Noncommutative gauge and gravity theories
and geometric Seiberg–Witten map

Paolo Aschieri and Leonardo Castellani

Dipartimento di Scienze e Innovazione Tecnologica
Università del Piemonte Orientale, viale T. Michel 11, 15121 Alessandria,
Italy
INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy

.

We give a pedagogical account of noncommutative gauge and gravity theories,
where the exterior product between forms is deformed into a $\star$-product
via an abelian twist (e.g. the Groenewold–Moyal twist). The Seiberg–Witten map
between commutative and noncommutative gauge theories is introduced. It allows
to express the action of noncommutative Einstein gravity coupled to spinor
fields in terms of the usual commutative action with commutative fields plus
extra interaction terms dependent on the noncommutativity parameter.

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###### Contents

  1. 1 Introduction
  2. 2 Noncommutative algebras, Weyl quantization and $\star$-products
  3. 3 $\star$-products from twists and noncommutative differential geometry
  4. 4 Noncommutative Yang-Mills actions
  5. 5 Noncommutative vierbein gravity coupled to fermions
    1. 5.1 Classical action and symmetries
    2. 5.2 Noncommutative gauge theory and Lorentz group
    3. 5.3 Noncommutative Gravity action and its symmetries
    4. 5.4 Classical limit and charge conjugation constraints
  6. 6 Seiberg–Witten map
    1. 6.1 Geometric Seiberg-Witten map
  7. 7 Noncommutative gauge theories with any gauge group
    1. 7.1 Expansion of gravity fields to first order in $\theta$
    2. 7.2 Expansion of noncommutative gravity action at first order in $\theta$
  8. 8 Conclusions
  9. A Ambiguities in the Seiberg–Witten map
  10. B Gamma matrices in $D=4$

##  1 Introduction

Noncommutativity of phase space is a core feature of quantum mechanics.
Noncommutativity of spacetime rather than phase space has been considered
since the early days of quantum mechanics as a possible way to reconcile
gravity with quantum theory. Indeed the dynamical variable in Einstein general
relativity is spacetime itself with its metric structure, and noncommutativity
of spacetime coordinates could lead to a regularization in the perturbative
treatment of gravity as a quantum field theory. Noncommutativity of spacetime
coordinates is further supported by Gedanken experiments that aim at probing
spacetime structures at very small distances. They show that due to
gravitational backreaction one cannot test spacetime under the Planck scale.
For example, in relativistic quantum mechanics the position of a particle can
be detected with a precision of at most the order of its Compton wavelength
$\lambda_{C}=\hbar/mc$. Probing spacetime at very short distances implies
extremely heavy particles, that in turn produce high spacetime curvature. When
$\lambda_{C}$ is of the order of the Planck length, the spacetime curvature
radius has the same order of magnitude of the Compton wavelength of the probe
particle, and the attempt to measure spacetime structure under the Planck
scale fails. Gedanken experiments of this type show that the description of
spacetime as a continuum of points (a smooth manifold) is an assumption no
more justified at the Planck scale. It is hence natural to relax this
assumption and conceive a more general noncommutative spacetime, where
uncertainty relations and discretization naturally arise. Space and time are
then described by a Noncommutative Geometry. In this way the impossibility of
testing spacetime under the Planck length, a dynamical feature due to
gravitational backreaction, is encoded at a deeper kinematical level.

In general spacetime discretization is expected in quantum gravity theories,
see the review [1]. For example in string theory the study of string
scatterings leads to generalized uncertainty principles where a minimal length
emerges. In other approaches, e.g. loop quantum gravity, minimal area and
volumes are predicted.

Spacetime noncommutativity also arises by considering an electron in a strong
magnetic field $B$. In this regime, due to the minimal coupling with the
background gauge field $(A_{x},A_{y})=(0,Bx)$ associated to the flux $B$, the
dynamics takes place in the reduced phase space $q=y,p=Bx$. Thus the
electron’s coordinates become noncommutative: $[x,y]=-\frac{i\hbar}{B}$ (cf.
[2] for an extended discussion). Hence quantum theory in the presence of a
magnetic field $B$ leads to a noncommutative spacetime. Similarly, low energy
effective actions of open strings in the presence of a background Neveu-
Schwarz $B$-field can be described by gauge theories on noncommutative spaces.
The study of Yang-Mills (and Born-Infeld) theories on noncommutative spaces
has proven very fruitful: it allows to realize string theory T-duality
symmetry within the low energy physics of noncommutative (super) Yang-Mills
theories [3] It provides exact low energy D-brane effective actions, in a
given $\alpha^{\prime}\to 0$ sector of string theory where closed strings
decouple, see ref. [4]. In that paper Seiberg and Witten provided an explicit
map (change of variables) between commutative and noncommutative gauge
theories.

In this paper we study gravity on noncommutative spacetime as a noncommutative
gauge theory of the above type. The noncommutative geometry is first
formulated in a geometric (coordinate independent) language, useful for
studying diffeomorphism invariant theories. Then the specific gauge theory
describing noncommutative gravity in first order formalism (with independent
vierbein and spin connection fields) is presented. As we discuss in Section
5.2, a generic feature of noncommutative gauge theories is that they are well
defined for $U(N)$ or $GL(N)$ gauge groups in the fundamental or the adjoint
representation but not for a generic representation, or for a generic gauge
group $G$ (like e.g. $SU(N)$). This general feature implies that the Lorentz
gauge invariance of the first order gravity action becomes a $\star$-gauge
invariance that enlarges the classical $SO(1,3)$ group to $GL(2,C)$.

The enlargement of the gauge group corresponds to an increase in the number of
fundamental fields of the theory. This increase can be mitigated by imposing
charge conjugation constraints on the noncommutative gauge action and can be
fully avoided by the use of the Seiberg-Witten map [4], that relates the
fields in the deformed action (the “quantum” fields) to the classical fields,
in such a way that the ordinary gauge variations of the classical fields
induce the $\star$-gauge variations on the quantum fields. We thus obtain a
gravity theory on noncommutative spacetime with the same degrees of freedom as
classical gravity.

Among other approaches to noncommutative gravity we mention gravity on fuzzy
spaces [5], emerging from matrix theory in the presence of fuzzy extra
dimensions [6], a metric approach [7], [8] where the noncommutative Levi-
Civita connection is constructed and the braided gauge theory approach of [9].

Finally, a noncommutative hamiltonian formalism for twisted geometric theories
has been developed in [10], and applied to noncommutative vierbein gravity. It
allows an algorithmic construction of the canonical $\star$-gauge generators.

The plan of the paper is as follows. Section 2 deals with the origin of
twisted products, i.e. Weyl quantization and Groenewold–Moyal product. Section
3 transfers noncommutativity of coordinates to noncommutativity of functions
and of exterior forms (via $\star$\- wedge products), and summarizes the basic
results of the corresponding noncommutative geometry. Section 4 illustrates
the procedure in the case of Yang-Mills theory, by deforming its classical
action. In Section 5 the $\star$-deformation of the gravity action is
discussed in detail reviewing [11]. Section 6 contains a discussion on the
Seiberg-Witten map. In Section 7 it is shown how the Seiberg–Witten map allows
to construct noncommutative gauge theories with any gauge group. The
Seiberg–Witten map for the noncommutative gravity action is then described in
detail, providing a noncommutative action with the same degrees of freedom as
the commutative one [12, 13, 14]. Expanding this action in power series of the
noncommutative deformation parameter we obtain an action on commutative
spacetime with interaction terms dictated by noncommmutativity of spacetime.
As we further elaborate in the conclusions of Section 8, we have thus
constructed a modified gravity action that is expected to capture some quantum
gravity aspects.

##  2 Noncommutative algebras, Weyl quantization and $\star$-products

The easiest way to describe a noncommutative spacetime is via the
noncommutative algebra of its coordinates, i.e., giving a set of generators
and relations. We list three typical examples of commutation relations:

$\displaystyle[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}~{}~{}~{}~{}~{}~{}~{}~{}$
canonical (2.1)
$\displaystyle~{}~{}~{}~{}~{}[x^{\mu},x^{\nu}]=if^{\mu\nu}_{~{}~{}\sigma}x^{\sigma}~{}~{}~{}~{}~{}~{}~{}~{}$
Lie algebra (2.2)
$\displaystyle~{}~{}~{}~{}~{}x^{\mu}x^{\nu}-qx^{\nu}x^{\mu}=0~{}~{}~{}~{}~{}~{}~{}~{}$
quantum (hyper)plane (2.3)

where $\theta^{\mu\nu}$ (a real antisymmetric matrix),
$f^{\mu\nu}_{~{}~{}\sigma}$ (real structure constants), $q$ (a complex number,
e.g. a phase) are the respective noncommutativity parameters. When the
noncommutativity parameters are turned off, the algebra becomes commutative
and is the algebra of polynomial functions on d-dimensional space
$\mathbb{R}^{d}$. We can also impose further constraints, for example
periodicity of the coordinates describing the canonical noncommutative
spacetime (2.1). This leads to a noncommutative torus rather than to a
noncommutative (hyper)plane. Similarly, constraining the coordinates of the
quantum (hyper)plane relations (2.3) we obtain a quantum (hyper)sphere.

This algebraic description should then be complemented by a topological
approach, leading for example to the notion of continuous functions. This is
achieved by completing the algebra generated by the noncommutative coordinates
to a $C^{\star}$-algebra. Typically $C^{\star}$-algebras arise as algebras of
operators on a Hilbert space. Connes noncommutative geometry [15] starts from
these notions and enriches the $C^{\star}$-algebra structure and its
representation on a Hilbert space, generalizing to the noncommutative case
also the notions of smooth functions and metric structure.

A complementary approach to noncommutative space is given by the
$\star$-product, retaining the usual space of functions but deforming the
pointwise product in a noncommutative one. Historically the $\star$-product
originated as a noncommutative product for functions on phase space. The
quantization of phase space coordinates $q$, $p$ with Poisson structure
$\\{q,p\\}=1$ to operators $\Phi(q)=\hat{q}$, $\Phi(p)=\hat{p}$ on Hilbert
space with $[\hat{q},\hat{p}]=i\hbar$ is extended à la Weyl to functions
$f(p,q)\to\Phi(f)(\hat{p},\hat{q})$. The operator product then induces a
$\star$-product, or Groenewold–Moyal product, on functions on phase space:

$f\star g=\Phi^{-1}(\Phi(f)\>\\!\Phi(g))~{}.$

On polynomial functions Weyl quantization amounts to replace $p,q$ with the
operators $\hat{p},\hat{q}$ and to symmetrize in $\hat{p}$ and $\hat{q}$:
$p^{m}q^{n}\mapsto\Phi(p^{m}q^{n})=Sym(\hat{p}^{m}\hat{q}^{n})$ where
$Sym(\hat{p}^{m}\hat{q}^{n})$ is the symmetrized polynomial in
$\hat{p},\hat{q}$ normalized so that $Sym(p^{m}q^{n})=p^{m}q^{n}$. It is
defined by
$(\hat{p}+\hat{q})^{\ell}=\sum_{m+n=\ell}\frac{(m+n)!}{m!n!}Sym(\hat{p}^{m}\hat{q}^{n}).$
For example, $Sym(\hat{p}^{m})=\hat{p}^{m}$, $Sym(\hat{q}^{n})=\hat{q}^{n}$,
$Sym(\hat{p}\hat{q})=\frac{1}{2}(\hat{p}\hat{q}+\hat{q}\hat{p})$,
$Sym(p^{2}q)=\frac{1}{3}(\hat{p}^{2}\hat{q}+\hat{p}\hat{q}\hat{p}+\hat{q}\hat{p}^{2})$.
The corresponding $\star$-product explicitly reads

$(f\star g)(p,q)=e^{\frac{i}{2}\hbar(\frac{\partial}{\partial
q}\frac{\partial}{\partial p^{\prime}}-\frac{\partial}{\partial
p}\frac{\partial}{\partial
q^{\prime}})}f(p,q)g(p^{\prime},q^{\prime})\big{|}_{p=p^{\prime},q=q^{\prime}}~{}.$

Since the operator product is associative so is the $\star$-product.

More in general (in formal deformation quantization) a $\star$-product on a
manifold $M$ with Poisson structure $\\{~{}\\!,~{}\\!\\}$ is a noncommutative
deformation of the usual pointwise product. It sends two smooth functions
$f,g$ to a third one $f\star g$ and is a differential operator on both its
arguments. It satisfies the associative property

$f\star(g\star h)=(f\star g)\star h~{},$

the normalization property $f\star 1=1\star f=f$ and

$f\star g=fg+\frac{i}{2}{\hbar}\\{f,g\\}+\mathcal{O}(\hbar^{2})$

so that in the semiclassical limit $\lim_{\hbar\to 0}\frac{-i}{\hbar}(f\star
g-g\star f)=\\{f,g\\}$, realizing the correspondence principle betweeen
quantum and classical mechanics. For further reading on the topics of this
section we refer to [16, Ch. 2, §3], [17, Ch. 6], [18].

##  3 $\star$-products from twists and noncommutative differential geometry

The $\star$-product on phase space of the previous section generalizes to
${\mathbb{R}}^{d}$ with coordinates $x^{\mu}$ as

$(f\star g)(x)={\rm e}^{\frac{i}{2}\theta^{\mu\nu}{\partial\over\partial
x^{\mu}}\otimes{\partial\over\partial y^{\nu}}}f(x)h(y)\big{|}_{x=y}~{}.$
(3.1)

Here the antisymmetric matrix $\hbar\theta^{\mu\nu}$ has been for short
denoted $\theta^{\mu\nu}$. Correspondingly, the classical limit $\hbar\to 0$
becomes $\theta\to 0$.

Notice that if we set

$\mathcal{F}^{-1}={\rm e}^{\frac{i}{2}\theta^{\mu\nu}{\partial\over\partial
x^{\mu}}\otimes{\partial\over\partial y^{\nu}}}$

then

$f\star g=\mu(\mathcal{F}^{-1}(f\otimes g))$ (3.2)

where $\mu$ is the usual product of functions $\mu(f\otimes g)=fg$. The
element $\mathcal{F}={\rm
e}^{-\frac{i}{2}\theta^{\mu\nu}{\partial\over\partial
x^{\mu}}\otimes{\partial\over\partial y^{\nu}}}$ is an example of a Drinfeld
twist. It is defined by the exponential series in powers of the
noncommutativity parameters $\theta^{\mu\nu}$,

$\mathcal{F}={\rm e}^{-\frac{i}{2}\theta^{\mu\nu}{\partial\over\partial
x^{\mu}}\otimes{\partial\over\partial y^{\nu}}}=1\otimes 1-{i\over
2}\theta^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}-{1\over
8}\theta^{\mu_{1}\nu_{1}}\theta^{\mu_{2}\nu_{2}}\partial_{\mu_{1}}\partial_{\mu_{2}}\otimes\partial_{\nu_{1}}\partial_{\nu_{2}}+\ldots$

It is easy to see that $x^{\mu}\star x^{\nu}-x^{\nu}\star
x^{\mu}=i\theta^{\mu\nu}$ thus recovering the noncommutative algebra
abstractly defined in (2.1).

The method of constructing $\star$-products using Drinfeld twists [20] (see
e.g. [21] for a quick introduction) is not the most general method (it does
not apply to an arbitrary Poisson manifold [22]), however it is quite
powerful, and the class of $\star$-products obtained is quite wide. For
example choosing the appropriate twist we can obtain the noncommutative
relations (2.1), (2.2) and also (depending on the explicit expression of the
structure constants) some of the Lie algebra type (2.3). It is also well
adapted to a coordinate free description of the $\star$-algebra of functions
on a manifold $M$ and to its differential geometry.

Let $M$ be a smooth manifold. A twist is an invertible element $\mathcal{F}\in
U\Xi\otimes U\Xi$ where $U\Xi$ is the universal enveloping algebra of vector
fields, (i.e. it is the algebra generated by vector fields on $M$ and where
the element $XY-YX$ is identified with the vector field $[X,Y]$). The element
$\mathcal{F}$ must satisfy some further conditions that we do not write here,
but that hold true if we consider abelian twists, i.e., twists of the form

${\mathcal{F}}={\rm e}^{-\frac{i}{2}\theta^{IJ}{X_{I}}\otimes{X_{J}}}=1\otimes
1-{i\over 2}\theta^{IJ}X_{I}\otimes X_{J}-{1\over
8}\theta^{I_{1}J_{1}}\theta^{I_{2}J_{2}}X_{I_{1}}X_{I_{2}}\otimes
X_{J_{1}}X_{J_{2}}+\ldots$ (3.3)

where the vector fields $X_{I}$ ($I=1,...s$ with $s$ not necessarily equal to
$d={\rm{dim}}\,M$) are mutually commuting $[X_{I},X_{J}]=0$ (hence the name
abelian twist).

It is convenient to introduce the following notation

$\begin{split}\mathcal{F}^{-1}&=1\otimes 1+{i\over 2}\theta^{IJ}X_{I}\otimes
X_{J}-{1\over
8}\theta^{I_{1}J_{1}}\theta^{I_{2}J_{2}}X_{I_{1}}X_{I_{2}}\otimes
X_{J_{1}}X_{J_{2}}+\ldots\\\ &=\bar{\rm f}^{\alpha}\otimes\bar{\rm
f}_{\alpha}\end{split}$

where a sum over the multi-index $\alpha$ is understood.

With a twist $\mathcal{F}$ we deform the whole differential geometry of $M$.
Let $\cal A$ be the algebra of smooth functions on the manifold $M$. We deform
$\cal A$ to a noncommutative algebra $\cal A_{\star}$ by defining the new
product of functions

$f\star g=\bar{\rm f}^{\alpha}(f)\,\bar{\rm f}_{\alpha}(g)\,.$

We see that this formula is a generalization of the Gronewold–Moyal star
product on ${\mathbb{R}}^{d}$ defined in (3.1) or (3.2). Since the vector
fields $X_{I}$ are mutually commuting, this $\star$-product is associative.
Note that only the algebra structure of $\cal A$ is changed to $\cal
A_{\star}$ while, as vector spaces, $\cal A$ and $\cal A_{\star}$ are the
same. We similarly consider the algebra of exterior forms $\Omega^{\bullet}$
with the wedge product $\wedge$, and deform it in the noncommutative exterior
algebra $\Omega_{\star}^{\bullet}$ that is characterized by the graded
noncommutative exterior product $\wedge_{\star}$ given by

$\tau\wedge_{\star}\tau^{\prime}=\bar{\rm f}^{\alpha}(\tau)\wedge\bar{\rm
f}_{\alpha}(\tau^{\prime})~{},$

where $\tau$ and $\tau^{\prime}$ are arbitrary exterior forms, and each vector
field $X_{I_{1}},X_{I_{2}},X_{J_{1}},X_{J_{2}}\ldots$ in (3) acts on forms via
the Lie derivative. Only the product is deformed and hence
$\Omega_{\star}^{\bullet}=\Omega^{\bullet}$ as (graded) vector spaces, in
particular $\Omega_{\star}^{n}=\Omega^{n}$ for any degree $n$.

It is easy to show that the usual exterior derivative is compatible with the
new $\wedge_{\star}$-product,

$d(\tau\wedge_{\star}\tau^{\prime})=d(\tau)\wedge_{\star}\tau^{\prime}+(-1)^{deg(\tau)}\tau\wedge_{\star}d\tau^{\prime}$
(3.4)

since the exterior derivative commutes with the Lie derivative.

We also have compatibility with the usual undeformed integral (graded
cyclicity property):

$\int\tau\wedge_{\star}\tau^{\prime}=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge_{\star}\tau$
(3.5)

with $deg(\tau)+deg(\tau^{\prime})\\!=d=\,$dim$\>\\!M$. In fact we have, up to
boundary terms,

$\int\tau\wedge_{\star}\tau^{\prime}=\int\tau\wedge\tau^{\prime}=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge\tau=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge_{\star}\tau$

For example at first order in $\theta$,

$\int\tau\wedge_{\star}\tau^{\prime}=\int\tau\wedge\tau^{\prime}-{i\over
2}\theta^{IJ}\int{\cal L}_{X_{I}}(\tau\wedge{\cal
L}_{X_{J}}\tau^{\prime})=\int\tau\wedge\tau^{\prime}-{i\over 2}\theta^{IJ}\int
d{i}_{X_{I}}(\tau\wedge{\cal L}_{X_{J}}\tau^{\prime})$

where we used the Cartan formula ${\cal L}_{X_{I}}=di_{X_{I}}+i_{X_{I}}d$ and
$\tau\wedge{\cal L}_{X_{J}}\tau^{\prime}$ being a $d$-form so that its
exterior derivative vanishes.

Finally, provided that the commuting vector fields $\\{X_{I}\\}$ defining an
abelian twist are all (anti)hermitian, we have compatibility with the
undeformed complex conjugation

$(\tau\wedge_{\star}\tau^{\prime})^{*}=(-1)^{deg(\tau)deg(\tau^{\prime})}\tau^{\prime*}\wedge_{\star}\tau^{*}~{}.$
(3.6)

Indeed, sending $i$ into $-i$ in the twist (3.3) amounts to send $\theta^{IJ}$
into $-\theta^{IJ}=\theta^{JI}$, i.e. to exchange the order of the factors in
the $\star$-product.

##  4 Noncommutative Yang-Mills actions

It is straightforward to write a $U(N)$ Yang-Mills theory on noncommutative
space given by Groenewold–Moyal star product,

$S_{NCYM}=\,{-1\,\over 2g^{2}}^{\,}\int
d^{4}x\>T^{\\!}r({\hat{F}}_{\mu\nu}\star{\hat{F}}^{\mu\nu})$ (4.1)

where the noncommutative field strength ${\hat{F}}$ is defined by

$\widehat{F}_{\mu\nu}=\partial_{\mu}\widehat{A}_{\nu}-\partial_{\nu}\widehat{A}_{\mu}-i(\widehat{A}_{\mu}\star\widehat{A}_{\nu}-\widehat{A}_{\nu}\star\widehat{A}_{\mu})$

This action is invariant under the noncommutative gauge transformations

$\hat{\delta}\hat{A}_{\mu}=\partial_{\mu}\hat{\epsilon}+i(\hat{\epsilon}\star\hat{A}_{\mu}-\hat{A}_{\mu}\star\hat{\epsilon})~{},$

which imply
$\hat{\delta}\widehat{F}_{\mu\nu}=i(\hat{\epsilon}\star\widehat{F}_{\mu\nu}-\widehat{F}_{\mu\nu}\star\hat{\epsilon})$.
Using the differential geometry developed in the previous section this action
can also be rewritten as

$S_{NCYM}=\,{-1\,\over 2g^{2}}^{\,}\int
T^{\\!}r({\hat{F}}\wedge_{\star}\ast_{H\\!}{\hat{F}})$

where $A=A_{\mu}\star dx^{\mu}$,
${\hat{F}}=d{\hat{A}}-i{\hat{A}}\wedge_{\star}{\hat{A}}$ and the Hodge star
operator $\ast_{H\\!}$ is the usual commutative one in flat Minkowski metric
(recall that as vector spaces $\Omega_{\star}^{2}=\Omega^{2}$). The
noncommutative gauge transformations now read

$\hat{\delta}\widehat{A}=d\hat{\epsilon}+i(\hat{\epsilon}\star\hat{A}-\hat{A}\star\hat{\epsilon})~{}.$

and imply
$\hat{\delta}\widehat{F}=i(\hat{\epsilon}\star\widehat{F}-\widehat{F}\star\hat{\epsilon})$,
$\hat{\delta}(\ast_{H}\widehat{F})=i\big{(}\hat{\epsilon}\star(\ast_{H}\widehat{F})-(\ast_{H}\widehat{F})\star\hat{\epsilon}\big{)}$.

In this action the gauge potential and the field strength are valued in
$n\times n$ hermitian matrices, that define the Lie algebra of $U(N)$. Other
representations of $U(N)$ and gauge groups are in general problematic. Indeed
consider an infinitesimal gauge transformation
$\epsilon=\varepsilon^{A}T^{A}$, where the generators $T^{A}$ belong to some
representation of a Lie group $G$. The commutator of two infinitesimal gauge
transformations is

$[\epsilon\,,~{}\epsilon^{\prime}]_{\star}\equiv\epsilon\star\epsilon^{\prime}-\epsilon^{\prime}\star\epsilon={1\over
2}\\{\varepsilon^{A}\,,~{}\varepsilon^{\prime
B}\\}_{\star}\,[T^{A},T^{B}]+{1\over
2}[\varepsilon^{A}\,,~{}\varepsilon^{\prime
B}]_{\star}\,\\{T^{A},T^{B}\\}~{},$ (4.2)

where $\\{U,V\\}_{\star}:=U\star V-V\star U$, $[U,V]_{\star}:=U\star V-V\star
U$. We see that also the anticommutator $\\{T^{A},T^{B}\\}$ appears. We thus
have two options:

i) Consider gauge groups like $U(N)$ or $GL(N)$ in the (anti)fundamental or in
the adjoint, since in this case $\\{T^{A},T^{B}\\}$ is again in the Lie
algebra.

ii) Allow for more general representations of $U(N)$ or $GL(N)$, or more
general Lie algebras (including all simple Lie algebras) with representations
that do not close under the anticommutator. In this case we have to enlarge
the Lie algebra to include also anticommutators besides commutators, i.e., we
have to consider all possible (symmetrized) products $T^{A}T^{B}\ldots T^{C}$
of generators. The gauge potential will correspondingly have components

$\hat{A}=\hat{A}^{A}T^{A}+\hat{A}^{AB}T^{AB}+\hat{A}^{ABC}T^{ABC}+\ldots$

and therefore infinite degrees of freedom. The Seiberg–Witten map discussed in
Section 6 allows to reduce them to the classical degrees of freedom, and
therefore to construct noncommutative Yang–Mills theories with any gauge
group.

##  5 Noncommutative vierbein gravity coupled to fermions

###  5.1 Classical action and symmetries

Here we apply the twist procedure to first order gravity in $d=4$ coupled to
fermions and obtain gravity on noncommutative spacetime. The usual action of
first-order gravity coupled to a spin ${1\over 2}$ field $\psi$ reads:

$S=\varepsilon_{abcd}\int R^{ab}\wedge V^{c}\wedge
V^{d}-i\bar{\psi}\gamma^{a}V^{b}\wedge V^{c}\wedge V^{d}\wedge
D\psi-i(D\bar{\psi})\gamma^{a}\wedge V^{b}\wedge V^{c}\wedge V^{d}\psi$ (5.1)

where the vierbein $V^{a}$ and the spin connection $\omega^{ab}$ are
independent one-forms:

$V^{a}=V^{a}_{\mu}dx^{\mu},~{}~{}~{}~{}~{}\omega^{ab}=\omega^{ab}_{\mu}dx^{\mu}$
(5.2)

and the two-form (Lorentz) curvature $R^{ab}$ is defined as

$R^{ab}=d\omega^{ab}-\omega^{a}_{~{}c}\wedge\omega^{cb}~{}.$ (5.3)

The Dirac conjugate is defined as usual:
$\bar{\psi}=\psi^{\dagger}\gamma_{0}$.

This action can be recast in an index-free notation [19], [11], convenient for
generalization to the noncommutative case:

$S=\int Tr\left(iR\wedge V\wedge V\gamma_{5}\right)+\bar{\psi}V\wedge V\wedge
V\gamma_{5}D\psi+D\bar{\psi}\wedge V\wedge V\wedge V\gamma_{5}\psi$ (5.4)

where

$R=d\Omega-\Omega\wedge\Omega,~{}~{}~{}~{}~{}D\psi=d\psi-\Omega\psi,~{}~{}~{}~{}~{}D\bar{\psi}=\overline{D\psi}=d\bar{\psi}+\bar{\psi}\Omega$
(5.5)

with

$V\equiv V^{a}\gamma_{a},~{}~{}~{}~{}~{}\Omega\equiv{1\over
4}\omega^{ab}\gamma_{ab},~{}~{}~{}~{}~{}R\equiv{1\over 4}R^{ab}\gamma_{ab}$
(5.6)

taking value in Dirac gamma matrices (recalled in Appendix B). Use of the
gamma matrix identities
$\gamma_{abc}=i\varepsilon_{abcd}\gamma^{d}\gamma_{5},$
$Tr(\gamma_{ab}\gamma_{c}\gamma_{d}\gamma_{5})=-4i\varepsilon_{abcd}$ in
computing the trace leads back to the usual action (5.1). Reality of the
component fields $V^{a}$, $\omega^{ab}$, is equivalent to the hermiticity
conditions

$\gamma_{0}V\gamma_{0}=V^{\dagger},~{}~{}~{}-\gamma_{0}\Omega\gamma_{0}=\Omega^{\dagger}.~{}$
(5.7)

The action (5.4) is real; for the proof compare it to its complex conjugate,
obtained by taking the Hermitian conjugate of the 4-form inside the trace in
the integral.

The action is invariant under local diffeomorphisms (it is the integral of a
4-form on a 4-manifold, hence the infinitesimal diffeomorphism ${\cal
L}_{X}=di_{X}+i_{X}d$ reduces to $di_{X}$, a total derivative). It is also
invariant under local Lorentz rotations. These latter read

$\delta_{\epsilon}V=-[V,\epsilon],~{}~{}~{}\delta_{\epsilon}\Omega=d\epsilon-[\Omega,\epsilon],~{}~{}~{}~{}\delta_{\epsilon}\psi=\epsilon\psi,~{}~{}~{}\delta_{\epsilon}\bar{\psi}=-\bar{\psi}\epsilon$
(5.8)

with $\epsilon={1\over 4}\varepsilon^{ab}\gamma_{ab}$. The local Lorentz
invariance of the index free action follows from
$\delta_{\epsilon}R=-[{R},\epsilon]$ and $\delta_{\epsilon}D\psi=\epsilon
D\psi$, the cyclicity of the trace $Tr$ and the fact that the gauge parameter
$\epsilon$ commutes with $\gamma_{5}$.

After substituting (5.6) and $\epsilon={1\over 4}\varepsilon^{ab}\gamma_{ab}$
into (5.8), simple gamma algebra yields the gauge variations of the component
fields:

$\delta_{\epsilon}V^{a}=\varepsilon^{a}_{~{}b}V^{b},~{}~{}~{}\delta_{\epsilon}\omega^{ab}=d\varepsilon^{ab}-\omega^{a}_{~{}c}\varepsilon^{cb}+\omega^{b}_{~{}c}\varepsilon^{ca}\equiv
D\varepsilon^{ab}~{}.$ (5.9)

Similarly, the variation of the curvature components is found to be

$\delta_{\epsilon}R^{ab}=\varepsilon^{a}_{~{}c}R^{cb}-\varepsilon^{b}_{~{}c}R^{ca}~{},$
(5.10)

while
$\delta_{\epsilon}(\bar{\psi}\gamma^{a}D\psi)=\varepsilon^{a}_{~{}b}\bar{\psi}\gamma^{b}D\psi$.
Thus all quantities in the action (5.4) transform homogeneously under Lorentz
local rotations, and since $\varepsilon_{abcd}$ is an invariant tensor of
$SO(1,3)$, the action is likewise invariant. Here the proof of invariance
looks simple both in the index-free and in the component formulation. Note
however that in general the index-free proof is much simpler.

###  5.2 Noncommutative gauge theory and Lorentz group

Before presenting the noncommutative version of the action (5.4), we discuss
the $\star$-deformation of the Lorentz symmetry variations (5.8). These are
generated by the gamma matrices ${1\over 4}\gamma_{ab}$, and the gauge
parameter is $\epsilon={1\over 4}\varepsilon^{ab}\gamma_{ab}$. The commutator
of two Lorentz transformations is again a Lorentz transformation,
corresponding to the fact that the commutator of two $\gamma_{ab}$ matrices
contains only $\gamma_{ab}$ matrices. The situation changes when considering
the $\star$-deformation of this symmetry: as discussed in Section 4, the
commutator of two $\star$-gauge transformations contains also anticommutators
of the generators. The anticommutator of two ${1\over 4}\gamma_{ab}$ matrices
yields the identity and the $\gamma_{5}$ matrices, so that the gauge parameter
must now include them in its expansion:

$\epsilon={1\over 4}\varepsilon^{ab}\gamma_{ab}+i\varepsilon
1\\!\\!1+\tilde{\varepsilon}\gamma_{5}~{}.$

The extra gauge parameters $\varepsilon,\tilde{\varepsilon}$ can be chosen to
be real (like $\varepsilon_{ab}$). Indeed the reality of $\varepsilon_{ab}$,
$\varepsilon,\tilde{\varepsilon}$ is equivalent to the hermiticity condition

$-\gamma_{0}\epsilon\gamma_{0}=\epsilon^{\dagger}$ (5.11)

and if the gauge parameters $\epsilon$, $\epsilon^{\prime}$ satisfy this
condition then also $[\epsilon\,\star\\!\\!\\!_{,}~{}\epsilon^{\prime}]$ is
easily seen to satisfy this hermiticity condition.

Thus we have centrally extended the Lorentz group to

$SO(3,1)\rightarrow SO(3,1)\times U(1)\times R^{+}~{},$

or more precisely, (since our manifold $M$ has a spin structure and we have a
gauge theory of the spin group $SL(2,C)$)

$SL(2,C)\rightarrow GL(2,C)\,.$

The Lie algebra generator $i1\\!\\!1$ is the anti-hermitian generator
corresponding to the $U(1)$ extension, while $\gamma_{5}$ is the hermitian
generator corresponding to the noncompact $R^{+}$ extension.

Since under noncommutative gauge transformations we have

$\delta_{\epsilon}\Omega=d\epsilon-\Omega\star\epsilon+\epsilon\star\Omega$
(5.12)

also the spin connection and the curvature will be valued in the $GL(2,C)$ Lie
algebra representation given by all the even gamma matrices,

$\Omega={1\over 4}\omega^{ab}\gamma_{ab}+i\omega
1\\!\\!1+\tilde{\omega}\gamma_{5},~{}~{}~{}~{}~{}R={1\over
4}R^{ab}\gamma_{ab}+ir1\\!\\!1+\tilde{r}\gamma_{5}~{}.$ (5.13)

Similarly the gauge transformation of the vierbein,

$\delta_{\epsilon}V=-V\star\epsilon+\epsilon\star V,$ (5.14)

closes in the vector space of odd gamma matrices (i.e. the vector space
linearly generated by $\gamma^{a},\gamma^{a}\gamma_{5}$) and not in the
subspace of just the $\gamma^{a}$ matrices. Hence the noncommutative vierbein
are valued in the odd gamma matrices

$V=V^{a}\gamma_{a}+\tilde{V}^{a}\gamma_{a}\gamma_{5}~{}.$ (5.15)

Reality of the component fields $V^{a}$, $\tilde{V}^{a}$, $\omega^{ab}$,
$\omega$, and $\tilde{\omega}$ is equivalent to the hermiticity conditions

$\gamma_{0}V\gamma_{0}=V^{\dagger},~{}~{}~{}-\gamma_{0}\Omega\gamma_{0}=\Omega^{\dagger}.~{}$
(5.16)

These hermiticity conditions are consistent with the gauge variations.

Finally, the infinitesimal gauge transformations of the fields considered
close the Lie algebra of $GL(2,C)$,

$[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}]_{\star}=-\delta_{[\epsilon_{1},\epsilon_{2}]_{\star}}~{}.$
(5.17)

###  5.3 Noncommutative Gravity action and its symmetries

The abelian twist, defining the star products and compatible with usual
integration on $M$, leads to the extension of the Lorentz gauge group to
$GL(2,C)$. It allows to generalize to the noncommutative case the gravity
action (5.4). The noncommutative action reads

$S=\int
Tr\left(i{R}\wedge_{\star}\\!V\\!\wedge_{\star}\\!V\gamma_{5}\right)\,+\,\bar{\psi}\star
V\\!\wedge_{\star}\\!V\\!\wedge_{\star}\\!V\\!\wedge_{\star}\\!\gamma_{5}D\psi\,+\,D\bar{\psi}\wedge_{\star}V\\!\wedge_{\star}\\!V\\!\wedge_{\star}\\!V\\!\star\gamma_{5}\psi$
(5.18)

with

$R=d\Omega-\Omega\wedge_{\star}\Omega~{},~{}~{}~{}~{}D\psi=d\psi-\Omega\star\psi~{},~{}~{}~{}~{}D\bar{\psi}=d\bar{\psi}+\bar{\psi}\star\Omega~{}.$
(5.19)

Reality of this noncommutative action follows by comparing it to its complex
conjugate (obtained by taking the Hermitian conjugate of the 4-form inside the
trace in the integral).

Gauge invariance of the noncommutative action (5.18) under the
$\star$-variations is proved in the same way as for the commutative case,
noting that all the fields in the action transform homogeneously, cf. (5.12),
(5.14) and

$\delta_{\epsilon}\psi=\epsilon\star\psi,~{}~{}~{}\delta_{\epsilon}\bar{\psi}=-\bar{\psi}\star\epsilon~{},~{}~{}\delta_{\epsilon}D\psi=\epsilon\star
D\psi~{},~{}~{}\delta_{\epsilon}D\bar{\psi}=-D\bar{\psi}\star\epsilon~{},~{}~{}\delta_{\epsilon}R=-R\star\epsilon+\epsilon\star
R~{}.$ (5.20)

Using that $\epsilon$ commutes with $\gamma_{5}$, and the cyclicity of the
trace together with the graded cyclicity of the integral, the invariance of
(5.18) follows.

Diffeomorphisms invariance of the action (up to boundary terms) is proved
straightforwardly by using the Cartan identity ${\cal L}_{X}=i_{X}d+di_{X}$
for any infinitesimal diffeomorfism generated by a vector field $X$ and
recalling that the action is the integral of a $4$-form. Under these
diffeomorphisms the vector fields $X_{I}$ transform covariantly:
$\delta_{X}X_{I}=[X,X_{I}]$. It is also possible to introduce
$\star$-infinitesimal diffeomorphisms [21], see discussion in [17], Section
8.2.4. They satisfy a deformed Leibniz rule and leave invariant the vector
fields $X_{I}$ and hence the $\star$-product.

Charge conjugation invariance. Noncommutative charge conjugation reads:

$\psi\to\psi^{\;C}\equiv
C(\bar{\psi})^{T}=-\gamma_{0}C\psi^{\ast}\\!~{},~{}~{}V\to V^{\,C}\equiv
C{\,V}^{\;T}C\\!~{},~{}~{}\Omega\to\Omega^{\,C}\equiv
C{\>\\!\Omega}^{\;T}C\\!~{}$ (5.21)

with $\star_{\theta}\to\star_{\theta}^{C}=\star_{-\theta}$ and consequently
$\wedge_{\star_{\theta}}\to\wedge_{\star_{\theta}}^{\,C}=\wedge_{\star_{-\theta}}$,
(see Appendix B for the properties of the charge conjugation matrix $C$). The
action (5.18) is invariant under charge conjugation.

$\displaystyle S^{C}_{bosonic}\\!\\!$ $\displaystyle=$ $\displaystyle i\int
Tr({R^{C}}\wedge_{-\theta}V^{C}\wedge_{-\theta}V^{C}\gamma_{5})^{T}=-i\int
Tr({R^{T}}\wedge_{-\theta}V^{T}\wedge_{-\theta}V^{T}C\gamma_{5}C^{-1})^{T}$
(5.22) $\displaystyle=$ $\displaystyle-i\int
Tr\left((V^{T}\wedge_{-\theta}V^{T}\gamma_{5}^{T})^{T}\wedge_{\star}R\right)=-i\int
Tr\left(-(V^{T}\gamma^{T}_{5})^{T}\wedge_{\star}V\wedge_{\star}R\right)$
$\displaystyle=$ $\displaystyle i\int
Tr(\gamma_{5}V\wedge_{\star}V\wedge_{\star}R)=i\int
Tr(R\wedge_{\star}\gamma_{5}V\wedge_{\star}V)=i\int
Tr(R\wedge_{\star}V\wedge_{\star}V\gamma_{5})$ $\displaystyle=$ $\displaystyle
S_{bosonic}$

A similar proof holds for the fermionic part of the action :
$S_{fermionic}^{C}=S_{fermionic}$.

Noncommutative action and gauge variations for the component fields. Finally,
we give the bosonic noncommutative action in terms of the component fields
$V^{a}$, $\omega^{ab}$, $\tilde{V}^{a}$, $\omega$, and $\tilde{\omega}$, and
write the gauge variations of these fields.

$\displaystyle S_{bosonic}=$ $\displaystyle\\!\\!\\!\\!\\!\\!\int
R^{ab}\wedge_{\star}(V^{c}\wedge_{\star}V^{d}-\tilde{V}^{c}\wedge_{\star}\tilde{V}^{d})\varepsilon_{abcd}-2i~{}R^{ab}\wedge_{\star}(V_{a}\wedge_{\star}\tilde{V}_{b}-\tilde{V}_{a}\wedge_{\star}V_{b})$
$\displaystyle~{}~{}~{}-4~{}r\wedge_{\star}(V^{a}\wedge_{\star}\tilde{V}_{a}-\tilde{V}^{a}\wedge_{\star}V_{a})+4i~{}\tilde{r}\wedge_{\star}(V^{a}\wedge_{\star}V_{a}-\tilde{V}^{a}\wedge_{\star}\tilde{V}_{a})$

with

$\displaystyle R^{ab}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!d\omega^{ab}-{1\over
2}\omega^{a}_{~{}c}\wedge_{\star}\omega^{cb}+{1\over
2}\omega^{b}_{~{}c}\wedge_{\star}\omega^{ca}-i(\omega^{ab}\wedge_{\star}\omega+\omega\wedge_{\star}\omega^{ab})-$
$\displaystyle~{}-{i\over
2}\varepsilon^{ab}_{~{}~{}cd}(\omega^{cd}\wedge_{\star}\tilde{\omega}+\tilde{\omega}\wedge_{\star}\omega^{cd})$
$\displaystyle r\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!d\omega-{i\over
8}\omega^{ab}\wedge_{\star}\omega_{ab}-i(\omega\wedge_{\star}\omega-\tilde{\omega}\wedge_{\star}\tilde{\omega})$
$\displaystyle\tilde{r}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!d\tilde{\omega}+{i\over
16}\varepsilon_{abcd}\omega^{ab}\wedge_{\star}\omega^{cd}-i(\omega\wedge_{\star}\tilde{\omega}+\tilde{\omega}\wedge_{\star}\omega)~{}.$

The noncommutative gauge variations read

$\displaystyle\delta_{\epsilon}V^{a}={1\over 2}(\varepsilon^{a}_{~{}b}\star
V^{b}+V^{b}\star\varepsilon^{a}_{~{}b})+{i\over
4}\varepsilon^{a}_{~{}bcd}(\tilde{V}^{b}\star\varepsilon^{cd}-\varepsilon^{cd}\star\tilde{V}^{b})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star
V^{a}-V^{a}\star\varepsilon)-\tilde{\varepsilon}\star\tilde{V}^{a}-\tilde{V}^{a}\star\tilde{\varepsilon}$
$\displaystyle\delta_{\epsilon}\tilde{V}^{a}={1\over
2}(\varepsilon^{a}_{~{}b}\star\tilde{V}^{b}+\tilde{V}^{b}\star\varepsilon^{a}_{~{}b})+{i\over
4}\varepsilon^{a}_{~{}bcd}(V^{b}\star\varepsilon^{cd}-\varepsilon^{cd}\star
V^{b})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star\tilde{V}^{a}-\tilde{V}^{a}\star\varepsilon)-\tilde{\varepsilon}\star
V^{a}-V^{a}\star\tilde{\varepsilon}$
$\displaystyle\delta_{\epsilon}\omega^{ab}=d\varepsilon^{ab}+{1\over
2}(\varepsilon^{a}_{~{}c}\star\omega^{cb}-\varepsilon^{b}_{~{}c}\star\omega^{ca}+\omega^{cb}\star\varepsilon^{a}_{~{}c}-\omega^{ca}\star\varepsilon^{b}_{~{}c})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon^{ab}\star\omega-\omega\star\varepsilon^{ab})+{i\over
2}\varepsilon^{ab}_{~{}~{}cd}(\varepsilon^{cd}\star\tilde{\omega}-\tilde{\omega}\star\varepsilon^{cd})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star\omega^{ab}-\omega^{ab}\star\varepsilon)+{i\over
2}\varepsilon^{ab}_{~{}~{}cd}(\tilde{\varepsilon}\star\omega^{cd}-\omega^{cd}\star\tilde{\varepsilon})$
$\displaystyle\delta_{\epsilon}\omega=d\varepsilon-{i\over
8}(\omega^{ab}\star\varepsilon_{ab}-\varepsilon_{ab}\star\omega^{ab})+i(\varepsilon\star\omega-\omega\star\varepsilon-\tilde{\varepsilon}\star\tilde{\omega}+\tilde{\omega}\star\tilde{\varepsilon})$
$\displaystyle\delta_{\epsilon}\tilde{\omega}=d\tilde{\varepsilon}+{i\over
16}\varepsilon_{abcd}(\omega^{ab}\star\varepsilon^{cd}-\varepsilon^{cd}\star\omega^{ab})+i(\varepsilon\star\tilde{\omega}-\tilde{\omega}\star\varepsilon+\tilde{\varepsilon}\star\omega-\omega\star\tilde{\varepsilon})$

###  5.4 Classical limit and charge conjugation constraints

In the classical limit $\theta\rightarrow 0$ the $\star$-product becomes the
usual pointwise product. The noncommutative gauge symmetry becomes a usual
gauge symmetry with gauge group $GL(2,C)$ and the noncommutative vierbein in
the classical limit leads to two independent vierbeins: $V^{a}$ and
$\tilde{V}^{a}$ transforming both only under the $SL(2,C)$ subgroup of
$GL(2,C)$. As observed in [19] this is problematic because we obtain two
massless gravitons and only one local Lorentz symmetry. That is not enough in
order to kill the unphysical degrees of freedom. Either we concoct a mechanism
such that the second graviton becomes massive or we further constrain the
noncommutative theory so that in the classical limit the extra vierbein
vanishes.

We present two methods of constraining the noncommutative fields. The first
one is based on charge conjugation conditions. The second one will be
presented in Section 7; it has a wider application and is based on the
Seiberg-Witten map.

The fields in the noncommutative gravity action are in general
$\theta$-dependent as is clear by observing that the $\star$-gauge
transformation of a field is $\theta$ dependent (because of the $\theta$
dependence of the $\star$-product). The vanishing of the $\tilde{V}^{a}$
components in the classical limit is achieved by imposing charge conjugation
constraints on the fields [11]:

$CV_{\theta}(x)C=V_{-\theta}(x)^{T},~{}~{}~{}C\Omega_{\theta}(x)C=\Omega_{-\theta}(x)^{T},~{}~{}~{}C\varepsilon_{\theta}(x)C=\varepsilon_{-\theta}(x)^{T}$
(5.24)

where we have explicitly written the $\theta$-dependence of the fields.
Conditions (5.24) are consistent with $\star$-gauge transformations. For
example, the field $CV_{\theta}(x)^{T}C$ can be shown to transform in the same
way as $V_{-\theta}(x)$.

These charge conjugation constraints imply that the fields and gauge parameter
components $V^{a},\omega^{ab},\varepsilon^{ab}$ are even in $\theta$, while
the components
$\tilde{V}^{a},\omega,\tilde{\omega},\varepsilon,\tilde{\varepsilon}$ are odd.

We then conclude that the noncommutative gravity action in (5.18) with fields
satisfying the charge conjugation constraints (5.24) is real, diffeomorphisms
invariant, invariant under $GL(2,C)$ $\star$-gauge transformations and in the
classical limit reduces to the usual gravity action with usual $SL(2,C)$ gauge
invariance. Indeed, only the fields and gauge parameter components
$V^{a},\omega^{ab},\varepsilon^{ab}$ differ from zero in the classical limit.

As already observed the action is also charge conjugation invariant. In the
presence of the charge conjugation constraints (5.24) the bosonic gravity
action is furthermore even in $\theta$. Indeed (5.24) implies
$V^{\,C}=V_{-\theta},~{}\Omega^{\,C}=\Omega_{-\theta},~{}R^{\,C}=R_{-\theta}$.
From the first equality in (5.22) we see that the bosonic action
$S_{bosonic}(\theta)$ is mapped into $S_{bosonic}(-\theta)$ under charge
conjugation, and since it is also invariant we conclude that it is even in
$\theta$.

##  6 Seiberg–Witten map

We first study the Seiberg-Witten map between commutative and noncommutative
gauge theories with noncommutativity given by the Groenewold–Moyal product.

In a gauge theory physical quantities are gauge invariant: they do not depend
on the gauge potential but on the gauge equivalence class of the potential
given. The Seiberg-–Witten map relates the noncommutative gauge fields to the
commutative ones by requiring the noncommutative fields to have the same gauge
equivalence classes as the commutative ones [4]. Explicitly, the
noncommutative gauge potential $\hat{A}=A_{\mu}{\rm d}x^{\mu}$ and the
noncommutative gauge parameters ${\hat{\epsilon}}$ depend on the ordinary $A$
and $\epsilon$ so to satisfy:

${\hat{A}}(A+\delta_{\epsilon}A)={\hat{A}}(A)+{\hat{\delta}}_{\hat{\epsilon}}{\hat{A}}(A)$
(6.1)

with

$\displaystyle\delta_{\epsilon}A_{\mu}=\partial_{\mu}\epsilon-
iA_{\mu}\epsilon+i\epsilon A_{\mu},$ (6.2)
$\displaystyle{\hat{\delta}}_{\hat{\epsilon}}{\hat{A}}_{\mu}=\partial_{\mu}{\hat{\epsilon}}-i{\hat{A}}_{\mu}\star{\hat{\epsilon}}+i{\hat{\epsilon}}\star{\hat{A}}_{\mu}.$
(6.3)

This equation can be solved order by order in powers of the noncommutativity
parameter $\theta$ yielding ${\hat{A}}$ and ${\hat{\epsilon}}$ as power series
in $\theta$:

$\displaystyle{\hat{A}}(A)=A+A^{1}(A)+A^{2}(A)+\cdots+A^{n}(A)+\cdots$ (6.4)
$\displaystyle{\hat{\epsilon}}(\epsilon,A)=\epsilon+\epsilon^{1}(\epsilon,A)+\epsilon^{2}(\epsilon,A)+\cdots+\epsilon^{n}(\epsilon,A)+\cdots$
(6.5)

where $A^{n}(A)$ and $\epsilon^{n}(\epsilon,A)$ are of order $n$ in $\theta$.
Note that ${\hat{\epsilon}}$ depends on the ordinary $\epsilon$ and also on
$A$. For example, up to first order it is readily checked that

$\displaystyle{\hat{A}}_{\kappa}=A_{\kappa}+A_{\kappa}^{1}(A)+O(\theta^{2})=A_{\kappa}-\frac{\theta^{\mu\nu}}{4}\\{A_{\mu},\partial_{\nu}A_{\kappa}+F_{\nu\kappa}\\}+O(\theta^{2})$
(6.6)
$\displaystyle{\hat{\epsilon}}=\epsilon+\epsilon^{1}+O(\theta^{2})=\epsilon-\frac{\theta^{\mu\nu}}{4}\\{A_{\mu},\partial_{\nu}\epsilon\\}+O(\theta^{2})$
(6.7)

with
$F_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-iA_{\mu}A_{\nu}+iA_{\nu}A_{\mu}$
and where $\\{U,V\\}=UV+VU$ is the anticommutator of two operators.

The Seiberg–Witten condition (6.1) holds for any value of the noncommutativity
parameter $\theta$. If we consider it at $\theta^{\prime}$ and at $\theta$ we
easily obtain that gauge equivalence classes of the
$\theta^{\prime}$-noncommutative theory have to correspond to gauge equivalent
classes of the $\theta$-noncommutative theory, i.e., we generalize (6.1) to

${\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A}+\hat{\delta}_{\hat{\epsilon}}\hat{A})={\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A})-{\hat{\delta}}^{{\\!\;\\!\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\\!\\!\;{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}}{\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A})~{},$
(6.8)

where we denoted by $\star^{\prime}$,
${\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}$,
${\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}$,
${\hat{\delta}}^{{\\!\;\\!\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\\!\\!\>{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}}$
the star product, the gauge potential, the gauge parameter and the gauge
variation:
${\hat{\delta}}^{{\\!\;\\!\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\\!\\!\;{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}}{\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\kappa}=\partial_{\kappa}{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}-i{\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\kappa}\star^{\prime}{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}+i{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}\star^{\prime}{\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{\kappa}$
at noncommutativity parameter $\theta^{\prime}$. By considering $\theta$ and
$\theta^{\prime}$ infinitesimally close, so that
$\theta^{\prime}=\theta+\delta\theta$ and
${\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}=\hat{A}+\delta\theta^{\mu\nu}\frac{\partial\hat{A}}{\partial\theta^{\mu\nu}}$
(we consider $\frac{\partial}{\partial\theta^{\mu\nu}}$ independent from
$\frac{\partial}{\partial\theta^{\nu\mu}}$ and hence sum over all $\mu$,$\nu$
indices) a rather straightfoward computation, generalizing that for (6.6) and
(6.7), shows that if ${\hat{A}}$ and ${\hat{\epsilon}}$ solve the differential
equations

$\displaystyle{\partial\over\partial\theta^{\mu\nu}}{\hat{A}}_{\kappa}=-\frac{1}{8}\Big{(}\\{{\hat{A}}_{\mu},\partial_{\nu}{\hat{A}}_{\kappa}+{\hat{F}}_{\nu\kappa}\\}_{\star}-\\{{\hat{A}}_{\nu},\partial_{\mu}{\hat{A}}_{\kappa}+{\hat{F}}_{\mu\kappa}\\}_{\star}\Big{)}$
(6.9)
$\displaystyle{\partial\over\partial\theta^{\mu\nu}}{\hat{\epsilon}}=-\frac{1}{8}\Big{(}\\{{\hat{A}}_{\mu},\partial_{\nu}{\hat{\epsilon}}\\}_{\star}-\\{{\hat{A}}_{\nu},\partial_{\mu}{\hat{\epsilon}}\\}_{\star}\Big{)}$
(6.10)

where $\\{U,V\\}_{\star}=U\star V+V\star U$ and

${\hat{F}}_{\mu\nu}:=\partial_{\mu}{\hat{A}}_{\nu}-\partial_{\nu}{\hat{A}}_{\mu}-i{\hat{A}}_{\mu}\star{\hat{A}}_{\nu}+i{\hat{A}}_{\nu}\star{\hat{A}}_{\mu}~{},\\\
$ (6.11)

then ${\hat{A}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A})$ and
${\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}(\hat{\epsilon},\hat{A})$ satisfy also
the Seiberg–Witten condition (6.8) for arbitrary values of $\theta^{\prime}$
and $\theta$. In particular, therefore, they solve the Seiberg–Witten
condition (6.1).

The differential equations (6.9) and (6.10) admit solutions in terms of formal
power series in $\theta$. These are given recursively by

$\displaystyle A^{n+1}_{\mu}=-{1\over
4(n+1)}\theta^{\rho\sigma}\\{{\hat{A}}_{\rho},\partial_{\sigma}{\hat{A}}_{\mu}+{\hat{F}}_{\sigma\mu}\\}^{n}_{\star}\,,$
(6.12) $\displaystyle\epsilon^{n+1}=-{1\over
4(n+1)}\theta^{\rho\sigma}\\{{\hat{A}}_{\rho},\partial_{\sigma}{\hat{\epsilon}}\\}^{n}_{\star}\,,$
(6.13)

where $\\{{\hat{f}},{\hat{g}}\\}^{n}_{\star}$ is the $n$-th order term in
$\\{{\hat{f}},{\hat{g}}\\}_{\star}$, so that for example

$\\{{\hat{A}}_{\rho},\partial_{\sigma}{\hat{\epsilon}}\\}^{n}_{\star}\equiv\sum_{r+s+t=n}(A^{r}_{\rho}\star^{s}\partial_{\sigma}\epsilon^{t}+\partial_{\sigma}\epsilon^{t}\star^{s}A^{r}_{\rho})\,.$
(6.14)

Here $\star^{s}$ indicates the $s$-th order term in the star product expansion
[23]. There is a simple proof of (6.12), (6.13) [11]: multiplying the
differential equations by $\theta^{\mu\nu}$ and analysing them order by order
yields

$\displaystyle\theta^{\mu\nu}{\partial\over\partial\theta^{\mu\nu}}A_{\rho}^{n+1}=(n+1)A_{\rho}^{n+1}=-\frac{1}{4}\theta^{\mu\nu}\\{{\hat{A}}_{\mu},\partial_{\nu}{\hat{A}}_{\rho}+{\hat{F}}_{\nu\rho}\\}_{\star}^{n}~{},~{}~{}$
$\displaystyle\theta^{\mu\nu}{\partial\over\partial\theta^{\mu\nu}}\epsilon^{n+1}=(n+1)\epsilon^{n+1}=-\frac{1}{4}\theta^{\mu\nu}\\{{\hat{A}}_{\mu},\partial_{\nu}{\hat{\epsilon}}\\}_{\star}^{n}$

since $A_{\rho}^{n+1}$ and $\epsilon^{n+1}$ are homogeneous functions of
$\theta$ of order $n+1$.

Similar considerations hold for matter fields $\phi$ transforming in the
fundamental or in the adjoint representation of the gauge group. The
Seiberg–Witten condition reads, cf. [24],

${\hat{\phi}}(A+\delta_{\epsilon}A,\phi+\delta_{\epsilon}\phi)={\hat{\phi}}(A,\phi)+\hat{\delta}_{\hat{\epsilon}}{\hat{\phi}}(A,\phi)\,,$
(6.15)

or more generally,

${\hat{\phi}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A}+\delta_{\hat{\epsilon}}\hat{A},\hat{\phi}+\delta_{\hat{\epsilon}}\hat{\phi})={\hat{\phi}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A},\hat{\phi})+{\hat{\delta}}^{{\\!\;\\!\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}_{{\hat{\epsilon}}^{\\!\\!\;{}^{\prime}}}{\hat{\phi}}^{{\\!\\!\>\\!\\!\\!\phantom{I}}^{{{}^{\prime}}}}(\hat{A},\hat{\phi})\,,$
(6.16)

and it is satisfied if the matter fields solve the differential equation

$\begin{split}&\delta\theta^{\mu\nu}\frac{\partial\hat{\phi}}{\partial\theta^{\mu\nu}}=-\frac{1}{4}\delta\theta^{\mu\nu}\hat{A}_{\mu}\star(\partial_{\nu}\hat{\phi}+D_{\nu}\hat{\phi})\;~{}~{}~{}~{}~{}~{}~{}{\rm
fundamental~{}rep.,~{}i.e.,~{}}{\hat{\delta}}_{\hat{\epsilon}}{\hat{\phi}}=i{\hat{\epsilon}}\star{\hat{\phi}}\,,~{}~{}~{}~{}\\\
&\delta\theta^{\mu\nu}\frac{\partial\hat{\Psi}}{\partial\theta^{\mu\nu}}=-\frac{1}{4}\delta\theta^{\mu\nu}\bigl{\\{}\hat{A}_{\mu}\,,(\partial_{\nu}\hat{\Psi}+D_{\nu}\hat{\Psi})\bigr{\\}}_{\star}\;~{}~{}{\rm
adjoint~{}rep.,~{}i.e.,~{}}{\hat{\delta}}_{\hat{\epsilon}}{\hat{\Psi}}=i{\hat{\epsilon}}\star{\hat{\Psi}}-i{\hat{\Psi}}\star{\hat{\epsilon}}\,.\end{split}$
(6.17)

The explicit solutions order by order in $\theta$ are

$\displaystyle\phi^{n+1}=-{1\over
4(n+1)}\theta^{\mu\nu}\big{(}{\hat{A}}_{\mu}\star(\partial_{\nu}{\hat{\phi}}+D_{\nu}{\hat{\phi}})\big{)}^{\\!\>\\!n}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm(fundamental)}$
(6.18) $\displaystyle\Psi^{n+1}=-{1\over
4(n+1)}\theta^{\mu\nu}\\{{\hat{A}}_{\mu},\partial_{\nu}{\hat{\Psi}}+D_{\nu}{\hat{\Psi}}\\}^{n}_{\star}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm(adjoint)}$
(6.19)

where

$D_{\nu}{\hat{\phi}}=\partial_{\nu}{\hat{\phi}}-i{\hat{A}}_{\nu}\star{\hat{\phi}}~{}~{},~{}~{}~{}~{}D_{\nu}{\hat{\Psi}}=\partial_{\nu}{\hat{\Psi}}-i[{\hat{A}}_{\nu},{\hat{\Psi}}]_{\star}$

are the covariant derivative in the fundamental and in the adjoint, with
$[S,T]_{\star}:=S\star T-T\star S$.

The Seiberg–Witten differential equations (6.9), (6.10), (6.17) are not the
most general solutions to the gauge equivalence condition (6.1). For example,
from the differential equation for the gauge potential it is easy to see that
the field strength $\hat{F}_{\mu\nu}$ satisfies the differential equation

$\delta\theta^{\rho\sigma}\frac{\partial{\hat{F}}_{\mu\nu}}{\partial\theta^{\rho\sigma}}=-{1\over
4}\theta^{\rho\sigma}\big{(}\\{{\hat{A}}_{\rho},\partial_{\sigma}{\hat{F}}_{\mu\nu}+D_{\sigma}{\hat{F}}_{\mu\nu}\\}_{\star}-2\\{{\hat{F}}_{\mu\rho},{\hat{F}}_{\nu\sigma}\\}_{\star}\big{)}$

which has an extra addend with respect to the differential equation (6.17) for
fields transforming in the adjoint. The most general Seiberg–Witten
differential equations are presented in Appendix A. The freedom in the
Seiberg–Witten differential equations may be useful for their integration, see
e.g. Appendix A.

###  6.1 Geometric Seiberg-Witten map

The Seiberg–Witten map considered for Groenewold–Moyal noncommutativity can be
generalized to the case of an abelian twist

${\cal F}=e^{-\frac{i}{2}\theta^{IJ}X_{I}\otimes X_{J}}$ (6.20)

where $\\{X_{I}\\}$ is a set of mutually commuting vector fields globally
defined on a manifold $M$ and $\theta^{IJ}$ is a constant antisymmetric
matrix. The corresponding $\star$-product is obtained composing the usual
pointwise multiplication $\mu(f\otimes g)=fg$ with the inverse twist ${\cal
F}^{-1}=e^{\frac{i}{2}\theta^{IJ}X_{I}\otimes X_{J}}$,

$\displaystyle f\star g$ $\displaystyle=$ $\displaystyle\mu({\cal
F}^{-1}(f\otimes g))~{}.$ (6.21)

This $\star$-product is in general position dependent because the vector
fields $X_{I}$ are in general $x$-dependent. Associativity of the
$\star$-product is guaranteed by mutual commutativity of the vector fields
$X_{I}$. In the special case that $M=\mathbb{R}^{d}$ and
$X_{I}={\partial\over\partial x^{I}}$, $I=1,...,d$ we recover the
Groenewold–Moyal $\star$-product (3.1).

The use of vector fields $\\{X_{I}\\}$ on a manifold $M$ suggests a coordinate
independent approach to the Seiberg–Witten map. The resulting NC gauge
potential $\hat{A}$ is then a $1$-form that depends on $A$, on the mutually
commuting vector fields $X_{I}$ and on the deformation matrix
$\theta=(\theta^{IJ})_{I,J=1,...,d}$. The coordinate independent expression of
the Seiberg–Witten differential eq.s (6.6),(6.10) reads

$\displaystyle{\partial\over\partial\theta^{IJ}}{\hat{A}}=-{1\over
4}\\{i_{X_{[I}}{\hat{A}},{\cal
L}_{X_{J]}}{\hat{A}}+i_{X_{J]}}{\hat{F}}\\}_{\star}$ (6.22)
$\displaystyle{\partial\over\partial\theta^{IJ}}{\hat{\epsilon}}=-{1\over
4}\\{i_{X_{[I}}{\hat{A}},{\cal L}_{X_{J]}}{\hat{\epsilon}}\\}_{\star}$ (6.23)

where ${\hat{F}}\equiv d{\hat{A}}-i{\hat{A}}\wedge_{\star}{\hat{A}}$ is a two-
form, $i_{X_{I}}$ and ${\cal L}_{X_{I}}$ are respectively the contraction and
the Lie derivative along the mutually commuting vector fields $X_{I}$. When
the abelian twist reduces to the Groenewold–Moyal case of the preceding
Section, the curvature becomes ${\hat{F}}={1\over
2}{\hat{F}}_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$, where the ${\hat{F}}_{\mu\nu}$
components are given in (6.11). Note that in the Groenewold–Moyal case
$dx^{\mu}\wedge_{\star}dx^{\nu}=dx^{\mu}\wedge dx^{\nu}$ since ${\cal
L}_{X_{I}}dx^{\mu}=d{\cal L}_{X_{I}}x^{\mu}=0$. Proceeding as in the
Groenewold–Moyal case the recursive solutions are given by:

$\displaystyle A^{n+1}=-{1\over 4(n+1)}\theta^{IJ}\\{i_{X_{I}}{\hat{A}},{\cal
L}_{X_{J}}{\hat{A}}+i_{X_{J}}{\hat{F}}\\}^{n}_{\star}$ (6.24)
$\displaystyle\epsilon^{n+1}=-{1\over
4(n+1)}\theta^{IJ}\\{i_{X_{I}}{\hat{A}},{\cal
L}_{X_{J}}{\hat{\epsilon}}\\}^{n}_{\star}$ (6.25)

Similarly one proves the generalization of eq.s (6.18)-(6.19):

$\displaystyle\phi^{n+1}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle-{1\over
4(n+1)}\theta^{IJ}\left(i_{X_{I}}{\hat{A}}\star(2{\cal
L}_{X_{J}}{\hat{\phi}}-i(i_{X_{J}}{\hat{A}})\star{\hat{\phi}})\right)^{n},~{}~{}{\hat{\delta}}_{\hat{\epsilon}}{\hat{\phi}}=i{\hat{\epsilon}}\star{\hat{\phi}}$
(6.26) $\displaystyle\Psi^{n+1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-{1\over 4(n+1)}\theta^{IJ}\\{i_{X_{I}}{\hat{A}}\,,\,2{\cal
L}_{X_{J}}\hat{\psi}-i(i_{X_{J}}{\hat{A}})\star\hat{\Psi}+i\hat{\Psi}\star(i_{X_{J}}{\hat{A}})\\}_{\star}^{n},~{}~{}{\hat{\delta}}_{\hat{\epsilon}}\hat{\Psi}=i{\hat{\epsilon}}\star\hat{\Psi}-i\hat{\Psi}\star{\hat{\epsilon}}$

In this subsection we have constructed the geometric Seiberg–Witten map for
noncommutative gauge theories with gauge group $U(N)$ or $GL(N)$ (or products
thereof) and with $\star$-product given by a general abelian twist (6.20). For
abelian gauge groups the Seiberg–Witten map can be constructed for any
$\star$-product associated with an arbitrary Poisson tensor. The map is
obtained in [25] using Kontsevich formality theorem [26]. The study of its
global geometric aspects shows that the Seiberg–Witten map quantizes line
bundles with connections on a Poisson manifold to quantum (noncommutative)
line bundles with noncommutative connections [27] The Seiberg–Witten map for
nonabelian gauge groups and with arbitray Poisson tensors is in general an
open problem, we refer to [28] for interesting insights. The global geometric
aspects of the Seiberg–Witten map are well understood for nonabelian
$U(n)$-gauge fields on noncommutative tori: the Seiberg–Witten map defined in
(A.9), (A.10), with $\gamma=-3$, $\rho=i$, quantizes vector bundles on tori
with connections to vector bundles on noncommutative tori with noncommutative
connections and the results are nonformal in the sense that they do not rely
on power series expansion in the noncommutativity parameter $\theta$.

We also mention that the Seiberg-Witten map for Chern–Simons gauge theories
can be studied to all orders in the noncommutativity parameter $\theta$ [29],
[30].

##  7 Noncommutative gauge theories with any gauge group

Up to now we have considered noncommutative gauge theories with gauge group
$U(N)$ or $GL(N)$ in the fundamental or adjoint, and more generally
representations of gauge groups such that the generators $T^{A}$ of the Lie
algebra close also in the usual matrix product. This is needed for the closure
of infinitesimal gauge transformations, cf. (4.2). If on the other hand we
consider an arbitrary gauge group $G$ equation (4.2) shows that infinitesimal
gauge transformations do not close in the Lie algebra Lie$(G)$, but in the
universal enveloping algebra ${\mathcal{U}}($Lie$(G))$. This latter is the
product of all generators $T^{A}$ modulo the relations
$T^{A}T^{B}-T^{B}T^{A}=[T^{A},T^{B}]$. When considering an arbitrary gauge
group $G$ the noncommutative gauge potential is therefore universal enveloping
algebra valued

${\hat{A}}={\hat{A}}^{A}T^{A}+{\hat{A}}^{AB}T^{A}T^{B}+{\hat{A}}^{ABC}T^{A}T^{B}T^{C}+\ldots~{}$

and hence with infinitely many (symmetric) components ${\hat{A}}^{A}$,
${\hat{A}}^{AB}$, ${\hat{A}}^{ABC}$, …. . The Seiberg–Witten map is well
defined also in this case. It then constrains these infinite components to
depend on the commutative ones $A^{A}=A^{A}_{\mu}dx^{\mu}$ in $A=A^{A}T^{A}$.
Similarly, the gauge parameters and the matter fields depend on the
commutative gauge parameters and matter fields, besides the commutative gauge
potential.

This Seiberg–Witten map approach to noncommutative gauge theories is called
universal enveloping algebra valued approach [24]. It has been used to propose
noncommutative standard and grand unified particle physics models [31, 32]
having the same degrees of freedom as in the commutative models.
Renormalizability and scattering amplitudes using the Seiberg–Witten map have
been studied e.g. in [33, 34, 35, 36], either considering a power series
expansion in $\theta$ or a $\theta$-exact approach (i.e. to all arders in
$\theta$) [34, 37], where the power series is instead in the gauge coupling
constant. For recent literature on scattering amplitudes of noncommutative
particle models using the Seiberg–Witten map see [38] and references therein.

We next study the Seiberg–Witten map for the gravity action (5.18).

###  7.1 Expansion of gravity fields to first order in $\theta$

Up to first order in $\theta$ the solution to the Seiberg-Witten conditions
for the gravity fields reads:

$\displaystyle\hat{\Omega}=\Omega-\frac{1}{4}\theta^{IJ}\\{\Omega_{I},{\cal
L}_{J}\Omega+R_{J}\\}+\mathcal{O}(\theta^{2})$ (7.1)
$\displaystyle{\hat{\epsilon}}=\epsilon-{1\over
4}\theta^{IJ}\\{\Omega_{I},{\cal L}_{J}\epsilon\\}+\mathcal{O}(\theta^{2})$
(7.2) $\displaystyle\hat{\psi}=\psi-{1\over 4}\theta^{IJ}\Omega_{I}({\cal
L}_{J}+L_{J})\psi+\mathcal{O}(\theta^{2})$ (7.3)
$\displaystyle\hat{R}=R-{1\over 4}\theta^{IJ}\left(\\{\Omega_{I},({\cal
L}_{J}+L_{J})R\\}-[R_{I},R_{J}]\right)+\mathcal{O}(\theta^{2})$ (7.4)

where $\Omega_{I}$, $R_{I}$ are defined as the contraction along the tangent
vector $X_{I}$ of the exterior forms $\Omega$, $R$, i.e., $\Omega_{I}\equiv
i_{I}\Omega$, $R_{I}\equiv i_{I}R$, ($i_{I}$ being the contraction along
$X_{I}$). We have also introduced the Lie derivative ${\cal L}_{I}$ along the
vector field $X_{I}$, and the covariant Lie derivative $L_{I}$ along the
vector field $X_{I}$. $L_{I}$ acts on $R$ and $\psi$ as $L_{I}R={\cal
L}_{I}R-i\Omega_{I}\star R+iR\star\Omega_{I}$ and $L_{I}\psi={\cal
L}_{I}\psi-\Omega_{I}\psi$. In fact the covariant Lie derivative $L_{I}$ has
the Cartan form:

$L_{I}=i_{I}D+Di_{I}~{}$

where $D$ is the covariant derivative. We refer to [12] for higher order in
$\theta$ expressions.

###  7.2 Expansion of noncommutative gravity action at first order in $\theta$

The Seiberg–Witten map allows to expand the noncommutative action in terms of
the commutative fields. Noncommutative gravity is therefore seen as
commutative gravity with further interaction terms due to noncommutativity of
space time.

The Seiberg–Witten solutions (7.1)-(7.4) are not $SO(1,3)$-gauge covariant,
due to the presence of the “naked" connection $\Omega$ and the non-covariant
Lie derivative ${\cal L}_{I}=i_{I}{{\rm d}}+{{\rm d}}i_{I}$. However, when
inserted in the NC action the resulting action is gauge invariant order by
order in $\theta$. Indeed usual gauge variations induce the $\star$-gauge
variations under which the noncommutative action is invariant. Therefore the
NC action, re-expressed in terms of ordinary fields via the SW map, is
invariant under usual gauge transformations. Moreover the action, once re-
expressed in terms of ordinary fields remains geometric, and hence invariant
under diffeomorphisms. This is the case because the noncommutative action and
the SW map are geometric: indeed only coordinate independent operations like
the contraction $i_{I}$ and the Lie derivatives ${\cal L}_{I}$ and $L_{I}$
appear in the Seiberg–Witten map.

We replace the noncommutative fields appearing in the action with their
expansions (7.1)-(7.4) in commutative fields, and integrating by parts we
obtain the following gravity action coupled to spinors

$\displaystyle S$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle\\!\\!\int
Tr\left(iRVV\gamma_{5}\right)+\bar{\psi}V^{3}\gamma_{5}D\psi+D\bar{\psi}V^{3}\gamma_{5}\psi$
$\displaystyle+\frac{i}{4}\theta^{IJ}\Big{(}\bar{\psi}\\{V^{3},R_{IJ}\\}\gamma_{5}D\psi+D\bar{\psi}\\{V^{3},R_{IJ}\\}\gamma_{5}\psi\Big{)}$
$\displaystyle+\frac{i}{2}\theta^{IJ}\Big{(}2L_{I}\bar{\psi}R_{J}V^{3}\gamma_{5}\psi-2\bar{\psi}V^{3}R_{I}\gamma_{5}L_{J}\psi-
L_{I}\bar{\psi}V^{3}\gamma_{5}L_{J}D\psi-
L_{I}D\bar{\psi}\,V^{3}\gamma_{5}L_{J}\psi$
$\displaystyle+\bar{\psi}(\\{L_{I}\\!VL_{J}\\!V,V\\}+L_{I}\\!V\,VL_{J}\\!V)\gamma_{5}D\psi+D\bar{\psi}(\\{L_{I}\\!VL_{J}\\!V,V\\}+L_{I}\\!V\,VL_{J}\\!V)\gamma_{5}\psi\Big{)}+O(\theta^{2})$

where we have omitted writing the wedge product, and $V^{3}=V\wedge V\wedge
V$. The expression of the gravity action, up to second order in $\theta$, in
terms of the commutative fields has been given in [13], after a propaedeutical
study of the Seiberg–Witten map for $\star$-products of fields.

##  8 Conclusions

We have constructed a gravity action in noncommutative spacetime –an expected
feature of quantum spacetime– and shown its equivalence to the usual gravity
action (in the first order formalism) on commutative spacetime with extra
interaction terms. These are obtained from spacetime noncommutativity using
the Seiberg–Witten map between commutative and noncommutative gauge theories.
This extended gravity action is invariant under local Lorentz transformations
because it is expressed solely in terms of gauge covariant operators
$L_{I},i_{I},D$, curvature $R$, vierbein $V$, spinor fields $\psi$, invariant
vector fields $\\{X_{I}\\}$ and noncommutativity parameter $\theta$. It is
diffeomorphic invariant and charge conjugation invariant. This noncommutative
gravity action can also be coupled to noncommutative scalar and gauge fields
[39, 40]. Choosing an appropriate kinetic term the vector fields $\\{X_{I}\\}$
can become dynamical, the idea being that both spacetime curvature and
noncommutativity should depend on matter distribution. It would be interesting
to study cosmological models as solutions of these extended gravity actions.

##  Appendix A Ambiguities in the Seiberg–Witten map

The solution to the Seiberg–Witten conditions (6.1), (6.15) is not unique. For
example if $\hat{A}_{\mu}$ is a solution, any noncommutative gauge
transformation of $\hat{A}_{\mu}$ gives another solution. Another source of
ambiguities is that of field redefinitions of the gauge potential (e.g., if
$\hat{A}_{\mu}$ is a solution then so is
$\hat{A}_{\mu}+\theta^{\rho\sigma}\theta^{\lambda\eta}\hat{F}_{\rho\lambda}\star
D_{\sigma}\hat{F}_{\eta\mu}$). We generalize the Seiberg–Witten equations
(6.6), (6.10) and (6.17) allowing for three extra terms
$\hat{D}_{\mu\nu\rho}(\hat{A}),\hat{E}_{\mu\nu}(\hat{A},\hat{\epsilon})$,
$\hat{C}_{\mu\nu}(\hat{A},\hat{\phi})$ and
$\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})$ that are a priori arbitrary functions
of their arguments and derivatives thereof, that are (formal) power series in
$\theta$ and that are antisymmetric in the $\mu,\nu$ indices. We consider the
equations

$\displaystyle\\!\\!\\!\\!\\!\delta^{\theta}\hat{A}_{\kappa}\,=\,\delta\theta^{\mu\nu}\frac{\partial\hat{A}_{\kappa}}{\partial\theta^{\mu\nu}}$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!-\frac{1}{4}\delta\theta^{\mu\nu}\Bigl{(}\\{\hat{A}_{\mu},\partial_{\nu}\hat{A}_{\kappa}+\hat{F}_{\nu\kappa}\\}_{\star}+\hat{D}_{\mu\nu\kappa}(\hat{A})\Bigr{)}\,,~{}~{}~{}~{}~{}~{}$
(A.1)
$\displaystyle\\!\\!\\!\\!\\!\delta^{\theta}\hat{\epsilon}\,=\,\delta\theta^{\mu\nu}\frac{\partial\hat{\epsilon}}{\partial\theta^{\mu\nu}}$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!-\frac{1}{4}\delta\theta^{\mu\nu}\Bigl{(}\\{\partial_{\mu}\hat{\epsilon},\hat{A}_{\nu}\\}_{\star}+\hat{E}_{\mu\nu}(\hat{A},\hat{\epsilon})\Bigr{)}\,,$
(A.2)
$\displaystyle\\!\\!\\!\\!\\!\delta^{\theta}\hat{\phi}\,=\,\delta\theta^{\mu\nu}\frac{\partial\hat{\phi}}{\partial\theta^{\mu\nu}}$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!-\frac{1}{4}\delta\theta^{\mu\nu}\Bigl{(}\hat{A}_{\mu}\star\partial_{\nu}\hat{\phi}+\hat{A}_{\mu}\star
D_{\nu}\hat{\phi}+\hat{C}_{\mu\nu}(\hat{A},\hat{\phi})\Bigr{)}\,,$ (A.3)
$\displaystyle\\!\\!\\!\\!\\!\delta^{\theta}\hat{\Psi}\,=\,\delta\theta^{\mu\nu}\frac{\partial\hat{\Psi}}{\partial\theta^{\mu\nu}}$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!-\frac{1}{4}\delta\theta^{\mu\nu}\Bigl{(}\\{\hat{A}_{\mu},\partial_{\nu}\hat{\Psi}+D_{\nu}\hat{\Psi}\\}_{\star}+\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})\Bigr{)}$
(A.4)

and observe that $\hat{E}_{\mu\nu}$ must be linear in $\hat{\epsilon}$ since
all terms in (A.2) but $\hat{E}_{\mu\nu}$ are linear in $\hat{\epsilon}$,
similarly $\hat{C}_{\mu\nu}$ must be linear in $\hat{\phi}$ because of the
linearity in $\hat{\phi}$ of all other terms in (A.3), and similarly for
$\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})$ in (A.4). Imposing the Seiberg–Witten
conditions (6.8), (6.16) we obtain the conditions

$\begin{split}\hat{D}_{\mu\nu\kappa}(\hat{A}+\hat{\delta}_{\hat{\epsilon}}\hat{A})-\hat{D}_{\mu\nu\kappa}(\hat{A})-i[\hat{\epsilon},\hat{D}_{\mu\nu\kappa}(\hat{A})]_{\star}&=-D_{\kappa}\hat{E}_{\mu\nu}(\hat{A},\hat{\epsilon})\;,\\\
\hat{C}_{\mu\nu}(\hat{A}+\hat{\delta}_{\hat{\epsilon}}\hat{A},\hat{\phi}+\hat{\delta}_{\hat{\epsilon}}\hat{\phi})-\hat{C}_{\mu\nu}(\hat{A},\hat{\phi})-i\hat{\epsilon}\star\hat{C}_{\mu\nu}(\hat{A},\hat{\phi})&=-i\hat{E}_{\mu\nu}(\hat{A},\hat{\epsilon})\star\hat{\phi}\;,\\\
\hat{C}_{\mu\nu}(\hat{A}+\hat{\delta}_{\hat{\epsilon}}\hat{A},\hat{\Psi}+\hat{\delta}_{\hat{\epsilon}}\hat{\Psi})-\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})-i[\hat{\epsilon},\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})]_{\star}&=-i[\hat{E}_{\mu\nu}(\hat{A},\hat{\epsilon}),\hat{\Psi}]_{\star}\;.\end{split}$
(A.5)

In particular we notice that any $\hat{D}_{\mu\nu\kappa}$ and
$\hat{C}_{\mu\nu}$ covariant under gauge transformations solve (A.5) with
$\hat{E}_{\mu\nu}=0$.

In summary, as discussed in [41], the most general solution $\hat{A}(A)$,
$\hat{\epsilon}(A,\varepsilon)$, $\hat{\phi}(A,\phi)$, $\hat{\Psi}(A,\Psi)$ of
the Seiberg–Witten conditions (6.1), (6.15) is given by the differential
equations (A.1)-(A.4) where $\hat{D},\hat{E},\hat{C}$ are constrained by
(A.5). Further constraints on the $\hat{D},\hat{E},\hat{C}$ terms are obtained
by requiring that the Seiberg–Witten map respects hermiticity and charge
conjugation in the sense that the hermiticity and charge conjugation
properties of the commutative fields imply those of the noncommutative fields
[32, 12]. If we ask $\hat{D},\hat{E},\hat{C}$ to have no explicit dependence
on $\theta$ and to be covariant under constant $GL(d,\mathbb{R})$ coordinate
transformations (the star product $f\star g$ is itself invariant under
constant $GL(d,\mathbb{R})$ coordinate transformations: $x^{\mu}\to
M^{\mu}{}_{\rho}x^{\rho}$, $\theta^{\mu\nu}\to
M^{\mu}{}_{\rho}M^{\nu}{}_{\sigma}\theta^{\rho\sigma}$) we recover the results
in [42] and in [43]:

$\displaystyle\hat{D}_{\mu\nu\kappa}=\alpha D_{\kappa}\hat{F}_{\mu\nu}+\beta
D_{\kappa}[\hat{A}_{\mu},\hat{A}_{\nu}]_{\star}~{}~{},~{}~{}~{}~{}\hat{E}_{\mu\nu}=2\beta[\partial_{\mu}\hat{\epsilon},\hat{A}_{\nu}]_{\star}\;,$
(A.6)
$\displaystyle\hat{C}_{\mu\nu}=-2i\beta[\hat{A}_{\mu},\hat{A}_{\nu}]_{\star}\star\hat{\phi}+\gamma\hat{F}_{\mu\nu}\star\hat{\phi}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm(fundamental)}\;$
(A.7)
$\displaystyle\hat{C}_{\mu\nu}=-2i\beta[[\hat{A}_{\mu},\hat{A}_{\nu}]_{\star},\hat{\Psi}]_{\star}+\gamma^{\prime}\hat{F}_{\mu\nu}\star\hat{\Psi}+\tilde{\gamma}\hat{\Psi}\star\hat{F}_{\mu\nu}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\>~{}~{}~{}~{}~{}~{}{\rm(adjoint)}\;$
(A.8)

with $\alpha,\beta,\gamma,\gamma^{\prime}$ and $\tilde{\gamma}$ arbitrary
constants.

An interesting Seiberg–Witten differential equation is obtained considering
(A.1)-(A.4) with $\hat{D}_{\mu\nu\kappa}=0$, $\hat{E}_{\mu\nu}=0$ and

$\displaystyle{\hat{C}_{\mu\nu}(\hat{A},\hat{\phi})=\gamma\hat{F}_{\mu\nu}\star\hat{\phi}+\rho
D_{\mu}D_{\nu}\hat{\phi}~{}~{}~{}~{}\mbox{ { for $\mu<\nu$,} and
}~{}\hat{C}_{\nu\mu}:=-\hat{C}_{\mu\nu}}$ (A.9)
$\displaystyle{\hat{C}_{\mu\nu}(\hat{A},\hat{\Psi})=\gamma[\hat{F}_{\mu\nu},\hat{\Psi}]_{\star}+\rho
D_{\mu}D_{\nu}\hat{\Psi}~{}\mbox{ { for $\mu<\nu$,} and
}~{}\hat{C}_{\nu\mu}:=-\hat{C}_{\mu\nu}}$ (A.10)

Here $\rho$ is a constant and $GL(d,\mathbb{R})$ covariance is broken because
$D_{\mu}D_{\nu}\hat{\Psi}$ is not antisymmetric in the $\mu,\nu$ indices,
i.e., $\hat{C}_{\nu\mu}$ does not contain also the term $\rho
D_{\nu}D_{\mu}\phi$. This choice, with $\gamma=-3$ and $\rho=i$, allows to
solve the Seiberg–Witten map on noncommutative tori (obtained from the
Groenewold–Moyal noncommutative plane) to all orders in $\theta$ for
topologically nontrivial $U(N)$-gauge potentials with constant field strengths
[41].

##  Appendix B Gamma matrices in $D=4$

We summarize in this Appendix our gamma matrix conventions in $D=4$.

$\displaystyle\eta_{ab}=(1,-1,-1,-1),~{}~{}~{}\\{\gamma_{a},\gamma_{b}\\}=2\eta_{ab},~{}~{}~{}[\gamma_{a},\gamma_{b}]=2\gamma_{ab},$
(B.1) $\displaystyle\gamma_{5}\equiv
i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3},~{}~{}~{}\gamma_{5}\gamma_{5}=1,~{}~{}~{}\varepsilon_{0123}=-\varepsilon^{0123}=1,$
(B.2)
$\displaystyle\gamma_{a}^{\dagger}=\gamma_{0}\gamma_{a}\gamma_{0},~{}~{}~{}\gamma_{5}^{\dagger}=\gamma_{5}$
(B.3)
$\displaystyle\gamma_{a}^{T}=-C\gamma_{a}C^{-1},~{}~{}~{}\gamma_{5}^{T}=C\gamma_{5}C^{-1},~{}~{}~{}C^{2}=-1,~{}~{}~{}C^{\dagger}=C^{T}=-C$
(B.4)

Useful identities

$\displaystyle\gamma_{a}\gamma_{b}=\gamma_{ab}+\eta_{ab}$ (B.5)
$\displaystyle\gamma_{ab}\gamma_{5}={i\over 2}\varepsilon_{abcd}\gamma^{cd}$
(B.6)
$\displaystyle\gamma_{ab}\gamma_{c}=\eta_{bc}\gamma_{a}-\eta_{ac}\gamma_{b}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.7)
$\displaystyle\gamma_{c}\gamma_{ab}=\eta_{ac}\gamma_{b}-\eta_{bc}\gamma_{a}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.8)
$\displaystyle\gamma_{a}\gamma_{b}\gamma_{c}=\eta_{ab}\gamma_{c}+\eta_{bc}\gamma_{a}-\eta_{ac}\gamma_{b}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.9)
$\displaystyle\gamma^{ab}\gamma_{cd}=-i\varepsilon^{ab}_{~{}~{}cd}\gamma_{5}-4\delta^{[a}_{[c}\gamma^{b]}_{~{}~{}d]}-2\delta^{ab}_{cd}$
(B.10) $\displaystyle
Tr(\gamma_{a}\gamma^{bc}\gamma_{d})=8~{}\delta^{bc}_{ad}$ (B.11)
$\displaystyle
Tr(\gamma_{5}\gamma_{a}\gamma_{bc}\gamma_{d})=-4i\,\varepsilon_{abcd}$ (B.12)

where
$\delta^{ab}_{cd}\equiv\frac{1}{2}(\delta^{a}_{c}\delta^{b}_{d}-\delta^{b}_{c}\delta^{a}_{d})$skyp
and indices antisymmetrization in square brackets has total weight $1$.

Acknowledgements
The authors acknowledge partial support from INFN, CSN4, Iniziativa Specifica
GSS. This research has a financial support from Università del Piemonte
Orientale. P.A. is affiliated to INdAM-GNFM.

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