Braided gauge symmetry.

Let $\mathfrak{g}$ be a Lie algebra with bracket $[-,-]_{\mathfrak{g}}$, and
apply the twist deformation procedure to the Lie algebra
$(\Omega^{0}(M,\mathfrak{g}),[-,-]_{\mathfrak{g}})$ to get the new bracket

$\displaystyle[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}\ :=\
[-,-]_{\mathfrak{g}}\circ\mathscr{F}^{-1}(\lambda_{1}\otimes\lambda_{2})\ =\
[\bar{\rm f}^{\alpha}(\lambda_{1}),\bar{\rm
f}_{\alpha}(\lambda_{2})]_{\mathfrak{g}}\ .$ (3.4)

The resulting structure $\Omega_{\star}^{0}(M,\mathfrak{g})$ is no longer a
Lie algebra but rather a _braided Lie algebra_ [31, 32]; that is, it is
braided antisymmetric and obeys the braided Jacobi identity:

$\displaystyle\begin{split}[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}&\
=\ -[\bar{\rm R}^{\alpha}(\lambda_{2}),\bar{\rm
R}_{\alpha}(\lambda_{1})]_{\star}\ ,\\\\[4.0pt]
[\lambda_{1},[\lambda_{2},\lambda_{3}]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}&\
=\
[[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star},\lambda_{3}]_{\mathfrak{g}}^{\star}+[\bar{\rm
R}^{\alpha}(\lambda_{2}),[\bar{\rm
R}_{\alpha}(\lambda_{1}),\lambda_{3}]_{\mathfrak{g}}^{\star}]_{\mathfrak{g}}^{\star}\
.\end{split}$ (3.5)

Note that for a matrix Lie algebra $\mathfrak{g}$, the braided commutator is
_not_ the same as the more conventional star-commutator:

$\displaystyle[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}\ =\
\lambda_{1}\star\lambda_{2}-\bar{\rm R}^{\alpha}(\lambda_{2})\star\bar{\rm
R}_{\alpha}(\lambda_{1})\ \neq\
\lambda_{1}\star\lambda_{2}-\lambda_{2}\star\lambda_{1}\ =\
[\lambda_{1}\\!\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\\!\lambda_{2}]_{\mathfrak{g}}\
.$ (3.6)

Let $W$ be a linear representation of the Lie algebra $\mathfrak{g}$. Braided
gauge fields $A\in\Omega_{\star}^{1}(M,\mathfrak{g})$ and braided matter
fields $\phi\in\Omega_{\star}^{p}(M,W)$ transform in (left) braided
representations

$\displaystyle\delta_{\lambda}^{\star}\phi\ =\
-\lambda\star\phi\qquad\mbox{and}\qquad\delta_{\lambda}^{\star}A\ =\ {\rm
d}\lambda-[\lambda,A]_{\mathfrak{g}}^{\star}\ .$ (3.7)

These braided gauge transformations are not derivations but rather follow a
_braided Leibniz rule_ , for instance

$\displaystyle\delta_{\lambda}^{\star}(\phi\otimes A)\ =\
\delta_{\lambda}^{\star}\phi\otimes A+\bar{\rm
R}^{\alpha}(\phi)\otimes\delta^{\star}_{\bar{\rm R}_{\alpha}(\lambda)}A\ ,$
(3.8)

and they close a braided Lie algebra defined by the braided commutator

$\displaystyle\big{[}\delta_{\lambda_{1}}^{\star},\delta_{\lambda_{2}}^{\star}\big{]}^{\star}\
:=\
\delta_{\lambda_{1}}^{\star}\,\delta_{\lambda_{2}}^{\star}-\delta_{\bar{\rm
R}^{\alpha}(\lambda_{2})}^{\star}\,\delta^{\star}_{\bar{\rm
R}_{\alpha}(\lambda_{1})}\ =\
\delta^{\star}_{[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}^{\star}}\ .$ (3.9)

Braided gauge invariant dynamics is built from the braided left and right
covariant derivatives

$\displaystyle{\rm d}_{\star{\textrm{\tiny L}}}^{A}\phi\ :=\ {\rm
d}\phi+A\wedge_{\star}\phi\qquad\mbox{and}\qquad{\rm d}_{\star{\textrm{\tiny
R}}}^{A}\phi\ :=\ {\rm d}\phi+\bar{\rm R}^{\alpha}(A)\wedge_{\star}\bar{\rm
R}_{\alpha}(\phi)\ ,$ (3.10)

which are braided covariant:

$\displaystyle\delta_{\lambda}^{\star}\big{(}{\rm d}_{\star{\textrm{\tiny
L}},{\textrm{\tiny R}}}^{A}\phi\big{)}\ =\ -\lambda\star\big{(}{\rm
d}_{\star{\textrm{\tiny L}},{\textrm{\tiny R}}}^{A}\phi\big{)}\ .$ (3.11)

In particular, the braided curvature of a gauge field is

$\displaystyle F_{A}^{\star}\ :=\ {\rm
d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}^{\star}\qquad\mbox{with}\quad\delta_{\lambda}^{\star}F_{A}^{\star}\
=\ -[\lambda,F_{A}^{\star}]_{\mathfrak{g}}^{\star}\ .$ (3.12)

In our applications to gravity later on, we are particularly interested in the
deformation of the Lie algebra of vector fields $(\Gamma(TM),[-,-])$, which
defines the braided Lie algebra of braided diffeomorphisms
$\Gamma_{\star}(TM)$. We construct a noncommutative differential geometry on
$M$ by requiring it to be covariant with respect to the twisted Hopf algebra
$U_{\mathscr{F}}\Gamma(TM)$ [33, 34, 17]. The action of a twisted
diffeomorphism, generated by a vector field $\xi\in\Gamma(TM)$, on a tensor
field ${\bm{T}}$ is given by the braided Lie derivative

$\displaystyle\mathcal{L}_{\xi}^{\star}{\bm{T}}\ :=\ \mathcal{L}_{\bar{\rm
f}^{\alpha}(\xi)}\bar{\rm
f}_{\alpha}{\bm{(}}{\bm{T}})\qquad\mbox{with}\quad\big{[}\mathcal{L}_{\xi_{1}}^{\star},\mathcal{L}_{\xi_{2}}^{\star}\big{]}^{\star}\
=\ \mathcal{L}_{[\xi_{1},\xi_{2}]^{\star}}^{\star}\ .$ (3.13)

### Braided ${\bm{L_{\infty}}}$-structure of braided field theory.

Let $(V,\\{\ell_{n}\\})$ be a classical $L_{\infty}$-algebra in the category
of $U\Gamma(TM)$-modules; this means that $U\Gamma(TM)$ acts on the graded
vector space $V$ and all brackets $\ell_{n}$ are $U\Gamma(TM)$-equivariant
maps. Via the twist deformation procedure we obtain new brackets
$\ell_{n}^{\star}$ defined by

$\displaystyle\ell_{n}^{\star}(v_{1}\wedge\cdots\wedge v_{n})\ :=\
\ell_{n}(v_{1}\wedge_{\star}\cdots\wedge_{\star}v_{n})\ .$ (3.14)

Note that the differential $\ell_{1}^{\star}\ =\ \ell_{1}$ is unchanged, and
hence so is the underlying cochain complex $(V,\ell_{1})$; this will translate
to the statement that the free classical field theory in unchanged by the
noncommutative deformation. The maps $\ell_{n}^{\star}$ for $n\geq 2$ are
_braided graded antisymmetric:_

$\displaystyle\ell_{n}^{\star}(\dots,v,v^{\prime},\dots)\ =\
-(-1)^{|v|\,|v^{\prime}|}\ \ell_{n}^{\star}(\dots,\bar{\rm
R}^{\alpha}(v^{\prime}),\bar{\rm R}_{\alpha}(v),\dots)\ ,$ (3.15)

and they satisfy _braided homotopy Jacobi identities_ , which are unchanged
for $n\ =\ 1,2$. In [3, 4, 1] the resulting algebraic structure
$(V,\\{\ell_{n}^{\star}\\})$ is called a _braided $L_{\infty}$-algebra_ in the
category of $U_{\mathscr{F}}\Gamma(TM)$-modules.

The braided $L_{\infty}$-structure can be used to build noncommutative field
theories with braided symmetries in the same way as in the classical case. The
braided gauge transformations

$\displaystyle\delta_{\lambda}^{\star}A\ =\
\ell^{\star}_{1}(\lambda)+\ell_{2}^{\star}(\lambda,A)-\tfrac{1}{2}\,\ell_{3}^{\star}(\lambda,A,A)+\cdots$
(3.16)

close a braided Lie algebra under the braided commutator ${[-,-]^{\star}}$.
The braided field equations

$\displaystyle\mathcal{F}_{A}^{\star}\ =\
\ell_{1}^{\star}(A)-\tfrac{1}{2}\,\ell_{2}^{\star}(A,A)-\tfrac{1}{3!}\,\ell_{3}^{\star}(A,A,A)+\cdots$
(3.17)

are covariant:

$\displaystyle\delta_{\lambda}^{\star}\mathcal{F}_{A}^{\star}\ =\
\ell_{2}^{\star}(\lambda,\mathcal{F}_{A}^{\star})+\tfrac{1}{2}\,\big{(}\ell_{3}^{\star}(\lambda,\mathcal{F}_{A}^{\star},A)-\ell_{3}^{\star}(\lambda,A,\mathcal{F}_{A}^{\star})\big{)}+\cdots\
.$ (3.18)

Although there is no conventional moduli space of solutions to
$\mathcal{F}_{A}^{\star}\ =\ 0$ [4, 1], a set of _braided Noether identities_
continue to hold from a weighted sum over all braided homotopy identities
evaluated on $(A^{n})$:

$\displaystyle\begin{split}{\mathcal{I}_{A}^{\star}\mathcal{F}_{A}^{\star}}&\
=\
{\ell_{1}^{\star}(\mathcal{F}_{A}^{\star})+\tfrac{1}{2}\,\big{(}\ell_{2}^{\star}(\mathcal{F}_{A}^{\star},A)-\ell_{2}^{\star}(A,\mathcal{F}_{A}^{\star})\big{)}}\\\
&\qquad{+\tfrac{1}{3!}\,\ell_{1}^{\star}\big{(}\ell_{3}^{\star}(A,A,A)\big{)}+\tfrac{1}{4}\,\big{(}\ell_{2}^{\star}(\ell_{2}^{\star}(A,A),A)-\ell_{2}^{\star}(A,\ell_{2}^{\star}(A,A))\big{)}+\cdots\
\equiv\ 0}\ .\end{split}$ (3.19)

If the classical $L_{\infty}$-algebra $(V,\\{\ell_{n}\\})$ is further equipped
with a $U\Gamma(TM)$-invariant cyclic inner product
$\langle-,-\rangle:V\otimes V\longrightarrow\mathbb{R}$ of degree $-3$, then

$\displaystyle\langle-,-\rangle_{\star}\ :=\
\langle-,-\rangle\circ\mathscr{F}^{-1}\ =\ \langle\bar{\rm
f}^{\alpha}(-),\bar{\rm f}_{\alpha}(-)\rangle$ (3.20)

is a cyclic inner product on the braided $L_{\infty}$-algebra
$(V,\\{\ell^{\star}_{n}\\})$ of degree $-3$. With it we construct the action

$\displaystyle S^{\star}\ =\ \tfrac{1}{2}\,\langle
A,\ell_{1}^{\star}(A)\rangle_{\star}-\tfrac{1}{3!}\,\langle
A,\ell_{2}^{\star}(A,A)\rangle_{\star}-\tfrac{1}{4!}\,\langle
A,\ell_{3}^{\star}(A,A,A)\rangle_{\star}+\cdots\ .$ (3.21)

As in the classical case, cyclicity implies that $\delta S^{\star}\ =\
\langle\delta A,\mathcal{F}_{A}^{\star}\rangle_{\star}$ for arbitrary field
variations $\delta A$, and that $\delta_{\lambda}^{\star}S^{\star}\ =\
-\langle\lambda,\mathcal{I}_{A}^{\star}\mathcal{F}_{A}^{\star}\rangle_{\star}$
for arbitrary braided gauge transformations. Hence the variational principle
for the action $S^{\star}$ leads to the braided field equations
$\mathcal{F}_{A}^{\star}\ =\ 0$, while braided gauge invariance of $S^{\star}$
is equivalent to the braided Noether identities
$\mathcal{I}_{A}^{\star}\mathcal{F}_{A}^{\star}\ =\ 0$. For the relation of
this construction to a braided version of the BV formalism, see [35, 1].

The formulation provided by braided $L_{\infty}$-algebras yields systematic
constructions of _new_ noncommutative field theories with no new degrees of
freedom and good classical limits, unlike other approaches to noncommutative
field theories with arbitrary gauge algebras and matter field representations
(see e.g. [1] for a recent review), as well as some “surprises”. Let us
briefly mention some standard field theory examples; for further details and
examples, see [1]. Aspects of the corresponding braided quantum field theories
are currently under investigation.

### Example I: Braided Chern-Simons theory.

The prototypical example is braided Chern-Simons theory on a three-dimensional
oriented manifold $M$ with quadratic Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}},\>{\rm Tr}_{\mathfrak{g}})$ which is built
on a differential graded braided Lie algebra with homogeneous subspaces
$V_{p}\ =\ \Omega^{p}(M,\mathfrak{g})$, $p=0,1,2,3$ and the non-vanishing
brackets

$\displaystyle\ell_{1}^{\star}\ =\ \ell_{1}\ =\ {\rm
d}\qquad\mbox{and}\qquad\ell_{2}^{\star}\ =\ -[-,-]_{\mathfrak{g}}^{\star}\ .$
(3.22)

The action

$\displaystyle S^{\star}\ =\ \int_{M}\>{\rm
Tr}_{\mathfrak{g}}\Big{(}\frac{1}{2}\,A\wedge_{\star}{\rm
d}A+\frac{1}{3!}\,A\wedge_{\star}[A,A]^{\star}_{\mathfrak{g}}\Big{)}$ (3.23)

is invariant under braided gauge transformations closing the braided Lie
algebra $\Omega^{0}_{\star}(M,\mathfrak{g})$. The field equations are
$F_{A}^{\star}\ =\ 0$ and they are braided covariant. The Bianchi identities
are modified, see [3, 1].

### Example II: Noncommutative scalar field theory.

The $L_{\infty}$-algebra formalism is also equipped to handle theories without
gauge symmetries [21, 1]. Consider, for example, massive scalar
$\kappa\,\phi^{3}$-theory on $d$-dimensional Minkowski space
$\mathbb{R}^{1,d-1}$. In this case $V_{0}\ =\ V_{3}\ =\ 0$ since there are no
gauge symmetries, while $V_{1}\ =\ V_{2}\ =\ C^{\infty}(\mathbb{R}^{1,d-1})$
with the non-vanishing brackets

$\displaystyle\ell_{1}^{\star}\ =\ \ell_{1}\ =\
\square-m^{2}\qquad\mbox{and}\qquad\ell^{\star}_{2}(\phi_{1},\phi_{2})\ =\
\kappa\,\phi_{1}\star\phi_{2}$ (3.24)

for $\phi_{1},\phi_{2}\in V_{1}$. Note that $\ell_{2}^{\star}$ is braided
symmetric, and that the braided $L_{\infty}$-relations hold trivially for
degree reasons. The braided field equations are

$\displaystyle\mathcal{F}_{\phi}^{\star}\ =\
\ell_{1}^{\star}(\phi)-\tfrac{1}{2}\,\ell_{2}^{\star}(\phi,\phi)\ =\
(\square-m^{2})\,\phi-\tfrac{\kappa}{2}\,\phi\star\phi\ .$ (3.25)

With the cyclic inner product

$\displaystyle\langle\phi,\phi^{+}\rangle_{\star}\ =\ \int{\rm d}^{d}x\
\phi\star\phi^{+}$ (3.26)

for $\phi\in V_{1}$ and $\phi^{+}\in V_{2}$, the action is

$\displaystyle S^{\star}\ =\ \int{\rm d}^{d}x\
\frac{1}{2}\,\phi\star(\square-m^{2})\,\phi-\frac{\kappa}{3!}\,\phi\star\phi\star\phi\
.$ (3.27)

It follows that standard noncommutative scalar field theory is organised by a
braided $L_{\infty}$-algebra. Like the other usual noncommutative field
theories [1], it also has a standard $L_{\infty}$-algebra formulation in terms
of the symmetric bracket

$\displaystyle\ell_{2}^{\star{\rm sym}}(\phi_{1},\phi_{2})\ =\
\tfrac{\kappa}{2}\,(\phi_{1}\star\phi_{2}+\phi_{2}\star\phi_{1})\ .$ (3.28)

The differences between the two homotopy algebraic formulations appear in the
respective quantum field theories, see [35].

### Example III: Braided noncommutative Yang-Mills theory.

Starting from the standard Yang-Mills action

$\displaystyle{S\ =\ \frac{1}{2}\,\int\>{\rm
Tr}_{\mathfrak{g}}(F_{A}\wedge*\,F_{A})}$ (3.29)

for a gauge field $A\in\Omega^{1}(\mathbb{R}^{1,d-1},\mathfrak{g})$, the
classical Yang-Mills $L_{\infty}$-algebra is not a differential graded Lie
algebra, because $\ell_{3}(A,A,A)\ \neq\ 0$ [21, 1], unlike the previous two
examples. The corresponding braided Yang-Mills $L_{\infty}$-algebra gives the
braided field equations

$\displaystyle\begin{split}&\tfrac{1}{2}\,\big{(}{\rm
d}^{A}_{\star{\textrm{\tiny L}}}\ast F_{A}^{\star}+{\rm
d}^{A}_{\star{\textrm{\tiny R}}}\ast F_{A}^{\star}\big{)}\\\
&\quad+\,\tfrac{1}{6}\,\big{[}\bar{\rm R}^{\alpha}(A),\,\ast\,[\bar{\rm
R}_{\alpha}(A),A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}-\tfrac{1}{12}\,\big{[}A,\,\ast\,[A,A]_{\mathfrak{g}}^{\star}\,\big{]}_{\mathfrak{g}}^{\star}+\tfrac{(-1)^{d}}{12}\,\big{[}\ast[A,A]_{\mathfrak{g}}^{\star}\,,A\big{]}_{\mathfrak{g}}^{\star}\
=\ 0\ .\end{split}$ (3.30)

Despite the appearence of the second line, these are covariant and the
classical limit gives the usual Yang-Mills equations ${\rm d}^{A}\ast F_{A}\
=\ 0$ [1]. The action of the braided gauge theory is

$\displaystyle\begin{split}S^{\star}&\ =\ \frac{1}{2}\,\int\>{\rm
Tr}_{\mathfrak{g}}\big{(}F_{A}^{\star}\wedge_{\star}\ast\,F_{A}^{\star}\big{)}\\\
&\qquad+\frac{1}{24}\,\int\>{\rm Tr}_{\mathfrak{g}}\big{(}[A,\bar{\rm
R}^{\alpha}(A)]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast\,[\bar{\rm
R}_{\alpha}(A),A]_{\mathfrak{g}}^{\star}\,\big{)}-[A,A]_{\mathfrak{g}}^{\star}\,\wedge_{\star}\,\ast\,[A,A]_{\mathfrak{g}}^{\star}\,\big{)}\
.\end{split}$ (3.31)

Again this is gauge invariant with a good classical limit. The braided Noether
identities are a complicated modification of the classical identities ${\rm
d}^{A}({\rm d}^{A}\ast F_{A})\ =\ 0$. This noncommutative field theory defines
a new deformation of Yang-Mills theory. For further details and discussion,
see [1].

##  4 $L_{\infty}$-algebras of general relativity in the first order formalism

In the remainder of this contribution we turn our attention to the
applications of the formalism from Sections 2 and 3 to gravity. The
$L_{\infty}$-structure of classical gravity without matter fields in the
Einstein-Cartan-Palatini (ECP) formulation was discussed in detail in [2], in
arbitrary dimensionality and spacetime signature. In this section we briefly
review this construction, focusing for concreteness on the gravity theories in
three and four dimensions as well as with Lorentzian signature.

### Einstein-Cartan-Palatini gravity.

Let $M$ be a $d$-dimensional oriented manifold. Let
$\mathcal{V}\longrightarrow M$ be an oriented vector bundle of rank $d$ which
is isomorphic to the tangent bundle $TM$ and is endowed with a fibrewise
Minkowski metric $\eta$; the bundle $\mathcal{V}$ is called the ‘fake tangent
bundle’. The field content of ECP theory on $M$ consists of a bundle
isomorphism $e:TM\longrightarrow\mathcal{V}$, which defines the coframe field
$e\in\Omega^{1}(M,\mathcal{V})$, and a connection on the associated principal
$SO(1,d-1)$-bundle $P\longrightarrow M$ to $\mathcal{V}$ called the spin
connection, which has a curvature two-form

$\displaystyle R\ =\ {\rm
d}\omega+\tfrac{1}{2}\,[\omega,\omega]\in\Omega^{2}(M,P\times_{\rm
ad}\mathfrak{so}(1,d-1))\ .$ (4.1)

These fields satisfy the Bianchi identities

$\displaystyle{\rm d}^{\omega}T\ =\ R\wedge e\qquad\mbox{and}\qquad{\rm
d}^{\omega}R\ =\ 0\ ,$ (4.2)

where

$\displaystyle T\ =\ {\rm d}^{\omega}e\ =\ {\rm d}e+\omega\wedge e$ (4.3)

is the torsion of $\omega$. The orientation of $\mathcal{V}$ induces a map
$\>{\rm Tr}:\Omega^{d}(M,\wedge^{d}\mathcal{V})\longrightarrow\Omega^{d}(M)$.
Locally, or globally if $M$ is parallelizable, we take
$e\in\Omega^{1}(M,\mathbb{R}^{1,d-1})$,
$\omega\in\Omega^{1}(M,\mathfrak{so}(1,d-1))$ with $\mathfrak{so}(1,d-1)\
\simeq\ \wedge^{2}(\mathbb{R}^{1,d-1})$, and $\>{\rm
Tr}:\wedge^{d}(\mathbb{R}^{1,d-1})\longrightarrow\mathbb{R}$. We shall always
work in this local situation.

In any dimension $d$, the action for ECP gravity with cosmological constant
$\Lambda\in\mathbb{R}$ is given by

$\displaystyle\begin{split}S&\ =\ \int_{M}\>{\rm
Tr}\Big{(}\frac{1}{d-2}\,e^{d-2}\curlywedge
R+\frac{\Lambda}{d}\,e^{d}\Big{)}\\\\[4.0pt] &\ =\
\int_{M}\,\varepsilon_{a_{1}\cdots a_{d}}\,\Big{(}e^{a_{1}}\wedge\cdots\wedge
e^{a_{d-2}}\wedge
R^{a_{d-1}a_{d}}+\frac{\Lambda}{d}\,e^{a_{1}}\wedge\cdots\wedge
e^{a_{d}}\Big{)}\ ,\end{split}$ (4.4)

where the $\curlywedge$-product is the double exterior product combining the
differential form exterior product in $\Omega^{\bullet}(M)$ with the
multivector exterior product in $\wedge^{\bullet}(\mathbb{R}^{1,d-1})$, while
the $\wedge$-product acts via multivector representations and exterior
multiplication of spacetime forms. The infinitesimal gauge symmetries of this
theory are given by the semi-direct product Lie algebra of diffeomorphisms and
local Lorentz transformations:

$\displaystyle\Gamma(TM)\ltimes\Omega^{0}(M,\mathfrak{so}(1,d-1))\ .$ (4.5)

The corresponding field equations are

$\displaystyle e^{d-3}\curlywedge T\ =\ 0\qquad\mbox{and}\qquad
e^{d-3}\curlywedge R+\Lambda\,e^{d-1}\ =\ 0\ .$ (4.6)

In three and four dimensions, these equations respectively read

$\displaystyle\begin{split}T\ =\ 0\ =\ R+\Lambda\,e\curlywedge e\ \ \
\mbox{\sf(d\,=\,3)}\qquad\mbox{,}\qquad e\curlywedge T\ =\ 0\ =\ e\curlywedge
R+\Lambda\,e\curlywedge e\curlywedge e\ \ \ \mbox{\sf(d\,=\,4)}\ .\end{split}$
(4.7)

For non-degenerate coframes $e$, these are respectively equivalent to the
torsion-free condition for the spin connection and the vacuum Einstein
equations with cosmological constant.

Note that in contrast to the (second order) Einstein-Hilbert formulation of
general relativity, the (first order) ECP theory makes sense for degenerate
coframes $e$, as required for its $L_{\infty}$-algebra formulation which is
based on linear spaces. In the following we present this
$L_{\infty}$-structure for the case of parallelizable spacetimes in three and
four dimensions. See [2] for a more general construction of ‘covariant
$L_{\infty}$-algebras’ for arbitrary spacetimes, which are quasi-isomorphic to
the $L_{\infty}$-algebras below in the case of parallelizable manifolds.

###  ${\bm{L_{\infty}}}$-structure in ${\bm{d=3}}$.

The cochain complex underlying three-dimensional gravity is

$\displaystyle V_{0}\xrightarrow{\ \ \ell_{1}\ \ }V_{1}\xrightarrow{\ \
\ell_{1}\ \ }V_{2}\xrightarrow{\ \ \ell_{1}\ \ }V_{3}\ ,$ (4.8)

where the various vector spaces encode the symmetries and dynamics as follows:

$\displaystyle\begin{split}\mbox{\sf Gauge transformations:}&\hskip
56.9055pt{(\xi,\lambda)}\in\ V_{0}\ =\
\Gamma(TM)\times\Omega^{0}(M,\mathfrak{so}(1,2))\\\\[4.0pt] \mbox{\sf Physical
fields:}&\hskip 56.9055pt{(e,\omega)}\in\ V_{1}\ =\
\Omega^{1}(M,\mathbb{R}^{1,2})\times\Omega^{1}(M,\mathfrak{so}(1,2))\\\\[4.0pt]
\mbox{\sf Field equations:}&\hskip 54.06006pt{(E,\varOmega)}\in\ V_{2}\ =\
\Omega^{2}(M,\wedge^{2}(\mathbb{R}^{1,2}))\times\Omega^{2}(M,\mathbb{R}^{1,2})\\\\[4.0pt]
\mbox{\sf Noether identities:}&\hskip 56.9055pt{(\varXi,\varLambda)}\in\
V_{3}\ =\
\Omega^{1}(M,\Omega^{3}(M))\times\Omega^{3}(M,\mathbb{R}^{1,2})\end{split}$
(4.9)

The differential is given by

$\displaystyle\ell_{1}(\xi,\lambda)\ =\ (0,{\rm d}\lambda)\
,\quad\ell_{1}(e,\omega)\ =\ ({\rm d}\omega,{\rm
d}e)\qquad\mbox{and}\qquad\ell_{1}(E,\varOmega)\ =\ (0,{\rm d}\varOmega)\ .$
(4.10)

The first few $2$-brackets are

$\displaystyle\begin{split}\ell_{2}((\xi_{1},\lambda_{1}),(\xi_{2},\lambda_{2}))&\
=\
([\xi_{1},\xi_{2}],-[\lambda_{1},\lambda_{2}]+\mathcal{L}_{\xi_{1}}\lambda_{2}-\mathcal{L}_{\xi_{2}}\lambda_{1})\
,\\\\[4.0pt] \ell_{2}((\xi,\lambda),(e,\omega))&\ =\ (-\lambda\cdot
e+\mathcal{L}_{\xi}e,-[\lambda,\omega]+\mathcal{L}_{\xi}\omega)\ ,\\\\[4.0pt]
\ell_{2}((e_{1},\omega_{1}),(e_{2},\omega_{2}))&\ =\
-([\omega_{1},\omega_{2}]+2\,\Lambda\,e_{1}\curlywedge e_{2},\omega_{1}\wedge
e_{2}+\omega_{2}\wedge e_{1})\ ,\end{split}$ (4.11)

along with three further $2$-brackets related to the field equations and
Noether identities. There are no higher brackets, and so three-dimensional
gravity is organised by a differential graded Lie algebra.

Explicitly, the gauge symmetry is encoded by

$\displaystyle\delta_{(\xi,\lambda)}(e,\omega)\ =\ (-\lambda\cdot
e+\mathcal{L}_{\xi}e\,,\,{\rm
d}\lambda-[\lambda,\omega]+\mathcal{L}_{\xi}\omega)\ =\
\ell_{1}(\xi,\lambda)+\ell_{2}((\xi,\lambda),(e,\omega))\ ,$ (4.12)

and the field equations by

$\displaystyle\mathcal{F}_{(e,\omega)}\ =\
(\mathcal{F}_{e},\mathcal{F}_{\omega})\ =\ (R+\Lambda\,e\curlywedge e,T)\ =\
\ell_{1}(e,\omega)-\tfrac{1}{2}\,\ell_{2}((e,\omega),(e,\omega))\ .$ (4.13)

The Noether identities are

$\displaystyle\begin{split}\mathcal{I}_{(\xi,\lambda)}&\ =\ \big{(}{\rm
d}x^{\mu}\otimes\>{\rm Tr}(\iota_{\partial_{\mu}}e\curlywedge{\rm
d}\mathcal{F}_{\omega}-\iota_{\partial_{\mu}}{\rm
d}e\curlywedge\mathcal{F}_{\omega})+(e\leftrightarrow\omega)\,,\,{\rm
d}^{\omega}T-R\wedge e\big{)}\\\\[4.0pt] &\ =\
\ell_{1}(\mathcal{F}_{(e,\omega)})-\ell_{2}((e,\omega),\mathcal{F}_{(e,\omega)})\
\equiv\ (0,0)\ ,\end{split}$ (4.14)

and they are identically equal to zero _off-shell_.

The cyclic inner product

$\displaystyle\langle(e,\omega),(E,\varOmega)\rangle\ :=\ \int_{M}\>{\rm
Tr}(e\curlywedge E+\varOmega\curlywedge\omega)$ (4.15)

on $V_{1}\otimes V_{2}$ encodes the ECP action

$\displaystyle S\ =\
\tfrac{1}{2}\,\langle(e,\omega)\,,\,\ell_{1}(e,\omega)-\tfrac{1}{3!}\,\ell_{2}((e,\omega),(e,\omega))\rangle\
.$ (4.16)

We can extend this pairing to $\langle-,-\rangle:V_{0}\otimes
V_{3}\longrightarrow\mathbb{R}$ using gauge invariance and integration by
parts. Then cyclicity on $V_{0}$ implies the Noether identities.

### Application to Chern-Simons gravity.

Three-dimensional gravity is equivalent to a Chern-Simons theory whose gauge
algebra $\mathfrak{g}$ is the Poincaré algebra $\mathfrak{iso}(1,2)\ =\
\mathbb{R}^{1,2}\rtimes\mathfrak{so}(1,2)$ for $\Lambda\ =\ 0$, the de Sitter
algebra $\mathfrak{so}(1,3)$ for $\Lambda>0$, or the anti-de Sitter algebra
$\mathfrak{so}(2,2)$ for $\Lambda<0$ [36]. The field content of ECP theory may
be assembled into a connection one-form $A\ =\
(e,\omega)\in\Omega^{1}(M,\mathfrak{g})$ with curvature two-form $F_{A}\ =\
(R,T)\in\Omega^{2}(M,\mathfrak{g})$ in this Chern-Simons formulation.

A key point of this correspondence is the manner in which the diffeomorphism
symmetry of general relativity is realised on the Chern-Simons side. For
invertible coframes $e$, diffeomorphisms generated by vector fields
$\xi\in\Gamma(TM)$ are equivalent _on-shell_ to field-dependent gauge
transformations by

$\displaystyle(\tau_{\xi},\lambda_{\xi})\ =\
(\iota_{\xi}e,\iota_{\xi}\omega)\in\Omega^{0}(M,\mathfrak{g})\ .$ (4.17)

Explicitly,

$\displaystyle\delta_{\xi}A\ =\ \mathcal{L}_{\xi}A\ =\
\delta_{(\tau_{\xi},\lambda_{\xi})}A+\iota_{\xi}F_{A}\ .$ (4.18)

However, if one allows for degenerate metrics, as is necessary in the
$L_{\infty}$-algebra formalism, then the gauge orbits under the “shifts”
$(\tau_{\xi},\lambda_{\xi})$ only strictly contain the diffeomorphism orbits
on-shell. It was shown by [2] how to extend the correspondence of [36] to an
_off-shell_ equivalence including arbitrary coframe fields.

Diffeomorphisms in Chern-Simons theory are _redundant symmetries_. We can
trivialise them via an extension $V_{\textrm{\tiny CS}}^{\rm ext}$ of the
graded vector space $V_{\textrm{\tiny CS}}$ of Chern-Simons theory which
augments the degree $0$ (resp. degree $3$) subspace by $\Gamma(TM)$ (resp.
$\Omega^{1}(M,\Omega^{3}(M))$). On the ECP side we extend the homogenous
subspace $V_{0}$ (resp. $V_{3}$) by the extra “shift” symmetries
$\Omega^{0}(M,\mathbb{R}^{1,2})$ (resp. $\Omega^{3}(M,\mathfrak{so}(1,2))$) to
get a new graded vector space $V^{\rm ext}_{\textrm{\tiny ECP}}$. In both
cases we then add the subspace $V_{-1}\ =\ \Gamma(TM)$ (resp. $V_{4}\ =\
\Omega^{1}(M,\Omega^{3}(M))$ with differential $\ell_{1}:V_{-1}\hookrightarrow
V_{0}$ (resp. $\ell_{1}:V_{3}\twoheadrightarrow V_{4}$). The underlying
cochain complexes $(V_{\textrm{\tiny CS}},\ell_{1})$ and $(V_{\textrm{\tiny
CS}}^{\rm ext},\ell_{1})$ then have the same cohomology, and altogether we get
an isomorphism

$\displaystyle H^{\bullet}(V_{\textrm{\tiny ECP}}^{\rm ext},\ell_{1})\ \simeq\
H^{\bullet}(V_{\textrm{\tiny CS}},\ell_{1})\ .$ (4.19)

In [2] it is shown that there is an (off-shell) cyclic $L_{\infty}$-quasi-
isomorphism $\\{\psi_{n}\\}_{n\geq 1}$ with $\psi_{n}\ =\ 0$ for $n\geq 3$
from the differential graded Lie algebra of Chern-Simons theory to the
differential graded Lie algebra of ECP theory.

###  ${\bm{L_{\infty}}}$-structure in ${\bm{d=4}}$.

The graded vector space underlying four-dimensional gravity is now given by

$\displaystyle\begin{split}V_{0}&\ =\
\Gamma(TM)\times\Omega^{0}\big{(}M,\mathfrak{so}(1,3)\big{)}\ ,\\\\[4.0pt]
V_{1}&\ =\
\Omega^{1}(M,\mathbb{R}^{1,3})\times\Omega^{1}(M,\mathfrak{so}(1,3))\
,\\\\[4.0pt] V_{2}&\ =\
\Omega^{3}(M,\wedge^{3}(\mathbb{R}^{1,3}))\times\Omega^{3}(M,\wedge^{2}(\mathbb{R}^{1,3}))\
,\\\\[4.0pt] V_{3}&\ =\
\Omega^{1}(M,\Omega^{4}(M))\times\Omega^{4}(M,\wedge^{2}(\mathbb{R}^{1,3}))\
.\end{split}$ (4.20)

The differential is

$\displaystyle\ell_{1}(\xi,\lambda)\ =\ (0,{\rm d}\lambda)\
,\quad\ell_{1}(e,\omega)\ =\ (0,0)\
\qquad\mbox{and}\qquad\ell_{1}(E,{\varOmega})\ =\ (0,-{\rm d}{\varOmega})\ .$
(4.21)

The only $2$-bracket which is modified from the list (4.11) is

$\displaystyle\ell_{2}((e_{1},\omega_{1}),(e_{2},\omega_{2}))\ =\
-(e_{1}\curlywedge{\rm d}\omega_{2}+e_{2}\curlywedge{\rm
d}\omega_{1},e_{1}\curlywedge{\rm d}e_{2}+e_{2}\curlywedge{\rm d}e_{1})\ .$
(4.22)

Finally, there is a single non-trivial $3$-bracket

$\displaystyle\begin{split}&\ell_{3}((e_{1},\omega_{1}),(e_{2},\omega_{2}),(e_{3},\omega_{3}))\\\\[4.0pt]
&\hskip 28.45274pt\ =\
-(e_{1}\curlywedge[\omega_{2},\omega_{3}]+e_{2}\curlywedge[\omega_{1},\omega_{3}]+e_{3}\curlywedge[\omega_{2},\omega_{1}]+6\,\Lambda\,e_{1}\curlywedge
e_{2}\curlywedge e_{3},\\\ &\hskip 56.9055pt\
e_{1}\curlywedge(\omega_{2}\wedge e_{3})+_{(2\leftrightarrow
3)}+e_{2}\curlywedge(\omega_{1}\wedge e_{3})+_{(1\leftrightarrow
3)}+e_{3}\curlywedge(\omega_{2}\wedge e_{1})+_{(2\leftrightarrow 1)})\
.\end{split}$ (4.23)

Thus four-dimensional ECP gravity is organised by an $L_{\infty}$-algebra
which is not a differential graded Lie algebra, as $\ell_{3}\ \neq\ 0$,
similarly to Yang-Mills theory. In [2] it is shown that this
$L_{\infty}$-algebra picture is dual to the BV–BRST formulation of ECP gravity
from [37].

##  5 $L_{\infty}$-algebras of noncommutative gravity

In this final section we consider noncommutative deformations of the ECP
theory of gravity from Section 4. After describing the issues surrounding its
formulation in terms of the standard $L_{\infty}$-algebra picture of gauge
symmetries and dynamics, we describe the braided $L_{\infty}$-algebra picture
which leads to a new noncommutative deformation of gravity in four dimensions.

###  ${\bm{L_{\infty}}}$-structure.

Let us start by reviewing the standard formulation of noncommutative Einstein-
Cartan-Palatini gravity, in order to elucidate the problems with formulating
it in the standard picture of $L_{\infty}$-algebras. While the general
features hold more generally, we shall focus here on the case of $d\ =\ 3$
spacetime dimensions without cosmological constant; this theory is discussed
in [38]. We will proceed in two steps: first we consider the noncommutative
generalization of the local Lorentz transformations, and subsequently that of
the diffeomorphism symmetry. We will then attempt to understand them together
within the setting of classical $L_{\infty}$-algebras along the same lines as
in Section 4.

In the standard noncommutative gauge theories based on star-gauge symmetries,
one can deform any theory which is based on a matrix gauge algebra that closes
under both commutators and anticommutators (such as $\mathfrak{u}(N)$ and
$\mathfrak{gl}(N,\mathbb{C})$), and with matter fields in the trivial,
(anti)fundamental or adjoint representations. These noncommutative field
theories are organised by classical $L_{\infty}$-algebras in an identical way
to those of classical non-abelian gauge theories, see [1]. For the local
Lorentz symmetry, this would amount to defining the noncommutative
transformation laws of the noncommutative coframe $\hat{e}$ and spin
connection $\hat{\omega}$ by a gauge parameter $\hat{\lambda}$ as

$\displaystyle\hat{\delta}_{\hat{\lambda}}\hat{e}\ =\
-\hat{\lambda}\star\hat{e}\qquad\mbox{and}\qquad\hat{\delta}_{\hat{\lambda}}\hat{\omega}\
=\ {\rm
d}\hat{\lambda}-[\hat{\lambda}\stackrel{{\scriptstyle\star}}{{,}}\hat{\omega}]\
=\ {\rm
d}\hat{\lambda}-(\hat{\lambda}\star\hat{\omega}-\hat{\omega}\star\hat{\lambda})\
,$ (5.1)

where the noncommutative fields and gauge parameters are designated with a
hat.

However, the Lorentz algebra $\mathfrak{so}(1,2)$ is not closed under the
star-commutator $[-\stackrel{{\scriptstyle\star}}{{,}}-]$. Generally, for an
arbitrary matrix Lie algebra $\mathfrak{g}$ one solves this problem by
enlarging $\mathfrak{g}$ to its universal enveloping algebra $U\mathfrak{g}$,
in which the star-commutator does close. For the special case $\mathfrak{g}\
=\ \mathfrak{so}(1,2)$, the situation is simpler, because the anticommutator
of two elements is always proportional to the identity matrix and so is a
central element of $U\mathfrak{g}$:
$\\{\mathfrak{g},\mathfrak{g}\\}\subset\mathbb{R}\,{\bm{1}}$. Hence it
suffices to enlarge $\mathfrak{so}(1,2)$ to a centrally extended Lie algebra

$\displaystyle\hat{\mathfrak{g}}\ =\ \mathfrak{so}(1,2)\oplus\mathbb{R}\ ,$
(5.2)

whose fundamental representation is

$\displaystyle\hat{W}\ =\ \mathbb{R}^{1,2}\oplus\mathbb{R}\ ,$ (5.3)

and take all noncommutative fields and gauge parameters to be valued in
$\hat{\mathfrak{g}}$ and $\hat{W}$. We denote these by $\hat{\lambda}\ =\
\lambda+\tilde{\lambda}$, where $\lambda\in\Omega^{0}(M,\mathfrak{so}(1,2))$
and $\tilde{\lambda}\in\Omega^{0}(M)$, and similarly $\hat{e}\ =\ e+\tilde{e}$
and $\hat{\omega}\ =\ \omega+\tilde{\omega}$; we designate the new centrally
extended variables with a tilde, while the classical variables from Section 4
remain unlabelled.

This approach, which operates in the central extension (5.2)–(5.3), introduces
new fields $\tilde{e}$ and $\tilde{\omega}$. They can be regarded as new
degrees of freedom which are not present in the classical theory. Another way
to understand them is through the Seiberg-Witten map [39], which enables one
to express all noncommutative fields as functionals of the corresponding
commutative fields [39, 40]; this was applied to noncommutative ECP gravity in
[41, 42].

One can now check that the centrally extended local Lorentz transformations
(5.1) close the gauge algebra

$\big{[}\hat{\delta}_{\hat{\lambda}_{1}},\hat{\delta}_{\hat{\lambda}_{2}}\big{]}\
=\
\hat{\delta}_{[{\hat{\lambda}}_{1}\stackrel{{\scriptstyle\star}}{{,}}{\hat{\lambda}}_{2}]}\
.$ (5.4)

The noncommutative torsion and curvature are

$\displaystyle\begin{split}\hat{T}&\ =\ {\rm
d}\hat{e}+\tfrac{1}{2}\,\big{(}\hat{\omega}\wedge_{\star}{\hat{e}}+\bar{\rm
R}^{\alpha}(\hat{\omega})\wedge_{\star}\bar{\rm R}_{\alpha}(\hat{e})\big{)}\
=\ T+\tilde{T}\ ,\\\\[4.0pt] \hat{R}&\ =\ {\rm
d}\hat{\omega}+\tfrac{1}{2}\,[\hat{\omega}\stackrel{{\scriptstyle\star}}{{,}}\hat{\omega}]\
=\ R+\tilde{R}\ .\end{split}$ (5.5)

In addition to (5.1), one can show that they transform covariantly under the
infinitesimal noncommutative local Lorentz transformations:

$\displaystyle\hat{\delta}_{\hat{\lambda}}\hat{T}\ =\
-\hat{\lambda}\star\hat{T}\qquad\mbox{and}\qquad\hat{\delta}_{\hat{\lambda}}\hat{R}\
=\ -[\hat{\lambda}\stackrel{{\scriptstyle\star}}{{,}}\hat{R}]\ .$ (5.6)

Calculating the star-operations explicitly, the components of
$\hat{T}\in\Omega^{2}(M,\hat{W})$ valued in $\mathbb{R}^{1,2}$ and
$\mathbb{R}$ are given by [38]

$\displaystyle\begin{split}T&\ =\ {\rm
d}e+\tfrac{1}{2}\,\big{(}{\omega}\wedge_{\star}{e}+\bar{\rm
R}^{\alpha}(\omega)\wedge_{\star}\bar{\rm
R}_{\alpha}(e)\big{)}+\tilde{\omega}\wedge_{\star}e+e\wedge_{\star}\tilde{\omega}+\tfrac{1}{2}\ast(\omega\wedge_{\star}\tilde{e}+\tilde{e}\wedge_{\star}\omega)\
,\\\\[4.0pt] \tilde{T}&\ =\ {\rm
d}\tilde{e}+\tilde{\omega}\wedge_{\star}\tilde{e}+\tilde{e}\wedge_{\star}\tilde{\omega}+\tfrac{1}{4}\>{\rm
Tr}(\omega\curlywedge_{\star}e+e\curlywedge_{\star}\omega)\ ,\end{split}$
(5.7)

where $\ast$ is the Hodge duality operator on
$\wedge^{\bullet}(\mathbb{R}^{1,2})$ induced by the Minkowski metric $\eta$.
The components of $\hat{R}\in\Omega^{2}(M,\hat{\mathfrak{g}})$ valued in
$\mathfrak{so}(1,2)$ and $\mathbb{R}$ are

$\displaystyle\begin{split}R&\ =\ {\rm
d}{\omega}+\tfrac{1}{2}\,[{\omega}\stackrel{{\scriptstyle\star}}{{,}}{\omega}]+\omega\wedge_{\star}\tilde{\omega}+\tilde{\omega}\wedge_{\star}\omega\
,\\\\[4.0pt] \tilde{R}&\ =\ {\rm
d}\tilde{\omega}+\tilde{\omega}\wedge_{\star}\tilde{\omega}+\tfrac{1}{4}\>{\rm
Tr}(\omega\curlywedge_{\star}\ast\,\omega)\ .\end{split}$ (5.8)

In the classical limit, $T$ and $R$ become the standard torsion and curvature
from Section 4, while the new fields $\tilde{e}$ and $\tilde{\omega}$
completely decouple from the classical fields with the abelian field strengths
$\tilde{T}\ =\ {\rm d}\tilde{e}$ and $\tilde{R}\ =\ {\rm d}\tilde{\omega}$.

The action (without cosmological constant) which is invariant under the
noncommutative local Lorentz transformations is given by [38]

$\displaystyle S\ =\ \int_{M}\>{\rm
Tr}\big{(}\hat{e}\curlywedge_{\star}\hat{R}\big{)}\ =\
\int_{M}\,\varepsilon_{abc}\,e^{a}\wedge_{\star}R^{ab}+4\,\tilde{e}\wedge_{\star}\tilde{R}\
.$ (5.9)

The field equations resulting from the variations of the action (5.9) are

$\displaystyle T\ =\ 0\ =\ \tilde{T}\qquad\mbox{and}\qquad R\ =\ 0\ =\
\tilde{R}\ .$ (5.10)

The $L_{\infty}$-structure of the noncommutative local Lorentz symmetry now
follows the standard rules of the classical theory from Section 4. The
underlying graded vector space

$\displaystyle\hat{V}\ =\
\hat{V}_{0}\oplus\hat{V}_{1}\oplus\hat{V}_{2}\oplus\hat{V}_{3}$ (5.11)

contains the field content

$\displaystyle\begin{split}\hat{\lambda}\in\hat{V}_{0}&\ =\
\Omega^{0}(M,\hat{\mathfrak{g}})\ ,\\\\[4.0pt]
(\hat{e},\hat{\omega})\in\hat{V}_{1}&\ =\
\Omega^{1}(M,\hat{W})\times\Omega^{1}(M,\hat{\mathfrak{g}})\ ,\\\\[4.0pt]
(\hat{E},\hat{\varOmega})\in\hat{V}_{2}&\ =\
\Omega^{2}(M,\wedge^{2}(\hat{W}))\times\Omega^{2}(M,\hat{W})\ ,\\\\[4.0pt]
\hat{\varLambda}\in\hat{V}_{3}&\ =\ \Omega^{3}(M,\hat{W})\ .\end{split}$
(5.12)

The differential is

$\displaystyle\hat{\ell}_{1}(\hat{\lambda})\ =\ {\rm d}\hat{\lambda}\
,\quad\hat{\ell}_{1}(\hat{e},\hat{\omega})\ =\ ({\rm d}\hat{\omega},{\rm
d}\hat{e})\qquad\mbox{and}\qquad\hat{\ell}_{1}(\hat{E},\hat{\varOmega})\ =\
{\rm d}\hat{\varOmega}\ ,$ (5.13)

while the first few non-vanishing $2$-brackets are

$\displaystyle\begin{split}\hat{\ell}_{2}(\hat{\lambda}_{1},\hat{\lambda}_{2})&\
=\ -[\hat{\lambda}_{1}\stackrel{{\scriptstyle\star}}{{,}}\hat{\lambda}_{2}]\
,\\\\[4.0pt] \hat{\ell}_{2}(\hat{\lambda},(\hat{e},\hat{\omega}))&\ =\
(-\hat{\lambda}\star\hat{e},-[\hat{\lambda}\stackrel{{\scriptstyle\star}}{{,}}\hat{\omega}])\
,\\\\[4.0pt]
\hat{\ell}_{2}((\hat{e}_{1},\hat{\omega}_{1}),(\hat{e}_{2},\hat{\omega}_{2}))&\
=\
-([\hat{\omega}_{1}\stackrel{{\scriptstyle\star}}{{,}}\hat{\omega}_{2}],\hat{\omega}_{1}\wedge_{\star}\hat{e}_{1}+\hat{\omega}_{2}\wedge_{\star}\hat{e}_{1})\
,\end{split}$ (5.14)

along with three others related to the field equations and Noether identities.
One can check that this defines a differential graded Lie algebra, i.e. that
the standard $L_{\infty}$-homotopy relations hold.

The noncommutative field equations (5.10) can be rewritten as
$\hat{\mathcal{F}}_{(\hat{e},\hat{\omega})}\ =\ 0$, where

$\displaystyle\hat{\mathcal{F}}_{(\hat{e},\hat{\omega})}\ =\
(\hat{R},\hat{T})\ =\
\hat{\ell}_{1}(\hat{e},\hat{\omega})-\tfrac{1}{2}\,\hat{\ell}_{2}((\hat{e},\hat{\omega}),(\hat{e},\hat{\omega}))\
.$ (5.15)

The cyclic inner product
$\langle-,-\rangle:\hat{V}_{1}\otimes\hat{V}_{2}\longrightarrow\mathbb{R}$
that encodes the action (5.9) is the generalization of (4.15) given by

$\displaystyle\langle(\hat{e},\hat{\omega}),(\hat{E},\hat{\varOmega})\rangle\
=\ \int_{M}\>{\rm
Tr}\big{(}\hat{e}\curlywedge_{\star}\hat{E}+\hat{\varOmega}\curlywedge_{\star}\hat{\omega}\big{)}\
.$ (5.16)

Let us now study if and how the noncommutative diffeomorphism symmetry can be
incorporated into this classical $L_{\infty}$-structure. For a vector field
$\xi\in\Gamma(TM)$, the action of noncommutative diffeomorphisms on the
various fields (5.12) of three-dimensional noncommutative gravity is via the
braided Lie derivative (3.13):

$\displaystyle\hat{\delta}_{\xi}\hat{{\bm{T}}}\ =\
\mathcal{L}_{\xi}^{\star}\hat{{\bm{T}}}\ .$ (5.17)

Indeed, the action (5.9) is invariant under braided diffeomorphisms [38].

The dynamical part of the $L_{\infty}$-algebra is not altered by the inclusion
of diffeomorphisms, only its symmetry part, which we modify by extending the
vector spaces $\hat{V}_{0}$ and $\hat{V}_{3}$ respectively by

$\displaystyle\xi\in\Gamma(TM)\subset\hat{V}_{0}\qquad\mbox{and}\qquad\varXi\in\Omega^{1}(M,\Omega^{3}(M))\subset\hat{V}_{3}\
,$ (5.18)

as in the classical case. The brackets (5.13) and (5.14) are correspondingly
supplemented with

$\displaystyle\hat{\ell}_{1}(\xi)\ =\
(0,0)\quad,\quad\hat{\ell}_{2}(\xi_{1},\xi_{2})\ =\
[\xi_{1},\xi_{2}]^{\star}\quad\mbox{,}\quad\hat{\ell}_{2}(\xi,(\hat{e},\hat{\omega}))\
=\ (\mathcal{L}_{\xi}^{\star}\hat{e},\mathcal{L}_{\xi}^{\star}\hat{\omega})\
.$ (5.19)

Of course, $\hat{\ell}_{2}(\xi_{1},\xi_{2})$ is the bracket of the braided Lie
algebra $\Gamma_{\star}(TM)$ of noncommutative vector fields on $M$, and so is
not antisymmetric, but rather braided antisymmetric:

$\displaystyle\hat{\ell}_{2}(\xi_{1},\xi_{2})\ =\ -\hat{\ell}_{2}(\bar{\rm
R}^{\alpha}(\xi_{2}),\bar{\rm R}_{\alpha}(\xi_{1}))\ .$ (5.20)

As in the case of the noncommutative scalar field theory from Section 3, we
can antisymmetrize the bracket by hand and define

$\displaystyle\hat{\ell}_{2}^{\,{\rm sym}}(\xi_{1},\xi_{2})\ =\
\tfrac{1}{2}\,\big{(}[\xi_{1},\xi_{2}]^{\star}-[\xi_{2},\xi_{1}]^{\star}\big{)}\
.$ (5.21)

However, while most of the homotopy relations hold, there is one relation
which is violated:

$\displaystyle\hat{\ell}_{2}^{\,{\rm sym}}(\hat{\ell}_{2}^{\,{\rm
sym}}(\xi_{1},\xi_{2}),\xi_{3})+\hat{\ell}_{2}^{\,{\rm
sym}}(\hat{\ell}_{2}^{\,{\rm
sym}}(\xi_{3},\xi_{1}),\xi_{2})+\hat{\ell}_{2}^{\,{\rm
sym}}(\hat{\ell}_{2}^{\,{\rm sym}}(\xi_{2},\xi_{3}),\xi_{1})\ \neq\ 0\ .$
(5.22)

The vanishing of this expression would be the Jacobi identity for the
antisymmetrized bracket (5.21), which is violated because of the _braided_
Jacobi identity (3.5) for the braided Lie algebra $\Gamma_{\star}(TM)$.
Generally, the right-hand side of the homotopy Jacobi identity (5.22) can be
equal to $-\hat{\ell}_{1}(\hat{\ell}_{3}(\xi_{1},\xi_{2},\xi_{3}))$, which
requires extending the cochain complex $(\hat{V},\hat{\ell}_{1})$ by a
homogeneous subspace $\hat{V}_{-1}$ in degree $-1$ (and dually by
$\hat{V}_{4}$ in degree $4$) in order to accomodate a non-zero $3$-bracket
$\hat{\ell}_{3}(\xi_{1},\xi_{2},\xi_{3})$.

This suggests that, from the perspective of standard symmetries and their
$L_{\infty}$-structure, the twisted diffeomorphism symmetry of noncommutative
gravity may provide a reducible or _higher_ gauge symmetry. While this is a
tantalizing feature, it is not clear to what extent this approach is
technically feasible, as it may require the introduction of infinitely many
higher brackets, as in the $L_{\infty}$-algebra approaches to generic
noncommutative gauge theories [22, 23, 24, 25], however here with additional
corresponding homogeneous subspaces in negative degrees. Moreover, the
physical meaning of such an extension is not immediately apparent, as it
vanishes in the classical limit. We conclude that the twisted diffeomorphism
symmetry does not fit naturally (if at all) into the standard
$L_{\infty}$-algebra picture.

### Braided ${\bm{L_{\infty}}}$-structure.

Motivated by the issues encountered above, we deform the
$L_{\infty}$-structure itself to make it compatible with the twisted
diffeomorphisms, leading to our novel mathematical notion of a braided
$L_{\infty}$-algebra and the corresponding noncommutative field theories with
braided gauge symmetries. In contrast to the approach reviewed above, the
transformations (5.1) are now replaced with the braided Lorentz
transformations

$\displaystyle\delta_{\lambda}^{\star}e\ =\ -\lambda\star
e\qquad\mbox{and}\qquad\delta_{\lambda}^{\star}\omega\ =\ {\rm
d}\omega-[\lambda,\omega]^{\star}\ =\ {\rm
d}\lambda-\big{(}\lambda\star\omega-\bar{\rm R}^{\alpha}(\omega)\star\bar{\rm
R}_{\alpha}(\lambda)\big{)}\ .$ (5.23)

These transformations close in the Lie algebra, and there is no need to
centrally extend the Lorentz algebra (or more generally to extend to the
universal enveloping algebra). No new degrees of freedom are introduced, and
the noncommutative theory reduces exactly to the standard commutative theory
in the classical limit.

The braided noncommutative deformation of three-dimensional ECP gravity
follows much the same route as the braided Chern-Simons theory from Section 3,
resulting in the ‘naive’ deformed gravity theory obtained by replacing
products of fields with star-products in the classical theory; this appears to
be a general feature of the braided deformation of classical field theories
whose $L_{\infty}$-algebra is a differential graded Lie algebra. See [3] for a
brief review of the $d=3$ theory and [4] for a more detailed account. In $d\
=\ 4$ dimensions, a genuinely new noncommutative theory of gravity arises,
similarly to the braided noncommutative Yang-Mills theory from Section 3. Let
us summarise the main features; further details can be found in [4].

We add a cosmological constant $\Lambda$. As usual the differential
$\ell_{1}^{\star}\ =\ \ell_{1}$ from Section 4 is unchanged and thus given by
(4.21). The $2$-brackets are twisted to

$\displaystyle\begin{split}\ell_{2}^{\star}((\xi_{1},\lambda_{1}),(\xi_{2},\lambda_{2}))&\
=\
([\xi_{1},\xi_{2}]^{\star},-[\lambda_{1},\lambda_{2}]^{\star}+\mathcal{L}^{\star}_{\xi_{1}}\lambda_{2}-\mathcal{L}^{\star}_{\bar{\rm
R}^{\alpha}(\xi_{2})}\bar{\rm R}_{\alpha}(\lambda_{1}))\ ,\\\\[4.0pt]
\ell^{\star}_{2}((\xi,\lambda),(e,\omega))&\ =\ (-\lambda\star
e+\mathcal{L}^{\star}_{\xi}e,-[\lambda,\omega]^{\star}+\mathcal{L}^{\star}_{\xi}\omega)\
,\\\\[4.0pt] \ell^{\star}_{2}((e_{1},\omega_{1}),(e_{2},\omega_{2}))&\ =\
-(e_{1}\curlywedge_{\star}{\rm d}\omega_{2}+e_{2}\curlywedge_{\star}{\rm
d}\omega_{1},e_{1}\curlywedge_{\star}{\rm d}e_{2}+e_{2}\curlywedge_{\star}{\rm
d}e_{1})\ ,\end{split}$ (5.24)

and again with three more $2$-brackets related to the field equations and
Noether identities. Finally, the remaining non-zero braided $3$-bracket is
given by

$\displaystyle\small\begin{split}&\ell^{\star}_{3}((e_{1},\omega_{1}),(e_{2},\omega_{2}),(e_{3},\omega_{3}))\\\\[4.0pt]
&\hskip 14.22636pt\ =\
-(e_{1}\curlywedge_{\star}[\omega_{2},\omega_{3}]^{\star}+\bar{\rm
R}^{\alpha}(e_{2})\curlywedge_{\star}[\bar{\rm
R}_{\alpha}(\omega_{1}),\omega_{3}]^{\star}+[\omega_{1},\omega_{2}]^{\star}\curlywedge_{\star}e_{3}+6\,\Lambda\,e_{1}\curlywedge_{\star}e_{2}\curlywedge_{\star}e_{3},\\\\[2.0pt]
&\hskip 42.67912pt\
e_{1}\curlywedge_{\star}(\omega_{2}\wedge_{\star}e_{3})+e_{1}\curlywedge_{\star}(\bar{\rm
R}^{\alpha}(\omega_{3})\wedge_{\star}\bar{\rm R}_{\alpha}(e_{2}))\\\ &\hskip
56.9055pt\ +\bar{\rm R}^{\alpha}(e_{2})\curlywedge_{\star}(\bar{\rm
R}_{\alpha}(\omega_{1})\wedge_{\star}e_{3})+\bar{\rm
R}^{\alpha}(e_{2})\curlywedge_{\star}(\bar{\rm
R}^{\beta}(\omega_{3})\wedge_{\star}\bar{\rm R}_{\beta}\,\bar{\rm
R}_{\alpha}(e_{1}))\\\ &\hskip 71.13188pt\
-(\omega_{1}\wedge_{\star}e_{2})\curlywedge_{\star}e_{3}-(\bar{\rm
R}^{\alpha}(\omega_{2})\wedge_{\star}\bar{\rm
R}_{\alpha}(e_{1}))\curlywedge_{\star}e_{3})\ .\end{split}\normalsize$ (5.25)

The braided gauge transformations

$\displaystyle\delta_{(\xi,\lambda)}^{\star}(e,\omega)\ =\
\ell_{1}^{\star}(\xi,\lambda)+\ell_{2}^{\star}((\xi,\lambda),(e,\omega))\ =\
(-\lambda\star e+\mathcal{L}^{\star}_{\xi}e,{\rm
d}\lambda-[\lambda,\omega]^{\star}+\mathcal{L}_{\xi}^{\star}\omega)$ (5.26)

close a braided Lie algebra with braided Lie bracket $-\ell_{2}^{\star}$ of
gauge parameters given in (5.24). It follows that braided gravity is invariant
under the braided semi-direct product

$\displaystyle\Gamma_{\star}(TM)\ltimes_{\star}\Omega^{0}_{\star}(M,\mathfrak{so}(1,3))$
(5.27)

of the braided Lie algebra $\Gamma_{\star}(TM)$ of vector fields on $M$ and
the braided Lie algebra $\Omega^{0}_{\star}(M,\mathfrak{g})$ from Section 3
with $\mathfrak{g}\ =\ \mathfrak{so}(1,3)$, $W\ =\ \mathbb{R}^{1,3}$ and $p\
=\ 1$. Thus our braided theory of noncommutative gravity preserves the semi-
direct product structure of the classical symmetry algebra (4.5).

The field equations $\mathcal{F}_{(e,\omega)}^{\star}\ =\ (0,0)$ of braided
ECP gravity in four dimensions are given by

$\displaystyle\mathcal{F}_{(e,\omega)}^{\star}\ =\
-\tfrac{1}{2}\,\ell_{2}^{\star}((e,\omega),(e,\omega))-\tfrac{1}{6}\,\ell_{3}^{\star}((e,\omega),(e,\omega),(e,\omega))\
=\ (\mathcal{F}_{e}^{\star},\mathcal{F}_{\omega}^{\star})\ .$ (5.28)

Defining the braided left/right torsion and curvature of the spin connection
$\omega$ as

$\displaystyle T_{{\textrm{\tiny L}},{\textrm{\tiny R}}}^{\star}\ :=\ {\rm
d}^{\omega}_{\star{\textrm{\tiny L}},{\textrm{\tiny
R}}}e\qquad\mbox{and}\qquad R^{\star}\ :=\ {\rm
d}\omega+\tfrac{1}{2}\,[\omega,\omega]^{\star}\ ,$ (5.29)

they can be written as

$\displaystyle\begin{split}&e\curlywedge_{\star}T_{\textrm{\tiny
L}}^{\star}-T^{\star}_{\textrm{\tiny R}}\curlywedge_{\star}e-\,{\rm
d}^{\omega}_{\star{\textrm{\tiny L}}}(e\curlywedge_{\star}e)-{\rm
d}^{\omega}_{\star{\textrm{\tiny R}}}(e\curlywedge_{\star}e)\ =\ 0\
,\\\\[4.0pt]
&2\,e\curlywedge_{\star}R^{\star}+2\,R^{\star}\curlywedge_{\star}e+6\,\Lambda\,e\curlywedge_{\star}e\curlywedge_{\star}e\\\
&\qquad+\,e\curlywedge_{\star}{\rm d}\omega+{\rm
d}\omega\curlywedge_{\star}e+\bar{\rm
R}^{\alpha}(e)\curlywedge_{\star}[\bar{\rm
R}_{\alpha}(\omega),\omega]^{\star}\ =\ 0\ .\end{split}$ (5.30)

Despite the appearance of individually non-covariant terms, taken together
these equations are covariant, as guaranteed on general grounds by the braided
$L_{\infty}$-structure:

$\displaystyle\delta^{\star}_{(\xi,\lambda)}\mathcal{F}^{\star}_{(e,\omega)}\
=\
\ell_{2}^{\star}((\xi,\lambda),(\mathcal{F}_{e}^{\star},\mathcal{F}_{\omega}^{\star}))\
,$ (5.31)

which leads to

$\displaystyle\delta_{(\xi,\lambda)}^{\star}\mathcal{F}_{e}^{\star}\ =\
-\lambda\star\mathcal{F}_{e}^{\star}+\mathcal{L}_{\xi}^{\star}\mathcal{F}_{e}^{\star}\qquad\mbox{and}\qquad\delta_{(\xi,\lambda)}^{\star}\mathcal{F}_{\omega}^{\star}\
=\
-\lambda\star\mathcal{F}_{\omega}^{\star}+\mathcal{L}_{\xi}^{\star}\mathcal{F}_{\omega}^{\star}\
.$ (5.32)

In the classical limit they reduce respectively to the torsion-free condition
for the spin connection and the four-dimensional vacuum Einstein equations
with cosmological constant.

The action follows from twisting the classical ECP inner product (4.15) to

$\displaystyle\langle(e,\omega),(E,\varOmega)\rangle_{\star}\ =\
\int_{M}\>{\rm Tr}(e\curlywedge_{\star}E-\omega\curlywedge_{\star}\varOmega)$
(5.33)

on $V_{1}\otimes V_{2}$. Then the action

$\displaystyle\begin{split}S^{\star}&\ =\
\tfrac{1}{2}\,\langle(e,\omega),\ell_{1}^{\star}(e,\omega)\rangle_{\star}-\tfrac{1}{6}\,\langle(e,\omega),\ell_{2}^{\star}((e,\omega),(e,\omega))\rangle_{\star}\\\
&\qquad-\tfrac{1}{24}\,\langle(e,\omega),\ell_{3}^{\star}((e,\omega),(e,\omega),(e,\omega))\rangle_{\star}\end{split}$
(5.34)

leads to

$\displaystyle\begin{split}S^{\star}&\ =\ \int_{M}\>{\rm
Tr}\Big{(}\frac{1}{2}\,e\curlywedge_{\star}e\curlywedge_{\star}R^{\star}+\frac{\Lambda}{4}\,e\curlywedge_{\star}e\curlywedge_{\star}e\curlywedge_{\star}e\Big{)}\\\
&\qquad-\,\frac{1}{24}\,\int_{M}\>{\rm
Tr}\Big{(}\omega\curlywedge_{\star}\big{(}2\,e\curlywedge_{\star}T_{\textrm{\tiny
L}}^{\star}-2\,T^{\star}_{\textrm{\tiny R}}\curlywedge_{\star}e+{\rm
d}_{\star{\textrm{\tiny L}}}^{\omega}(e\curlywedge_{\star}e)+{\rm
d}_{\star{\textrm{\tiny R}}}^{\omega}(e\curlywedge_{\star}e)\big{)}\Big{)}\
.\end{split}$ (5.35)

Again, despite the appearance of the second line, this is gauge invariant with
a good classical limit that does not involve spurious degrees of freedom. The
associated Noether identities in this case are quite complicated, and are
discussed in detail in [4]. This theory defines a new deformation of general
relativity.

## Acknowledgments

The author warmly thanks Marija Dimitrijević Ćirić, Grigorios Giotopoulos,
Hans Nguyen, Voja Radovanović and Alexander Schenkel for collaborations and
discussions over the past few years, upon which this contribution is based. He
would also like to thank the organisors of the Corfu Summer Institute 2021 and
the Humboldt Kolleg on “Quantum Gravity and Fundamental Interactions” for the
stimulating meeting and the invitation to deliver a talk. This work was
supported by the Consolidated Grant ST/P000363/1 from the UK Science and
Technology Facilities Council.

## References

  * [1] G. Giotopoulos and R. J. Szabo, “Braided symmetries in noncommutative field theory,” arXiv:2112.00541 [hep-th].
  * [2] M. Dimitrijević Ćirić, G. Giotopoulos, V. Radovanović and R. J. Szabo, “$L_{\infty}$-algebras of Einstein-Cartan-Palatini gravity,” J. Math. Phys. 61 (2020) 112502 [arXiv:2003.06173 [hep-th]].
  * [3] M. Dimitrijević Ćirić, G. Giotopoulos, V. Radovanović and R. J. Szabo, “Homotopy Lie algebras of gravity and their braided deformations,” Proc. Sci. 376 (2020) 198 [arXiv:2005.00454 [hep-th]].
  * [4] M. Dimitrijević Ćirić, G. Giotopoulos, V. Radovanović and R. J. Szabo, “Braided $L_{\infty}$-algebras, braided field theory and noncommutative gravity,” Lett. Math. Phys. 111 (2021) 148 [arXiv:2103.08939 [hep-th]].
  * [5] R. Blumenhagen and E. Plauschinn, “Nonassociative gravity in string theory?,” J. Phys. A 44 (2011) 015401 [arXiv:1010.1263 [hep-th]].
  * [6] D. Lüst, “T-duality and closed string noncommutative (doubled) geometry,” JHEP 1012 (2010) 084 [arXiv:1010.1361 [hep-th]].
  * [7] R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, “Non-geometric fluxes, asymmetric strings and nonassociative geometry,” J. Phys. A 44 (2011) 385401 [arXiv:1106.0316 [hep-th]].
  * [8] D. Mylonas, P. Schupp and R. J. Szabo, “Membrane sigma-models and quantization of non-geometric flux backgrounds,” JHEP 1209 (2012) 012 [arXiv:1207.0926 [hep-th]].
  * [9] R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst and R. Sun, “Nonassociative deformations of geometry in double field theory,” JHEP 1404 (2014) 141 [arXiv:1312.0719 [hep-th]].
  * [10] R. J. Szabo, “Higher quantum geometry and non-geometric string theory,” Proc. Sci. 318 (2018) 151 [arXiv:1803.08861 [hep-th]].
  * [11] R. J. Szabo, “An introduction to nonassociative physics,” Proc. Sci. 347 (2019) 100 [arXiv:1903.05673 [hep-th]].
  * [12] D. Mylonas, P. Schupp and R. J. Szabo, “Non-geometric fluxes, quasi-Hopf twist deformations and nonassociative quantum mechanics,” J. Math. Phys. 55 (2014) 122301 [arXiv:1312.1621 [hep-th]].
  * [13] G. E. Barnes, A. Schenkel and R. J. Szabo, “Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms,” J. Geom. Phys. 89 (2015) 111–152 [arXiv:1409.6331 [math.QA]].
  * [14] P. Aschieri and R. J. Szabo, “Triproducts, nonassociative star products and geometry of $R$-flux string compactifications,” J. Phys. Conf. Ser. 634 (2015) 012004 [arXiv:1504.03915 [hep-th]].
  * [15] G. E. Barnes, A. Schenkel and R. J. Szabo, “Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature,” J. Geom. Phys. 106 (2016) 234–255 [arXiv:1507.02792 [math.QA]].
  * [16] R. Blumenhagen and M. Fuchs, “Towards a theory of nonassociative gravity,” JHEP 1607 (2016) 019 [arXiv:1604.03253 [hep-th]].
  * [17] P. Aschieri, M. Dimitrijević Ćirić and R. J. Szabo, “Nonassociative differential geometry and gravity with non-geometric fluxes,” JHEP 1802 (2018) 036 [arXiv:1710.11467 [hep-th]].
  * [18] G. E. Barnes, A. Schenkel and R. J. Szabo, “Working with nonassociative geometry and field theory,” Proc. Sci. 263 (2016) 081 [arXiv:1601.07353 [hep-th]].
  * [19] B. Zwiebach, “Closed string field theory: Quantum action and the BV master equation,” Nucl. Phys. B 390 (1993) 33–152 [arXiv:hep-th/9206084].
  * [20] O. Hohm and B. Zwiebach, “$L_{\infty}$-algebras and field theory,” Fortsch. Phys. 65 (2017) 1700014 [arXiv:1701.08824 [hep-th]].
  * [21] B. Jurčo, L. Raspollini, C. Sämann and M. Wolf, “$L_{\infty}$-algebras of classical field theories and the Batalin-Vilkovisky formalism,” Fortsch. Phys. 67 (2019) 1900025 [arXiv:1809.09899 [hep-th]].
  * [22] R. Blumenhagen, I. Brunner, V. G. Kupriyanov and D. Lüst, “Bootstrapping noncommutative gauge theories from $L_{\infty}$-algebras,” JHEP 1805 (2018) 097 [arXiv:1803.00732 [hep-th]].
  * [23] V. G. Kupriyanov, “$L_{\infty}$-bootstrap approach to noncommutative gauge theories,” Fortsch. Phys. 67 (2019) 1910010 [arXiv:1903.02867 [hep-th]].
  * [24] V. G. Kupriyanov and R. J. Szabo, “Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry,” J. Phys. A 55 (2022) 035201 [arXiv:2101.12618 [hep-th]].
  * [25] O. Abla, V. G. Kupriyanov and M. Kurkov, “On the $L_{\infty}$-structure of Poisson gauge theory,” arXiv:2202.10227 [hep-th].
  * [26] A. Nützi and M. Reiterer, “Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization,” arXiv:1812.06454 [math-ph].
  * [27] M. Reiterer and E. Trubowitz, “The graded Lie algebra of general relativity,” arXiv:1812.11487 [math-ph].
  * [28] T. Lada and J. Stasheff, “Introduction to sh Lie algebras for physicists,” Int. J. Theor. Phys. 32 (1993) 1087–1104 [arXiv:hep-th/9209099].
  * [29] T. Lada and M. Markl, “Strongly homotopy Lie algebras,” Commun. Algebra 23 (1995) 2147–2161 [arXiv:hep-th/9406095].
  * [30] B. Jurčo, T. Macrelli, L. Raspollini, C. Sämann and M. Wolf, “$L_{\infty}$-algebras, the BV formalism, and classical fields,” Fortsch. Phys. 67 (2019) 1910025 [arXiv:1903.02887 [hep-th]].
  * [31] S. L. Woronowicz, “Differential calculus on compact matrix pseudogroups (quantum groups),” Commun. Math. Phys. 122 (1989) 125–170.
  * [32] S. Majid, “Quantum and braided Lie algebras,” J. Geom. Phys. 13 (1994) 307–356 [arXiv:hep-th/9303148].
  * [33] P. Aschieri, M. Dimitrijević, F. Meyer and J. Wess, “Noncommutative geometry and gravity,” Class. Quant. Grav. 23 (2006) 1883–1912 [arXiv:hep-th/0510059].
  * [34] P. Aschieri, M. Dimitrijević, P. Kulish, F. Lizzi and J. Wess, “Noncommutative spacetimes: Symmetries in noncommutative geometry and field theory,” Lect. Notes Phys. 774 (2009) 1–199.
  * [35] H. Nguyen, A. Schenkel and R. J. Szabo, “Batalin-Vilkovisky quantization of fuzzy field theories,” Lett. Math. Phys. 111 (2021) 149 [arXiv:2107.02532 [hep-th]].
  * [36] E. Witten, “$(2+1)$-dimensional gravity as an exactly soluble system,” Nucl. Phys. B 311 (1988) 46–78.
  * [37] A. S. Cattaneo and M. Schiavina, “BV–BFV approach to general relativity: Palatini-Cartan-Holst action,” Adv. Theor. Math. Phys. 23 (2019) 1801–1835 [arXiv:1707.06328 [math-ph]].
  * [38] P. Aschieri and L. Castellani, “Noncommutative supergravity in $D=3$ and $D=4$,” JHEP 0906 (2009) 087 [arXiv:0902.3823 [hep-th]].
  * [39] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 9909 (1999) 032 [arXiv:hep-th/9908142].
  * [40] B. Jurčo, S. Schraml, P. Schupp and J. Wess, “Enveloping algebra valued gauge transformations for non-abelian gauge groups on noncommutative spaces,” Eur. Phys. J. C 17 (2000) 521–526 [arXiv:hep-th/0006246].
  * [41] P. Aschieri and L. Castellani, “Noncommutative gravity coupled to fermions: Second order expansion via Seiberg-Witten map,” JHEP 1207 (2012) 184 [arXiv:1111.4822 [hep-th]].
  * [42] P. Aschieri, L. Castellani and M. Dimitrijević, “Noncommutative gravity at second order via Seiberg-Witten map,” Phys. Rev. D 87 (2013) 024017 [arXiv:1207.4346 [hep-th]].