EMPG–21–14

Braided Symmetries in Noncommutative Field Theory

Grigorios Giotopoulos${}^{\,(a),(b),}$***Email<EMAIL_ADDRESS>and Richard J. Szabo${}^{\,(b),(c),}$†††Email<EMAIL_ADDRESS>
(a) Mathematics, Division of Science, New York University Abu Dhabi, U.A.E.

(b) Department of Mathematics, Heriot-Watt University
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.
and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K.
(c) Higgs Centre for Theoretical Physics, Edinburgh, U.K.

We give a pedagogical introduction to $L_{\infty}$-algebras and their uses in
organising the symmetries and dynamics of classical field theories, as well as
of the conventional noncommutative gauge theories that arise as low-energy
effective field theories in string theory. We review recent developments which
formulate field theories with braided gauge symmetries as a new means of
overcoming several obstacles in the standard noncommutative theories, such as
the restrictions on gauge algebras and matter fields. These theories can be
constructed by using techniques from Drinfel’d twist deformation theory, which
we review in some detail, and their symmetries and dynamics are controlled by
a new homotopy algebraic structure called a ‘braided $L_{\infty}$-algebra’. We
expand and elaborate on several novel theoretical issues surrounding these
constructions, and present three new explicit examples: the standard
noncommutative scalar field theory (regarded as a braided field theory), a
braided version of $BF$ theory in arbitrary dimensions (regarded as a higher
gauge theory), and a new braided version of noncommutative Yang–Mills theory
for arbitrary gauge algebras.

Contribution to the Special Issue of Journal of Physics A on ‘Noncommutative
Geometry in Physics’

###### Contents

  1. 1 Introduction and summary
  2. 2 Gauge symmetry and $L_{\infty}$-algebras
    1. 2.1 What is a gauge symmetry?
    2. 2.2 What is an $L_{\infty}$-algebra?
    3. 2.3 Batalin–Vilkovisky formalism
    4. 2.4 $L_{\infty}$-algebras of classical field theories
    5. 2.5 Example: Yang–Mills theory in the $L_{\infty}$-algebra formalism
  3. 3 Noncommutative gauge symmetry and $L_{\infty}$-algebras
    1. 3.1 Star-gauge theories in the $L_{\infty}$-algebra formalism
    2. 3.2 Fuzzy field theories in the $L_{\infty}$-algebra formalism
    3. 3.3 Working with non-trivial fluxes
  4. 4 Braided gauge symmetry and braided $L_{\infty}$-algebras
    1. 4.1 Closing noncommutative gauge transformations
    2. 4.2 Drinfel’d twist deformation theory
    3. 4.3 Braided gauge symmetry and noncommutative kinematics
    4. 4.4 Example: Braided Chern–Simons theory
    5. 4.5 Braided $L_{\infty}$-algebras
    6. 4.6 Braided $L_{\infty}$-algebras of noncommutative field theories
    7. 4.7 Reality conditions
    8. 4.8 Braided BV formalism
    9. 4.9 Moduli spaces of classical solutions
  5. 5 Noncommutative scalar field theory
    1. 5.1 Scalar field theory in the $L_{\infty}$-algebra formalism
    2. 5.2 Noncommutative scalar field theory in the braided $L_{\infty}$-algebra formalism
    3. 5.3 Global braided symmetry
    4. 5.4 Example: Braided Lorentz invariance
  6. 6 Braided $BF$ theory
    1. 6.1 $BF$ theory in the $L_{\infty}$-algebra formalism
    2. 6.2 Noncommutative $BF$ theory in the braided $L_{\infty}$-algebra formalism
    3. 6.3 Higher braided gauge symmetry
  7. 7 Braided noncommutative Yang–Mills theory
    1. 7.1 Noncommutative Yang–Mills theory in the braided $L_{\infty}$-algebra formalism
    2. 7.2 Braided gauge symmetry
    3. 7.3 Global braided symmetry

##  1 Introduction and summary

In this paper we review some recent developments which offer a new perspective
on symmetries in noncommutative field theory; in this paper, by a
‘noncommutative field theory’ we will mean a field theory that is a
_deformation_ of a classical field theory (and in almost all instances via a
star-product on the relevant algebras of fields). To motivate and introduce
these investigations, we begin with a flash review of one of the best
understood natural occurrence of noncommutative geometry in a candidate theory
of quantum gravity, namely open string theory on spacetimes with non-trivial
background NS–NS ($\text{NS}=\text{Neveu--Schwarz}$) fields.

#### D-branes and noncommutative gauge theory

Consider open string theory on flat spacetime in the background of a constant
Kalb–Ramond $B$-field. With suitable boundary conditions the open strings are
constrained to move in directions transverse to a submanifold
$\mathbbm{R}^{1,p}\subseteq\mathbbm{R}^{1,9}$ which is the worldvolume wrapped
by a stack of coincident flat D$p$-branes. It is a classic and remarkable
result that the effects of the $B$-field on open string interactions in
worldsheet conformal field theory correlation functions on the disk are
completely captured by the noncommutative Moyal–Weyl star-product111Notation:
$x=(x^{\mu})$ are (local) coordinates on the worldvolume $\mathbbm{R}^{1,p}$,
and $\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$ are vector fields of
the corresponding (local) holonomic frame.

$\displaystyle\begin{split}f\star
g&=\mu\circ\exp\big{(}\mbox{$\frac{\mathrm{i}\,\hbar}{2}$}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}(f\otimes
g)\\\\[4.0pt] &=f\cdot
g+\sum_{n=1}^{\infty}\,\Big{(}\frac{\mathrm{i}\,\hbar}{2}\Big{)}^{n}\,\frac{1}{n!}\,\theta^{\mu_{1}\nu_{1}}\cdots\theta^{\mu_{n}\nu_{n}}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}f\cdot\partial_{\nu_{1}}\cdots\partial_{\nu_{n}}g\
,\end{split}$ (1.1)

where $\mu(f\otimes g)=f\cdot g$ is the pointwise product of fields $f$ and
$g$, $\hbar$ is treated as a formal deformation parameter222Recall that there
are also convergent integral expressions for the Moyal–Weyl star-product,
defined on the space of Schwartz functions on $\mathbbm{R}^{1,p}$, whose
asymptotic series expansions in $\hbar$ coincide with (1.1), see e.g. [84].
and the constant Poisson bivector
$\theta=\frac{1}{2}\,\theta^{\mu\nu}\,\partial_{\mu}\wedge\partial_{\nu}$ is
related to the pullback of the $2$-form $B$ on $\mathbbm{R}^{1,9}$ to
$\mathbbm{R}^{1,p}$ [48, 5, 39, 82, 83]. In a suitable low-energy scaling
limit, the boundary operator product expansion of tachyon vertex operators
becomes exactly the Moyal–Weyl star-product.

The massless bosonic modes of the open strings consist of gauge fields
$A=A_{\mu}\,\mathrm{d}x^{\mu}$ on the worldvolume and scalar fields $X^{a}$
describing the transverse fluctuations of the D-branes. Their low-energy
dynamics is described by a _noncommutative gauge theory_ on
$\mathbbm{R}^{1,p}$ with gauge group $\mathsf{U}(n)$, where $n$ is the number
of coincident D$p$-branes. This noncommutative field theory is defined by
deformation of the standard (dimensionally reduced) Yang–Mills action on
$\mathbbm{R}^{1,p}\subseteq\mathbbm{R}^{1,9}$ via replacements of ordinary
pointwise matrix products of fields by the Moyal–Weyl star-product (1.1)
composed with matrix multiplication. See [49, 84] for extensive reviews of
this subject.

This story extends to curved D-branes in flat space, where the Poisson
structure $\theta$ is no longer constant and the Moyal–Weyl star-product (1.1)
is replaced by the more general Kontsevich star-product [63], and also to
curved backgrounds with a non-zero pullback of the NS–NS $3$-form flux
$H=\mathrm{d}B$ to the D-brane worldvolume [40, 54]. A standard class of
examples comes from D-branes in Wess–Zumino–Witten (WZW) models [1, 2], and
more recently as holographic dual gauge theories to backgrounds obtained via
integrable deformations of string sigma-models whose target space is
$\mathbbm{A}\mathbbm{d}\mathbbm{S}_{5}{\times}\mathbbm{S}^{5}$ [89, 4].

Despite their intensive investigation for over 20 years by various communities
in physics and mathematics, there are still many open general problems
surrounding the construction and description of these gauge theories on curved
D-branes. Foremost amongst these problems is the structure of the algebra of
star-gauge transformations. Consider for definiteness $\mathsf{U}(1)$
noncommutative gauge theory on a manifold $M$ with the star-commutator

$\displaystyle[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]:=\lambda\star
A-A\star\lambda$ (1.2)

among a gauge parameter $\lambda\in\Omega^{0}(M)=C^{\infty}(M,\mathbbm{R})$
and a gauge field $A\in\Omega^{1}(M)$. Then the naive definition of a star-
gauge transformation

$\displaystyle\delta_{\lambda}^{\star}A=\mathrm{d}\lambda+[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]$
(1.3)

will generally obstruct the closure of the gauge algebra:

$\displaystyle\big{[}\delta^{\star}_{\lambda_{1}},\delta^{\star}_{\lambda_{2}}\big{]}^{\phantom{{\dagger}}}_{\circ}A:=\big{(}\delta^{\star}_{\lambda_{1}}\circ\delta^{\star}_{\lambda_{2}}-\delta^{\star}_{\lambda_{2}}\circ\delta^{\star}_{\lambda_{1}}\big{)}A\
\neq\
\delta^{\star}_{[\lambda_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda_{2}]}A\
.$ (1.4)

This problem is related to the failure of the Leibniz rule of the de Rham
differential $\mathrm{d}$ for a general non-constant Poisson bivector
$\theta$. For instance

$\displaystyle\mathrm{d}(f\star g)\ \neq\ \mathrm{d}f\star
g+f\star\mathrm{d}g$ (1.5)

for functions $f,g\in C^{\infty}(M)$. This means that in general there is no
well-defined noncommutative differential calculus on $M$ from which to obtain
the standard ingredients of a field theory.

Both of these issues are related to the problem of firstly extending the
corresponding Poisson algebra on $C^{\infty}(M,\mathbbm{R})$ to the exterior
algebra $\Omega^{\bullet}(M)$ of differential forms on $M$ in such a way as to
extend the de Rham complex $\big{(}\Omega^{\bullet}(M),\mathrm{d}\big{)}$ to a
differential graded Poisson algebra, and subsequently quantizing this to a
noncommutative differential graded algebra. There is a fairly extensive
literature of investigations into this problem through different approaches
during the last two decades, most of which involve a choice of an auxiliary
(symplectic) connection and work only at the semi-classical (Poisson) level
without achieving the second step of quantization; see e.g. [55]. Here we
focus instead on recent approaches to this problem which are tied to more
modern perspectives on classical and quantum field theory.

#### Homotopy algebras

Higher structures in string theory and quantum field theory have become more
and more prominent in the last few years. In this paper we study occurrences
of ‘higher algebras’, and in particular homotopy Lie algebras. Their role in
physics makes nice contact with recent mathematical developments in
deformation theory and derived geometry. Homotopy algebraic techniques and
concepts have also recently been shown to offer a natural arena and physically
motivated tools for formulating the closure of the noncommutative gauge
algebra, as well as constructing a differential graded algebra structure on
the fields and their dynamics. As these developments are central to the
discussion of this paper, let us offer a brief and somewhat sketchy history of
the appearance of $L_{\infty}$-algebras in physics and mathematics,
emphasising those occurrences which fit contextually into our treatment. More
precise definitions and explanations will be given later on in this article.

The origins of $L_{\infty}$-algebras in physics can be traced back to early
work from the 1980s on higher spin gauge theories [25], where it was observed
that the closure of the gauge algebra takes place in an extended space of
field dependent gauge parameters:

$\displaystyle\big{[}\delta_{\lambda_{1}},\delta_{\lambda_{2}}\big{]}^{\phantom{{\dagger}}}_{\circ}\Phi=\delta_{C(\lambda_{1},\lambda_{2};\Phi)}\Phi\
,$ (1.6)

where here and below $\Phi$ symbolically denotes the collection of fields of
the theory, and $\lambda$ the corresponding gauge parameters. The first full
structural occurrence of $L_{\infty}$-algebras appeared in the early 1990s in
the work of Zwiebach on closed string field theory [93], where the
generalized333Here and in the following by ‘generalized’ we mean local
symmetries of a field theory which do not necessarily correspond to
infinitesimal automorphisms of a principal bundle formulation. gauge
symmetries were shown to involve infinitely-many higher bracket operations
$\ell_{n}$ for $n\geq 1$:

$\displaystyle\delta_{\lambda}\Phi=\ell_{1}(\lambda)+\sum_{n=1}^{\infty}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n+1}(\lambda,\Phi^{\otimes n})\ .$ (1.7)

Inspired by these algebraic structures from string field theory, it has been
recently emphasised by [56, 60] that the dynamics of any perturbative444By
‘perturbative’ here we mean theories with only polynomial interactions among
the fields. classical field theory with generalized gauge symmetries is
naturally organised by an $L_{\infty}$-algebra.555Our conventions for the
brackets $\ell_{n}$ agree with [56], whereas the operations $\mu_{n}$ of [60]
are related to ours by a sign: $\mu_{n}=(-1)^{n\choose 2}\,\ell_{n}$. This
observation is useful because it enables one to develop new techniques for
studying perturbative quantum field theories in a unifed setting using the
powerful tools of homotopical algebra, see e.g. [72, 70, 6, 31] for some
recent applications.

The connection between homotopy algebras and field theories arises because
$L_{\infty}$-algebras are the natural algebraic structure underlying the
Batalin–Vilkovisky (BV) formalism [60]. Mathematically, this is due to the
duality between $L_{\infty}$-algebras and differential graded commutative
algebras [69]. These are also useful tools in deformation theory. For example,
Kontsevich’s formality theorem [63], which lies at the heart of deformation
quantization of Poisson manifolds, is based on $L_{\infty}$-quasi-isomorphisms
between differential graded Lie algebras, namely the Schouten–Nijenhuis
algebra of multivector fields and the Gerstenhaber algebra of differential
operators.

These developments have recently inspired systematic descriptions of
noncommutative gauge theories on curved D-branes based on
$L_{\infty}$-algebras [29, 65] (homotopy coherent weakenings of Lie algebras)
and also on $P_{\infty}$-algebras [67] (homotopy coherent weakenings of
Poisson algebras). However, while these approaches are superior to earlier
ones in that they do not involve auxiliary choices such as a symplectic
connection, they have also so far not been understood beyond the semi-
classical level. The purpose of this paper is to review more recent
developments which use homotopy algebras to go beyond the Poisson
approximation, and to use these techniques to present new examples of
noncommutative gauge theories that are based on fully fledged noncommutative
differential graded algebras for the first time.

#### Non-geometric backgrounds and noncommutative gravity

The new homotopy algebraic techniques which we review are in part inspired by
much less concrete and understood appearances of noncommutative geometry in
_closed_ string theory. On general grounds, it is expected that noncommutative
theories of gravity should serve as low-energy limits of any theory of quantum
gravity, retaining some features of a quantized spacetime. In particular, in
certain non-geometric flux backgrounds, it has been conjectured that the low-
energy dynamics of closed strings may be described by a theory of
noncommutative gravity [27, 71, 76] (see [87] for a review). One of the hopes
here is that noncommutative geometry may serve as a more precise description
of the notion of ‘non-geometry’; see e.g. [57] for how this works in the
_open_ string sector, and [12] for a recent rigorous perspective, where the
target space geometry probed by D-branes is a noncommutative torus bundle.
This is difficult to check explicitly, physically because the low-energy
limits are much harder to control as compared to the open string sector, and
mathematically because the metric aspects of noncommutative differential
geometry are only partially developed so far [19, 16, 26, 14, 9]. In
particular, no general version of the Einstein–Hilbert action functional is
currently known.

This has led to attempts to treat noncommutative gravity as a deformation of a
‘gauge theory’, that is, in the first order formalism for general relativity:
One uses the Einstein–Cartan principal bundle formulation, with star-gauge
symmetry, where the corresponding variational principle is based on the
Palatini action functional [38, 34, 10, 22, 42]. However, diffeomorphisms can
never be implemented as star-gauge symmetries, in any formalism, and instead
the noncommutative theory of gravity is invariant under a ‘twisted’ action of
infinitesimal diffeomorphisms [19, 16]. This dichotomy between the local
Lorentz and diffeomorphism symmetries presents a serious obstruction to the
formulation of noncommutative gravity for general backgrounds in the setting
of homotopy algebras, because the twisted diffeomorphism symmetry does not
appear to fit naturally into the standard $L_{\infty}$-algebra picture of
star-gauge symmetries.

These issues prompt another approach. Firstly, we resolve the dichotomy
between the different types of symmetries by letting the local Lorentz
rotations instead act in a ‘twisted’ manner, like the diffeomorphisms, which
replaces the notion of star-gauge symmetry with _braided gauge symmetry_.
Secondly, we do not attempt to package this symmetry into a standard
$L_{\infty}$-algebra, which indeed does not seem possible to do in a closed or
algorithmic manner, but rather deform the underlying $L_{\infty}$-algebra
itself to make it compatible with the new type of symmetry. This results in a
systematic construction of new theories of noncommutative gravity, which were
formulated in detail by [44, 45].

#### Summary and outlook

The procedure discussed above works without any reference to gravity (even
though this is what motivated the construction), and leads to both a new
symmetry principle for noncommutative field theories from a physics
perspective, as well as a new homotopy algebraic structure from a mathematics
perspective, that was called a _braided $L_{\infty}$-algebra_ in [44, 45].
Noncommutative field theories whose symmetries and dynamics are organised by
braided $L_{\infty}$-algebras are correspondingly called _braided field
theories_. The purpose of this contribution is to review the definition of
noncommutative field theories with braided gauge symmetries, together with
their formulation through a general notion of braided ${L_{\infty}}$-algebras,
and to explain how the mathematical machinery constructs novel examples with
exciting new features. We further expand and elucidate various details of the
general theory which were not addressed in [45]. In addition to reviewing the
basic example of braided Chern–Simons theory from [45], we present three new
examples: the standard noncommutative scalar field theory (regarded as a
braided field theory), braided $BF$ theory in any dimension (which serves as
an elementary example of a higher gauge theory), and a new braided version of
noncommutative Yang–Mills theory for an arbitrary gauge algebra. The reader
interested in the applications to noncommutative gravity is invited to consult
[45] where a detailed exposition can be found.

We should stress that the notion of ‘braided gauge symmetry’ that we review
here is not really new: kinematical aspects of this idea have appeared long
before (see e.g. [33]), while some concepts and techniques are borrowed from
twisted noncommutative gravity, for which there is a fairly extensive
literature. What is new however is the application of these techniques to
ordinary (non-gravitational) gauge theories, the full treatment of braided
noncommutative dynamics through the organising principle of a braided
$L_{\infty}$-algebra, and the new examples of noncommutative field theories
with braided symmetries that this produces. We further emphasise that the
braided noncommutative gauge and gravity theories considered here and in [45]
are _not_ the same as the ‘twisted’ noncommutative gauge and gravity theories
considered in the mid-2000s (see e.g. [19, 18]). In particular, braided
symmetries are based on braided derivations closing a braided Lie algebra,
whereas twisted symmetries are based on a classical Lie algebra and a deformed
Leibniz rule following from twist deformation of the coproduct of the
associated enveloping Hopf algebra.

There are several obvious open questions which we do not address in this
contribution, as these are still under development in this very new subject.
For instance, what do the corresponding quantum field theories look like? The
quantization of our noncommutative field theories are related to Oeckl’s
symmetric braided quantum field theory [78, 79], which however does not
consider theories with gauge symmetries. Preliminary investigations in this
direction within the context of fuzzy field theories are found in [77], which
should be related to the lattice regularization of braided quantum field
theory discussed in [41]. A simple example of a continuum field theory with
gauge symmetry is under development in [47].

Another immediate question concerns the explicit realizations of our braided
noncommutative field theories in string theory (or in other theories of
quantum gravity). For this, one could consider the Hopf algebraic symmetries
of string worldsheet conformal field theory correlation functions discussed by
[7, 8], and then mimick the route used derive the standard noncommutative
field theories. This approach should be made feasible by comparing with the
Ward–Takahashi identities of braided quantum field theory developed in [81].
Such a development then further has the exciting potential of providing the
very first precise derivation of a (braided) noncommutative theory of gravity
from string theory, bypassing previous no-go results concerning the
realisation of twisted diffeomorphisms as symmetries of string theory in the
low-energy limit [3, 26]. This will rely on a deeper understanding of the Hopf
algebra symmetry of string theory with a $B$-field background, which can be
understood by viewing quantisation of the string using Drinfel’d twist
deformations [8]. Braided diffeomorphisms which are consistent with the closed
string sector are necessarily different from their actions in the low-energy
effective theory on D-branes, as the latter is known to be insufficient for
describing symmetries in brane-induced gravity [3]. The twisted Hopf algebras
of [8] enable extensions to couplings of D-branes to the closed string sector,
or even to closed string amplitudes. Braided noncommutative deformations of
string theory are also argued by [89] to underlie the AdS/CFT dual gauge
theories to Yang–Baxter deformations of
$\mathbbm{AdS}_{5}{\times}\mathbbm{S}^{5}$ string sigma-models. Understanding
these approaches better may realise how the noncommutative geometry probed by
D-branes carries over naturally to the bulk closed string sector.

#### Outline of the paper

Throughout this paper we have endeavoured to give a pedagogical and relatively
self-contained presentation of both the relevant physical and mathematical
concepts. We provide a fairly in-depth review of $L_{\infty}$-algebras and
their uses in formulating classical field theory, and how the formalism
immediately adapts to some standard examples of conventional noncommutative
gauge theories. We review only the salient mathematical aspects of homotopy
algebras and Drinfel’d twist deformation techniques that we need, but
formulated in a language that we hope is palatable to both novices and experts
alike. We elaborate and expand on various aspects of the notion of braided
$L_{\infty}$-algebra that was introduced by [45], as well as of their
applications to braided field theories. In particular, we expand the
repertoire of examples presented in [45] beyond the realm of diffeomorphism
invariant theories.

We begin in Section 2 by giving a fairly detailed review of the
$L_{\infty}$-algebra formulation of classical field theories, starting with a
pedagogical review of gauge symmetries through the prototypical example of
Chern–Simons gauge theory. We then explain the notion of an
$L_{\infty}$-algebra and its connection with the BV formalism, in a way that
is immediately adaptable to the braided generalizations that we consider later
on in the paper. We further elucidate on the role of higher gauge
transformations in field theories with reducible gauge symmetries, and in
particular their dual notion of ‘higher Noether identities’ which are not
usually stressed in the literature, but which are of paramount importance in
braided field theory, as we discuss. We also discuss the basic example of
Yang–Mills theory whose homotopy algebraic structure goes beyond the
differential graded Lie algebra organising Chern–Simons theory.

In Section 3 we revisit some standard examples of noncommutative gauge
theories which arise as low-energy effective field theories on D-branes in
string theory, and how the classical $L_{\infty}$-algebra formulation easily
adapts to them using the examples considered in Section 2. These include the
conventional noncommutative field theories on flat Minkowski spacetime with
Moyal–Weyl star-gauge symmetry, as well as the fuzzy field theories supported
by curved D-branes in WZW models, treating the examples of gauge theories on
the fuzzy sphere in detail. We further discuss the issues involved in adapting
the standard $L_{\infty}$-algebra picture to generic examples of
noncommutative gauge theories on D-branes in curved backgrounds, which serves
as a (partial) motivation for moving outside of the framework of classical
homotopy algebras.

With the classical framework and its limitations in noncommutative field
theory understood, we proceed in Section 4 to our main topic of braided gauge
symmetry and the accompanying notion of a braided $L_{\infty}$-algebra. We
start by reviewing some well-known issues with the closure of standard
noncommutative gauge transformations and diffeomorphisms, which motivates the
use of Drinfel’d twist deformation techniques that we explain in detail. These
methods naturally lead to the notions of braided gauge symmetry and associated
noncommutative kinematics. Like we did in the classical case, we illustrate
the novel ensuing properties through the simplest example of a braided version
of Chern–Simons gauge theory, which inspires a natural braided deformation of
the definition of an $L_{\infty}$-algebra. We expand on several technical
points surrounding the mathematical presentation of braided
$L_{\infty}$-algebras in [45], and their applications to the unambiguous
construction of noncommutative field theories, such as their reality
properties. Following [77] we explain the relation to a braided version of the
BV formalism, and we further discuss some preliminary ideas on how to make
sense of moduli spaces of classical solutions in braided field theories, which
seems to resolve some paradoxical observations in a way that is common to the
usual thinking in noncommutative geometry, but which requires a great deal of
further technical development to make precise.

In the remaining three sections we proceed to analyse new explicit examples in
the braided $L_{\infty}$-algebra formalism. We revisit the standard
noncommutative scalar field theory with polynomial interactions in Section 5
and show that it is naturally encoded in a braided $L_{\infty}$-algebra. As a
_braided_ field theory, this example exhibits a number of interesting features
compared to its incarnation in the usual noncommutative field theory
framework. In particular, we address the well-known problem of implementation
of global symmetries in noncommutative field theory and show that they too
have a natural realisation in the language of braided $L_{\infty}$-algebras,
treating the traditional example of Lorentz transformations in detail.

In Section 6 we turn to the natural extensions of Chern–Simons gauge theory to
arbitrary dimensions provided by $BF$ theories. After reviewing their
classical $L_{\infty}$-algebra formulation, we describe their braided
noncommutative deformations following the braided $L_{\infty}$-algebra
prescription. These theories provide simple illustrative examples of gauge
symmetries that are reducible, which are naturally described in the
$L_{\infty}$-algebra framework. We discuss the conceptual issues surrounding
the braided versions of these higher gauge symmetries, and how they may be
resolved along the lines of our sketch of the description of the classical
moduli spaces of braided field theories.

Finally, in Section 7 we proceed to our main example. By twist deforming the
$L_{\infty}$-algebra structure of classical Yang–Mills theory, we obtain a new
braided version of noncommutative Yang–Mills theory for arbitrary gauge
algebras. Unlike the examples of scalar field theory, or the topological gauge
theories which are based on differential graded Lie algebras, the braided
noncommutative Yang–Mills theory does not follow the obvious deformation of
its classical counterpart, like it does in conventional noncommutative field
theories. We derive the modified field equations, Noether identities and
action functional in detail, and we further discuss the implementation of
global symmetries of the theory in the braided $L_{\infty}$-algebra framework.
We provide all necessary details that we hope will inspire further
investigations of this somewhat complicated but interesting gauge theory, and
in particular a detailed comparison with its (unbraided) standard
noncommutative counterpart, but we leave this for future work. Of particular
immediate interest are basic questions concerning the corresponding braided
quantum field theory, such as unitarity and the behaviour of perturbation
theory.

#### Acknowledgments

We thank Marija Dimitrijević Ćirić, Larisa Jonke, Voja Radovanović, Alexander
Schenkel and Francesco Toppan for helpful discussions and correspondence.
R.J.S. thanks the editors Paolo Aschieri, Edwin Beggs, Francesco D’Andrea,
Emil Prodan and Andrzej Sitarz for the invitation to contribute to this
special issue. The work of G.G. was supported by a Doctoral Training Grant
from the UK Engineering and Physical Sciences Research Council. The work of
R.J.S. was supported by the Consolidated Grant ST/P000363/1 from the UK
Science and Technology Facilities Council.

##  2 Gauge symmetry and $L_{\infty}$-algebras

In this section we will give a basic and pedagogical introduction to the
$L_{\infty}$-algebra formulation of classical field theories with generalized
gauge symmetries, reviewing the necessary mathematical concepts which we will
later generalize to the case of noncommutative field theories with braided
gauge symmetries.

###  2.1 What is a gauge symmetry?

To motivate the uses of $L_{\infty}$-algebras in classical field theory, as
well as to highlight some salient features of the notion of gauge symmetry
that we will need later on, we begin with a simple prototypical model which
illustrates most essential features of more general field theories, like the
ones discussed in Section 1. We consider the well-known example of
Chern–Simons theory on a closed oriented three-dimensional manifold $M$. Let
$\mathfrak{g}$ be a quadratic Lie algebra, that is, $\mathfrak{g}$ is a real
vector space equipped with a Lie bracket
$[-,-]_{\mathfrak{g}}:\mathfrak{g}\wedge\mathfrak{g}\to\mathfrak{g}$ and a
non-degenerate bilinear pairing
$\mathrm{Tr}_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\to\mathbbm{R}$
which is invariant under the natural adjoint action of $\mathfrak{g}$ on
itself:

$\displaystyle\mathrm{Tr}_{\mathfrak{g}}\big{(}[X,Z]_{\mathfrak{g}}\otimes
Y+Z\otimes[X,Y]_{\mathfrak{g}}\big{)}=0\ ,$ (2.1)

for all $X,Y,Z\in\mathfrak{g}$. We denote by
$\Omega^{\bullet}(M,\mathfrak{g}):=\Omega^{\bullet}(M)\otimes\mathfrak{g}$ the
graded Lie algebra of $\mathfrak{g}$-valued differential forms on $M$ with Lie
bracket given by the tensor product of the Lie bracket of $\mathfrak{g}$ with
exterior multiplication; by a slight abuse of notation we also denote this
extended Lie bracket by $[-,-]_{\mathfrak{g}}$. The de Rham differential acts
non-trivially only on the form part of $\Omega^{\bullet}(M,\mathfrak{g})$, and
we denote $\mathrm{d}\otimes\mathbbm{1}$ simply by $\mathrm{d}$ here and in
the following.

The Chern–Simons action functional
$S:\Omega^{1}(M,\mathfrak{g})\to\mathbbm{R}$ for a gauge field
$A\in\Omega^{1}(M,\mathfrak{g})$ is defined by

$\displaystyle
S(A)=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\Big{(}\frac{1}{2}\,A\wedge\mathrm{d}A+\frac{1}{3!}\,A\wedge[A,A]_{\mathfrak{g}}\Big{)}\
.$ (2.2)

Despite its seemingly non-covariant dependence on $A$, this action functional
is invariant under the standard infinitesimal gauge transformations

$\displaystyle\delta_{\lambda}A=\mathrm{d}\lambda+[A,\lambda]_{\mathfrak{g}}$
(2.3)

for $\lambda\in\Omega^{0}(M,\mathfrak{g})$, that is, $\delta_{\lambda}S(A)=0$.
Using the graded derivation property of the de Rham differential $\mathrm{d}$
with respect to the Lie bracket $[-,-]_{\mathfrak{g}}$ and the Jacobi identity
for $[-,-]_{\mathfrak{g}}$, one easily verifies that the gauge variations
close off-shell to the Lie algebra $\Omega^{0}(M,\mathfrak{g})$:

$\displaystyle\big{[}\delta_{\lambda_{1}},\delta_{\lambda_{2}}\big{]}_{\circ}A:=\big{(}\delta_{\lambda_{1}}\circ\delta_{\lambda_{2}}-\delta_{\lambda_{2}}\circ\delta_{\lambda_{1}}\big{)}A=\delta_{[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}}A\
.$ (2.4)

Varying (2.2) with respect to arbitrary variations $\delta A$ of the gauge
fields, one easily derives the corresponding Euler–Lagrange equations
$\mathfrak{F}_{A}=0$, which are given in terms of the curvature of the
connection $A$ as

$\displaystyle\mathfrak{F}_{A}=F_{A}:=\mathrm{d}A+\tfrac{1}{2}\,[A,A]_{\mathfrak{g}}=0\
\in\ \Omega^{2}(M,\mathfrak{g})\ .$ (2.5)

The critical locus of the Chern–Simons functional thus consists of flat
$\mathfrak{g}$-connections on the three-manifold $M$. Unlike the gauge field
$A$, the equations of motion transform covariantly under gauge transformations
by virtue of the covariance of the curvature $2$-form $F_{A}$:

$\displaystyle\delta_{\lambda}\mathfrak{F}_{A}=-[\lambda,\mathfrak{F}_{A}]_{\mathfrak{g}}\
.$ (2.6)

This implies that the gauge symmetry acts on the classical solutions of
Chern–Simons theory because

$\displaystyle\mathfrak{F}_{A+\delta_{\lambda}A}=\mathfrak{F}_{A}+\delta_{\lambda}\mathfrak{F}_{A}+O(\lambda^{2})\
,$ (2.7)

and so the first order gauge variation of a solution $A$ to
$\mathfrak{F}_{A}=0$ produces another solution $A+\delta_{\lambda}A$. Thus
there are gauge redundancies in the description of the classical degrees of
freedom, and the physical states are gauge orbits of classical solutions. The
space of physical states of the classical field theory is therefore the moduli
space of classical solutions modulo gauge transformations, which in this case
is the moduli space $\mathscr{M}_{\rm flat}(M,\mathfrak{g})$ of flat
$\mathfrak{g}$-connections on $M$.666For the time being we do not go into any
technical details on how one should make rigorous sense of such a moduli
space, which is possible but we are solely interested in a motivational
discussion here. We will return to a more precise description of this and
other related moduli spaces in Section 4.9.

What is important for us is an equivalent but perhaps less widely appreciated
perspective on the gauge redundancies. The key observation, that has already
been implicitly used, is that conventional gauge transformations
$\delta_{\lambda}A$ are just special instances of general field variations
$\delta A$ along specified directions in the tangent space to the space of
fields $\Omega^{1}(M,\mathfrak{g})$. The gauge variation of the action
functional (2.2) for an arbitrary gauge parameter
$\lambda\in\Omega^{0}(M,\mathfrak{g})$ can be computed as

$\displaystyle\delta_{\lambda}S(A)=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\delta_{\lambda}A\wedge\mathfrak{F}_{A}\big{)}=-\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}\big{(}\lambda\wedge\mathrm{d}^{A}\mathfrak{F}_{A}\big{)}\
,$ (2.8)

where

$\displaystyle\mathrm{d}^{A}\mathfrak{F}_{A}=\mathrm{d}\mathfrak{F}_{A}+[A,\mathfrak{F}_{A}]_{\mathfrak{g}}\
\in\ \Omega^{3}(M,\mathfrak{g})$ (2.9)

is the gauge covariant derivative of the Euler–Lagrange equation
$\mathfrak{F}_{A}=F_{A}$. The first equality follows from an integration by
parts as well as ${\rm ad}(\mathfrak{g})$-invariance (2.1) using the Leibniz
rule for the gauge variations $\delta_{\lambda}$, and is the same calculation
one does when deriving the equations of motion (2.5). The second equality
similarly follows using (2.3).

Of course, we know that

$\displaystyle\mathrm{d}^{A}\mathfrak{F}_{A}=\mathrm{d}^{A}F_{A}=0$ (2.10)

by the standard Bianchi identity for the curvature $F_{A}$, and this is what
guarantees the gauge invariance of the action functional. On the other hand,
if we didn’t know about the Bianchi identity, we would have deduced it from
the requirement of gauge invariance of the Chern–Simons functional. Thus the
Bianchi identity $\mathrm{d}^{A}\mathfrak{F}_{A}=0$ is _equivalent_ to gauge
invariance of the action functional $\delta_{\lambda}S(A)=0$ for all
$\lambda\in\Omega^{0}(M,\mathfrak{g})$.

What we have just explained is a simple example of the more general statement
of

######  Theorem 2.11 (Noether’s second theorem).

Gauge symmetries of a classical field theory are in a one-to-one
correspondence with differential identities among its Euler–Lagrange
derivatives.

The proof is a generalization of the computation in (2.8). We stress that
Noether’s second theorem asserts the existence of _identities_ , which hold
off-shell, i.e. when $\mathfrak{F}_{A}\neq 0$.777This is not to be confused
with the more commonly known Noether’s _first_ theorem, which is an on-shell
statement asserting the existence of weakly conserved currents (and
corresponding weakly conserved charges) for every continuous global symmetry
of a classical field theory. These identities are called _Noether identities_
and they exhibit the interdependence of the classical degrees of freedom due
to the gauge symmetries, offering a different perspective on the gauge
redundancies of the field theory. In particular, the converse of Noether’s
second theorem is a means of recovering gauge symmetries of an action
functional $S$ which may be unknown _a priori_. Notice that Chern–Simons
theory is a special case where the geometric Bianchi identity coincides with
the Noether identity, but in general Noether identities do not coincide with
Bianchi identities.

Let us now recap what mathematically went into the description of classical
Chern–Simons theory. To describe the classical moduli space we relied on two
ingredients:

  * •

The graded vector space

$\displaystyle V=\Omega^{\bullet}(M,\mathfrak{g})=V^{0}\oplus V^{1}\oplus
V^{2}\oplus V^{3}\ ,$ (2.12)

where $V^{p}=\Omega^{p}(M,\mathfrak{g})$ is the space of $\mathfrak{g}$-valued
$p$-forms on the three-manifold $M$. The homogeneous subspace for $p=0$
contains the gauge parameters $\lambda$, for $p=1$ the fields $A$, for $p=2$
the field equations $\mathfrak{F}_{A}$, and for $p=3$ the Noether identities
$\mathrm{d}^{A}\mathfrak{F}_{A}=0$.

  * •

The ‘brackets’

$\displaystyle\ell_{1}=\mathrm{d}\qquad\mbox{and}\qquad\ell_{2}=-[-,-]_{\mathfrak{g}}\
.$ (2.13)

The linear map $\ell_{1}:V\to V$ of degree $1$ is a differential making $V$
into a cochain complex, which in this case is just the de Rham complex
$\big{(}\Omega^{\bullet}(M,\mathfrak{g}),\mathrm{d}\big{)}$ of
$\mathfrak{g}$-valued differential forms on $M$. The bilinear map
$\ell_{2}:V\otimes V\to V$ of degree $0$ is a graded Lie bracket on $V$, i.e.
it is graded antisymmetric and satisfies the graded Jacobi identity, and
$\ell_{1}$ is a derivation of $\ell_{2}$. With these brackets, the gauge
transformations, field equations and Noether identites can be written in the
form

$\displaystyle\begin{split}\delta_{\lambda}A=\ell_{1}(\lambda)+\ell_{2}(\lambda,A)&\qquad,\qquad\mathfrak{F}_{A}=\ell_{1}(A)-\tfrac{1}{2}\,\ell_{2}(A,A)\
,\\\\[4.0pt] \mathrm{d}^{A}\mathfrak{F}_{A}=&\
\ell_{1}(\mathfrak{F}_{A})+\ell_{2}(\mathfrak{F}_{A},A)\ ,\end{split}$ (2.14)

with gauge closure and covariance represented by

$\displaystyle\big{[}\delta_{\lambda_{1}},\delta_{\lambda_{2}}\big{]}_{\circ}A=\delta_{-\ell_{2}(\lambda_{1},\lambda_{2})}A\qquad\mbox{and}\qquad\delta_{\lambda}\mathfrak{F}_{A}=\ell_{2}(\lambda,\mathfrak{F}_{A})\
.$ (2.15)

In order to derive the dynamics and Noether identities from a variational
principle, we further required:

  * •

The ‘cyclic inner product’

$\displaystyle\langle\alpha,\beta\rangle=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(\alpha\wedge\beta)$
(2.16)

on $V$, which pairs differential forms $\alpha$ and $\beta$ on $M$ in
complementary degrees; in other words, $\langle-,-\rangle:V\otimes
V\to\mathbbm{R}$ is a bilinear map of degree $-3$, and so is non-vanishing
only on the homogeneous subspaces $V^{0}\otimes V^{3}$ and $V^{1}\otimes
V^{2}$. With it, the action functional is constructed by pairing a gauge field
$A\in V^{1}$ with $\ell_{1}(A)\in V^{2}$ and $\ell_{2}(A,A)\in V^{2}$ through

$\displaystyle S(A)=\tfrac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\tfrac{1}{3!}\,\langle A,\ell_{2}(A,A)\rangle\ .$ (2.17)

The cyclic property in this case is just the statement of
$\ell_{1}$-invariance of the pairing, which follows from integration by parts,
and of ${\rm ad}(\mathfrak{g})$-invariance with respect to the brackets
(2.13), both of which were used when varying (2.17) to derive the equations of
motion as well as the Noether identities.

The first two items simply say that the triple
$\big{(}\Omega^{\bullet}(M,\mathfrak{g}),\ell_{1},\ell_{2})$ is a differential
graded Lie algebra, while the third item equips this with a cyclic structure.
In other words, the symmetries and dynamics of Chern–Simons gauge theory are
governed in an algebraic way by the objects and relations of a _cyclic
differential graded Lie algebra_. This is the simplest prototypical example of
a more general statement: Any perturbative classical field theory with
generalized gauge symmetries is organised by a _cyclic $L_{\infty}$-algebra_
[56, 60]. In the remainder of this section we will explain this general
statement in detail, beginning with an introduction to the mathematical
concepts involved.

###  2.2 What is an $L_{\infty}$-algebra?

We begin with a simple and concise version of the notion of an
‘$L_{\infty}$-algebra’, following [69, 68], which also nicely ties in with its
field theory realisations. Let $V=\bigoplus_{k\in\mathbbm{Z}}\,V^{k}$ be a
$\mathbbm{Z}$-graded real vector space with (reduced) graded symmetric algebra
$\mathrm{Sym}(V[1]):=\bigoplus_{n\geq 1}\,V[1]^{\odot n}$, where $V[1]$ is the
same underlying vector space as $V$ but with the degrees of its homogeneous
subspaces shifted by $1$, i.e. $V[1]^{k}:=V^{k+1}$ for all $k\in\mathbbm{Z}$.
Then $\mathrm{Sym}(V[1])$ can be regarded as a free cocommutative coalgebra
with coproduct
$\Delta_{\textrm{\tiny$V$}}:\mathrm{Sym}(V[1])\to\mathrm{Sym}(V[1])\otimes\mathrm{Sym}(V[1])$
defined on homogeneous subspaces by

$\displaystyle\begin{split}\Delta_{\textrm{\tiny$V$}}\big{|}_{V[1]^{\odot
n}}:=\sum_{i=1}^{n-1}\ \sum_{\sigma\in{\rm
Sh}(i;n-i)}\,\big{(}\mathbbm{1}^{\odot i}\otimes\mathbbm{1}^{\odot
n-i}\big{)}\circ\tau^{\sigma}\ ,\end{split}$ (2.18)

where ${\rm Sh}(i;n-i)\subset S_{n}$ is the set of $(i;n-i)$-shuffled
permutations of degree $n$, which are ordered as $\sigma(1)<\cdots<\sigma(i)$
and $\sigma(i+1)<\cdots<\sigma(n)$, and $\tau^{\sigma}:V[1]^{\otimes n}\to
V[1]^{\otimes n}$ denotes the action of the permutation $\sigma$ via the
trivial transposition isomorphism $\tau$ times the Koszul sign multiplication
which interchanges factors in a tensor product of graded vector spaces.

######  Definition 2.19.

An _$L_{\infty}$ -algebra_ is a graded vector space $V$ together with a
coderivation

$\displaystyle D:\mathrm{Sym}(V[1])\longrightarrow\mathrm{Sym}(V[1])$ (2.20)

of degree $1$ which is a differential, that is, $D^{2}=0$.

The coderivation property

$\displaystyle\Delta_{\textrm{\tiny$V$}}\circ
D=(D\otimes\mathbbm{1}+\mathbbm{1}\otimes D)\circ\Delta_{\textrm{\tiny$V$}}$
(2.21)

implies that the differential $D:\mathrm{Sym}(V[1])\to\mathrm{Sym}(V[1])$ is
completely determined by its ‘Taylor components’ defined by projecting its
image to the generating subspace to give a map
${}_{\textrm{\tiny$V[1]$}}D:\mathrm{Sym}(V[1])\to V[1]$ with the expansion

$\displaystyle{}_{\textrm{\tiny$V[1]$}}D=\sum_{n\geq 1}\,b_{n}\ ,$ (2.22)

where each $b_{n}:V[1]^{\odot n}\to V[1]$ is a graded symmetric multilinear
map of degree $1$. We can restore the original grading by introducing the
‘suspension’ isomorphism $s:V\to V[1]$ which is simply the identity map but
decreases the grading by $1$, and defining multilinear graded antisymmetric
maps $\ell_{n}:\text{\Large$\wedge$}^{n}V\to V$ of degree $2-n$ by

$\displaystyle\ell_{n}:=s^{-1}\circ b_{n}\circ s^{\otimes n}\ ,$ (2.23)

for each $n\geq 1$. The maps $\ell_{n}$ are called the _$n$ -brackets_ of the
$L_{\infty}$-algebra.

Graded antisymmetry is encoded by

$\displaystyle\ell_{n}(v_{1},\dots,v_{n})=-(-1)^{|v_{i}|\,|v_{i+1}|}\
\ell_{n}(v_{1},\dots,v_{i-1},v_{i+1},v_{i},v_{i+2},\dots,v_{n})$ (2.24)

for $i=1,\dots,n-1$, where $|v|$ denotes the degree of a homogenous element
$v\in V$. Nilpotency $D^{2}=0$ translates into an infinite set of
identities888The peculiar extra signs here and in the following are due to
shifting degree and using $(s^{-1})^{\otimes n}\circ s^{\otimes
n}=(-1)^{n\choose 2}\,\mathbbm{1}$ (see e.g. [60]).

$\displaystyle\sum_{i=1}^{n}\,(-1)^{i\,(n-i)}\,\ell_{n-i+1}\circ\big{(}\ell_{i}\otimes\mathbbm{1}^{\otimes
n-i}\big{)}\ \circ\ \sum_{\sigma\in{\rm Sh}(i;n-i)}\,{\rm
sgn}(\sigma)\,\tau^{\sigma}=0$ (2.25)

for each $n\geq 1$. These identities are called the _homotopy Jacobi
identities_ of the $L_{\infty}$-algebra.

Let us unravel the first few homotopy Jacobi identities to get an idea of
their meaning. For $n=1$ the identity (2.25) reads

$\displaystyle(\ell_{1})^{2}=0\ ,$ (2.26)

which states that the linear map $\ell_{1}:V\to V$ is a differential. Thus
every $L_{\infty}$-algebra has an underlying cochain complex $(V,\ell_{1})$.
The identity (2.25) for $n=2$ states that $\ell_{2}:V\otimes V\to V$ is a
cochain map. This is equivalent to saying that the differential $\ell_{1}$ is
a graded derivation of the $2$-bracket $\ell_{2}$; explicitly, when evaluated
on elements $v_{1},v_{2}\in V$:

$\displaystyle\ell_{1}\big{(}\ell_{2}(v_{1},v_{2})\big{)}=\ell_{2}\big{(}\ell_{1}(v_{1}),v_{2}\big{)}+(-1)^{|v_{1}|}\,\ell_{2}\big{(}v_{1},\ell_{1}(v_{2})\big{)}\
.$ (2.27)

For $n=3$ the identity (2.25) states that the $2$-bracket $\ell_{2}$ obeys the
Jacobi identity up to the cochain homotopy $\ell_{3}:V\otimes V\otimes V\to
V$; explicitly, when evaluated on elements $v_{1},v_{2},v_{3}\in V$:

$\displaystyle\begin{split}&\ell_{2}\big{(}\ell_{2}(v_{1},v_{2}),v_{3}\big{)}-(-1)^{|v_{2}|\,|v_{3}|}\,\ell_{2}\big{(}\ell_{2}(v_{1},v_{3}),v_{2}\big{)}+(-1)^{(|v_{2}|+|v_{3}|)\,|v_{1}|}\,\ell_{2}\big{(}\ell_{2}(v_{2},v_{3}),v_{1}\big{)}\\\\[4.0pt]
&\hskip
28.45274pt=-\ell_{3}\big{(}\ell_{1}(v_{1}),v_{2},v_{3}\big{)}-(-1)^{|v_{1}|}\,\ell_{3}\big{(}v_{1},\ell_{1}(v_{2}),v_{3}\big{)}-(-1)^{|v_{1}|+|v_{2}|}\,\ell_{3}\big{(}v_{1},v_{2},\ell_{1}(v_{3})\big{)}\\\
&\hskip 113.81102pt-\ell_{1}\big{(}\ell_{3}(v_{1},v_{2},v_{3})\big{)}\
.\end{split}$ (2.28)

This continues in general for all $n\geq 4$ to higher coherence conditions for
the $n$-brackets. In the special instance when $\ell_{n}=0$ for all $n\geq 3$,
an $L_{\infty}$-algebra is simply a differential graded Lie algebra. Thus
$L_{\infty}$-algebras are homotopy coherent generalizations of Lie algebras;
for this reason, $L_{\infty}$-algebras are also sometimes called (_strong_)
_homotopy Lie algebras_. In particular, the cohomology
$H^{\bullet}(V,\ell_{1})$ of an $L_{\infty}$-algebra is a $\mathbbm{Z}$-graded
Lie algebra.

The natural notion of an invariant inner product on an $L_{\infty}$-algebra is
called a _cyclic pairing_.

######  Definition 2.29.

A _cyclic $L_{\infty}$-algebra_ is an $L_{\infty}$-algebra
$(V,\\{\ell_{n}\\})$ together with a non-degenerate graded symmetric cochain
map $\langle-,-\rangle:V\otimes V\to\mathbbm{R}$ that satisfies the _cyclicity
condition_

$\displaystyle\langle-,-\rangle\,\circ\,(\mathbbm{1}\otimes\ell_{n})={\rm
sgn}(\sigma)\,\langle-,-\rangle\,\circ\,(\mathbbm{1}\otimes\ell_{n})\,\circ\,\tau^{\sigma}\
,$ (2.30)

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

Explicitly, when evaluated on elements $v_{0},v_{1},\dots,v_{n}\in V$,
cyclicity translates to

$\displaystyle\langle
v_{0},\ell_{n}(v_{1},v_{2},\dots,v_{n})\rangle=\pm\,\langle
v_{n},\ell_{n}(v_{0},v_{1},\dots,v_{n-1})\rangle\ ,$ (2.31)

where the $\pm$ sign is determined by the grading of the elements
$v_{i}$.999See [60, 45] for the explicit cumbersome sign factors. Thus cyclic
$L_{\infty}$-algebras are homotopy coherent generalizations of quadratic Lie
algebras.

The dualization of the differential graded cocommutative coalgebra $(V,D)$
leads to an equivalent formulation of an $L_{\infty}$-algebra in terms of a
differential graded commutative algebra with respect to the symmetric tensor
product of $\mathrm{Sym}(V[1])$.

######  Definition 2.32.

The _Chevalley–Eilenberg algebra_ of an $L_{\infty}$-algebra $(V,D)$ is the
differential graded commutative algebra
$\big{(}\mathrm{Sym}(V[1])^{*},Q\big{)}$, where

$\displaystyle
Q=D^{*}:\mathrm{Sym}(V[1])^{*}\longrightarrow\mathrm{Sym}(V[1])^{*}$ (2.33)

is a graded derivation of degree $1$ such that $Q^{2}=0$.

This is a homotopy coherent generalization of the Chevalley–Eilenberg algebra
of a Lie algebra. In this dual language, a cyclic pairing translates to a
graded symplectic $2$-form $\omega\in\Omega^{2}(V[1])$ which is $Q$-invariant.
Evidently, one could equivalently use Definition 2.32 as the starting point
for the definition of an $L_{\infty}$-algebra, and this is the point of view
which naturally makes contact between the objects and relations of
$L_{\infty}$-algebras and the symmetries and dynamics of field theories, as we
now briefly sketch; see e.g. [60] for a more detailed and precise review of
the correspondence.

###  2.3 Batalin–Vilkovisky formalism

The Batalin–Vilkovisky (BV) formalism constructs a differential graded
commutative algebra101010The field observables actually form a much bigger
space of functionals on $V[1]$ which are smooth in a suitable sense, but for
our illustrative purposes here it suffices to consider only polynomial field
observables in the dual symmetric algebra $\mathrm{Sym}(V[1])^{*}$.
$\big{(}\mathrm{Sym}(V[1])^{*},Q_{\textrm{\tiny BV}}\big{)}$ for a graded
vector space $V=\bigoplus_{k\in\mathbbm{Z}}\,V^{k}$ which encodes the BV
fields, namely ghosts and fields associated to a generalized gauge field
theory, together with their antifields. More precisely, the space of BRST
fields is $V^{\leq 1}[1]$ where $V^{\leq 1}:=\bigoplus_{k\leq 1}\,V^{k}$ with
$V^{1}$ encoding the fields and $V^{0}$ the ghosts, while the negatively
graded homogeneous subspaces are non-zero only if there are higher gauge
redundancies so that they encode ghosts-for-ghosts, etc. The BV field space
extends these by their corresponding antifields to the $-1$-shifted cotangent
bundle $V[1]\simeq T^{*}[-1]V^{\leq 1}[1]$. This data is supplemented by the
BV antibracket $\\{-,-\\}_{\textrm{\tiny BV}}$, which is the canonical graded
Poisson bracket compatible with the differential $Q_{\textrm{\tiny BV}}$ and
whose inverse is the BV symplectic form $\omega_{\textrm{\tiny BV}}$ on $V$ of
degree $-1$; using this non-degenerate symplectic pairing we identify the dual
$V^{*}\simeq V[1]$ and111111We do not delve into precise technical details of
defining duals and tensor products of infinite-dimensional vector spaces, such
as those which typically arise in field theories, as these involve subtle
topological considerations.

$\displaystyle\mathrm{Sym}(V[1])^{*}\simeq\mathrm{Sym}(V[2])\ .$ (2.34)

For illustrative purposes, let us first sketch how to build the corresponding
$L_{\infty}$-algebra, and then make a more precise statement afterwards. The
action of the BV differential is given by taking the BV antibracket with the
BV action functional $S_{\textrm{\tiny BV}}$, $Q_{\textrm{\tiny
BV}}=\\{S_{{\textrm{\tiny BV}}},-\\}_{\textrm{\tiny BV}}$. The action
functional can be expanded as

$\displaystyle S_{\textrm{\tiny BV}}=\sum_{m\geq 2}\,S_{\textrm{\tiny
BV}}^{(m)}\ ,$ (2.35)

where $S_{\textrm{\tiny BV}}^{(m)}$ is the part of $S_{\textrm{\tiny BV}}$
which is a polynomial of degree $m$ in the BV fields. Then the brackets
$\ell_{n}$ of the $L_{\infty}$-algebra are given by $\big{\\{}S_{\textrm{\tiny
BV}}^{(n+1)},-\big{\\}}_{\textrm{\tiny BV}}$ for $n\geq 1$. Nilpotency
$(Q_{\textrm{\tiny BV}})^{2}=0$, or equivalently the classical master equation
$\\{S_{\textrm{\tiny BV}},S_{\textrm{\tiny BV}}\\}_{\textrm{\tiny BV}}=0$,
then translates to the homotopy Jacobi identities (2.25) for the brackets
$\ell_{n}$, and $(V,\\{\ell_{n}\\})$ is an $L_{\infty}$-algebra. The
$(-1)$-shifted symplectic structure $\omega_{\textrm{\tiny BV}}$ induces a
cyclic pairing of degree $-3$ on $V$, making it into a cyclic
$L_{\infty}$-algebra.

The action of the BV differential $Q_{\textrm{\tiny BV}}$ on fields $A\in
V^{1}$ and ghosts $c_{-k}\in V^{-k}$ for $k\geq 0$ encodes the kinematical
gauge symmetry of the field theory, that is, the (higher) gauge
transformations and the closure of the gauge algebra. Its action on the
antifields $A^{+}\in V^{2}$ incorporates the dynamical brackets of the
$L_{\infty}$-algebra, while the BV transformations of the antifields
$c_{k}^{+}\in V^{k+3}$ correspond to the (higher) Noether identities and the
corresponding actions of the gauge parameters. The cohomology of
$Q_{\textrm{\tiny BV}}$ in degree $0$ thus simultaneously encodes the
quotients of the space of fields by the ideal of Euler–Lagrange derivatives
and by the Lie algebra action of gauge transformations, that is, the classical
observables of the field theory.

Altogether, the natural algebraic structure underlying the BV formalism
organises the gauge symmetries and dynamics of a field theory in a cochain
complex that can be interpreted in the form

$\displaystyle\cdots\ \xrightarrow{\quad\ \ }\ \begin{matrix}{\scriptstyle
V^{0}}\\\ \Big{\\{}\begin{matrix}\textrm{\footnotesize gauge}\\\\[-4.30554pt]
\textrm{\footnotesize parameters}\end{matrix}\Big{\\}}\end{matrix}\
\xrightarrow{\quad\ \ }\ \begin{matrix}{\scriptstyle V^{1}}\\\
\begin{matrix}\big{\\{}\textrm{\footnotesize fields}\big{\\}}\\\\[-4.30554pt]
\phantom{\textrm{\footnotesize fields}}\end{matrix}\end{matrix}\
\xrightarrow{\quad\ \ }\ \begin{matrix}{\scriptstyle V^{2}}\\\
\Big{\\{}\begin{matrix}\textrm{\footnotesize equations}\\\\[-4.30554pt]
\textrm{\footnotesize of motion}\end{matrix}\Big{\\}}\end{matrix}\
\xrightarrow{\quad\ \ }\ \begin{matrix}{\scriptstyle V^{3}}\\\
\Big{\\{}\begin{matrix}\textrm{\footnotesize Noether}\\\\[-4.30554pt]
\textrm{\footnotesize identities}\end{matrix}\Big{\\}}\end{matrix}\
\xrightarrow{\quad\ \ }\ \cdots$ (2.36)

The homogeneous subspaces $V^{-k}$ for $k\geq 1$ to the left encode ‘higher’
gauge transformations for reducible symmetries, while $V^{k+3}$ to the right
dually encode ‘higher’ Noether identities (differential relations among the
Noether identities, and so on).

Conversely, starting from an $L_{\infty}$-algebra $(V,\\{\ell_{n}\\})$ with a
cyclic structure $\langle-,-\rangle$ of degree $-3$, choose a basis
$\\{\tau_{\alpha}\\}\subset V$ of the $L_{\infty}$-algebra with corresponding
dual basis $\\{\tau^{\alpha}\\}\subset V^{*}\simeq V[3]$ relative to the
cyclic pairing, that is,
$\langle\tau^{\alpha},\tau_{\beta}\rangle=\delta^{\alpha}_{\beta}$ for all
$\alpha,\beta$. Following [60], we introduce the ‘contracted coordinate
functions’ as the elements121212Throughout this paper we adhere to the
Einstein summation convention over repeated upper and lower indices.

$\displaystyle\xi:=\tau^{\alpha}\otimes\tau_{\alpha}\ \in\
\mathrm{Sym}(V[2])\otimes V$ (2.37)

of degree $1$. The $L_{\infty}$-algebra structure on $V$ naturally extends to
the tensor product $\mathrm{Sym}(V[2])\otimes V$ through the extended
brackets131313More generally, the tensor product of any differential graded
commutative algebra with an $L_{\infty}$-algebra admits an
$L_{\infty}$-algebra structure with grading defined by the total degree, see
Section 4.9; this generalizes the corresponding statement for Lie algebras, a
prominent example being the Lie algebra
$\Omega^{\bullet}(M,\mathfrak{g})=\Omega^{\bullet}(M)\otimes\mathfrak{g}$ of
differential forms valued in a Lie algebra $\mathfrak{g}$ that we encountered
before. Here we view $\mathrm{Sym}(V[2])$ as equipped with the zero
differential.

$\displaystyle\ell_{n}^{\,\rm ext}(\zeta_{1}\otimes
v_{1},\dots,\zeta_{n}\otimes
v_{n}):=\pm\,(\zeta_{1}\odot\cdots\odot\zeta_{n})\otimes\ell_{n}(v_{1},\dots,v_{n})\
,$ (2.38)

for all $n\geq 1$, $\zeta_{1},\dots,\zeta_{n}\in\mathrm{Sym}(V[2])$ and
$v_{1},\dots,v_{n}\in V$; the explicit Koszul sign factors $\pm$ depend on the
gradings and can be found in [60, Section 2.3]. Similarly, the cyclic
structure naturally extends to a symmetric non-degenerate pairing
$\langle-,-\rangle^{\rm ext}:\big{(}\mathrm{Sym}(V[2])\otimes
V\big{)}\otimes\big{(}\mathrm{Sym}(V[2])\otimes V\big{)}\to\mathrm{Sym}(V[2])$
of degree $-3$ given by

$\displaystyle\langle\zeta_{1}\otimes v_{1},\zeta_{2}\otimes v_{1}\rangle^{\rm
ext}:=\pm\,(\zeta_{1}\odot\zeta_{2})\,\langle v_{1},v_{2}\rangle\ .$ (2.39)

With the contracted coordinate functions (2.37), we now set

$\displaystyle Q_{\textrm{\tiny BV}}\xi=-\sum_{n\geq 1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n}^{\,\rm ext}(\xi^{\otimes n})\ ,$ (2.40)

which we interpret as acting on elements of $\mathrm{Sym}(V[1])$. The homotopy
Jacobi identities (2.25) imply that $(Q_{\textrm{\tiny BV}})^{2}=0$. The
extended pairing $\langle-,-\rangle^{\rm ext}$ induces a $(-1)$-shifted
symplectic structure

$\displaystyle\omega_{\textrm{\tiny
BV}}:=-\tfrac{1}{2}\,\langle\delta\xi,\delta\xi\rangle^{\rm ext}\ \in\
\Omega^{2}(V[1])\ .$ (2.41)

In this way we recover the differential $Q_{\textrm{\tiny BV}}$ and
antibracket $\\{-,-\\}_{\textrm{\tiny BV}}$ of the BV formalism directly from
the cyclic $L_{\infty}$-algebra of a field theory. Moreover, the BV action
functional is obtained from the element [60, Section 4.3]

$\displaystyle S_{\textrm{\tiny BV}}=\sum_{n\geq 1}\,\frac{(-1)^{n\choose
2}}{(n+1)!}\,\langle\xi,\ell_{n}^{\,\rm ext}(\xi^{\otimes n})\rangle^{\rm
ext}\ \in\ \mathrm{Sym}(V[2])$ (2.42)

of degree $0$. Then

$\displaystyle Q_{\textrm{\tiny BV}}=\\{S_{\textrm{\tiny
BV}},-\\}_{\textrm{\tiny BV}}\ ,$ (2.43)

and nilpotency $(Q_{\textrm{\tiny BV}})^{2}=0$ is equivalent to the classical
master equation

$\displaystyle\\{S_{\textrm{\tiny BV}},S_{\textrm{\tiny BV}}\\}_{\textrm{\tiny
BV}}=0\ .$ (2.44)

###  2.4 $L_{\infty}$-algebras of classical field theories

We can turn the formalism sketched in Section 2.3 into a concrete prescription
for explicitly developing the classical dynamics of any generalized gauge
field theory entirely from its underlying $L_{\infty}$-algebra
$(V,\\{\ell_{n}\\})$; in this framework the equations of motion of the theory
are the Maurer–Cartan equations of the $L_{\infty}$-algebra. For a field
theory on a manifold $M$ we always assume that its $L_{\infty}$-algebra is
_local_ , that is, all bracket operations are given by polydifferential
operators. Focusing momentarily on field theories with only irreducible gauge
symmetries, i.e. with independent gauge transformations, the $4$-term cochain
complex

$\displaystyle V^{0}\xrightarrow{\ \ \ell_{1}\ \ }V^{1}\xrightarrow{\ \
\ell_{1}\ \ }V^{2}\xrightarrow{\ \ \ell_{1}\ \ }V^{3}$ (2.45)

encodes the free field dynamics, that is, the linearized gauge
transformations, equations of motion and Noether identities. In particular,
the free fields (solutions to the linearized equations of motion modulo
linearized gauge transformations) live in the degree $1$ cohomology
$H^{1}(V,\ell_{1})$ of this complex. The higher brackets $\ell_{n}$ with
$n\geq 2$ encode the interactions, and recover the full symmetries and
dynamics of the generalized gauge theory. The homotopy Jacobi identities
ensure covariance of the field equations, (on-shell) closure of gauge
transformations, and the Noether identities, as we now explain.

The vector spaces $V^{0}$ and $V^{1}$ are respectively the spaces of gauge
parameters and dynamical fields. Given $\lambda\in V^{0}$ and $A\in V^{1}$, a
gauge transformation is the map $A\mapsto A+\delta_{\lambda}A$ where the
infinitesimal gauge variation is given by141414Here and in the following we
use the convention ${k\choose 2}:=0$ for $k<2$.

$\displaystyle\delta_{\lambda}A=\ell_{1}(\lambda)+\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell_{n+1}(\lambda,A^{\otimes n})\ .$ (2.46)

The gauge variations extend to maps $\delta_{\lambda}:V\to V$ of degree $0$ by
acting trivially on $V^{0}$, by commuting with arbitrary maps $V\to V$, and as
derivations of operations $\mu$ defined on $V\otimes V$, that is, they satisfy
the Leibniz rule

$\displaystyle\delta_{\lambda}\mu(v_{1}\otimes
v_{2})=\mu(\delta_{\lambda}v_{1}\otimes
v_{2})+\mu(v_{1}\otimes\delta_{\lambda}v_{2})$ (2.47)

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

The vector space $V^{2}$ is the space that the field equations of the theory
belong to. They are encoded as the _Maurer–Cartan equations_
$\mathfrak{F}_{A}=0$, where

$\displaystyle\mathfrak{F}_{A}:=\ell_{1}(A)+\sum_{n\geq
2}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell_{n}(A^{\otimes n})\ ,$ (2.48)

with the covariant gauge variations

$\displaystyle\delta_{\lambda}\mathfrak{F}_{A}=\ell_{2}(\lambda,\mathfrak{F}_{A})+\sum_{n\geq
1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n+2}(\lambda,\mathfrak{F}_{A},A^{\otimes n})\ ,$ (2.49)

which follow from homotopy Jacobi identity (2.25) evaluated on
$(\lambda,A^{\otimes n})$ and summed over $n\geq 1$. Using symmetry of the
brackets under exchange of degree $1$ elements along with the Leibniz rule, it
is easy to see that
$\mathfrak{F}_{A+\delta_{\lambda}A}=\mathfrak{F}_{A}+\delta_{\lambda}\mathfrak{F}_{A}+O(\lambda^{2})$,
and so the gauge symmetry acts on the subspace of classical solutions $A\in
V^{1}$. The quotient of the space of solutions to the equations of motion
$\mathfrak{F}_{A}=0$ by the action of gauge transformations is the
_Maurer–Cartan moduli space_ $\mathscr{M}\mathscr{C}(V,\\{\ell_{n}\\})$, which
is the space of classical physical states of the field theory. The BV complex
$\big{(}\mathrm{Sym}(V[1])^{*},Q_{\textrm{\tiny BV}}\big{)}$ of Section 2.3
provides a rigorous definition of this moduli space as a homological
resolution of the classical observables. On the other hand, the form of the
dynamical equations (2.48) suggest that $L_{\infty}$-algebras are a dual
formulation of the free differential algebras which play a central role in the
geometric formulation of supergravity theories [35].

The closure relation for the gauge algebra then has the form

$\displaystyle[\delta_{\lambda_{1}},\delta_{\lambda_{2}}]_{\circ}A=\delta_{C(\lambda_{1},\lambda_{2};A)}A+\triangle_{\lambda_{1},\lambda_{2}}A\
,$ (2.50)

where
$[\delta_{\lambda_{1}},\delta_{\lambda_{2}}]_{\circ}:=\delta_{\lambda_{1}}\circ\delta_{\lambda_{2}}-\delta_{\lambda_{2}}\circ\delta_{\lambda_{1}}$
is the commutator, while

$\displaystyle
C(\lambda_{1},\lambda_{2};A)=-\ell_{2}(\lambda_{1},\lambda_{2})-\sum_{n\geq
1}\,\frac{(-1)^{n\choose
2}}{n!}\,\ell_{n+2}(\lambda_{1},\lambda_{2},A^{\otimes n})$ (2.51)

and

$\displaystyle\triangle_{\lambda_{1},\lambda_{2}}A=\sum_{n\geq
0}\,\frac{(-1)^{n-2\choose
2}}{n!}\,\ell_{n+3}(\lambda_{1},\lambda_{2},\mathfrak{F}_{A},A^{\otimes n})\
.$ (2.52)

This follows from the homotopy Jacobi identity (2.25) evaluated on
$(\lambda_{1},\lambda_{2},A^{\otimes n})$ and summed over $n\geq 1$. The
algebra of gauge transformations thus closes on-shell, that is, when
$\mathfrak{F}_{A}=0$, and on this locus the gauge algebra generally depends on
the fields $A$; in other words, the gauge symmetries generally form an _open_
gauge algebra. The homotopy relations of the $L_{\infty}$-algebra guarantee
that the Jacobi identity is generally satisfied for any triple of maps
$\delta_{\lambda_{1}}$, $\delta_{\lambda_{2}}$ and $\delta_{\lambda_{3}}$. The
field theories considered in this paper will always have _closed_ field-
independent gauge algebras generated by gauge transformations that are linear
in the fields; that is, (2.52) vanishes identically at each order in
$A^{\otimes n}$, while the sums in (2.46) and (2.51) terminate at the
$2$-bracket $\ell_{2}$.

Finally, the vector space $V^{3}$ accomodates Noether’s second theorem: It
contains the image of a local differential operator ${\sf d}_{A}$ on $V^{2}$,
which may depend on the fields. The Noether identities are then encoded by

$\displaystyle{\mathsf{d}}_{A}\mathfrak{F}_{A}:=\ell_{1}(\mathfrak{F}_{A})+\sum_{n\geq
1}\,\frac{(-1)^{n\choose 2}}{n!}\,\ell_{n+1}(\mathfrak{F}_{A},A^{\otimes n})\
=\ 0\ ,$ (2.53)

which vanishes identically as a consequence of the homotopy relations of the
$L_{\infty}$-algebra: evaluating (2.25) on $A^{\otimes n}$, using symmetry of
the brackets under interchange of degree $1$ elements, shows that

$\displaystyle\sum_{k=0}^{n}\,\frac{(-1)^{n\choose
2}}{k!\,(n-k)!}\,\ell_{k+1}\big{(}\ell_{n-k}(A^{\otimes n-k}),A^{\otimes
k}\big{)}=0\ ,$ (2.54)

and summing over $n\geq 1$ yields (2.53). This expresses local differential
relations among the Euler–Lagrange derivatives $\mathfrak{F}_{A}$ which hold
_off-shell_.

The action functional of the field theory is encoded via a symmetric non-
degenerate bilinear pairing $\langle-,-\rangle:V\otimes V\to\mathbbm{R}$ of
degree $-3$, which makes $V$ into a cyclic $L_{\infty}$-algebra. The only non-
trivial pairings are

$\displaystyle\langle-,-\rangle:V^{1}\otimes
V^{2}\longrightarrow\mathbbm{R}\qquad\mbox{and}\qquad\langle-,-\rangle:V^{0}\otimes
V^{3}\longrightarrow\mathbbm{R}\ .$ (2.55)

It is easy to see that the equations of motion $\mathfrak{F}_{A}=0$ follow
from varying the _Maurer–Cartan action functional_

$\displaystyle S(A):=\frac{1}{2}\,\langle A,\ell_{1}(A)\rangle+\sum_{n\geq
2}\,\frac{(-1)^{n\choose 2}}{(n+1)!}\,\langle A,\ell_{n}(A^{\otimes
n})\rangle\ ,$ (2.56)

since then cyclicity and the Leibniz rule imply $\delta S(A)=\langle\delta
A,\mathfrak{F}_{A}\rangle$. Note that $\ell_{1}(A)$ is associated with the
free field equations of motion, while $\ell_{n}(A^{\otimes n})$ for $n\geq 2$
contribute interaction vertices in the action functional.

Cyclicity and the Leibniz rule also imply

$\displaystyle\begin{split}\delta_{\lambda}S(A)=\langle\delta_{\lambda}A,\mathfrak{F}_{A}\rangle&=\sum_{n\geq
0}\,\frac{(-1)^{n\choose 2}}{n!}\,\langle\ell_{n+1}(\lambda,A^{\otimes
n}),\mathfrak{F}_{A}\rangle\\\\[4.0pt] &=-\sum_{n\geq 0}\,\frac{(-1)^{n\choose
2}}{n!}\,\langle\lambda,\ell_{n+1}(A^{\otimes
n},\mathfrak{F}_{A})\rangle=-\langle\lambda,{\mathsf{d}}_{A}\mathfrak{F}_{A}\rangle\
.\end{split}$ (2.57)

Thus gauge invariance of the action functional $\delta_{\lambda}S(A)=0$, for
all $\lambda\in V^{0}$, is equivalent to the Noether identities
${\mathsf{d}}_{A}\mathfrak{F}_{A}=0$ by non-degeneracy of the pairing. This is
simply a reformulation of Noether’s second theorem. From this perspective, the
Noether operator ${\sf d}_{A}:V^{2}\to V^{3}$ is the adjoint, with respect to
the cyclic inner product, of $-\delta_{\lambda}$ viewed as a differential
operator $-\delta_{(-)}A:V^{0}\to V^{1}$ acting on a gauge parameter
$\lambda\in V^{0}$ with image in $V^{1}$.

#### Higher gauge symmetries

If the symmetries themselves have non-trivial symmetries, that is, there are
further gauge redundancies in the description and the gauge symmetries are
reducible, then the cochain complex (2.45) should be extended into negative
degrees $V^{-k}$ for $k\geq 1$, which are the spaces of ‘higher’ gauge
parameters, together with their duals $V^{k+3}$ which contain the
corresponding ‘higher’ Noether identities. The vector space $V^{-1}$
parametrizes gauge transformations of the gauge parameters, $V^{-2}$
parametrizes gauge variations of the gauge transformations of gauge
parameters, and so on. These higher gauge symmetries are encoded as

$\displaystyle\delta_{(\lambda_{-k-1},A)}\lambda_{-k}=\ell_{1}(\lambda_{-k-1})+\sum_{n\geq
1}\,\frac{(-1)^{{n+1\choose
2}+n\,k}}{n!}\,\ell_{n+1}(\lambda_{-k-1},A^{\otimes n})\ ,$ (2.58)

where $\lambda_{-k}\in V^{-k}$ for $k\geq 0$.

Like the commutator of gauge transformations, these close only on-shell in
general, as

$\displaystyle\delta_{(\lambda_{-k-2},A)}\big{(}\delta_{(\lambda_{-k-1},A)}\lambda_{-k}\big{)}=\ell_{2}(\lambda_{-k-2},\mathfrak{F}_{A})+\sum_{n\geq
1}\,\frac{(-1)^{{n+2\choose
2}+n\,k}}{n!}\,\ell_{n+2}(\lambda_{-k-2},\mathfrak{F}_{A},A^{\otimes n})\ ,$
(2.59)

for all $k\geq-1$ with the convention $\lambda_{+1}:=A$, which follow from the
homotopy Jacobi identity (2.25) evaluated on $(\lambda_{-k-2},A^{\otimes n})$
and summed over $n\geq 1$. Thus gauge transformations of level $k$ gauge
parameters leave level $k$ gauge transformations unchanged up to the equations
of motion $\mathfrak{F}_{A}=0$. At the classical level, their inclusion is
purely algebraic and only serves to alter the cohomology
$H^{\bullet}(V,\ell_{1})$ of the underlying cochain complex at its
extremities, leaving the moduli space
$\mathscr{M}\mathscr{C}(V,\\{\ell_{n}\\})$ of classical states unchanged. The
inclusion of higher gauge parameters is necessary so that the resulting BV
complex $\big{(}\mathrm{Sym}(V[1])^{*},Q_{\textrm{\tiny BV}}\big{)}$ of
Section 2.3 properly forms a resolution of the space of classical observables.

Higher gauge redundancies are equivalent to higher Noether identities. Taking
into account level 1 higher gauge parameters extends the underlying graded
vector space $V$ with a non-trivial homogeneous subspace $V^{-1}$ and its dual
$V^{4}$, and hence the duality pairing

$\displaystyle\langle-,-\rangle:V^{-1}\otimes V^{4}\longrightarrow\mathbbm{R}\
.$ (2.60)

Using cyclicity, we define the level 1 Noether operator
${\mathsf{d}}_{A}^{{\textrm{\tiny(1)}}}:V^{3}\to V^{4}$ as the adjoint of
$\delta_{(\lambda_{-1},A)}$ from (2.58) viewed as an operator
$\delta_{(-,A)}\lambda:V^{-1}\to V^{0}$ on $V^{-1}$ with values in $V_{0}$,
that is,

$\displaystyle\begin{split}\langle\delta_{(\lambda_{-1},A)}\lambda,\varLambda\rangle=:\langle\lambda_{-1},{\sf
d}_{A}^{{\textrm{\tiny(1)}}}\varLambda\rangle\ ,\end{split}$ (2.61)

for all $\varLambda\in V^{3}$. Explicitly

$\displaystyle{\sf
d}_{A}^{\textrm{\tiny(1)}}\varLambda=\ell_{1}(\varLambda)+\sum_{n\geq
1}\,\frac{(-1)^{{n+1\choose 2}+n}}{n!}\,\ell_{n+1}(\varLambda,A^{\otimes n})\
.$ (2.62)

Applying $\delta_{(\lambda_{-1},A)}$ on both sides of
$\langle\delta_{\lambda}A,\mathcal{A}\rangle=-\langle\lambda,{\mathsf{d}}_{A}\mathcal{A}\rangle$
for arbitrary $\mathcal{A}\in V^{2}$, we get

$\displaystyle\langle\delta_{(\lambda_{-1},A)}(\delta_{\lambda}A),\mathcal{A}\rangle=-\langle\delta_{(\lambda_{-1},A)}\lambda,{\mathsf{d}}_{A}\mathcal{A}\rangle\
.$ (2.63)

Using the closure property (2.59), the definition (2.62) of the level 1
Noether operator, and finally cyclicity and non-degeneracy of the pairing, we
obtain the level 1 Noether identities

$\displaystyle{\mathsf{d}}_{A}^{{\textrm{\tiny(1)}}}\circ{\mathsf{d}}_{A}=-\ell_{2}(\mathfrak{F}_{A},-)-\sum_{n\geq
1}\,\frac{(-1)^{{n\choose 2}}}{n!}\,\ell_{n+2}(\mathfrak{F}_{A},A^{\otimes
n},-)\ ,$ (2.64)

which state that, up to the equations of motion $\mathfrak{F}_{A}=0$, not all
components of the level 0 Noether operator ${\mathsf{d}}_{A}$ are
independent.151515Of course, this can also be derived directly from the
homotopy Jacobi identities of the extended $L_{\infty}$-algebra. This is
naturally ‘dual’ to the statement that the level 0 gauge parameters generate
reducible gauge symmetries, i.e. that not all gauge directions are independent
in the tangent space to the space of fields $V^{1}$. This is to be compared
with the level 0 Noether identities ${\mathsf{d}}_{A}\circ\mathfrak{F}_{-}=0$,
which state that not all components of the Maurer–Cartan operator
$\mathfrak{F}_{-}:V^{1}\to V^{2}$ are independent due to the existence of
level 0 gauge symmetries.

Proceeding inductively, the existence of level $k$ gauge transformations
extends the underlying graded vector space $V$ with a non-trivial homogeneous
subspace $V^{-k}$ and its dual $V^{k+3}$, with the duality pairing
$\langle-,-\rangle:V^{-k}\otimes V^{k+3}\to\mathbbm{R}$. The adjoint of
$\delta_{(\lambda_{-k},A)}$ defines the level $k$ Noether operator
${\mathsf{d}}_{A}^{\textrm{\tiny$(k)$}}:V^{k+2}\to V^{k+3}$ which encodes the
interdependence of the level $k{-}1$ Noether operator through the level $k$
Noether identities

$\displaystyle{\mathsf{d}}_{A}^{\textrm{\tiny$(k)$}}\circ{\mathsf{d}}^{\textrm{\tiny$(k{-}1)$}}_{A}=-\ell_{2}(\mathfrak{F}_{A},-)-\sum_{n\geq
1}\,\frac{(-1)^{{n\choose 2}}}{n!}\,\ell_{n+2}(\mathfrak{F}_{A},A^{\otimes
n},-)\ .$ (2.65)

###  2.5 Example: Yang–Mills theory in the $L_{\infty}$-algebra formalism

Yang–Mills theory is the basic example of a field theory with gauge symmetries
where the underlying $L_{\infty}$-algebra is not a differential graded Lie
algebra. The higher bracket arises from the quartic interaction vertex in the
Yang–Mills action functional. The Yang–Mills $L_{\infty}$-algebra has appeared
many times in the literature. It was first explicitly noted in [92] and has
more recently reappeared in [56, 60, 50]. We close this section with a review
of the $L_{\infty}$-algebra formulation of Yang–Mills theory, which along with
the Chern–Simons gauge theory of Section 2.1 will play a prominent role in
some of our later examples.

Let $(M,g)$ be a $d$-dimensional oriented Lorentzian manifold,161616The
discussion below is presented for Lorentzian signature in light of our later
considerations, but it is easily adapted to more general pseudo-Riemannian
manifolds. and $\ast_{\textrm{\tiny H}}:\Omega^{k}(M)\to\Omega^{d-k}(M)$ the
corresponding Hodge duality operator. Let
$(\mathfrak{g},[-,-]_{\mathfrak{g}},\mathrm{Tr}_{\mathfrak{g}})$ be a
quadratic Lie algebra. As in Section 2.1, we extend the Lie bracket on
$\mathfrak{g}$ to a Lie bracket on $\mathfrak{g}$-valued differential forms
$\Omega^{\bullet}(M,\mathfrak{g}):=\Omega^{\bullet}(M)\otimes\mathfrak{g}$ via
the tensor product with exterior multiplication, and we denote this extended
Lie bracket with the same symbol $[-,-]_{\mathfrak{g}}$.

The Yang–Mills action functional for a gauge field
$A\in\Omega^{1}(M,\mathfrak{g})$ is defined by

$\displaystyle
S(A)=\int_{M}\,\text{Tr}_{\mathfrak{g}}(F_{A}\wedge\ast_{\textrm{\tiny
H}}F_{A})\ ,$ (2.66)

where as before
$F_{A}:=\mathrm{d}A+\frac{1}{2}\,[A,A]_{\mathfrak{g}}\in\Omega^{2}(M,\mathfrak{g})$
is the curvature 2-form. It is invariant under the usual gauge transformations

$\displaystyle\delta_{\lambda}A=\mathrm{d}\lambda+[A,\lambda]_{\mathfrak{g}}$
(2.67)

for $\lambda\in\Omega^{0}(M,\mathfrak{g})$. The Yang–Mills field equations
state that the Hodge dual of the curvature is covariantly constant:

$\displaystyle\mathfrak{F}_{A}=\mathrm{d}^{A}\ast_{\textrm{\tiny
H}}F_{A}=\mathrm{d}\ast_{\textrm{\tiny H}}F_{A}+[A,\ast_{\textrm{\tiny
H}}F_{A}]_{\mathfrak{g}}=0\ \in\ \Omega^{d-1}(M,\mathfrak{g})\ ,$ (2.68)

whereas the corresponding Noether identities are

$\displaystyle\mathrm{d}^{A}\mathfrak{F}_{A}=\big{(}\mathrm{d}^{A}\big{)}^{2}\ast_{\textrm{\tiny
H}}F_{A}=[F_{A},\ast_{\textrm{\tiny H}}F_{A}]_{\mathfrak{g}}=0\ \in\
\Omega^{d}(M,\mathfrak{g})\ .$ (2.69)

These may be also verified directly by symmetry and antisymmetry of the
$d$-form $[F_{A},\ast_{\textrm{\tiny H}}F_{A}]_{\mathfrak{g}}$ valued in the
Lie algebra $\mathfrak{g}$. From these expressions the $L_{\infty}$-algebra
organising Yang–Mills theory may be determined.

The underlying graded vector space is $V=V^{0}\oplus V^{1}\oplus V^{2}\oplus
V^{3}$, where

$\displaystyle\begin{split}V^{0}:=\Omega^{0}(M,\mathfrak{g})\ ,\quad
V^{1}:=\Omega^{1}(M,\mathfrak{g})\ ,\quad
V^{2}&:=\Omega^{d-1}(M,\mathfrak{g})\qquad\mbox{and}\qquad
V^{3}:=\Omega^{d}(M,\mathfrak{g})\ .\end{split}$ (2.70)

Denoting the corresponding elements by $\lambda$, $A$, $\mathcal{A}$ and
$\varLambda$ respectively, the $L_{\infty}$-algebra structure is given by the
differential

$\displaystyle\ell_{1}(\lambda)=\mathrm{d}\lambda\
,\quad\ell_{1}(A)=\mathrm{d}\ast_{\textrm{\tiny
H}}\mathrm{d}A\qquad\mbox{and}\qquad\ell_{1}(\mathcal{A})=\mathrm{d}\mathcal{A}\
,$ (2.71)

along with the 2-brackets

$\displaystyle\begin{split}\ell_{2}(\lambda_{1},\lambda_{2})=-[\lambda_{1},\lambda_{2}]_{\mathfrak{g}}\qquad&,\qquad\ell_{2}(\lambda,A)=-[\lambda,A]_{\mathfrak{g}}\
,\\\\[4.0pt]
\ell_{2}(\lambda,\mathcal{A})=-[\lambda,\mathcal{A}]_{\mathfrak{g}}\qquad,\qquad\ell_{2}(\lambda,\varLambda)&=-[\lambda,\varLambda]_{\mathfrak{g}}\qquad,\qquad\ell_{2}(A,\mathcal{A})=-[A,\mathcal{A}]_{\mathfrak{g}}\
,\\\\[4.0pt] \ell_{2}(A_{1},A_{2})=-\mathrm{d}\ast_{\textrm{\tiny
H}}[A_{1},A_{2}]_{\mathfrak{g}}-&[A_{1},\ast_{\textrm{\tiny
H}}\mathrm{d}A_{2}]_{\mathfrak{g}}+(-1)^{d}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}A_{1},A_{2}]_{\mathfrak{g}}\ ,\end{split}$ (2.72)

and finally the non-zero 3-bracket

$\displaystyle\begin{split}\ell_{3}(A_{1},&A_{2},A_{3})=-\big{[}A_{1},\ast_{\textrm{\tiny
H}}[A_{2},A_{3}]_{\mathfrak{g}}\big{]}_{\mathfrak{g}}-\big{[}A_{2},\ast_{\textrm{\tiny
H}}[A_{1},A_{3}]_{\mathfrak{g}}\big{]}_{\mathfrak{g}}+(-1)^{d}\,\big{[}\ast_{\textrm{\tiny
H}}[A_{1},A_{2}]_{\mathfrak{g}},A_{3}\big{]}_{\mathfrak{g}}\ ,\end{split}$
(2.73)

owing to the quartic interaction vertex. Note that the underlying cochain
complex differs from that of Chern–Simons theory (see Section 2.1), and that
now the $L_{\infty}$-algebra is no longer a differential graded Lie algebra
because $\ell_{3}\neq 0$.

The resulting cyclic inner product is given by

$\displaystyle\langle
A,\mathcal{A}\rangle:=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(A\wedge\mathcal{A})\qquad\mbox{and}\qquad\langle\lambda,\varLambda\rangle:=\int_{M}\,\mathrm{Tr}_{\mathfrak{g}}(\lambda\wedge\varLambda)\
,$ (2.74)

with cyclicity following from ${\rm ad}(\mathfrak{g})$-invariance of the
pairing
$\mathrm{Tr}_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\to\mathbbm{R}$,
integration by parts (for fields with suitable asymptotic decay) and symmetry
properties of the Hodge duality operator. By design, this cyclic
$L_{\infty}$-algebra encodes the full data of classical Yang–Mills theory as
described in Section 2.4.

##  3 Noncommutative gauge symmetry and $L_{\infty}$-algebras

As a first step towards understanding noncommutative field theories in the
modern context of this paper, in this section we look at some basic examples,
mentioned in Section 1, of noncommutative gauge theories arising in string
theory which fit into the standard $L_{\infty}$-algebra framework discussed in
Section 2. These are the examples of flat D-branes in a constant $B$-field,
which lead to the standard Moyal–Weyl deformations of gauge theories, and also
curved D-branes in WZW models based on compact Lie groups, which lead to fuzzy
field theories (i.e. matrix models). We will then discuss the limitations of
this perspective when confronted with the noncommutative gauge theories on
D-branes wrapping curved submanifolds with non-trivial worldvolume fluxes in a
generic background.

###  3.1 Star-gauge theories in the $L_{\infty}$-algebra formalism

Let us start with the conventional noncommutative field theories on flat
$d$-dimensional Minkowski spacetime $\mathbbm{R}^{1,d-1}$ with a constant
Poisson bivector
$\theta=\frac{1}{2}\,\theta^{\mu\nu}\,\partial_{\mu}\wedge\partial_{\nu}$. We
consider for illustration a noncommutative gauge theory with gauge algebra
$\mathfrak{u}(1)$ and fields multiplied with the Moyal–Weyl star-product
(1.1).171717The extension to nonabelian gauge algebras $\mathfrak{u}(n)$ with
$n>1$ is a straightforward modification (with no essential novelties) obtained
by composing the star-product (1.1) with matrix multiplication and all
integrations with the trace in the fundamental representation of
$\mathfrak{u}(n)$.

Let us start by introducing some notation: Given functions
$\lambda,\rho\in\Omega^{0}(\mathbbm{R}^{1,d-1})$, we denote their star-
commutator by

$\displaystyle[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\rho]:=\lambda\star\rho-\rho\star\lambda\
,$ (3.1)

which makes $\Omega^{0}(\mathbbm{R}^{1,d-1})$ into a Lie algebra due to
associativity of the Moyal–Weyl star-product. Given differential forms
$\alpha,\beta\in\Omega^{\bullet}(\mathbbm{R}^{1,d-1})$, we denote their star-
exterior product by $\alpha\wedge_{\star}\beta$, which is defined by regarding
the graded commutative exterior algebra
$\Omega^{\bullet}(\mathbbm{R}^{1,d-1})$ as a module over the noncommutative
algebra of functions $\big{(}\Omega^{0}(\mathbbm{R}^{1,d-1}),\star\big{)}$; in
particular, the holonomic coframe on $\mathbbm{R}^{1,d-1}$ obeys
$\mathrm{d}x^{\mu}\wedge_{\star}\mathrm{d}x^{\nu}=\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}=-\mathrm{d}x^{\nu}\wedge_{\star}\mathrm{d}x^{\mu}$
and
$\lambda\star\mathrm{d}x^{\mu}=\lambda\cdot\mathrm{d}x^{\mu}=\mathrm{d}x^{\mu}\star\lambda$
for $\lambda\in\Omega^{0}(\mathbbm{R}^{1,d-1})$. Similarly to (3.1), we define
the graded star-commutator of arbitrary homogeneous differential forms by

$\displaystyle[\alpha\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\beta]:=\alpha\wedge_{\star}\beta-(-1)^{|\alpha|\,|\beta|}\,\beta\wedge_{\star}\alpha\
.$ (3.2)

Then
$\big{(}\Omega^{\bullet}(\mathbbm{R}^{1,d-1}),[-\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}-],\mathrm{d}\big{)}$
is a differential graded Lie algebra, and noncommutative field theory with the
Moyal–Weyl star-product has a simple noncommutative differential calculus.

With these preliminaries, it is clear that the conventional noncommutative
$\mathsf{U}(1)$ gauge theories are formally identical to their classical
counterparts with nonabelian gauge symmetry, as is well-known. In particular,
the _star-gauge transformation_ of a gauge field
$A\in\Omega^{1}(\mathbbm{R}^{1,d-1})$ by a gauge parameter
$\lambda\in\Omega^{0}(\mathbbm{R}^{1,d-1})$ is given by the usual expression

$\displaystyle\delta_{\lambda}^{\star}A=\mathrm{d}\lambda-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]\
.$ (3.3)

Using the Leibniz rule for the star-commutator
$[-\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}-]$ with respect
to the de Rham differential $\mathrm{d}$, together with its Jacobi identity,
one easily checks the off-shell gauge closure

$\displaystyle\big{[}\delta^{\star}_{\lambda_{1}},\delta^{\star}_{\lambda_{2}}\big{]}^{\phantom{{\dagger}}}_{\circ}A=\delta^{\star}_{[\lambda_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda_{2}]}A$
(3.4)

to the Lie algebra
$\big{(}\Omega^{0}(\mathbbm{R}^{1,d-1}),[-\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}-]\big{)}$.

Under these star-gauge transformations, the noncommutative field strength

$\displaystyle
F_{A}^{\star}:=\mathrm{d}A+\tfrac{1}{2}\,[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]=\mathrm{d}A+A\wedge_{\star}A\
\in\ \Omega^{2}(\mathbbm{R}^{1,d-1})$ (3.5)

transforms covariantly:

$\displaystyle\delta_{\lambda}^{\star}F_{A}^{\star}=-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}F_{A}^{\star}]\
.$ (3.6)

This similarity to the analysis of ordinary nonabelian gauge theories was used
in [29] to cast noncommutative $\mathsf{U}(1)$ Chern–Simons and Yang–Mills
theories into the framework of standard $L_{\infty}$-algebras, which we now
describe.

#### Noncommutative Chern–Simons theory

The $L_{\infty}$-algebra formulation of noncommutative Chern–Simons theory in
three dimensions follows the analysis of the usual nonabelian Chern–Simons
gauge theory presented in Section 2.1. In this case the organising
$L_{\infty}$-algebra is simply the differential graded Lie algebra with
underlying graded vector space

$\displaystyle V=\Omega^{\bullet}(\mathbbm{R}^{1,2})$ (3.7)

and the non-zero brackets

$\displaystyle\ell_{1}(\alpha)=\mathrm{d}\alpha\qquad\mbox{and}\qquad\ell_{2}(\alpha,\beta)=-[\alpha\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\beta]$
(3.8)

for $\alpha,\beta\in V$. Note that the underlying cochain complex (2.45) (the
de Rham complex in this case) is unchanged by the noncommutative deformation;
that is, the free fields are the same and only the interactions are modified
by the noncommutativity, as is well-known for Moyal–Weyl type noncommutative
field theories.

Applying the standard machinery from Section 2.4, one easily verifies that the
noncommutative gauge symmetry (3.3) is reproduced in the expected way as

$\displaystyle\delta_{\lambda}^{\star}A=\ell_{1}(\lambda)+\ell_{2}(\lambda,A)\
,$ (3.9)

with the closure (3.4) expressed through

$\displaystyle\big{[}\delta^{\star}_{\lambda_{1}},\delta^{\star}_{\lambda_{2}}\big{]}_{\circ}A=\delta^{\star}_{-\ell_{2}(\lambda_{1},\lambda_{2})}A\
.$ (3.10)

The Maurer–Cartan equations $\mathfrak{F}_{A}=0$ reproduce the noncommutative
Chern–Simons equation of motion

$\displaystyle\mathfrak{F}_{A}=\ell_{1}(A)-\tfrac{1}{2}\,\ell_{2}(A,A)=F_{A}^{\star}\
,$ (3.11)

with the covariance (3.6) expressed as

$\displaystyle\delta_{\lambda}^{\star}\mathfrak{F}_{A}=\ell_{2}(\lambda,\mathfrak{F}_{A})\
.$ (3.12)

The Noether identities

$\displaystyle 0\ =\
{\mathsf{d}}_{A}\mathfrak{F}_{A}=\ell_{1}(\mathfrak{F}_{A})+\ell_{2}(\mathfrak{F}_{A},A)=\mathrm{d}F_{A}^{\star}+[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}F_{A}^{\star}]=:\mathrm{d}_{\star}^{A}F_{A}^{\star}$
(3.13)

are simply the star-gauge Bianchi identities
$\mathrm{d}_{\star}^{A}F_{A}^{\star}=0$ satisfied by the noncommutative field
strength $F_{A}^{\star}$, where $\mathrm{d}_{\star}^{A}$ is star-gauge
covariant derivative.

The inner product

$\displaystyle\langle\alpha,\beta\rangle=\int_{\mathbbm{R}^{1,2}}\,\alpha\wedge_{\star}\beta=\int_{\mathbbm{R}^{1,2}}\,\alpha\wedge\beta$
(3.14)

on $V$ pairs differential forms $\alpha$ and $\beta$ on $\mathbbm{R}^{1,2}$ in
complementary degrees, where the second equality follows from an integration
by parts and is sometimes expressed as saying that the Moyal–Weyl star-product
is _cyclic_. This makes $V$ into a cyclic $L_{\infty}$-algebra, and the
corresponding Maurer–Cartan action functional then reproduces the standard
noncommutative $\mathsf{U}(1)$ Chern–Simons functional:

$\displaystyle S(A)=\frac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\frac{1}{3!}\,\langle
A,\ell_{2}(A,A)\rangle=\frac{1}{2}\,\int_{\mathbbm{R}^{1,2}}\,A\wedge_{\star}\mathrm{d}A+\frac{2}{3}\,A\wedge_{\star}A\wedge_{\star}A\
,$ (3.15)

where we used
$[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]=2\,A\wedge_{\star}A$.
Thus, like the classical theory, noncommutative $\mathsf{U}(1)$ Chern–Simons
theory fits nicely into the algebraic framework of a differential graded Lie
algebra.

#### Noncommutative Yang–Mills theory

A similar analysis can be done for noncommutative Yang–Mills theory in any
spacetime dimension $d$, using the $L_{\infty}$-algebra of usual nonabelian
Yang–Mills theory from Section 2.5. Let $\ast_{\textrm{\tiny
H}}:\Omega^{k}(\mathbbm{R}^{1,d-1})\to\Omega^{d-k}(\mathbbm{R}^{1,d-1})$ be
the Hodge duality operator induced by the standard flat Minkowski metric of
$\mathbbm{R}^{1,d-1}$.

The underlying graded vector space is $V=V^{0}\oplus V^{1}\oplus V^{2}\oplus
V^{3}$, where

$\displaystyle\begin{split}V^{0}=\Omega^{0}(\mathbbm{R}^{1,d-1})\ ,\quad
V^{1}=\Omega^{1}(\mathbbm{R}^{1,d-1})\ ,\quad
V^{2}=\Omega^{d-1}(\mathbbm{R}^{1,d-1})\ ,\quad
V^{3}=\Omega^{d}(\mathbbm{R}^{1,d-1})\ .\end{split}$ (3.16)

Denoting the corresponding elements by $\lambda\in V^{0}$, $A\in V^{1}$,
$\mathcal{A}\in V^{2}$ and $\varLambda\in V^{3}$, the differential is given by

$\displaystyle\ell_{1}(\lambda)=\mathrm{d}\lambda\
,\quad\ell_{1}(A)=\mathrm{d}\ast_{\textrm{\tiny
H}}\mathrm{d}A\qquad\mbox{and}\qquad\ell_{1}(\mathcal{A})=\mathrm{d}\mathcal{A}\
,$ (3.17)

the 2-brackets are

$\displaystyle\begin{split}\ell_{2}(\lambda_{1},\lambda_{2})=-[\lambda_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda_{2}]\qquad&,\qquad\ell_{2}(\lambda,A)=-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]\
,\\\\[4.0pt]
\ell_{2}(\lambda,\mathcal{A})=-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\mathcal{A}]\qquad,\qquad\ell_{2}(\lambda,\varLambda)&=-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\varLambda]\qquad,\qquad\ell_{2}(A,\mathcal{A})=-[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\mathcal{A}]\
,\\\\[4.0pt] \ell_{2}(A_{1},A_{2})=-\mathrm{d}\ast_{\textrm{\tiny
H}}[A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{2}]-&[A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\ast_{\textrm{\tiny
H}}\mathrm{d}A_{2}]+(-1)^{d}\,[\ast_{\textrm{\tiny
H}}\mathrm{d}A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{2}]\
,\end{split}$ (3.18)

and finally there is a non-zero 3-bracket

$\displaystyle\begin{split}\ell_{3}(A_{1},&A_{2},A_{3})=-\big{[}A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\ast_{\textrm{\tiny
H}}[A_{2}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{3}]\big{]}-\big{[}A_{2}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\ast_{\textrm{\tiny
H}}[A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{3}]\big{]}+(-1)^{d}\,\big{[}\ast_{\textrm{\tiny
H}}[A_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{2}]\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A_{3}\big{]}\
.\end{split}$ (3.19)

The noncommutative gauge transformations and their closure again follow as in
(3.9) and (3.10), while the Maurer–Cartan equations $\mathfrak{F}_{A}=0$
reproduce the vacuum equation of motion for noncommutative $\mathsf{U}(1)$
Yang–Mills theory:

$\displaystyle\mathfrak{F}_{A}=\ell_{1}(A)-\tfrac{1}{2}\,\ell_{2}(A,A)-\tfrac{1}{3!}\,\ell_{3}(A,A,A)=\mathrm{d}_{\star}^{A}\ast_{\textrm{\tiny
H}}F_{A}^{\star}\ ,$ (3.20)

with covariance again captured by (3.12). The Noether identities

$\displaystyle 0\ =\
{\mathsf{d}}_{A}\mathfrak{F}_{A}=\ell_{1}(\mathfrak{F}_{A})+\ell_{2}(\mathfrak{F}_{A},A)=\mathrm{d}_{\star}^{A}\mathfrak{F}_{A}=\big{(}\mathrm{d}_{\star}^{A}\big{)}^{2}\ast_{\textrm{\tiny
H}}F_{A}^{\star}=[F_{A}^{\star}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\ast_{\textrm{\tiny
H}}F_{A}^{\star}]$ (3.21)

may be checked directly using symmetry properties of the $d$-form
$[F_{A}^{\star}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\ast_{\textrm{\tiny
H}}F_{A}^{\star}]\in\Omega^{d}(\mathbbm{R}^{1,d-1})$.

The inner product on $V$ given by

$\displaystyle\langle
A,\mathcal{A}\rangle=\int_{\mathbbm{R}^{1,d-1}}\,A\wedge_{\star}\mathcal{A}\qquad\mbox{and}\qquad\langle\lambda,\varLambda\rangle=\int_{\mathbbm{R}^{1,d-1}}\,\lambda\wedge_{\star}\varLambda$
(3.22)

defines a cyclic structure on the noncommutative Yang–Mills
$L_{\infty}$-algebra, and the corresponding Maurer–Cartan action functional
reproduces (after integration by parts) the standard noncommutative
$\mathsf{U}(1)$ Yang–Mills functional:

$\displaystyle S(A)=\frac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\frac{1}{3!}\,\langle
A,\ell_{2}(A,A)\rangle-\frac{1}{4!}\,\langle
A,\ell_{3}(A,A,A)\rangle=\frac{1}{2}\,\int_{\mathbbm{R}^{1,d-1}}\,F_{A}^{\star}\wedge_{\star}\ast_{\textrm{\tiny
H}}F_{A}^{\star}\ .$ (3.23)

###  3.2 Fuzzy field theories in the $L_{\infty}$-algebra formalism

A particularly well understood and tractable class of noncommutative gauge
theories on D-branes wrapping curved submanifolds with a non-constant gauge
flux is provided by D-branes in WZW models. These involve target spaces which
are group manifolds $\mathsf{G}$ with non-trivial NS–NS $3$-form fluxes, and
they contain D-branes that wrap conjugacy classes of $\mathsf{G}$ carrying
non-constant $2$-form flux [1]. Here we focus on the simplest example
$\mathsf{G}=\mathsf{SU}(2)$ for illustration, where the conjugacy classes are
$2$-spheres $\mathbbm{S}^{2}\subset\mathbbm{S}^{3}$ labelled by the spin $j$
of a representation of the $\mathfrak{su}(2)_{k}$ current algebra at WZW level
$k\in\mathbbm{Z}_{>0}$. In the semi-classical limit with $k\to\infty$, the
space of open string ground states forms a finite-dimensional matrix algebra,
in contrast to the infinite-dimensional flat space case from Section 3.1, and
the noncommutative matrix product $f\,g$ now plays the analogue role of the
Moyal–Weyl star-product.

The noncommutative gauge theory supported by the two-dimensional worldvolume
of the D-brane is thus no longer a continuum field theory but rather a matrix
model [2], called a ‘fuzzy’ field theory, or alternatively a field theory on
the ‘fuzzy sphere’ which is a matrix approximation of the standard $2$-sphere
$\mathbbm{S}^{2}$ obtained by quantizing conjugacy classes in
$\mathsf{SU}(2)$, and which is covariant under $\mathsf{SU}(2)$. Being finite-
dimensional systems, these noncommutative field theories are particularly well
tailored to the formalism developed in Section 2, as they avoid the many
functional analytic complications brought in by continuum field theories.

Let us first explain how to set up a differential calculus on the fuzzy sphere
suitable for the formulation of fuzzy field theories. The algebra of functions
on the fuzzy sphere $\mathbbm{S}_{N}^{2}$ is defined by

$\displaystyle\mathcal{C}_{N}:=(j)\otimes(j)^{*}\simeq{\sf Mat}(N)\ ,$ (3.24)

where $(j)$ denotes the irreducible spin $j=(N-1)/2$ representation of
$\mathfrak{su}(2)$ for an integer $N>1$ and $(j)^{*}$ denotes the dual
representation; the isomorphism with $N{\times}N$ complex matrices follows
from the fact that $(j)$ is $N$-dimensional. Fix a basis of the Lie algebra
$\mathfrak{su}(2)$ where the structure constants are given by the Levi–Civita
symbol $\epsilon_{ab}{}^{c}$, for $a,b,c=1,2,3$, and denote by
$\frac{1}{r_{N}}\,X_{a}\in\mathcal{C}_{N}$ the generators of
$\mathfrak{su}(2)$ in the spin $j$ representation, where
$r_{N}:=\frac{1}{\sqrt{j\,(j+1)}}$. Then the matrices $X_{a}$ generate the
algebra $\mathcal{C}_{N}$ and satisfy the relations

$\displaystyle[X_{a},X_{b}]=\mathrm{i}\,r_{N}\,\epsilon_{ab}{}^{c}\,X_{c}\
,\quad\delta^{ab}\,X_{a}\,X_{b}=\mathbbm{1}\qquad\mbox{and}\qquad
X_{a}^{\ast}=X_{a}$ (3.25)

of the fuzzy unit sphere, where

$\displaystyle[\lambda,\rho]:=\lambda\,\rho-\rho\,\lambda$ (3.26)

denotes the matrix commutator of $\lambda,\rho\in\mathcal{C}_{N}$, which makes
$\mathcal{C}_{N}$ into a Lie algebra, and ∗ denotes Hermitian conjugation.

The fuzzy $2$-sphere $\mathbbm{S}_{N}^{2}$ has a three-dimensional
$\mathfrak{su}(2)$-covariant differential calculus given by the
Chevalley–Eilenberg algebra

$\displaystyle\Omega^{\bullet}(\mathcal{C}_{N}):=\mathcal{C}_{N}\otimes\text{\Large$\wedge$}^{\bullet}\mathfrak{su}(2)^{*}$
(3.27)

of $\mathfrak{su}(2)$ with coefficients in the fuzzy sphere algebra (3.24),
which as an $\mathfrak{su}(2)$-module decomposes into irreducible
representations as

$\displaystyle\mathcal{C}_{N}\simeq\bigoplus_{J=0}^{N-1}\,(J)\ .$ (3.28)

Let $\vartheta^{a}\in\mathfrak{su}(2)^{*}\subset\Omega^{1}(\mathcal{C}_{N})$
for $a=1,2,3$ be a compatible coframe for the $\mathcal{C}_{N}$-module of
$1$-forms defined by the dual of the chosen Lie algebra basis, with
$\vartheta^{a}\wedge\vartheta^{b}=-\vartheta^{b}\wedge\vartheta^{a}$, which
generates the entire differential calculus $\Omega^{\bullet}(\mathcal{C}_{N})$
and commutes with $\Omega^{0}(\mathcal{C}_{N})=\mathcal{C}_{N}$. The
Chevalley–Eilenberg differential $\mathrm{d}_{\textrm{\tiny CE}}$ is then
defined by

$\displaystyle\mathrm{d}_{\textrm{\tiny
CE}}\lambda:=\tfrac{1}{r_{N}}\,[X_{a},\lambda]\,\vartheta^{a}$ (3.29)

for $\lambda\in\Omega^{0}(\mathcal{C}_{N})=\mathcal{C}_{N}$, together with the
Maurer–Cartan equations

$\displaystyle\mathrm{d}_{\textrm{\tiny
CE}}\vartheta^{a}=-\tfrac{\mathrm{i}}{2}\,\epsilon^{a}{}_{bc}\,\vartheta^{b}\wedge\vartheta^{c}\
,$ (3.30)

and extended to the whole of $\Omega^{\bullet}(\mathcal{C}_{N})$ by the graded
Leibniz rule. We extend the matrix commutator on $\mathcal{C}_{N}$ to
arbitrary homogeneous differential forms
$\alpha,\beta\in\Omega^{\bullet}(\mathcal{C}_{N})$ by

$\displaystyle[\alpha,\beta]:=\alpha\wedge\beta-(-1)^{|\alpha|\,|\beta|}\,\beta\wedge\alpha\
,$ (3.31)

which makes
$\big{(}\Omega^{\bullet}(\mathcal{C}_{N}),[-,-],\mathrm{d}_{\textrm{\tiny
CE}}\big{)}$ into a differential graded Lie algebra, enabling the construction
of field theories on the fuzzy sphere $\mathbbm{S}_{N}^{2}$.

With this noncommutative differential calculus it is now straightforward to
formulate noncommutative gauge symmetries in the standard way: The _fuzzy
gauge transformation_ of a gauge field $A\in\Omega^{1}(\mathcal{C}_{N})$ by a
gauge parameter $\lambda\in\Omega^{0}(\mathcal{C}_{N})$ is given by the usual
formula

$\displaystyle\delta_{\lambda}^{\textrm{\tiny CE}}A=\mathrm{d}_{\textrm{\tiny
CE}}\lambda-[\lambda,A]\ ,$ (3.32)

and using the by now familiar arguments one shows that they close to the Lie
algebra $(\mathcal{C}_{N},[-,-])$:

$\displaystyle\big{[}\delta_{\lambda_{1}}^{\textrm{\tiny
CE}},\delta_{\lambda_{2}}^{\textrm{\tiny
CE}}\big{]}_{\circ}=\delta^{\textrm{\tiny CE}}_{[\lambda_{1},\lambda_{2}]}\ .$
(3.33)

The field strength defined by

$\displaystyle F_{A}^{\textrm{\tiny CE}}:=\mathrm{d}_{\textrm{\tiny
CE}}A+\tfrac{1}{2}\,[A,A]=\mathrm{d}_{\textrm{\tiny CE}}A+A\wedge A\ \in\
\Omega^{2}(\mathcal{C}_{N})$ (3.34)

transforms covariantly under this gauge symmetry:

$\displaystyle\delta_{\lambda}^{\textrm{\tiny CE}}F_{A}^{\textrm{\tiny
CE}}=-[\lambda,F_{A}^{\textrm{\tiny CE}}]\ .$ (3.35)

The low-energy effective field theory on a D-brane of type $j$ wrapping this
fuzzy $2$-sphere was derived in [2] as a noncommutative gauge theory whose
action functional is a sum of a Yang–Mills term and a Chern–Simons term, each
of which is individually invariant under the gauge transformations
(3.32).181818One can extend this description to stack of $n>1$ D-branes of
type $j$ by considering the differential graded noncommutative algebra
$\Omega^{\bullet}(\mathcal{C}_{N},\mathfrak{u}(n)):=\Omega^{\bullet}(\mathcal{C}_{N})\otimes\mathfrak{u}(n)$
of $\mathfrak{u}(n)$-valued forms on the fuzzy $2$-sphere, with the
differential $\mathrm{d}_{\textrm{\tiny CE}}\otimes\mathbbm{1}$, and the
(graded) Lie bracket extended to the matrix commutator on ${\sf
Mat}(N)\otimes{\sf Mat}(n)$, and similarly for the normalised trace below. The
underlying $L_{\infty}$-algebra was described in [29] in a flat holonomic
coframe. Here we give an alternative treatment which uses the more natural
covariant coframe $\vartheta^{a}$ appropriate to a curved space; Yang–Mills
and Chern–Simons gauge theories in this formulation were first discussed in
[53]. The advantage of this description is that, like the flat space
Moyal–Weyl type noncommutative gauge theories of Section 3.1, the two theories
can be treated separately and the $L_{\infty}$-algebra structure exactly
parallels the case of ordinary nonabelian gauge theories.

#### Chern–Simons theory on the fuzzy sphere

The $L_{\infty}$-algebra formulation of Chern–Simons theory on
$\mathbbm{S}_{N}^{2}$ in the coframe formalism was discussed in [77]. As usual
it is based on a differential graded Lie algebra whose underlying graded
vector space is

$\displaystyle V=\Omega^{\bullet}(\mathcal{C}_{N})$ (3.36)

with the non-zero brackets

$\displaystyle\ell_{1}(\alpha)=\mathrm{d}_{\textrm{\tiny
CE}}\alpha\qquad\mbox{and}\qquad\ell_{2}(\alpha,\beta)=-[\alpha,\beta]$ (3.37)

for $\alpha,\beta\in V$. Notably, in this framework the underlying cochain
complex is markedly different from the classical case, as its differential is
the Chevalley–Eilenberg differential $\mathrm{d}_{\textrm{\tiny CE}}$, rather
than the de Rham differential $\mathrm{d}$. This is not so surprising since
the geometry of the fuzzy sphere $\mathbbm{S}_{N}^{2}$ involves a three-
dimensional differential calculus on a fuzzy version of a two-dimensional
space. This is due to the imposition of $\mathfrak{su}(2)$-covariance, which
requires an embedding of the classical $2$-sphere in three-dimensional flat
space: the extra dimension comes from the pullback of the tangent bundle
$T_{\mathbbm{R}^{3}}\to\mathbbm{R}^{3}$ to $\mathbbm{S}^{2}$, which decomposes
as the Whitney sum
$T_{\mathbbm{R}^{3}}|_{\mathbbm{S}^{2}}=T_{\mathbbm{S}^{2}}\oplus
N_{\mathbbm{S}^{2}}$ with $N_{\mathbbm{S}^{2}}\to\mathbbm{S}^{2}$ the normal
bundle to the embedding $\mathbbm{S}^{2}\hookrightarrow\mathbbm{R}^{3}$.
Higher-dimensional covariant calculi are a common feature of a broad range of
examples in noncommutative geometry.

The gauge symmetry (3.32)–(3.33) and field equation $F_{A}^{\textrm{\tiny
CE}}=0$ arise in the standard way from these brackets, as does the usual
Bianchi identity

$\displaystyle\mathrm{d}^{A}_{\textrm{\tiny CE}}F_{A}^{\textrm{\tiny
CE}}:=\mathrm{d}_{\textrm{\tiny CE}}F_{A}^{\textrm{\tiny
CE}}+[A,F_{A}^{\textrm{\tiny CE}}]=0$ (3.38)

from the corresponding Noether identity of the $L_{\infty}$-algebra. We
introduce a cyclic structure in the usual way by the pairing

$\displaystyle\langle\alpha,\beta\rangle=\int_{\mathbbm{S}_{N}^{2}}\,\alpha\wedge\beta$
(3.39)

of forms $\alpha,\beta\in\Omega^{\bullet}(\mathcal{C}_{N})$ in complementary
degrees, where the integration of top-forms $\Omega^{3}(\mathcal{C}_{N})$ is
defined via the normalized $N{\times}N$ matrix trace

$\displaystyle\int_{\mathbbm{S}_{N}^{2}}\,\lambda\
\vartheta^{1}\wedge\vartheta^{2}\wedge\vartheta^{3}:=\frac{4\pi}{N}\,\mathrm{Tr}(\lambda)\
,$ (3.40)

where $\lambda\in\mathcal{C}_{N}$. The Maurer–Cartan action functional

$\displaystyle S(A)=\frac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\frac{1}{3!}\,\langle
A,\ell_{2}(A,A)\rangle=\frac{1}{2}\,\int_{\mathbbm{S}_{N}^{2}}\,A\wedge\mathrm{d}_{\textrm{\tiny
CE}}A+\frac{2}{3}\,A\wedge A\wedge A$ (3.41)

is just the standard Chern–Simons functional on the fuzzy sphere as written in
[53]. It has a natural global $\mathsf{SU}(2)$ symmetry given by rotations of
the coframe $\vartheta^{a}$, and in the continuum limit $N\to\infty$ it
describes a $BF$-type topological field theory on the classical $2$-sphere
$\mathbbm{S}^{2}$. We shall give a more thorough account of general $BF$
theories in the $L_{\infty}$-algebra formalism in Section 6.1.

We can write the action functional (3.41) in a more transparent form of a
matrix model. For this, we write the gauge field
$A\in\Omega^{1}(\mathcal{C}_{N})$ explicitly in component form
$A=A_{a}\,\vartheta^{a}$ with $A_{a}\in{\sf Mat}(N)$ and define
$\mathrm{L}_{a}A_{b}:=\frac{1}{r_{N}}\,[X_{a},A_{b}]$. Then we can write
(3.41) as

$\displaystyle S(A)=\tfrac{2\pi}{N}\,\mathrm{Tr}\big{(}\epsilon^{abc}\,{\tt
CS}_{abc}\big{)}\ ,$ (3.42)

where

$\displaystyle{\tt
CS}_{abc}=\mathrm{L}_{a}A_{b}\,A_{c}+\tfrac{1}{3}\,A_{a}\,[A_{b},A_{c}]-\tfrac{\mathrm{i}}{2}\,\epsilon_{ab}{}^{d}\,A_{d}\,A_{c}\
.$ (3.43)

This is the form of the fuzzy Chern–Simons functional derived in [2] from a
worldsheet conformal field theory analysis.191919This form of the Chern–Simons
functional can be extended to any semi-simple Lie algebra $\mathfrak{g}$ of
higher rank by replacing the Levi–Civita symbol $\epsilon_{ab}{}^{c}$ with the
structure constants $f_{ab}{}^{c}$ of a basis of $\mathfrak{g}$, in which case
it is part of the low-energy effective field theory on a D-brane wrapping
conjugacy classes in the corresponding compact semi-simple Lie group
$\mathsf{G}$ [2]. However, the noncommutative geometric origin of the action
functional in these extensions is not clear.

#### Yang–Mills theory on the fuzzy sphere

To define Yang–Mills theory on $\mathbbm{S}_{N}^{2}$ we need an analogue of
the Hodge operator $\ast_{\textrm{\tiny
CE}}:\Omega^{k}(\mathcal{C}_{N})\to\Omega^{3-k}(\mathcal{C}_{N})$, which is
defined on the compatible coframe of $\Omega^{\bullet}(\mathcal{C}_{N})$ by

$\displaystyle\ast_{\textrm{\tiny
CE}}\mathbbm{1}:=\tfrac{1}{3!}\,\epsilon_{abc}\,\vartheta^{a}\wedge\vartheta^{b}\wedge\vartheta^{c}\qquad\mbox{and}\qquad\ast_{\textrm{\tiny
CE}}\vartheta^{a}:=\tfrac{1}{2}\,\epsilon^{a}{}_{bc}\,\vartheta^{b}\wedge\vartheta^{c}\
,$ (3.44)

together with involutivity $(\ast_{\textrm{\tiny CE}})^{2}=\mathbbm{1}$. The
underlying graded vector space of the $L_{\infty}$-algebra is again

$\displaystyle V=\Omega^{\bullet}(\mathcal{C}_{N})\ ,$ (3.45)

and, with similar notation as in Section 3.1, the differential is given by

$\displaystyle\ell_{1}(\lambda)=\mathrm{d}_{\textrm{\tiny CE}}\lambda\
,\quad\ell_{1}(A)=\mathrm{d}_{\textrm{\tiny CE}}\ast_{\textrm{\tiny
CE}}\mathrm{d}_{\textrm{\tiny
CE}}A\qquad\mbox{and}\qquad\ell_{1}(\mathcal{A})=\mathrm{d}_{\textrm{\tiny
CE}}\mathcal{A}\ ,$ (3.46)

the 2-brackets are

$\displaystyle\begin{split}\ell_{2}(\lambda_{1},\lambda_{2})=-[\lambda_{1},\lambda_{2}]\qquad&,\qquad\ell_{2}(\lambda,A)=-[\lambda,A]\
,\\\\[4.0pt]
\ell_{2}(\lambda,\mathcal{A})=-[\lambda,\mathcal{A}]\qquad,\qquad\ell_{2}(\lambda,\varLambda)&=-[\lambda,\varLambda]\qquad,\qquad\ell_{2}(A,\mathcal{A})=-[A,\mathcal{A}]\
,\\\\[4.0pt] \ell_{2}(A_{1},A_{2})=-\mathrm{d}_{\textrm{\tiny
CE}}\ast_{\textrm{\tiny CE}}[A_{1},A_{2}]-&[A_{1},\ast_{\textrm{\tiny
CE}}\mathrm{d}_{\textrm{\tiny CE}}A_{2}]-[\ast_{\textrm{\tiny
CE}}\mathrm{d}_{\textrm{\tiny CE}}A_{1},A_{2}]\ ,\end{split}$ (3.47)

and the non-zero 3-bracket is

$\displaystyle\begin{split}\ell_{3}(A_{1},A_{2},A_{3})=-\big{[}A_{1},\ast_{\textrm{\tiny
CE}}[A_{2},A_{3}]\big{]}-\big{[}A_{2},\ast_{\textrm{\tiny
CE}}[A_{1},A_{3}]\big{]}-\big{[}\ast_{\textrm{\tiny
CE}}[A_{1},A_{2}],A_{3}\big{]}\ .\end{split}$ (3.48)

The Maurer–Cartan equation $\mathfrak{F}_{A}=0$ leads to the usual covariant
equation of motion given by $\mathrm{d}_{\textrm{\tiny
CE}}^{A}\ast_{\textrm{\tiny CE}}F_{A}^{\textrm{\tiny CE}}=0$, and, with the
same cyclic inner product (3.39), the Maurer–Cartan action functional

$\displaystyle S(A)=\frac{1}{2}\,\langle
A,\ell_{1}(A)\rangle-\frac{1}{3!}\,\langle
A,\ell_{2}(A,A)\rangle-\frac{1}{4!}\,\langle
A,\ell_{3}(A,A,A)\rangle=\frac{1}{2}\,\int_{\mathbbm{S}_{N}^{2}}\,F_{A}^{\textrm{\tiny
CE}}\wedge\ast_{\textrm{\tiny CE}}F_{A}^{\textrm{\tiny CE}}$ (3.49)

is precisely the $\mathsf{SU}(2)$-symmetric Yang–Mills functional on the fuzzy
sphere as presented in [53]. Written explicitly in component form
$F_{A}^{\textrm{\tiny
CE}}=\frac{1}{2}\,F_{ab}\,\vartheta^{a}\wedge\vartheta^{b}$ with
$F_{ab}\in{\sf Mat}(N)$, this becomes explicitly a matrix model

$\displaystyle S(A)=\tfrac{\pi}{N}\,\mathrm{Tr}\big{(}F_{ab}\,F^{ab}\big{)}\
,$ (3.50)

where

$\displaystyle
F_{ab}=\mathrm{L}_{a}A_{b}-\mathrm{L}_{b}A_{a}+[A_{a},A_{b}]-\epsilon_{ab}{}^{c}\,A_{c}$
(3.51)

and $F^{ab}:=\delta^{ac}\,\delta^{bd}\,F_{cd}$, which is precisely the form of
the fuzzy Yang–Mills functional derived in [2].

###  3.3 Working with non-trivial fluxes

The two special classes of noncommutative field theories which we have
considered thus far in this section have the special feature that they each
admit a natural noncommutative differential calculus, which enabled a
straightforward adaptation of techniques from ordinary nonabelian gauge
theories to reformulate them algebraically in the formalism of standard
$L_{\infty}$-algebras, even in the curved space example of the fuzzy sphere.
As discussed in Section 1, this will not be the case in general for curved
D-branes and for curved backgrounds. For a non-constant Poisson bivector
$\theta=\frac{1}{2}\,\theta^{\mu\nu}\,\partial_{\mu}\wedge\partial_{\nu}$, the
deformation quantization of the worldvolume $M$ is described by the Kontsevich
star-product [63] which up to second order in the deformation parameter
$\hbar$ can be written in the form

$\displaystyle f\circledast g=f\star
g-\tfrac{\hbar^{2}}{12}\,\theta^{\mu\rho}\,\partial_{\rho}\theta^{\nu\lambda}\,\big{(}\partial_{\mu}\partial_{\nu}f\,\partial_{\lambda}g+\partial_{\mu}\partial_{\nu}g\,\partial_{\lambda}f\big{)}+O(\hbar^{3})$
(3.52)

for fields $f$ and $g$ on $M$, where $f\star g$ is given by the standard
Moyal–Weyl formula (1.1).

The usual de Rham differential $\mathrm{d}$ is evidently not a derivation of
this star-product unless $\theta$ is constant. This also implies that the
naive definition (3.3) of noncommutative gauge transformations no longer
satisfies the gauge closure property (3.4). Moreover, in the case that the
pullback of the NS–NS $3$-form flux to the D-brane worldvolume is non-zero,
the bivector $\theta$ defines a _twisted_ Poisson structure and the star-
product (3.52) is no longer associative; in this case the star-commutator
$[-,-]_{\circledast}$ fails to define a Lie bracket on $\Omega^{0}(M)$.

In [29] a homotopy algebraic approach to constructing generic noncommutative
gauge theories was proposed and dubbed the ‘$L_{\infty}$-bootstrap’. According
to this prescription, one starts with the natural brackets

$\displaystyle\ell_{1}(\lambda)=\mathrm{d}\lambda\qquad\mbox{and}\qquad\ell_{2}(\lambda_{1},\lambda_{2})=-[\lambda_{1},\lambda_{2}]_{\circledast}$
(3.53)

on the vector space $V^{0}=\Omega^{0}(M)$ of gauge parameters, and attempts to
construct the rest of the $L_{\infty}$-algebra structure by consistently
solving the homotopy Jacobi identities order by order in the deformation
parameter $\hbar$. The approach of [67, 66], based on symplectic embeddings of
(almost) Poisson structures, improves this perspective because it can in
principle compute all orders expressions, which are sometimes asymptotic
expansions of analytic functions known in closed form. However, while these
approaches suggest novel algebraic constructions which go beyond earlier
approaches, they suffer from a more serious problem: so far they have only
been developed in detail for the semi-classical limit of a full noncommutative
gauge theory, where the star-commutator is approximated by the underlying
almost Poisson bracket. It is not currently clear how these methods can be
extended to construct complete noncommutative gauge theories, beyond their
semi-classical approximations.

In the remainder of this contribution we will explain an alternative homotopy
algebraic construction of noncommutative field theories, which does not handle
generic (almost) Poisson structures, but is suggestive of a more systematic
and elegant approach to dealing with the general problems discussed above.
Rather than attempting to fit noncommutative gauge symmetry into the standard
structure of an $L_{\infty}$-algebra, we adapt the $L_{\infty}$-algebra itself
by deforming it in a way which is compatible with the underlying
noncommutative algebra of differential forms. This yields a new perspective on
the implementation of general symmetries in noncommutative field theories
while producing new deformations of the standard gauge theories, and it
introduces a new type of homotopy algebraic structure which to our knowledge
has not been previously discussed in the mathematics literature.

##  4 Braided gauge symmetry and braided $L_{\infty}$-algebras

In this section we come to the main topic of this paper. We shall introduce
the notion of braided gauge symmetry using Drinfel’d twist deformation
techniques, comparing it to the more conventional star-gauge symmetries that
were discussed in Section 3.1. After going back to our simple prototypical
example of Chern–Simons gauge theory from Section 2.1 and describing its
braided noncommutative deformation, we introduce braided $L_{\infty}$-algebras
as the organising algebraic principle behind noncommutative field theories
with braided gauge symmetry. Throughout we stress the novelties behind these
‘braided field theories’ as compared to the standard noncommutative field
theories, and we discuss a braided version of the BV formalism which should
serve as a useful tool for the quantization of these theories.

###  4.1 Closing noncommutative gauge transformations

In Section 3.3 we pointed out issues involved in attempting to close an
algebra of star-gauge transformations (3.3) for generic (almost) Poisson
structures. Let us discuss a somewhat different but equally well-known closure
issue with these noncommutative gauge symmetries, which will serve to motivate
the constructions which follow later on in this section.

As in Section 3.1, consider a noncommutative field theory defined with the
Moyal–Weyl star-product (1.1) for a constant Poisson bivector $\theta$ on flat
space $M=\mathbbm{R}^{1,d-1}$. Let $\mathfrak{g}$ be a matrix Lie algebra, and
endow the exterior algebra
$\Omega^{\bullet}(M,\mathfrak{g})=\Omega^{\bullet}(M)\otimes\mathfrak{g}$ of
$\mathfrak{g}$-valued forms with the graded star-commutator

$\displaystyle[\alpha\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\beta]_{\mathfrak{g}}:=\alpha\wedge_{\star}\beta-(-1)^{|\alpha|\,|\beta|}\,\beta\wedge_{\star}\alpha$
(4.1)

for homogeneous differential forms
$\alpha,\beta\in\Omega^{\bullet}(M,\mathfrak{g})$, where here the star-
products are composed with the matrix multiplication in $\mathfrak{g}$. Again
we define the _star-gauge transformation_ of a gauge field
$A\in\Omega^{1}(M,\mathfrak{g})$ by a gauge parameter
$\lambda\in\Omega^{0}(M,\mathfrak{g})$ as the naive deformation of a classical
gauge transformation:

$\displaystyle\delta_{\lambda}^{\star}A=\mathrm{d}\lambda-[\lambda\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]_{\mathfrak{g}}\
.$ (4.2)

The problem now is that these star-gauge variations do not generally close on
the Lie algebra $\mathfrak{g}$: whereas the commutator of two gauge variations
still closes onto the star-commutator as
$\big{[}\delta_{\lambda_{1}}^{\star},\delta_{\lambda_{2}}^{\star}\big{]}_{\circ}=\delta_{[\lambda_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda_{2}]_{\mathfrak{g}}}^{\star}$,
the star-commutator itself does not close:

$\displaystyle[\lambda_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\lambda_{2}]_{\mathfrak{g}}=\lambda_{1}\star\lambda_{2}-\lambda_{2}\star\lambda_{1}\
\notin\ \Omega^{0}(M,\mathfrak{g})\ .$ (4.3)

Moreover, the naive deformation of the field strength

$\displaystyle
F_{A}^{\star}=\mathrm{d}A+\tfrac{1}{2}\,[A\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}A]_{\mathfrak{g}}=\mathrm{d}A+A\wedge_{\star}A\
\notin\ \Omega^{2}(M,\mathfrak{g})$ (4.4)

is no longer valued in $\mathfrak{g}$. In other words, the star-commutator
(4.1) does not generally define a Lie bracket on the exterior algebra
$\Omega^{\bullet}(M,\mathfrak{g})$. An easy calculation in a basis of
$\mathfrak{g}$ shows that the exception is when $\mathfrak{g}$ is a matrix Lie
algebra which is also closed under the anticommutator of matrices, such as
$\mathfrak{g}=\mathfrak{gl}(n)$ or $\mathfrak{g}=\mathfrak{u}(n)$; if the
gauge theory is coupled to matter fields then those must further be restricted
to the trivial, (anti-)fundamental or adjoint representations of
$\mathfrak{g}$. This is a well-known issue in noncommutative gauge theory,
noted originally in [88] and subsequently elucidated by [30, 37]. We note that
exactly the same problem arises in the fuzzy field theories from Section 3.2:
the extension of the graded matrix commutator (3.31) to ${\sf
Mat}(N)\otimes\mathfrak{g}$ does not close in $\mathfrak{g}$ unless
$\mathfrak{g}$ is also closed under anticommutators.

Let us briefly discuss the two most common workarounds of the problem of
defining Moyal–Weyl star-gauge theories for generic Lie algebras
$\mathfrak{g}$, each of which has its own drawbacks. Firstly, one can notice
that the star-gauge closure takes place not in the Lie algebra $\mathfrak{g}$
but rather in its universal enveloping algebra $\mathsf{U}\mathfrak{g}$, which
is the free unital algebra over $\mathbbm{R}$ generated by $\mathfrak{g}$
modulo the two-sided ideal generated by $X\,Y-Y\,X-[X,Y]_{\mathfrak{g}}$ for
all $X,Y\in\mathfrak{g}$. This leads to the notion of _enveloping algebra-
valued gauge symmetry_ [61, 10], generated by extended gauge parameters
$\lambda\in\Omega^{0}(M,\mathsf{U}\mathfrak{g})$ acting on extended gauge
fields $A\in\Omega^{1}(M,\mathsf{U}\mathfrak{g})$. While this extension is
appealing because it makes sense for arbitrary (not necessarily matrix) Lie
algebras $\mathfrak{g}$, its drawback is that it introduces infinitely many
new degrees of freedom for the gauge fields $A$ due to the infinite
dimensionality of the algebra $\mathsf{U}\mathfrak{g}$. These new fields have
no clear meaning in the underlying gauge theory at $\hbar=0$, and so these
noncommutative gauge theories do not possess good classical limits.

Alternatively, one can use the _Seiberg–Witten map_ [83], which was originally
found as an equivalence of low-energy effective descriptions of open strings
in a constant $B$-field background in terms of ordinary as well as
noncommutative gauge theories. This identifies the noncommutative and
classical gauge orbits through a field redefinition which maps ordinary gauge
parameters and fields $(\lambda,a)$ to noncommutative gauge parameters and
fields $\big{(}\hat{A}(a),\hat{\Lambda}(\lambda,a)\big{)}$ defined by

$\displaystyle\hat{A}(a+\delta_{\lambda}a)=\hat{A}(a)+\delta_{\hat{\Lambda}(\lambda,a)}^{\star}\hat{A}(a)\
.$ (4.5)

If the gauge theory is coupled to matter fields, then there is a further map
from ordinary matter fields $\phi$ to noncommutative matter fields
$\hat{\Phi}(\phi,a)$. In this case no new degrees of freedom are introduced,
while the noncommutative variables are functions of the corresponding ordinary
variables and the background deformation. Interesting new interaction vertices
appear which enable the construction of noncommutative gauge theories [62];
this has been used extensively over the years in phenomenological explorations
of noncommutative field theories (see e.g. [86] for a review).

The Seiberg–Witten map has a clear geometric interpretation which enables its
extension to arbitrary Poisson structures [58, 59] and even to twisted Poisson
structures [15, 76]. It can also be interpreted as an $L_{\infty}$-quasi-
isomorphism of the underlying $L_{\infty}$-algebras [28]. However, the
drawback of this approach is that it does not produce the intrinsically
noncommutative differential graded algebra underlying a noncommutative gauge
theory, along the lines we anticipated in Section 1, but rather maps the
problem into a non-local classical field theory.

In the following we review a more recent approach to the construction of
noncommutative gauge theories which circumvents these problems: these field
theories are defined analytically in closed form for arbitrary gauge algebras
$\mathfrak{g}$ and matter field representations without the introduction of
spurious degrees of freedom, and they possess good classical limits to the
corresponding classical field theories. Their construction is based on a
precise and systematic homotopy algebraic organising principle which avoids
the sometimes cumbersome ordering ambiguities in definitions of interaction
vertices due to noncommutativity of fields (e.g. $\Phi_{1}\star
A\star\Phi_{2}\neq\Phi_{1}\star\Phi_{2}\star A$): the entire noncommutative
field theory is uniquely defined by the structure of an underlying ‘braided
$L_{\infty}$-algebra’ which is constructed as an unambiguous and precise
noncommutative deformation of the standard $L_{\infty}$-algebra underlying the
classical field theory.

#### Noncommutative diffeomorphisms

To help motivate how the braided algebraic structure appears, let us bring
gravity into the present discussion and attempt to analyse its noncommutative
deformation through the methods just explained (recall the motivation from
Section 1). The symmetries underlying general relativity are spacetime
diffeomorphisms which are generated by vector fields in
$\mathfrak{v}:={\sf\Gamma}(TM)$, which is a Lie algebra with the usual Lie
bracket of vector fields which we denote by $[-,-]_{\mathfrak{v}}$. In the
Moyal–Weyl deformation, we define the star-bracket
$[-\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}-]_{\mathfrak{v}}$
in the enveloping algebra $\mathsf{U}\mathfrak{v}$ by regarding the vector
space ${\sf\Gamma}(TM)$ as a module over the noncommutative algebra of
functions $\big{(}C^{\infty}(M,\mathbbm{R}),\star\big{)}$, with the holonomic
frame on $M=\mathbbm{R}^{1,d-1}$ obeying
$f\star\partial_{\mu}=f\cdot\partial_{\mu}$ and $\partial_{\mu}\star
f=\partial_{\mu}f$ for $f\in C^{\infty}(M,\mathbbm{R})$, and
$\partial_{\mu}\star\partial_{\nu}=\partial_{\mu}\,\partial_{\nu}=\partial_{\nu}\star\partial_{\mu}$
in $\mathsf{U}\mathfrak{v}$. As above, if $\xi_{1},\xi_{2}\in\mathfrak{v}$ are
vector fields, then an easy calculation in components reveals that the star-
bracket
$[\xi_{1}\stackrel{{\scriptstyle\scriptstyle\star}}{{\scriptstyle,}}\xi_{2}]_{\mathfrak{v}}\in\mathsf{U}\mathfrak{v}$
is no longer a vector field in general. However, no analogue of the
Seiberg–Witten map is generally known for deformed diffeomorphisms, which are
instead most naturally defined through Drinfel’d twist techniques [19, 16]. We
now proceed to review these techniques, and then demonstrate how they can be
fruitfully applied to internal gauge symmetries as well.

###  4.2 Drinfel’d twist deformation theory

We shall now briefly review the basics of the Drinfel’d twist deformation
formalism, which is a method to consistenly deform symmetries (described by a
Hopf algebra). Here we only recall the salient aspects that we will use
explicitly in the following; see [74, 24] for a more detailed and complete
account. A lot of what we say in the remainder of this section can be applied
in a much more general context, and in particular to incorporate the fuzzy
field theories discussed in Section 3.2 (see [77]). For the examples we treat
in the present paper, however, we will always start from a continuum field
theory on a manifold $M$ and suitably deform it analogously to the Moyal–Weyl
type noncommutative field theories. Hence we start from the Lie algebra of
vector fields $\mathfrak{v}:={\sf\Gamma}(TM)$, which generate infinitesimal
diffeomorphisms of $M$, and later on specialize to suitable subalgebras
appropriate to the spacetime symmetries of a given field theory example at
hand.

The universal enveloping algebra $\mathsf{U}\mathfrak{v}$ of the Lie algebra
$\mathfrak{v}$ is naturally a cocommutative Hopf algebra with coproduct
$\Delta:\mathsf{U}\mathfrak{v}\to\mathsf{U}\mathfrak{v}\otimes\mathsf{U}\mathfrak{v}$
defined on generators by

$\displaystyle\begin{split}\Delta(\xi)&=\xi\otimes
1+1\otimes\xi\qquad\mbox{and}\qquad\Delta(1)=1\otimes 1\ ,\end{split}$ (4.6)

for all $\xi\in\mathfrak{v}$. The map $\Delta$ is extended as an algebra
homomorphism to all of $\mathsf{U}\mathfrak{v}$ (analogously to (2.18)).
Cocommutativity is the property $\tau\circ\Delta=\Delta$, where $\tau$ is the
transposition which interchanges the factors in a tensor product of vector
spaces; this is equivalent to saying that $\mathsf{U}\mathfrak{v}$ has a
triangular structure with trivial ${\mathcal{R}}$-matrix $1\otimes 1$. The
Hopf algebra structure is completed by defining a counit and antipode whose
explicit forms will not be needed in this paper. These structure maps together
satisfy a compatibility condition with the product and the unit $1$ of
$\mathsf{U}\mathfrak{v}$ as a unital algebra [74, 24].

There is a symmetric monoidal category
${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ whose objects are left
$\mathsf{U}\mathfrak{v}$-modules and whose morphisms are equivariant maps (see
e.g. [20]), that is, linear maps which commute with the action of
$\mathfrak{v}={\sf\Gamma}(TM)$ via the trivial coproduct $\Delta$. Since
$\mathsf{U}\mathfrak{v}$ is a cocommutative Hopf algebra, the braiding
isomorphism of ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$ is just the trivial
transposition $\tau$. A _$\mathsf{U}\mathfrak{v}$ -module algebra_ is an
algebra in the category ${}_{\mathsf{U}\mathfrak{v}}\mathscr{M}$. By this we
mean an associative algebra $(\mathcal{A},\mu)$ with a left
$\mathsf{U}\mathfrak{v}$-action
$\triangleright:\mathsf{U}\mathfrak{v}\otimes\mathcal{A}\rightarrow\mathcal{A}$
which is compatible with the algebra multiplication via the coproduct
$\Delta$, that is

$\displaystyle X\triangleright\mu(a\otimes
b)=\mu\big{(}\Delta(X)\triangleright(a\otimes b)\big{)}$ (4.7)

for all $X\in\mathsf{U}\mathfrak{v}$ and $a,b\in\mathcal{A}$, where
$\mu:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}$ is the product on
$\mathcal{A}$. In the following we will drop the symbol $\triangleright$ to
simplify the notation, and usually omit the adjective ‘left’ with the
understanding that only left modules are considered in this paper.

######  Example 4.8.

In our concrete applications to field theory later on, $(\mathcal{A},\mu)$
will be, for instance, the space of functions
$\mathcal{A}=C^{\infty}(M,\mathbbm{R})$ on the manifold $M$ with
$\mu=\,\cdot\,$ the pointwise product, the space of differential forms
$\mathcal{A}=\Omega^{\bullet}(M)$ with $\mu=\wedge$ the exterior product, or
more generally the space of tensor fields with the tensor product. These are
all modules over the Lie algebra of vector fields $\mathfrak{v}$, where the
(left) action of $\xi\in\mathfrak{v}$ on $a\in\mathcal{A}$ is given by the Lie
derivative $\mathrm{L}_{\xi}$ along $\xi$:

$\displaystyle\xi(a):=\mathrm{L}_{\xi}a\ .$ (4.9)

This makes $(\mathcal{A},\mu)$ into a $\mathsf{U}\mathfrak{v}$-module algebra,
by virtue of the Leibniz rule for the Lie derivative:

$\displaystyle\xi\big{(}\mu(a\otimes
b)\big{)}=\mathrm{L}_{\xi}\circ\mu(a\otimes b)=\mu(\mathrm{L}_{\xi}a\otimes
b)+\mu(a\otimes\mathrm{L}_{\xi}b)=\mu\circ\Delta(\xi)(a\otimes b)\ .$ (4.10)

This is of course just the basis of diffeomorphism symmetry: The algebras
($\mathcal{A},\mu)$ are covariant under the action of the universal enveloping
algebra of infinitesimal diffeomorphisms $\mathsf{U}\mathfrak{v}$.

If $V$ is a real vector space, we denote by $V[[\hbar]]$ the vector space of
formal power series in a deformation parameter $\hbar$ with coefficients in
the complexification $V_{\mathbbm{C}}=V\otimes\mathbbm{C}$;202020We should
more accurately write $V_{\mathbbm{C}}[[\hbar]]$, but we drop the subscript C
to simplify the notation, as no confusion should arise. it is naturally a
module over $\mathbbm{C}[[\hbar]]$. If $V$ and $W$ are real vector spaces,
then $V[[\hbar]]\otimes W[[\hbar]]\simeq(V\otimes W)[[\hbar]]$.212121To keep
the notation simple, we do not distinguish between the appropriate topological
tensor product needed for formal power series extensions and the usual tensor
product of vector spaces over $\mathbbm{C}$. With these conventions, we denote
by $\mathsf{U}\mathfrak{v}[[\hbar]]$ the formal power series extension of the
cocommutative Hopf algebra $\mathsf{U}\mathfrak{v}$, with operations extended
linearly over $\mathbbm{C}$ and applied term by term to the coefficients of
series.

######  Definition 4.11.

A _Drinfel’d twist_ on a manifold $M$ is an invertible element
$\mathcal{F}\in\mathsf{U}\mathfrak{v}[[\hbar]]\otimes\mathsf{U}\mathfrak{v}[[\hbar]]$
satisfying the cocycle condition