[math-ph]

# Covariant Star Product on Semi-Conformally Flat Noncommutative
Calabi-Yau Manifolds and Noncommutative Topological Index Theorem

Amir Abbass Varshovi
ab.varshovi@sci.ui.ac.ir/amirabbassv<EMAIL_ADDRESS>Faculty of Mathematics and Statistics, Department of Applied Mathematics and
Computer Science, University of Isfahan, Isfahan, IRAN.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395-5746, Tehran, IRAN.

###### Abstract

Abstract: A differential geometric statement of the noncommutative topological
index theorem is worked out for covariant star products on noncommutative
vector bundles. For start, a noncommutative manifold is considered as a
product space $X=Y\times Z$, wherein $Y$ is a closed manifold, and $Z$ is a
flat Calabi-Yau $m$-fold. Also a semi-conformally flat metric is considered
for $X$ which leads to a dynamical noncommutative spacetime from the viewpoint
of noncommutative gravity. Based on the Kahler form of $Z$ the noncommutative
star product is defined covariantly on vector bundles over $X$. This covariant
star product leads to the celebrated Groenewold-Moyal product for trivial
vector bundles and their flat connections, such as $C^{\infty}(X)$. Hereby,
the noncommutative characteristic classes are defined properly and the
noncommutative Chern-Weil theory is established by considering the covariant
star product and the superconnection formalism. Finally, the index of the
$\star$-noncommutative version of elliptic operators is studied and the
noncommutative topological index theorem is stated accordingly.
Keywords: Covariant Star Product, Semi-Conformally Flat Metric, Kahler Form,
Calabi-Yau manifold, Superconnection, Chern-Weil Theory, Noncommutative
Topological Index Theorem.

###### pacs:

math-phys

##  I Introduction

Intimate correlations of Einstein-Yang-Mills theories and topological
structures of their underlying manifolds have been understood thoroughly over
the last four decades.111See for example donaldson ; don ; witten ; witten' .
Physical effects such as instantons, consistent and covariant anomalies,
holonomy structures, Schwinger terms, Chern-Simons theory, etc. in Yang-Mills
theories, together with topological features of gravity and string theory
including BTZ topological black holes, topological strings, mirror symmetry,
topological dyons in anti-de Sitter Einstein-Yang-Mills theories, Witten-Vafa
twist, magnetic monopoles and the Seiberg-Witten equations, etc.222See for
example nakahara ; tillmann2 ; labastida ; nash ; schwarz ; shuryak ; ring ;
katz ; hollands ; shnir  and the references therein. are established to
contain significant information about the geometry and the topology of
spacetime. However, the situation is somehow unclear when the spacetime $X$ is
a noncommutative space. This noncommutativity is mostly inspired by
superstring developments, in which the $B$ background field induces a
Groenewold-Moyal star product for the spacetime.

Despite of the elegant achievements of noncommutative geometry via the
spectacular machineries of cyclic cohomology, spectral triples, Connes-Chern
characters and the fascinating descriptions of the index theorem for
noncommutative manifolds via a beautiful local index formula connes , it seems
there is a serious lack of topological interpretation for Groenewold-Moyal
noncommutative Yang-Mills theories. In principle, although some of the
physical effects of noncommutative Yang-Mills theories have been formulated
within noncommutative framework, but their topological meanings have not been
understood thoroughly yet.

Actually, this failure is due to the ambiguity of the de Rham cohomological
counterparts of such algebro-geometric formulations in noncommutative
geometry. But, actually, for the Groenewold-Moyal noncommutative Yang-Mills
theories, wherein we essentially deal with substantially ordinary
(commutative) manifolds, the noncommutative (cyclic) and the commutative (de
Rham) settings are expected to be connected via a clear mechanism including
the topological features. Such a formulation has not been clearly developed
yet via a definit topological method. Actually, the deRham and cyclic
cohomologies for commutative algebra are dual objects, but for noncommutative
algebras there is not a clear definition for the de Rham cohomology as one may
expect for noncommutative Yang-Mills theories.

The Seiberg-Witten map is in fact an attempt to overcome this problem via a
string theoretic approach given for the Dirac-Born-Infeld actions of open
strings seiberg-witten .333See varshovi2 ; varshovi3  for some cohomological
approach to figure out the problem. Meanwhile, it has already been established
cattaneo  that this map is also in intimate correlation with the celebrated
Kontsevich’s quantization formula for Poisson manifolds kontsevich , hence
possesses some significant information of the background geometry/topology.
Although in seiberg-witten  the authors have demonstrated that the emergence
of the Groenewold-Moyal or the ordinary product is related to the chosen
Pauli-Villars or the point splitting regularization methods, the topological
information of the noncommutative curvature and its Yang-Mills theory has not
been disclosed entirely. On the other hand, some exact formulations of the
Seiberg-Witten map show that when one transfers from noncommutative to the
ordinary Yang-Mills fields, the spacetime geometry admits some smooth
deformation which is manifestly figured out via a non-flat metric or an
emergent gravity yang .444See also banerjee2 ; banerjee1 .

Thus, the topological features of the underlying geometry of noncommutative
Yang-Mills theories are still obscure and need to be discovered or interpreted
by some well understood framework such as the de Rham cohomology. On the other
hand, the formulation of the Seiberg-Witten map is only well-defined over flat
vector bundles. In fact, whenever the Yang-Mills geometry is nontrivial the
Groenewold-Moyal product loses its consistency due to the appearance of
transition functions or gauge transformations. In such situations the
noncommutative curvature and its corresponding Yang-Mills action would not be
well-defined. Consequently, it seems that although the Seiberg-Witten map
could be considered as an appropriate machinery for formulating noncommutative
Yang-Mills on flat manifolds (e.g. $\mathbb{R}^{n}$), but it must be modified
covariantly for nontrivial geometries. Indeed, the emergence of gravity via a
non-flat metric over a deformed underlying manifold reveals some clear
evidences for this theoretical refinement.

Actually, as we will show below, it can be seen that the noncommutativity of
quantum fields has to be defined upon the Yang-Mills geometry over the
spacetime manifold. Thus, the noncommutativity would be deformed from the
Groenewold-Moyal product for the vacuum field to some fairly rigorous
covariant star product in presence of the matter fields.555See vitale et al.
for some related evidences. By this modified quantization method we would have
a well-defined quantum theory which carries all possible topological
information as the ordinary Yang-Mills theory do. Therefore, it seems that a
covariant redefinition of the Seiberg-Witten map is appropriate and mandatory
from both physical and geometric viewpoints.

More generally, the correlation between topological aspects of a
noncommutative manifold and the de Rham cohomology of the underlying
(commutative) geometry is still unclear. As the most significant feature, the
noncommutative index theorem has not have a bright interpretation via the
differential geometric structures on the underlying (commutative) manifold
yet. Hence, the noncommutative versions of characteristic classes given by
replacing the ordinary (wedge) product of differential forms with
noncommutative star product suffer from our ignorance of their topological
interpretations, if there exist any. However, as we will show in the
following, it seems that the covariant star product which leads to covariant
noncommutative curvature and well-defined characteristic classes gives rise to
a beautiful solution for the problem.

In principle, employing the covariant star product (which is roughly defined
by replacing the partial derivative by the covariant derivative in the
Groenewold-Moyal star product) a well-defined framework for noncommutative
differential geometry is constructed which inherits the most important
possible geometric/topological aspects of the de Rham cohomology. In
principle, the essential motivation to extend the definition of Groenewold-
Moyal product to the covariant star product for the space of smooth sections
of an arbitrary vector bundle, is to find a concrete formulation of
noncommutative gravity which admits the most possible symmetries of spacetime.
It seems that such a formulation must involve a dynamical noncommutative
spacetime due to a non-constant noncommutativity matrix which carry some
intrinsic structure of spacetime. Such formulations has recently been provided
via a beautiful approach of matrix model based on the string features of IKKT
and IIB models steinacker .

Following the main idea of steinacker , the initial objective of this research
is to define a covariant star product by rescaling the noncommutativity matrix
with the connection form (and consequently with the curvature) at each point
of spacetime. In fact, the main mathematical idea of covariant star product in
this perusal stems from bayen1 ; bayen2 , which put an associative covariant
star product on the tangent bundle of a flat Poisson manifold. However, the
procedure fails whenever the vector bundle is nontrivial, hence non-flat.
Hereon, we employ the flatness of a Calabi-Yau manifold $Z$ together with a
semi-conformally flat structure on $X$ to generalize the formulation of bayen1
to any given vector bundle over $Z$.666There are other definitions of
covariant star products. For example see chaichian-zet ; chaichian-zet2 ;
vassilevich ; zumino ; blohman ; masmoudi ; manolakos ; sperling ; heckman ;
steinacker2  and the references therein. The algebraic structure is
effectively associative, gauge invariant and diffeomorphism invariant on the
(co)tangent bundle. Therefore, it will provide a consistent setting for
noncommutative Yang-Mills and noncommutative gravity.777There are other
methods for replacing the non-dynamical noncommutative spacetime with matrix
model or fuzzy spaces of extra dimensions. See for example the introduction of
manolakos0  for a full list of and extensive description for alternative
noncommutative models for gravity.

In fact, as we will show below both the noncommutative Yang-Mills and the
noncommutative Einstein-Hilbert actions given by this covariant star product
remain invariant after any gauge and general coordinate transformation. Thus,
the covariant star product carry the most symmetries of the ordinary Yang-
Mills and gravity. Hence, it is expected that the covariant star product would
keep the topological information of the underlying geometries. Within this
consistent setting, as we will show in the following discussions, it could be
firmly established that the ordinary characteristic classes are surprisingly
cohomologous to their noncommutative versions given with the covariant star
product. Afterward, a noncommutative Chern-Weil theorem will be obtained which
would provide a spectacular topological achievement for noncommutative
geometry. This will lead naturally to reliable evidence for the existence of
some noncommutative topological index theorem.

In this article we try to work out the entire expression of the noncommutative
topological index theorem on noncommutative manifolds. Our strategy is first
to generalize the Groenewold-Moyal star product for any given vector bundle
over a noncommutative flat Calabi-Yau $m$-fold. We accomplish the process by
replacing the partial derivative with the covariant derivative, coupled with
the Kahler form of the base manifold, and furnishing the base manifold with a
semi-conformally flat noncommutative structure. Next, by employing the
superconnection method we will prove a noncommutative version of the
celebrated Chern-Weil theorem which demonstrates an intimate correlation
between topological properties of ordinary and noncommutative manifolds.
Finally, the index of the noncommutative version of elliptic operators is
studied and the full expression of the noncommutative topological index
theorem is worked out on semi-conformally flat noncommutative Calabi-Yau
manifolds.

##  II Covariant Star Product on Associative Vector Bundles Over A Semi-
Conformally Flat Noncommutative Calabi-Yau Manifold

Let $X$ be a $2n$-dimensional closed manifold which is described as a product
space as $X=Y\times Z$, wherein $Y$, the so called _commutative_ component is
itself an even dimensional closed orientable manifold, and $Z$, the
_noncommutative_ part, is a flat Calabi-Yau $m$-fold equipped with a flat
Kahler metric, say $h$, an almost complex structure $J$ and the Kahler form
$\hat{\omega}$. In principle, to work out a noncommutative version of quantum
gravity upon the viewpoints of the superstring theory or the supergravity, $X$
could be assumed as the spacetime with extra dimensions in $Z$. Here, $Z$
becomes the substructure of $X$ which is in fact, unobservable and responsible
for the noncommutativity of quantum fields. Hence, $Y$ would be the observable
four-dimensional spacetime. However, the fluctuations of the metric via the
quantum gravity effects are not only considered for the commutative part $Y$.

Actually, the definition of a Calabi-Yau manifold varies according to the
documents, thus it must be specified that in the following we identify it as a
Ricci flat Kahler manifold hubsh ; gross . Here, we generally put no condition
on the canonical line bundle $\mathcal{O}_{Z}$ (e.g. its positive powers to be
trivial) or on the holonomy group $\text{Hol}(h)$ (e.g. to be $SU(m)$).
However, since $Z$ is a flat manifold it is seen that $\text{Hol}(h)$ is a
discrete group, i.e. the group of deck transformations of its simply connected
covering space hatcher ; poor .

Hence, in our investigation, a flat Calaby-Yau manifold is in fact a flat
Kahler manifold. In fact, the collection of such spaces includes, of course,
even dimensional tori and some special kinds of Kummer surfaces $K_{3}$
kronheimer ; ohsawa . But, in principle, flat Kahler manifolds could also be
of more varied topologies with different algebro-geometric properties such as
the Hodge numbers.888See flat-kahler  for more discussion about different
types of flat Kahler manifolds of 4 and 6 dimensions. In general,
$Z=\mathfrak{G}\setminus E(2m)/O(2m)$, where $E(k)=O(k)\ltimes\mathbb{R}^{k}$
is the Euclidean group of $k$-dimensional motions and $\mathfrak{G}$ is a
compact, discrete and torsion free subgroup of $E(2m)$. In this case, upon the
Bieberbach’s theorems we see that $\pi_{1}(Z)=\mathfrak{G}$ and
$\text{Hol}(h)=\mathfrak{G}/T$, for $T\subset\mathfrak{G}$ the subgroup of all
pure translations bieberbach1 ; bieberbach2 ; flat-kahler .999See ratcliffe
for more discussion about the Bieberbach’s theorems.

To specify the spin structure of $X$ it is enough to assume that $Y$ is a spin
manifold, since in fact, any Calaby-Yau manifold $Z$ is itself a spin manifold
due to the equality of: $w_{2}(Z)=c_{1}(Z)~{}(\text{mod}~{}2)$, for $w_{2}(Z)$
the second Stiefel-Whitney class of $TZ$, and for $c_{1}(Z)=0$, the first
Chern class of $TZ$ spin-geom . Therefore, since $w_{2}(Y\times
Z)=w_{2}(Y)+w_{2}(Z)+w_{1}(Y)\wedge w_{1}(Z)~{}(\text{mod}~{}2)$, then $X$
would be a spin manifold, whenever the first Stiefel-Whitney class of $TY$
(resp. $TZ$) vanishes identically, i.e. $w_{1}(Y)=0$ (resp. $w_{1}(Z)=0$).

We should continue with a statement of our main conventions about the
noncommutative manifold $X$. Set $\dim Y=D$ and let
$\mathcal{A}^{X}=\\{(U^{X}_{\alpha}=U^{Y}_{\alpha}\times U^{Z}_{\alpha}\subset
Y\times Z,\\{x^{\mu}_{\alpha}\\}_{\mu=0}^{2n-1})\\}_{\alpha\in\mathfrak{I}}$
be a covering atlas on $X$, whose the coordinate functions $x^{\mu}_{\alpha}$s
split $X$; that is $x^{a}_{\alpha}$ belongs to $Y$ for $a=0,\cdots,D-1$, and
$x^{i}_{\alpha}$s are local coordinates on $Z$ for $D\leq\mu\leq 2n-1$. As a
convention, the local coordinates of $Y$ (resp. $Z$) are labeled with letters
$a,b,c,\cdots$ (resp. $i,j,k,\cdots$) and referred to as the _commutative
coordinates_ (resp. _noncommutative coordinates_). However, the local
coordinate functions of $X$ are generally labeled with Greek letters
$0\leq\mu,\nu,\sigma,\cdots<2n$. Thus, $(U^{Y}_{\alpha},\\{x^{a}_{\alpha}\\})$
(resp. $(U^{Z}_{\alpha},\\{x^{i}_{\alpha}\\})$) is a local coordinate system
on $Y$ (resp. $Z$). But, however, if there is no confusion we drop the
subscript $\alpha$ and simply write $x^{\mu}$, otherwise we keep $\alpha$.

We assume that each $(U^{Z}_{\alpha},\\{x^{i}_{\alpha}\\})$ is a Darboux’s
chart for the symplectic form $\hat{\omega}$. This, ensures that the Kahler
form
$\hat{\omega}=\theta^{\alpha}_{ij}\text{d}x^{i}_{\alpha}\wedge\text{d}x_{\alpha}^{j}$
has constant coefficients $\theta_{ij}^{\alpha}$ on each
$(U_{\alpha},\\{x^{i}_{\alpha}\\})$. Indeed,
$\hat{\omega}=\sum_{i=1}^{m}\text{d}x^{i}_{\alpha}\wedge\text{d}x^{i+m}_{\alpha}$.
In general, when there is no confusion we may drop the superscript $\alpha$
and simply write $\theta_{ij}$. However, we can always choose
$x^{i}_{\alpha}$s so that
$\theta_{ij}^{\alpha}=\theta^{\beta}_{ij}\in\mathbb{R}$, for each
$\alpha,\beta\in\mathfrak{I}$ with $U^{Z}_{\alpha}\bigcap
U^{Z}_{\beta}\neq\emptyset$. Actually, according to the Bieberbach’s theorems
bieberbach1 ; bieberbach2  the flatness of $Z$ ensures that it is finitely
covered by $\mathbb{T}^{2m}$. More precisely, since $Z$ is flat it is a
quotient of $\mathbb{T}^{2m}$ modulo some finte group, i.e. the deck
transformation group. Therefore, $Z$ is itself a Lie group, and hence
possesses a trivial tangent (resp. cotangent) bundle;
$TZ=Z\times\mathbb{R}^{2m}$ (resp. $T^{*}Z=Z\times\mathbb{R}^{2m}$). Thus,
since $Z$ is Abelian it admits $2m$ numbers of globally defined nonvanishing
vector fields $\\{\partial_{i}\\}_{i=1}^{2m}$ (resp. one-forms
$\\{\text{d}x^{i}\\}_{i=1}^{2m})$ with constant symplectic form $\hat{\omega}$
on the second component $\mathbb{R}^{2m}$. In fact, these $2m$ vector fields
(resp. one-forms) are integrable locally on each contractible
$U^{Z}_{\alpha}$, producing local coordinate functions
$\\{x^{i}_{\alpha}\\}_{i=1}^{2m}$. Then, $\frac{\partial}{\partial
x^{i}_{\alpha}}=\frac{\partial}{\partial x^{i}_{\beta}}=\partial_{i}$ (resp.
$\text{d}x^{i}_{\alpha}=\text{d}x^{i}_{\beta}=\text{d}x^{i}$) on
$U^{Z}_{\alpha}\bigcap U^{Z}_{\beta}\neq\emptyset$, and we obtain the globally
defined $\hat{\omega}=\theta_{ij}\text{d}x^{i}\wedge\text{d}x^{j}$ with
constant coefficients $\theta_{ij}$.101010Conversely, one can similarly figure
out the topology of $X$ as a principal $Z$-bundle over $Y$. In fact,
$\\{\partial_{i}\\}$ generates a smooth involutive distribution $\mathcal{D}$
on $X$ and its maximal connected integral manifolds provide the fibers of $X$
according to the Frobenius theorem warner . The triviality of the topology of
$X$, as a fibration over the base manifold $Y$, is indeed an immediate
consequence of definition of $\mathcal{D}$ which admits a global basis. This
fixes the unique differential and topologic structure of $X$ taubes .

The open sets $U^{Y}_{\alpha}$ and $U^{Z}_{\alpha}$ (hence $U^{X}_{\alpha}$)
are assumed to be contractible. The flat Kahler metric $h$, as introduced
above, is the unique flat $J$ compatible Riemannian metric on $Z$ due to the
Yau’s theorem yau . The points of $X$ are simply shown with $(y,z)\in Y\times
Z$. The metric of $Y$, shown with $g_{ab}(y,z)$, is considered to be _semi-
conformally flat via noncommutative part $Z$_, i.e.
$g_{ab}(y,z)=e^{\sigma(z)}G_{ab}(y)$, where $\sigma(z)$ belongs to
$C^{\infty}(Z)$, and $G_{ab}(y)$ is a (pseudo-) Riemannian metric on $Y$.
Actually, a _semi-conformally flat noncommutative structure_ on noncommutative
manifold $X=Y\times Z$ is a pseudo-Riemannian metric
$g_{\mu\nu}(y,z)=g_{ab}(y,z)\oplus h_{ij}(z)$ so that $h_{ij}(z)$ is the flat
Kahler-Yau metric on $Z$, and $g_{ab}(y,z)=e^{\sigma(z)}G_{ab}(y)$, for
$G_{ab}(y)$ an arbitrary (pseudo-) Riemannian metric on $Y$. In general, $X$
is considered to be Riemannian, but however, for significance of its
applications in physics it can be assumed to admit the Lorentz signature
$(-,+,\cdots,+)$, whilst the minus sign belongs to $g_{ab}$. Moreover,
$h_{ij}$s are constant in the mentioned Darboux’s charts due to the structure
of the Kahler form: $h_{ij}=-\theta_{ik}J^{k}_{j}$, wherein $J^{i}_{j}$ is the
almost complex structure.

Let $\nabla^{X}$ be the Levi-Civita connections for $g_{\mu\nu}=g_{ab}\oplus
h_{ij}$. It is seen that
$\Gamma^{a}_{ij}=\Gamma^{i}_{aj}=\Gamma^{i}_{ja}=\Gamma^{i}_{jk}=0$,
$\Gamma^{a}_{ib}=\frac{\partial_{i}\sigma}{2}\delta^{a}_{b}$, and
$\Gamma^{i}_{ab}=-\frac{h^{ij}g_{ab}}{2}\partial_{j}\sigma$. Also, the
Christoffel symbols on $Y$ are given for the metric $G_{ab}$;
$\Gamma^{a}_{bc}=\frac{G^{ad}}{2}(G_{bd,c}+G_{dc,b}-G_{bc,d})$. Hence;

$\nabla_{i}^{X}\text{d}x^{j}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{i}^{X}\text{d}x^{a}\propto\text{d}x^{a}.$
(II.1)

All in all, we find:

$[\nabla^{X}_{i},\nabla^{X}_{j}]=0,~{}~{}~{}~{}~{}\text{or;}~{}~{}~{}~{}~{}{R^{X}_{ij\mu}}^{\nu}=0.$
(II.2)

We also read; ${R^{X}_{aij}}^{k}={R^{X}_{abi}}^{j}={R^{X}_{abi}}^{c}=0$, while
the other components are generally nontrivial. Specifically,

$\begin{split}{R^{X}_{abc}}^{d}={R^{Y}_{abc}}^{d}+\frac{1}{4}\left(\delta^{d}_{b}g_{ac}-\delta^{d}_{a}g_{bc}\right)\partial_{i}\sigma\partial^{i}\sigma,\\\
{R^{X}_{aij}}^{b}=-\frac{\delta^{b}_{a}}{2}\left(\partial_{i}\partial_{j}\sigma-\frac{1}{2}\partial_{i}\sigma\partial_{j}\sigma\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
{R^{X}_{aib}}^{j}=\frac{g_{ab}}{2}\left(\partial_{i}\partial^{j}\sigma-\frac{1}{2}\partial_{i}\sigma\partial^{j}\sigma\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{split}$
(II.3)

wherein ${R^{Y}_{abc}}^{d}$ is the curvature of the Levi-Civita connection
$\nabla^{Y}$ of $g_{ab}$. In principle, the nontrivial topological features of
noncommutative gravity for a semi-conformally flat noncommutative structure
are subject to a Weyl rescaling on $Z$. One can easily see that
$R^{X}_{ab}=R^{Y}_{ab}-\frac{g_{ab}}{2}\left(\frac{1}{2}\partial^{i}\sigma\partial_{i}\sigma-\partial^{i}\partial_{i}\sigma\right)$,
$R^{X}_{ai}=0$, and
$R^{X}_{ij}=-n\left(\partial_{i}\partial_{j}\sigma+\frac{1}{2}\partial_{i}\sigma\partial_{j}\sigma\right)$,
for $R^{X}_{\mu\nu}$ (resp. $R^{Y}_{ab}$) the Ricci tensor of connection
$\nabla^{X}$ (resp. $\nabla^{Y}$). In addition, we obtain
$\mathcal{R}^{X}=\mathcal{R}^{Y}-n\partial^{i}\partial_{i}\sigma$, where
$\mathcal{R}^{X}$ (resp. $\mathcal{R}^{Y}_{ab}$) is the Ricci scalar of
$\nabla^{X}$ (resp. $\nabla^{Y}$). Thus, even before introducing a star
product on $X$, the ordinary Einstein-Hilbert action of the vacuum is affected
by the extra dimensions of $Z$ with a Klein-Gordon theory. However, this
scalar field would not give rise to nontrivial effects for commutative fields,
since it behaves like some Weyl rescaling. Indeed, via the mentioned formalism
the nontrivial features of the Einstein-Hilbert theory arise only when one
considers the contributions of the matter fields and their interactions with
some noncommutative star product.

Now, we are ready to put a noncommutative structure on $X$ by defining a
covariant noncommutative structure for general vector bundles over $X$. Set
$\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ and let $\mathbb{F}^{K_{i}}\to
E_{i}\to X$, $i=1,2,3$, be three arbitrary vector bundles endowed with a
smooth bundle structure as; $\mathfrak{m}:E_{1}\otimes E_{2}\to E_{3}$.
Actually, $\mathfrak{m}$ induces a multiplication of smooth sections, also
denoted with the same symbol $\mathfrak{m}$ as;
$\mathfrak{m}:\Gamma(E_{1})\otimes\Gamma(E_{2})\to\Gamma(E_{3})$. To produce
the most general formulation of star product we quantize the multiplication
map $\mathfrak{m}$ in a covariant framework. In principle, to quantize
$\mathfrak{m}$ consistently we should follow the well-understood methods of
deformation quantization explained in bayen1 ; bayen2  and define a
generalized Moyal-Weyl-Wigner quantization on $X$ with employing the
connections on $E_{i}$s. Let $\nabla^{i}$ be a connection on $E_{i}$ and
assume that $\hat{\omega}^{\sharp}=\theta^{ij}\partial_{i}\otimes\partial_{j}$
be the Poisson bivector as the dual of the Kahler form $\hat{\omega}$ obtained
via converting the noncommutativity matrix $\theta_{ij}$, i.e.
$\theta^{ij}\theta_{jk}=\delta^{i}_{k}$. We introduce a bilinear differential
operator on $\Gamma(E_{1})\otimes\Gamma(E_{2})$ with;

$\Theta=\theta^{ij}\nabla^{1}_{i}~{}\otimes\nabla^{2}_{j},$ (II.4)

which is the contraction of $\hat{\omega}^{\sharp}$ and
$\nabla^{1}\otimes\nabla^{2}=\nabla^{1}_{\mu}\text{d}x^{\mu}\otimes\nabla^{2}_{\nu}\text{d}x^{\nu}$.
Obviously $\Theta$ is globally well-defined on $X$. Now, we introduce the
_covariant star product_ $\star$ as a generalized Groenewold-Moyal star
product for smooth sections $\omega_{i}\in\Gamma(E_{i})$, $i=1,2$, as:

$\begin{split}\omega_{1}\star\omega_{2}:=\mathfrak{m}\left(e^{i\Theta}\left(\omega_{1}\otimes\omega_{2}\right)\right)=\mathfrak{m}\left(e^{i\theta^{ij}\nabla^{1}_{i}\otimes\nabla^{2}_{j}}\left(\omega_{1}\otimes\omega_{2}\right)\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\sum_{N\geq
0}\frac{i^{N}}{N!}\theta^{i_{1}j_{1}}\cdots\theta^{i_{N}j_{N}}\mathfrak{m}\left(\left(\nabla^{1}_{i_{N}}\cdots\nabla^{1}_{i_{1}}\omega_{1}\right)\otimes\left(\nabla^{2}_{j_{N}}\cdots\nabla^{2}_{j_{1}}\omega_{2}\right)\right)\\\
=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}}{N!}\theta^{(i)(j)}\mathfrak{m}\left(\left(\nabla^{1}_{(i)}\omega_{1}\right)\otimes\left(\nabla^{2}_{(j)}\omega_{2}\right)\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{split}$
(II.5)

wherein $(i)$ is a short notation for multi-indices $(i_{1},\cdots,i_{N})$
with degree $|(i)|=N$, and $\nabla_{(i)}$ stands for multiple covariant
derivatives $\nabla_{i_{N}}\cdots\nabla_{i_{1}}$. Also, we assume
$\nabla_{(i)}\omega=\omega$ for $N=0$. As we read from (II.5) the covariant
star product after the second order of $\theta$ is essentially rescaled by the
curvature of connections $\nabla^{i}$s. This is the geometric realization of
dynamical noncommutative spacetime for noncommutative gravity.111111See
steinacker  for more discussions.

It is worth to note that the star product (II.5) is smoothly defined
everywhere on $X$, and for the flat connection
$\nabla^{i}_{\mu}=\partial_{\mu}$, $i=1,2,3$, on trivial vector bundles
$E_{i}=X\times\mathbb{F}^{K_{i}}$, the covariant star product $\star$
naturally leads to the well-known Groenewold-Moyal product $\star_{G-M}$.
Indeed, this situation arises for example when the base manifold is
contractible, e.g. $X=\mathbb{R}^{4}$, which appears in quantum field theory.
Specially, when $E_{1}=E_{2}=E_{3}=X\times\mathbb{R}$, one finds:
$\Gamma(E_{i})=C^{\infty}(X)$, $i=1,2,3$, thus we would have;
$\Theta=\theta^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}$, which readily
results in: $\star=\star_{G-M}$. In this case $\star$ becomes an associative
product. In fact, if $E_{1}=E_{2}=E_{3}=E$ is trivial and
$\nabla^{1}_{\mu}=\nabla^{2}_{\mu}=\nabla^{3}_{\mu}=\partial_{\mu}$, the
covariant star product $\star$ would be an associative structure on
$\Gamma(E)$, whenever $\mathfrak{m}$ is associative too.

When $E_{1}=E_{2}=E_{3}=\oplus_{k=0}^{2m}\Lambda^{k}T^{*}X$ the mentioned
quantization method is accomplished with differential forms on $X$ for
$\Gamma(E_{i})=\Omega^{*}(X)$, $i=1,2,3$, whereas $\mathfrak{m}$ is the
ordinary wedge product. In this case, as we consider $\nabla^{i}=\nabla^{X}$
for each $i$, then the star product $\star$ would be still an associative
multiplication; i.e.
$\omega_{1}\star(\omega_{2}\star\omega_{3})=(\omega_{1}\star\omega_{2})\star\omega_{3}$.
This is in fact, the immediate consequence of the Baker-Campbell-Hausdorff
formula and (II.2). Moreover, for any arbitrary topology and connection of
$E\to X$ the scalar multiplication is always associative, i.e. for each
$f,g\in C^{\infty}(X)$ and $\zeta\in\Gamma(E)$, we read:
$f\star(g\star\zeta)=(f\star g)\star\zeta$. Indeed, it is also an apparent
result of (II.5).

Substantially, if $E_{1}=E_{2}=E_{3}=E$ is not nontrivial, the covariant star
product $\star:\Gamma(E)\otimes\Gamma(E)\to\Gamma(E)$ is not generally
associative, even for associative $\mathfrak{m}$. However, if $E\to X$ admits
a connection $\nabla$ with vanishin curvature on $Z$, then $\star$ would be
associative too. In fact, if $[\nabla_{i},\nabla_{j}]=0$, then upon the Baker-
Campbell-Hausdorff formula the covariant star product $\star$ as the
quantization of $\mathfrak{m}$ would be still associative. Such vector bundles
(resp. connections) are referred to as _associative vector bundles_ (resp.
_associative connections_). Indeed, whenever $E$ is associative we consider an
associative connection in quantizing $\mathfrak{m}$.

Actually, the star product quantization of differential structures over $X$
could be consistently implemented by means of (II.5). Also, the inner products
on vector bundles could be quantized via the covariant star product (II.5).
Let $\mathbb{F}^{K}\to E\to X$ be a vector bundle endowed with a fiberwise
smooth metric $\mathfrak{g}_{pq}$, defined locally for the local frame
$\\{e_{p}\\}$. This induces a multiplication as;

$\mathfrak{g}:\Gamma(E)\otimes\Gamma(E)\to
C^{\infty}(X),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathfrak{g}(\zeta^{p}e_{p}\otimes\xi^{q}e_{q})=\mathfrak{g}_{pq}\zeta^{p}\xi^{q}.$
(II.6)

Therefore, upon (II.5) the quantization of $\mathfrak{g}$, as a covariant star
product is locally given as:

$\zeta\star\xi=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}}{N!}\theta^{(i)(j)}\mathfrak{g}_{pq}\left(\nabla_{(i)}\zeta^{p}\right)\left(\nabla_{(j)}\xi^{q}\right).$
(II.7)

where $\nabla$ is the connection of $E$. Specially, if $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ are two vector fields on $X$, then their star inner
product is:

$\mathcal{X}\star\mathcal{X}^{\prime}=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}}{N!}\theta^{(i)(j)}g_{\mu\nu}\left(\nabla^{X}_{(i)}\mathcal{X}^{\mu}\right)\left(\nabla^{X}_{(j)}\mathcal{X}^{\prime\nu}\right).$
(II.8)

It can be seen that (II.7) is independent of the chosen local frame
$\\{e_{p}\\}$ for $E$, and hence is a well-defined structure. To see this let
$\\{e^{\prime}_{p}\\}$ be another local frame for $E$ defined over some
trivialization chart. Then, for any smooth section $\zeta\in\Gamma(E)$ we
have; $\zeta^{\prime p}=U^{p}_{q}\zeta^{q}$, for some invertible matrix $U$.
On the other hand, in the basis of $\\{e^{\prime}_{p}\\}$ the connection
$\nabla$ becomes; $\nabla^{\prime}=U\nabla U^{-1}$. Thus, we read:
$\nabla^{\prime}_{(i)}\zeta^{\prime p}=U^{p}_{q}\nabla_{(i)}\zeta^{q}$.
Moreover, in the basis $\\{e^{\prime}_{p}\\}$ the fiber metric is:
$\mathfrak{g}^{\prime}_{pq}={U^{-1}}_{p}^{r}\mathfrak{g}_{rs}{U^{-1}}^{s}_{q}$.
Consequently;

$\mathfrak{g}^{\prime}_{pq}\left(\nabla^{\prime}_{(i)}\zeta^{\prime
p}\right)\left(\nabla^{\prime}_{(i)}\xi^{\prime
q}\right)=\mathfrak{g}_{pq}\left(\nabla_{(i)}\zeta^{p}\right)\left(\nabla_{(i)}\xi^{q}\right),$
(II.9)

which confirms the consistency of (II.7). Similarly, we see that if $E$ is a
complex vector bundle endowed with a Hermitian metric
$\mathfrak{h}:\Gamma(E)\otimes\Gamma(E)\to C^{\infty}(X)$, and the connection
$\nabla$, the quantization of $\mathfrak{h}$ via

$\zeta\star\xi=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}}{N!}\theta^{(i)(j)}\mathfrak{h}_{pq}\left(\nabla_{(i)}\zeta^{p}\right)\overline{\left(\nabla_{(j)}{\xi}^{q}\right)}$
(II.10)

is well-defined and independent of the chosen local frame. However, as we see
from (II.7) the fiber metric components $\mathfrak{g}_{pq}$s would not be
covariantly differentiated in the process of multiplying two sections with
$\star$. This is, in principle, the main reason for the consistency of (II.7).
Nonetheless, the star inner product (II.7) is not bilinear for the scalar star
product; i.e. if $f\in C^{\infty}(X)$ and $\zeta,\xi\in\Gamma(E)$, then
$f\star(\zeta\star\xi)$, $(f\star\zeta)\star\xi$, and $\zeta\star(f\star\xi)$
are generally different.

##  III Some Aspects of Noncommutative Physics on Semi-Conformally Flat
Noncommutative Spacetime with Covariant Star Product

Obviously, although the star inner product is well-defined it is not a local
structure. However, for non-smooth objects on $X$ we naively assume the
covariant derivative to vanish identically. For instance, for two tangent
vectors $\mathcal{V},\mathcal{V}^{\prime}\in T_{p}X$, we readily suppose:
$\nabla^{X}\mathcal{V}=\nabla^{X}\mathcal{V}^{\prime}=0$, hence;
$\mathcal{V}\star\mathcal{V}^{\prime}=\mathcal{V}.\mathcal{V}^{\prime}$. Also,
we obtain $\nabla^{X}_{\mathcal{V}}\mathcal{X}=0$, whenever $\mathcal{X}$ is a
vector field along a smooth curve in $X$, and $\mathcal{V}$ is a tangent
vector not proportional to $\mathcal{X}$. Therefore, $\star$ will not appear
in the geodesic equations on $X$. In principle, a geodesic on $X$ is a
solution of equations:

$\begin{split}\frac{\text{d}^{2}x^{a}}{\text{d}t^{2}}+\Gamma^{a}_{bc}\frac{\text{d}x^{b}}{\text{d}t}\frac{\text{d}x^{c}}{\text{d}t}+\partial_{i}\sigma\frac{\text{d}x^{a}}{\text{d}t}\frac{\text{d}x^{i}}{\text{d}t}=0,\\\
\frac{\text{d}^{2}x^{i}}{\text{d}t^{2}}-\frac{\partial^{i}\sigma}{2}g_{ab}\frac{\text{d}x^{a}}{\text{d}t}\frac{\text{d}x^{b}}{\text{d}t}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{split}$
(III.1)

Actually, from the viewpoint of general relativity the first equation of
(III.1) implies that the observable trajectory of an inertial body is affected
by the Weyl rescaling factor of the extra dimensions of $Z$ and their
corresponding (unobservable) components of the velocity. On the other hand,
the second equation of (III.1) states that the extra dimensional components of
the velocity could not be vanished unless $\frac{\text{d}\gamma}{\text{d}t}=0$
on $Y$, i.e. for light like geodesics in $Y$ whenever $g_{ab}$ is Lorentzian.
Hence, although $\star$ does not take part in geodesic equations, the
geometries of the geodesics are affected by the extra dimensions of $Z$ for
massive particles. More precisely, as far as we use semi-conformally flat
noncommutative structures the noncommutative effects of gravity would appear
only in presence of matter fields.

The covariant star product $\star$ on differential structures of $X$ has some
other interesting features with significance in noncommutative gravity. For
instance, since $\nabla^{X}g_{\mu\nu}=0$, we easily see that the contraction
of any arbitrary tensor field with $g_{\mu\nu}$ receives no contribution from
the noncommutativity. Specially, rising and lowering the indices of tensors
taken place ordinarily on noncommutative manifold $X$. Therefore, the
determinant of the metric $g_{\mu\nu}$ is calculated via the usual way in
presence of $\star$, i.e.;

$g_{\star}=\frac{1}{(2n)!}\epsilon^{\mu_{1}\cdots\mu_{2n}}\epsilon^{\nu_{1}\cdots\nu_{2n}}g_{\mu_{1}\nu_{1}}\star\cdots\star
g_{\mu_{2n}\nu_{2n}}=\frac{1}{(2n)!}\epsilon^{\mu_{1}\cdots\mu_{2n}}\epsilon^{\nu_{1}\cdots\nu_{2n}}g_{\mu_{1}\nu_{1}}\cdots
g_{\mu_{2n}\nu_{2n}}=g.$ (III.2)

Hence, the volume form of $X$ is unchanged as we transfer from ordinary to
noncommutative structures on $X$, i.e. $\text{dvol}_{\star}=\text{dvol}$.
Meanwhile, the Christoffel symbols remain invariant via transferring from the
ordinary product to the covariant star product $\star$;
$\Gamma_{\star\mu\nu}^{\sigma}=\Gamma^{\sigma}_{\mu\nu}$. This, confirms the
mentioned claims about the geodesic equations on $X$. Nonetheless, the
noncommutative Riemann curvature tensor
${R^{X}_{\star\mu\nu\sigma}}^{\lambda}$, produced by employing $\star$, will
not coincide with its ordinary version ${R^{X}_{\mu\nu\sigma}}^{\lambda}$
generally. The main difference here is that the commutator of Christoffel
symbols via the star product $\star$ would add nontrivial terms to
${R^{X}_{\star\mu\nu\sigma}}^{\lambda}$. However, the structure of $\star$ is
so that ${R^{X}_{\star\mu\nu\sigma}}^{\lambda}$ remains well-defined and gauge
(general coordinate transformation) covariance. Moreover, the noncommutative
Ricci tensor $R^{X}_{\star\mu\nu}={R^{X}_{\star\sigma\mu\nu}}^{\sigma}$ and
the noncommutative Ricci scalar
$\mathcal{R}_{\star}^{X}=R^{X}_{\star\mu\nu}g^{\mu\nu}$ both have only one
star product included.

In order to compare the noncommutative version of the Einstein-Hilbert action
with its ordinary formula we need to prove an important lemma. However, let us
give some preliminaries first: Suppose that the vector bundle $E\to X$ and the
connection $\nabla$ are associative. We also assume that $E$ admits a
fiberwise metric $\mathfrak{g}_{pq}$, the so called _associative metric_ ,
which is compatible with $\nabla$, i.e. $\nabla_{i}\mathfrak{g}_{pq}=0$ for
each $D\leq i<2n$. Thus, we can simply write (II.7) as;

$\zeta\star\xi=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}}{N!}\theta^{(i)(j)}\left(\nabla_{(i)}\zeta_{p}\right)\left(\nabla_{(j)}\xi^{p}\right),$
(III.3)

for $\zeta_{p}=\mathfrak{g}_{pq}\zeta^{q}$. In principle, from now on for all
associative vector bundles, we would assume associative connections and
metrics accordingly. Now, we can prove an important lemma that has a
significant role in our study.

Lemma 1; _Assume that $E\to X$ is associative. Then;_

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int_{X}*(\zeta\star\xi)=\int_{X}*(\zeta.\xi),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta,\xi\in\Gamma(E),$
(III.4)

_for $*$ the Hodge star operator and for $\zeta.\xi$ the ordinary inner
product of $\zeta$ and $\xi$ in $\Gamma(E)$._

Proof; At first we should note that if $\mathcal{X}$ is a vector field on $X$,
then:

$*(\nabla^{X}_{\mu}\mathcal{X}^{\mu})=\text{d}(\mathfrak{i}_{\mathcal{X}}\text{dvol}),$
(III.5)

for $\mathfrak{i}_{\mathcal{X}}$ the interior product with the vector field
$\mathcal{X}$. We should recall that $\partial_{i}$s are globally defined
vector fields on $X$. Therefore, if $\mathcal{X}=\mathcal{X}^{i}\partial_{i}$
is a vector field on $X$ we have;

$*(\nabla^{X}_{i}\mathcal{X}^{i})=\text{d}\left(\mathcal{X}^{i}(\mathfrak{i}_{\partial_{i}}\text{dvol})\right).$
(III.6)

Moreover, apart from the leading term the inner star product $\zeta\star\xi$
consists of the following terms;

$\theta^{ij}\left(\nabla_{i}\zeta^{\prime}_{p}\right)\left(\nabla_{j}\xi^{\prime
p}\right)=\partial_{i}\left(\theta^{ij}\zeta^{\prime}_{p}\nabla_{j}\xi^{\prime
p}\right)-\frac{\theta^{ij}}{2}\left(\zeta^{\prime}_{p}\right)\left([\nabla_{i},\nabla_{j}]\xi^{\prime
p}\right)=\nabla^{X}_{i}\left(\theta^{ij}\zeta^{\prime}_{p}\partial_{j}\xi^{\prime
p}\right).$ (III.7)

Hence, if we set
$\mathcal{X}^{i}=\theta^{ij}\zeta^{\prime}_{p}\nabla_{j}\xi^{\prime p}$, then
(III.6) and (III.7) lead to;

$*(\zeta\star\xi)=*(\zeta.\xi)+\text{d}\beta,$ (III.8)

for $\text{d}\beta$ an exact form on $X$. Thus, the lemma follows. Q.E.D.

The above lemma leads to two important properties of $\star$ of
$C^{\infty}(X)$ which are in common with the Groenewold-Moyal star product and
have significant applications in noncommutative gravity: The Triviality and
the Traciality;

a) Triviality; _If $f_{1},f_{2}\in C^{\infty}(X)$, then; $*(f_{1}\star
f_{2})=*(f_{1}f_{2})+\text{d}\beta$, for $\emph{d}\beta$ an exact form on $X$.
Hence, if $f_{1},f_{2}\in C^{\infty}(X)$, then;_

$\int_{X}*(f_{1}\star f_{2})=\int_{X}*(f_{1}f_{2}).$ (III.9)

_In principle, $*(f_{1}\star f_{2})$ and $*(f_{1}f_{2})$ belong to the same
cohomology class in $H^{2n}_{\emph{dR}}(X,\mathbb{R})$._

b) Traciality; _If $\\{f_{i}\\}_{i=1}^{s}$ is a set of smooth functions on
$X$, then;_

$*(f_{1}\star\cdots\star f_{s})=*(f_{s}\star f_{1}\star\cdots\star
f_{s-1})+\text{d}\beta.$ (III.10)

_Hence, if $f_{1},f_{2},f_{3}\in C^{\infty}(X)$, then;_

$\int_{X}*(f_{1}\star f_{2}\star f_{3})=\int_{X}*(f_{3}\star f_{1}\star
f_{2}).$ (III.11)

Therefore, upon the statement of Triviality, the noncommutative Einstein-
Hilbert action has no $\star$ included, although it contains a Weyl rescaling
factor $e^{\sigma}(z)\in C^{\infty}(Z)$ of the (unobservable) noncommutative
part of spacetime. Actually, we have;

$S_{\star E-H}=\frac{1}{16\pi
G}\int_{X}\mathcal{*}R^{X}_{\star}=\frac{1}{16\pi
G}\int_{X}*\mathcal{R}^{X}=S_{E-H},$ (III.12)

wherein we used the identity $f\star\text{dvol}=f\text{dvol}$, for any $f\in
C^{\infty}(X)$. In principle, the vacuume solution of Einstein’s field
equation on a noncommutative spacetime, receives no contribution from
covariant star product $\star$. As a special case the Schwarzchild black hole
has exactly the same gravitational features as its noncommutative version
provided a semi-conformally flat noncommutative structure is considered on
$X$. But, however, when the matter fields are included, the noncommutative
gravitational features will appear accordingly. Thus, it seems that upon the
semi-conformally flat noncommutative structures the noncommutativity of
spacetime would be a dynamical entity; the idea that was noted in steinacker
.121212See also nicolini  for more discussions.

It is worth to note that if $Y$ is a $4$-manifold, then the black hole
solution on some contractible $U^{X}_{\alpha}$ is simply given by considering
$G_{ab}$ to be the Schwarzschild metric, i.e.;

$\text{d}s^{2}=e^{\sigma(x^{4},\cdots,x^{2n-1})}\left(-\left(1-\frac{r_{s}}{r_{\alpha}}\right)c^{2}\text{d}t_{\alpha}^{2}+\left(1-\frac{r_{s}}{r_{\alpha}}\right)^{-1}\text{d}r_{\alpha}^{2}+r_{\alpha}^{2}\left(\text{d}\theta_{\alpha}^{2}+\sin^{2}\theta_{\alpha}\text{d}\phi_{\alpha}^{2}\right)\right)+\sum_{i}{\text{d}x^{i}_{\alpha}}^{2},$
(III.13)

wherein $t_{\alpha}=x^{0}_{\alpha}$, and
$(r_{\alpha},\theta_{\alpha},\phi_{\alpha})$ is the spherical coordinate on
the space like submanifold of $U^{Y}_{\alpha}$. However, the Weil rescaling
power $\sigma\in C^{\infty}(Z)$ is the solution of the following Euler-
Lagrange equation:

$\partial_{i}\partial^{i}\sigma-\frac{1}{2n}\mathcal{R}^{Y}+\frac{D}{4}\partial_{i}\sigma\partial^{i}\sigma=0.$
(III.14)

Let $Z$ be $2m$-dimensional torus. Then, the Fourier transformation of
(III.14) is;

$p^{2}a_{p}+\frac{1}{2n}R^{Y}\delta_{p,0}+\frac{D}{4}(p-q)_{i}q^{i}a_{p-q}a_{q}=0,$
(III.15)

with $a_{p}$ the Fourier coefficient of $\sigma$ for the Fourier mode
$p=(p_{1},\cdots,p_{2m})\in\mathbb{Z}^{2m}$. The solution for $p=0$ leads to
$|a_{q}|^{2}=\frac{\mathcal{R}^{Y}}{2nq^{2}}$, $q\neq 0$, which yield the
following equation;

$\mathcal{R}^{Y}\geq 0.$ (III.16)

Hence, for the vacuum spacetime of the Schwarzchild black hole the classical
solution results in constant $\sigma$. However, in presence of matter fields,
$\sigma$ cannot be trivial and gives rise to a nontrivial topological
constraint on $Y$ via (III.16) as an obstruction for admitting negative Ricci
scalar. Actually, $\sigma$ can be compared with gravitational dilatons such as
the Kaluza-Klein radion. Therefore, if $Y$ is a $4p$-dimensional, then upon
the Lichnerowicz obstruction theorem lic , in order to fulfill (III.16) the de
Rham cohomology class of $\hat{A}_{p}(TY)$ must be vanished identically, where
$\hat{A}_{p}(TY)$ is the $p$-th Dirac genus of $TY$.

Another immediate consequence of Triviality is that the kinetic action of
noncommutative scalar field theory in presence of gravity coincides exactly
with that of the ordinary commutative case. In fact, the noncommutative Klein-
Gordon theory includes no $\star$, i.e.;

$S_{\star
K-G}=\frac{1}{2}\int_{X}*(\text{d}\phi\star\text{d}\phi+m^{2}\phi\star\phi)=\frac{1}{2}\int_{X}*(\text{d}\phi.\text{d}\phi+m^{2}\phi^{2})=S_{K-G},$
(III.17)

wherein the equality of
$\text{d}\phi.\text{d}\phi=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi=\partial_{\mu}\phi\partial^{\mu}\phi$,
is the inner product in associative bundle $T^{*}X$. Hence, even in presence
of a background noncommutative gravity, the quantum effects are affected by
$\star$ only for the interaction of matter fields. Therefore, the expectation
value of the effective noncommutativity matrix must be substantially a
dynamical entity. This is in fact, an immediate consequence of the semi-
conformally flat noncommutative structure and the consistent construction of
covariant star product.

It can be easily seen that the kinetic terms of noncommutative gauge theories,
including the Yang-Mills and the Maxwell theory, are both out of $\star$.
Actually, if $A$ is a local Yang-Mills or Abelian gauge field, then by Lemma 1
the kinetic term of the corresponding noncommutative gauge theory is locally
given by $*(\text{d}A\star\text{d}A)$ which equals with
$*(\text{d}A.\text{d}A)$ up to some exact term. Here, $\star$ is the
quantization of the inner product of two-forms in $\Omega^{2}(X)$ given as:
$\omega.\omega^{\prime}=\omega_{\mu\nu}\omega^{\prime\mu\nu}$. Actually, we
read:
$\text{d}A.\text{d}A=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})$.

We conclude this section with a notable remark about $\star$. Indeed, it is
easy to see that the covariant star product $\star$ provides an involutive
structure via complex conjugation. Particularly, since the Kahler form
$\hat{\omega}$ is real, then $\star$ behaves similar to the Groenewold-Moyal
product versus complex conjugation, i.e. if $\omega_{i}\in\Omega^{n_{i}}(X)$,
with $i=1,2$, then;

$\overline{\omega_{1}\star\omega_{2}}=(-1)^{n_{1}n_{2}}\overline{\omega_{2}}\star\overline{\omega_{1}}.$
(III.18)

Therefore, the noncommutative Dirac action would be real and well-defined
whenever a semi-conformally flat noncommutative structure is assumed on $X$.
Indeed, similarly, we can see that the kinetic term of the Dirac action
contains no $\star$.

##  IV Covariant Star Product, Noncommutative Characteristic Classes and
Noncommutative Chern-Weil Theorem

Assume that $\mathbb{F}^{K_{i}}\to E_{i}\to X$, $i=1,2$, are vector bundles
and $\Psi:\Gamma(E_{1})\to\Gamma(E_{2})$ be a pseudo-differential linear
operator of order $r$. Therefore, $\Psi$ induces a vector bundle homomorphism
as; $\Psi:\mathfrak{J}^{r}(E_{1})\to E_{2}$, wherein $\mathfrak{J}^{r}(E_{1})$
is the $r$-th jet bundle of $E_{1}$. In other words, $\Psi$ is a section of
vector bundle $\mathbb{F}^{K}\to\Xi=E_{2}\otimes\mathfrak{J}^{r}(E_{1})^{*}\to
X$, with $K=K_{1}K_{2}{2n+r\choose r}$. Actually, $\Xi$ is endowed with the
induced connection $\nabla^{\Xi}=\nabla^{2}\otimes 1+1\otimes\nabla^{1}$. The
noncommutative version of $\Psi$, denoted with $\Psi_{\star}$, is defined by
quantizing the following map:

$\mathfrak{m}:\Gamma(\Xi)\otimes\Gamma(E_{1})\to\Gamma(E_{2}).$ (IV.1)

Hence, if $\zeta\in\Gamma(E_{1})$, then we obtain:

$\Psi_{\star}(\zeta)=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}\theta^{(i)(j)}}{N!}\nabla^{\Xi}_{(i)}\Psi\left(\nabla^{1}_{(j)}\zeta\right),$
(IV.2)

with $\nabla^{\Xi}_{(i)}\Psi=\Psi$ for $N=0$,
$\nabla^{\Xi}_{(i)}\Psi=\nabla^{2}_{i}\Psi-\Psi\nabla^{1}_{i}$ for $N=1$, and

$\nabla^{\Xi}_{(i)}\Psi=\nabla^{2}_{i_{N}}\left(\nabla^{\Xi}_{(i^{\prime})}\Psi\right)-\left(\nabla^{\Xi}_{(i^{\prime})}\Psi\right)\nabla^{1}_{i_{N}},$
(IV.3)

for $N>1$, where $(i)=(i_{1},\cdots,i_{N})$ and
$(i^{\prime})=(i_{1},\cdots,i_{N-1})$. Specially, when $E_{1}=E_{2}=E$, and
$\nabla^{1}=\nabla^{2}=\nabla$, we have;

$\Psi_{\star}(\zeta)=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}\theta^{(i)(j)}}{N!}[\nabla_{(i)},\Psi]\left(\nabla_{(j)}\zeta\right).$
(IV.4)

Also, the quantization of composition of two pseudo-differential operators is
thus given by (II.5) as;

$\Psi_{1}\star\Psi_{2}=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=N}\frac{i^{N}\theta^{(i)(j)}}{N!}[\nabla_{(i)},\Psi_{1}]\circ[\nabla_{(j)},\Psi_{2}].$
(IV.5)

Hence, if $E$ is associative and $\Psi_{1},\Psi_{2}:\Gamma(E)\to\Gamma(E)$ are
two pseudo-differential operators, then:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{1\star}\left(\Psi_{2\star}(\zeta)\right)=\left(\Psi_{1}\star\Psi_{2}\right)_{\star}(\zeta),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta\in\Gamma(E).$
(IV.6)

In principle, we obtain:

$\begin{split}[\nabla_{\mu},\nabla_{\nu}]_{\star}=\nabla_{\mu}\star\nabla_{\nu}-\nabla_{\nu}\star\nabla_{\mu}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=2N}\frac{(-1)^{N}\theta^{(i)(j)}}{(2N)!}\left[[\nabla_{(i)},\nabla_{\mu}],[\nabla_{(j)},\nabla_{\nu}]\right]~{}~{}~{}~{}\\\
+i\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=2N+1}\frac{(-1)^{N}\theta^{(i)(j)}}{(2N+1)!}\\{[\nabla_{(i)},\nabla_{\mu}],[\nabla_{(j)},\nabla_{\nu}]\\}.\end{split}$
(IV.7)

The noncommutative curvature tensor is defined by coupling
$[\nabla_{\mu},\nabla_{\nu}]_{\star}$ with
$\text{d}x^{\mu}\wedge\text{d}x^{\nu}$ as
$F_{\star}=F_{\star\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$. Therefore,
the anti-commutator terms of (IV.7) would be removed and $F_{\star}$ is given
as;

$F_{\star\mu\nu}=\sum_{N\geq
0}\sum_{~{}~{}|(i)|,|(j)|=2N}\frac{(-1)^{N}\theta^{(i)(j)}}{(2N)!}\left[[\nabla_{(i)},\nabla_{\mu}],[\nabla_{(j)},\nabla_{\nu}]\right].$
(IV.8)

Actually, we obtain:

$\begin{split}F_{\star\mu\nu}=F_{\mu\nu}-\frac{1}{2!}\theta^{i_{1}j_{1}}\theta^{i_{2}j_{2}}[[\nabla_{i_{1}},F_{i_{2}\mu}],[\nabla_{j_{1}},F_{j_{2}\nu}]]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+\frac{1}{4!}\theta^{i_{1}i_{1}}\theta^{i_{2}j_{2}}\theta^{i_{3}j_{3}}\theta^{i_{4}j_{4}}[[\nabla_{i_{1}},[\nabla_{i_{2}},[\nabla_{i_{3}},F_{i_{4}\mu}]]],[\nabla_{j_{1}},[\nabla_{j_{2}},[\nabla_{j_{3}},F_{j_{4}\nu}]]]]+\cdots.\end{split}$
(IV.9)

In principle, since $F_{\star}$ contains no anti-commutator, it would be an
element of the Lie algebra of the structure group. In other words, if $G$ is
the structure group of $\mathbb{F}^{K}\to E\to X$ with connection $\nabla$,
then $F_{\star}\in\Omega^{2}(X)\otimes\mathfrak{g}$, for $\mathfrak{g}$ the
Lie algebra of $G$. Moreover, if $E$ is an associative vector bundle then;
$F_{\star ij}=F_{ij}=0$. But, however, there are more similarities between
$F_{\star\mu\nu}$ and $F_{\mu\nu}$. The following lemma proves that
$F_{\star\mu\nu}$ is a gauge covariant, and consequently a well-defined
tensor.

Lemma 2; _A change of local frame (i.e. a gauge transformation) in $E\to X$,
given as_

$\nabla=\text{d}+A~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla^{\prime}=U^{-1}\nabla
U=\text{d}+U^{-1}AU+U^{-1}\text{d}U$

_leads to: $F^{\prime}_{\star}=U^{-1}F_{\star}U$._

Proof; If $A$ and $B$ are two pseudo-differential operators, then:
$[U^{-1}AU,U^{-1}BU]=U^{-1}[A,B]U$. Therefore, since
$\nabla^{\prime}=U^{-1}\nabla U$ and $F^{\prime}_{\mu\nu}=U^{-1}F_{\mu\nu}U$,
by employing induction on consecutive terms of sequence (IV.8) (or (IV.9)), it
is seen that $F^{\prime}_{\star\mu\nu}=U^{-1}F_{\mu\nu}U$. Q.E.D.

The following lemma establishes one of the most important identities for
$F_{\star}$.

Lemma 3; _The noncommutative curvature $F_{\star}$ fulfills the Bianchi
identity, i.e.;_

$\nabla_{\mu}F_{\star\nu\sigma}+\nabla_{\nu}F_{\star\sigma\mu}+\nabla_{\sigma}F_{\star\mu\nu}=0,$
(IV.10)

_or; $[\hat{\nabla},F_{\star}]_{s}=0$, wherein
$\hat{\nabla}=\emph{d}x^{\mu}\wedge\nabla_{\mu}$, and $[,]_{s}$ is the
supercommutator of differential forms._

Proof; According to (IV.9) the noncommutative curvature $F_{\star}$ is
expressed as;

$F_{\star}=\sum_{(i),(j)}\Phi^{(i)(j)}[T_{(i)\mu},T_{(j)\nu}]~{}\text{d}x^{\mu}\wedge\text{d}x^{\nu},$
(IV.11)

wherein $(i)$ and $(j)$ are multi-indices of even degrees, $T_{(i)\mu}$ is the
commutator of $\nabla_{(i)}$ and $\nabla_{\mu}$, and $\Phi^{(i)(j)}$ is
proportional to $\theta^{(i)(j)}$, hence a symmetric tensor. On the other
hand, we have;

$\begin{split}[\hat{\nabla},[T_{(i)\mu},T_{(j)\nu}]]_{s}~{}\Phi^{(i)(j)}~{}\text{d}x^{\mu}\wedge\text{d}x^{\nu}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\left([[\nabla_{\sigma},T_{(i)\mu}],T_{(j)\nu}]+[[\nabla_{\sigma},T_{(j)\nu}],T_{(i)\mu}]\right)~{}\Phi^{(i)(j)}~{}\text{d}x^{\sigma}\wedge\text{d}x^{\mu}\wedge\text{d}x^{\nu},\end{split}$
(IV.12)

which vanishes identically, since it is a contraction of a symmetric and an
antisymmetric tensor. This proves the noncommutative Bianchi identity. Q.E.D.

In principle, $F_{\star}$ is the most significant differential form for semi-
conformally flat noncommutative manifolds. Actually, despite the naive
noncommutative formula of the curvature $F_{\star_{G-M}}$, given by the
standard form of Groenewold-Moyal product $\star_{G-M}$, the noncommutative
curvature $F_{\star}$ introduced by (IV.9) gives rise to a numbers of
interesting and well-defined differential geometric achievements. To see them,
first of all one should note that $F_{\star}$, produced by covariant star
product, in contrast to $F_{\star_{G-M}}$, is a well-defined smooth
differential form on $X$. This is the main reason of its gauge covariance. On
the other hand, this property helps us to create some well-defined cohomology
classes by means of $F_{\star}$. Actually, if $\mathbb{F}^{K}\to E\to X$ is a
vector bundle over $X$ with noncommutative curvature $F_{\star}$, and $P$ is a
symmetric invariant polynomial on the Lie algebra
$\mathfrak{gl}(K,\mathbb{F})=\mathbb{M}_{K\times K}(\mathbb{F})$, then
$P(F_{\star})$ is a well-defined differential form on $X$ which as we will
establish in below carries some significant pieces of geometric/topological
information about $X$.

Actually, $P(F_{\star})$ and its commutative version $P(F)$, provided by the
ordinary curvature $F$, have several common significant properties. However,
if $E$ is associative, then we can replace the matrix product in the
polynomial $P$ with $\star$ due to (II.5). Actually, we can see that this
noncommutative version of $P(F_{\star})$, denoted as $P_{\star}(F_{\star})$,
also produces a well-defined (i.e. gauge invariant) differential form. But,
however, $P_{\star}(F_{\star})$ is not closed in general, hence would not
define a de Rham cohomology class. Therefore, we may use $P(F_{\star})$ in
studying the noncommutative characteristic classes. The next theorem is in
fact, the noncommutative version of the celebrated Chern-Weil theorem.

Theorem 1; _Assume that $\mathbb{F}^{K}\to E\to X$ is an associative vector
bundle with connection $\nabla$ and $\star$ is its star product. Also, let $P$
be a symmetric invariant polynomial on $\mathbb{M}_{K\times K}(\mathbb{F})$.
Then, we have:_

a) _$P(F_{\star})$ is a closed differential form._

b) _The cohomology class of $P(F_{\star})$ is independent of the connection
$\nabla$._

c) _$P(F)$ and $P(F_{\star})$ are cohomologous, i.e.
$P(F)=P(F_{\star})+\emph{d}\lambda$ for some exact form $\emph{d}\lambda$._

Proof; a) Let $P$ be of degree $r$. Then, since $P$ is symmetric and invariant
it is seen that;

$\text{d}P_{\star}(F_{\star})=rP_{\star}(\text{d}F_{\star},F_{\star},\cdots,F_{\star})=rP_{\star}([\hat{\nabla},F_{\star}]_{s},F_{\star},\cdots,F_{\star})=0,$
(IV.13)

where we have added a vanishing commutator term in the second equality,
meanwhile use of the noncommutative Bianchi identity have been made.

b) In fact, according to the invariant theory any invariant polynomial of
$K\times K$ matrices with entries in a field of characteristic zero (e.g.
$\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$) is actually composed of trace
functions, hence given in terms of a polynomial of eigenvalues of matrices
rasmyslev ; procesi .131313It was firstly conjectured by Artin artin .
Therefore, it is enough to prove the claim only for trace function.

However, as we will show below each trace polynomial could be considered as a
supertrace function on some $\mathbb{Z}_{2}$-graded algebra or superalgebra.
As Quillen has been shown such supertraces will provide a beautiful framework
for producing many important characteristic classes within the formulation of
superconnections quillen . Hereon, we employ the spectacular machinery of
Quillen’s supper connection to prove the theorem.

Assume that $E\to X$ is a complex vector bundle, otherwise we can simply
replace $E$ by $E^{\mathbb{C}}$. Suppose that $\nabla^{t}$, $t\in[0,1]$, is a
continuous family of associative connections on $E$. Assume that $V$ is a
complex superalgebra generated by $1$ (even) and
$\\{\mathfrak{e}^{i}\\}_{i=1}^{2m}$ (odd), with
$\mathfrak{e}^{i}.\mathfrak{e}^{i}=i\theta^{ij}$, and let
$\mathfrak{T}=\oplus_{k=0}^{\infty}V^{k\otimes}$, $V^{0\otimes}=\mathbb{C}$,
be its tensor superalgebra with total degree. Now assume that $\mathfrak{S}$
is another superalgebra generated by $\mathbb{Z}_{2}=\\{1,\varepsilon\\}$,
where $\varepsilon$ is a nilpotent odd element $\varepsilon^{2}=1$. Hence,
$S=\mathfrak{S}\otimes\mathfrak{T}$ is a superalgebra by the induced degree.

Set, $T=X\times S$ and consider $\mathfrak{E}=E\otimes T$ as a super vector
bundle over $X$. Put a superconnection on $\mathfrak{E}$ as
$\nabla_{s}=\nabla^{t}+{\mathcal{A}}^{\theta}$, with

${\mathcal{A}}^{\theta}=\varepsilon
F^{t}_{i}{\mathfrak{e}}^{i}+\frac{1}{\sqrt{2!}}[\nabla^{t}_{i},F^{t}_{j}]{\mathfrak{e}}^{i}{\mathfrak{e}}^{j}+\frac{1}{\sqrt{3!}}\varepsilon[\nabla^{t}_{i},[\nabla^{t}_{j},F^{t}_{k}]]{\mathfrak{e}}^{i}\mathfrak{e}^{j}\mathfrak{e}^{k}+\cdots,$
(IV.14)

wherein $F^{t}_{i}=F^{t}_{i\mu}\text{d}x^{\mu}$ and $F^{t}$ is the curvature
tensor of $\nabla^{t}$. Here we omitted the tensor product for simplicity. For
example we use the short notation of
$\mathfrak{e}^{i_{1}}\mathfrak{e}^{i_{2}}\cdots\mathfrak{e}^{i_{r}}$ instead
of
$\mathfrak{e}^{i_{1}}\otimes\mathfrak{e}^{i_{2}}\otimes\cdots\otimes\mathfrak{e}^{i_{r}}$
in (IV.14). Obviously, $\nabla_{s}$ is of odd degree and hence a
superconnection. The curvature of $\nabla_{s}$ is given by;

$\begin{split}F_{s}=F^{t}-\frac{1}{2!}\theta^{i_{1}j_{1}}\theta^{i_{2}j_{2}}[[\nabla^{t}_{i_{2}},F^{t}_{i_{1}}],[\nabla^{t}_{j_{2}},F^{t}_{j_{1}}]]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+\frac{1}{4!}\theta^{i_{1}i_{1}}\theta^{i_{2}j_{2}}\theta^{i_{3}j_{3}}\theta^{i_{4}j_{4}}[[\nabla^{t}_{i_{1}},[\nabla^{t}_{i_{2}},[\nabla^{t}_{i_{3}},F^{t}_{i_{4}}]]],[\nabla^{t}_{j_{1}},[\nabla^{t}_{j_{2}},[\nabla^{t}_{j_{3}},F^{t}_{j_{4}\nu}]]]]+\cdots+\mathcal{O}(\mathfrak{e}^{i};\varepsilon),\\\
\end{split}$ (IV.15)

wherein $\mathcal{O}(\mathfrak{e}^{i};\varepsilon)$ includes some numbers of
$\mathfrak{e}_{i}$s or contains $\varepsilon$. Thus, upon (IV.8);

$F_{s}=F^{t}_{\star^{t}}+\mathcal{O}(\mathfrak{e}^{i};\varepsilon),$ (IV.16)

for $F^{t}_{\star^{t}}$ the noncommutative curvature of $\nabla^{t}$ and its
covariant star product $\star^{t}$. We obtain:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}tr_{s}(F_{s}^{k})=tr_{s}({F^{t}_{\star^{t}}}^{k})=tr({F^{t}_{\star^{t}}}^{k}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k\geq
1,$ (IV.17)

wherein $tr_{s}$ is the supertrace. Here, we used the relations
$tr_{s}(\varepsilon)=tr_{s}(\mathfrak{e}^{i})=0$ and $tr_{s}(A\otimes
B)=tr_{s}(A)tr_{s}(B)$. According to quillen  $tr({F^{t}_{\star^{t}}}^{k})$ is
a closed differential form on $X$, and its de Rham cohomology class is
independent of $t$. Hence, if we put
$\nabla^{t}=(1-t)\nabla^{\prime}+t\nabla^{\prime\prime}$, for associative
connections $\nabla^{\prime}$ and $\nabla^{\prime\prime}$, we conclude that;

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}tr(F^{\prime
k}_{\star^{\prime}})=tr(F^{\prime\prime
k}_{\star^{\prime\prime}})+\text{d}\lambda,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k\geq
1,$ (IV.18)

for $F^{\prime}_{\star^{\prime}}$ (resp.
$F^{\prime\prime}_{\star^{\prime\prime}}$) the noncommutative curvature of
$\nabla^{\prime}$ (resp. $\nabla^{\prime\prime}$) and its covariant star
product $\star^{\prime}$ (resp. $\star^{\prime\prime}$). This proves the claim
(b).

c) To prove the claim one should employ the above machinery within some
different formulation. Let $\nabla_{s}=\nabla+t{\mathcal{A}}^{\theta}$,
$t\in[0,1]$. Following the above machinery with this superconnection leads to:

$tr_{s}(F_{s}^{k})=tr(F^{k}),~{}~{}~{}~{}~{}(\text{for}~{}t=0),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}tr_{s}(F_{s}^{k})=tr(F_{\star}^{k}),~{}~{}~{}~{}~{}(\text{for}~{}t=1),$
(IV.19)

for each $k\geq 1$. Hence;

$tr(F^{k})=tr(F_{\star}^{k})+\text{d}\lambda.$ (IV.20)

Therefore, upon the invariant theory the de Rham cohomology classes provided
by symmetric invariant polynomials for $F_{\star}$ coincide entirely with
those obtained for $F$. This finishes the theorem.

Another Proof by Employing the Seiberg-Witten Map;

In this case we initially prove (c), then (b).

c) To establish this claim we use the elegant results of the Seiberg-Witten
map. Here, we may refer to its exact formula which has significant topological
features.141414See banerjee1 ; yang  for more details. Let
$\hat{\nabla}^{\theta}$ be the Seiberg-Witten connection with local connection
form $\hat{A}^{\theta}$ and the ordinary (commutative) curvature
$\hat{F}^{\theta}$, which the later coincides with the noncommutative
curvature $F_{\star}$ seiberg-witten . Hence, by the Chern-Weil theorem we
have: $P(F_{\star})=P(\hat{F}^{\theta})=P(F)+\text{d}\lambda$. This finishes
the claim (c).

b) Let $\nabla^{\prime}$ be another connection on $E$. We show the covariant
star product and the noncommutative curvature of $\nabla^{\prime}$
respectively with $\star^{\prime}$ and $F^{\prime}_{\star^{\prime}}$.
According to (c) one reads;

$P(F_{\star})-P(F)=\text{d}\lambda,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P(F^{\prime}_{\star^{\prime}})-P(F^{\prime})=\text{d}\lambda^{\prime}.$
(IV.21)

Moreover, upon the Chern-Weil theorem we have;
$P(F^{\prime})-P(F)=\text{d}\xi$. All in all, it is seen that:

$P(F^{\prime}_{\star^{\prime}})-P(F_{\star})=\text{d}\left(\lambda^{\prime}-\lambda+\xi\right).$
(IV.22)

This finishes the theorem. Q.E.D.

##  V Noncommutative Chern-Weil Theorem and Some Topological Information
Carried by the Noncommutative Curvature

Through this and the next sections all vector bundles are assumed to be
associative. Suppose that $\mathbb{C}^{K}\to E\to X$ is a complex vector
bundle and $F_{\star}$ is its noncommutative curvature. Then, the
_noncommutative total Chern class_ of $E$ is defined as;

$c_{\star}(E)={\det}\left(1+\frac{iF_{\star}}{2\pi}\right):=\oplus_{k=0}^{K}c_{\star
k}(E).$ (V.1)

Actually, $c_{\star k}(E)$, the so called _$k$ -th noncommutative Chern
class_, is defined by;

$c_{\star
k}(E_{\star})=\frac{1}{k!}\frac{\text{d}^{k}}{\text{d}s^{k}}|_{s=0}~{}{\det}\left(1+\frac{isF_{\star}}{2\pi}\right).$
(V.2)

In fact, upon to Theorem 1 we readily find:

Corollary 1; _For each complex vector bundle $\mathbb{C}^{K}\to E\to X$ the
$k$-th noncommutative Chern class $c_{\star k}(E_{\star})$, $0\leq k\leq 2n$,
is a topological invariant and cohmologous to $c_{k}(E)$ in
$H^{2k}_{\emph{dR}}(X,\mathbb{Z})$, i.e._

$\int_{X}c_{\star k}(E)=\int_{X}c_{k}(E)\in\mathbb{Z}.$ (V.3)

_Moreover, if $E_{i}$s, $1\leq i\leq N$, are complex vector bundles on $X$,
then:_

$c_{\star}(E_{1}\oplus\cdots\oplus E_{N})=c_{\star}(E_{1})\cdots
c_{\star}(E_{N}),$ (V.4)

_wherein $E_{1}\oplus\cdots\oplus E_{N}$ is the Whitney sum of vector bundles
$E_{i}$s milnor ; kobayashi ._

A similar conclusion holds for the Chern character. In fact, if
$\mathbb{F}^{K}\to E\to X$ is an associative vector bundle, then the _$k$ -th
noncommutative Chern character_ of $E$, $0\leq k\leq n$, is defined as:

$\text{ch}_{\star
k}(E)=\frac{1}{k!}\text{tr}\left(\frac{iF_{\star}}{2\pi}\right)^{k}.$ (V.5)

According to Theorem 1 once again we conclude:

Corollary 2; _For each vector bundle $\mathbb{F}^{K}\to E\to X$,
$\emph{ch}_{\star k}(E)$ is a topological invariant and belongs to the
integral cohomology class of $\emph{ch}_{k}(E)$ in
$H^{2k}_{\emph{dR}}(X,\mathbb{Z})$. Specially;_

$\int_{X}\text{ch}_{\star n}(E)=\int_{X}\text{ch}_{n}(E)\in\mathbb{Z}.$ (V.6)

_Moreover, if $\\{E_{i}\\}_{i=1}^{N}$ is a set of complex vector bundles on
$X$, then:_

$\begin{split}\text{ch}_{\star k}(E_{1}\oplus\cdots\oplus
E_{N})=\sum_{i=1}^{N}\text{ch}_{\star k}(E_{i})=\text{ch}_{\star
k}(E_{1})+\cdots+\text{ch}_{\star k}(E_{N}),~{}~{}\\\ \text{ch}_{\star
k}(E_{1}\otimes\cdots\otimes E_{N})=\sum\prod_{i=1}^{N}\text{ch}_{\star
k_{i}}(E_{1})=\text{ch}_{\star k_{1}}(E_{1})\cdots\text{ch}_{\star
k_{N}}(E_{N}),\end{split}$ (V.7)

_wherein in the second equality the summation is over all $N$-plets
$(k_{1},\cdots,k_{N})$ with $\sum_{i}k_{i}=k$._

To study more noncommutative characteristic classes we need some more
algebraic precisions. Let $\mathbb{F}^{K}\to E\to X$ be a vector bundle
equipped with connection $\nabla$ so that its curvature $F$ and the
noncommutative curvature $F_{\star}$ are both diagonalizable on $X$. For
example we can assume that $E$ is a complex vector bundle and its structure
group is $SU(K)$. Then, $F$ and $F_{\star}$ both are $\mathfrak{su}(K)$-valued
two forms and hence diagonalizable. Moreover, suppose that
$\mathfrak{F}(a_{1},\cdots,a_{K})$ is a symmetric polynomial (analytic
function) of $K$ variables $a_{i}$s. Let $\\{x_{\star i}\\}_{i=1}^{K}$ (resp.
$\\{x_{i}\\}_{i=1}^{K}$) be the collection of the eigenvalues of
$\frac{iF_{\star}}{2\pi}$ (resp. $\frac{iF}{2\pi}$). Therefore,
$\mathfrak{F}(x_{\star 1},\cdots,x_{\star K})$ (resp.
$\mathfrak{F}(x_{1},\cdots,x_{K})$) is simply decomposed to homogeneous
components by the degree of differential forms as;

$\begin{split}\mathfrak{F}(x_{\star 1},\cdots,x_{\star
K}):=\oplus_{k=0}^{n}\mathfrak{F}_{k}(F_{\star})=\oplus_{k=0}^{n}\mathfrak{F}_{k}(x_{\star
1},\cdots,x_{\star K}),\\\
\mathfrak{F}(x_{1},\cdots,x_{K}):=\oplus_{k=0}^{n}\mathfrak{F}_{k}(F)=\oplus_{k=0}^{n}\mathfrak{F}_{k}(x_{1},\cdots,x_{K}),~{}~{}~{}~{}~{}\end{split}$
(V.8)

with
$\mathfrak{F}_{k}(a_{1},\cdots,a_{K})=\frac{1}{k!}\frac{\text{d}^{k}}{\text{d}s^{k}}|_{s=0}~{}\mathfrak{F}(sa_{1},\cdots,sa_{K})$.
Therefore, since $\mathfrak{F}$ is in fact a symmetric invariant polynomial on
$\mathbb{M}_{K\times K}(\mathbb{F})$, then the noncommutative Chern-Weil
theorem would be translated to the corresponding formulation based on the
eigenvalues of $F_{\star}$.151515See kobayashi  for more discussion. For
example, for the noncommutative setting we have;

$c_{\star}(E)=\prod_{i=1}^{K}(1+x_{\star
i})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{ch}_{\star}(E)=\oplus_{k=0}^{n}\sum_{i=1}^{K}\frac{x_{\star
i}^{k}}{k!}=\sum_{i=1}^{K}e^{x_{\star i}}.$

The following corollary provides a noncommutative formulation of Chern-Weil
theorem given in terms of eigenvalues of $F_{\star}$.

Corollary 3; _For each vector bundle $\mathbb{F}^{K}\to E\to X$, and for any
symmetric polynomial $\mathfrak{F}(a_{1},\cdots,a_{K})$, the homogeneous
polynomials $\mathfrak{F}_{k}(F)$ and $\mathfrak{F}_{k}(F_{\star})$, $0\leq
k\leq n$, defined by (V.8), are closed and belong to the same de Rham
cohomology class of $H^{2k}_{\emph{dR}}(X,\mathbb{R})$. In fact,
$\mathfrak{F}_{k}(F_{\star})$ is a topological invariant and its cohomology
class is independent of the connection of $E$._

Now, by employing Corollary 3 we can provide the noncommutative versions of
more characteristic classes. For instance, the _noncommutative total Todd
class_ is defined as;

$\text{Td}_{\star}(E)=\frac{1}{K!}\sum_{\sigma\in
S_{K}}\left(\frac{x_{\star\sigma(1)}}{1-e^{-x_{\star\sigma(1)}}}\right)\cdots\left(\frac{x_{\star\sigma(K)}}{1-e^{-x_{\star\sigma(K)}}}\right),$
(V.9)

wherein $S_{K}$ is the permutation group of $\\{1,\cdots,K\\}$. In principle,
the noncommutative Todd class is decomposed as;
$\text{Td}_{\star}(E)=\oplus_{k=0}^{n}\text{Td}_{\star k}(E)$, wherein
$\text{Td}_{\star k}(E)$ is referred to as the _$k$ -th noncommutative Todd
class_ of $E$.

Corollary 4; _For each vector bundle $\mathbb{F}^{K}\to E\to X$ the $k$-th
noncommutative Todd class of $E$, $\text{Td}_{\star k}(E)$, $0\leq k\leq n$,
is a topological invariant and belongs to the rational cohomology class of
$\text{Td}_{k}(E)$ in $H^{2k}_{\text{dR}}(X,\mathbb{Q})$ hirz-book . On the
other hand, if $\\{E_{i}\\}_{i=1}^{N}$ is a set of vector bundles over $X$,
then;_

$\text{Td}_{\star}(E_{1}\oplus\cdots\oplus
E_{N})=\prod_{i=1}^{N}\text{Td}_{\star}(E_{i})=\text{Td}_{\star}(E_{1})\cdots\text{Td}_{\star}(E_{N}).$
(V.10)

_Moreover, if $X$ is a complex manifold, then the arithmetic genus of $X$ is
given as;_

$\sum_{k=0}^{n}(-1)^{k}b^{0,k}=\int_{X}\text{Td}_{\star n}(TX^{+}),$ (V.11)

_wherein $b^{0,k}=\dim_{\mathbb{C}}H^{0,k}(X,\mathbb{C})$, $0\leq k\leq n$,
are the Hodge numbers, and $TX^{+}$ is the holomorphic tangent bundle of $X$._

Similarly, the _noncommutative Hirzebruch $L$-polynomial_ is defined as;

$L_{\star}(E)=\frac{1}{K!}\sum_{\sigma\in
S_{K}}\left(\frac{x_{\star\sigma(1)}}{\tanh({x_{\star\sigma(1)})}}\right)\cdots\left(\frac{x_{\star\sigma(K)}}{\tanh({x_{\star\sigma(K)})}}\right):=\oplus_{k=0}^{[n/2]}L_{\star
k}(E).$ (V.12)

As a convention $L_{\star k}(E)$ is referred to as the _$k$ -th noncommutative
Hirzebruch $L$-genus_, while the highest term of (V.12), is called the _top
noncommutative Hirzebruch $L$-genus_. It can be seen that $L_{\star k}(E)$ is
a $4k$-form with $0\leq k\leq n/2$. Hence, to obtain a volume form from the
Hirzebruch polynomial $n$ must be an even number, say $n=2p$.

Corollary 5; _For each vector bundle $\mathbb{F}^{K}\to E\to X$, the $k$-th
noncommutative Hirzebruch genus $L_{\star k}(E)$, $0\leq k\leq n/2$, is a
topological invariant and cohomologous to $L_{k}(E)$ in
$H_{dR}^{4k}(X,\mathbb{Z})$. Specially, if $n=2p$, then the top noncommutative
Hirzebruch $L$-genus $L_{\star p}(E)$ is a topological invariant and_

$\int_{X}L_{\star p}(E)=\int_{X}L_{p}(E)\in\mathbb{Z}.$ (V.13)

_On the other hand, if $X$ is a 4-manifold and $E=TX$, then;_

$\tau(X)=\int_{X}L_{\star 1}(TX)\in\mathbb{Z},$ (V.14)

_wherein $\tau(X)$ is the Hirzebruch signature. Moreover, if $X$ is also a
spin manifold, then by Rochlin’s theorem rokhlin  the noncommutative $L$ genus
$L_{\star 1}(TX)$ (i.e. the result of (V.14)) is divided by 16 freedman ._

We emphasize that upon Lemma 1 the Hirzebruch signature of $X$ as a
$2n$-dimensional manifold could be calculated in the $\star$ formulation of
pairing
$\left\langle{.}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{.}\right\rangle_{\star}:H^{n}_{\text{dR}}(X,\mathbb{R})\otimes
H^{n}_{\text{dR}}(X,\mathbb{R})\to\mathbb{R}$, as;

$\left[\omega_{1}\right]\otimes[\omega_{2}]\mapsto\left\langle{\omega_{1}}\mathrel{\left|{\vphantom{\omega_{1}\omega_{2}}}\right.\kern-1.2pt}{\omega_{2}}\right\rangle_{\star}:=\int_{X}\omega_{1}\star\omega_{2}=\int_{X}\omega_{1}\wedge\omega_{2}=\left\langle{\omega_{1}}\mathrel{\left|{\vphantom{\omega_{1}\omega_{2}}}\right.\kern-1.2pt}{\omega_{2}}\right\rangle.$
(V.15)

In this sense the noncommutative Hirzebruch signature $\tau_{\star}(X)$
coincides with the ordinary Hirzebruch signature $\tau(X)$, and hence, (V.14)
could be interpreted as a significant evidence of a noncommutative index
theorem on noncommutative manifolds.

The _noncommutative Dirac genus_ (or the _noncommutative $\hat{A}$ genus_) is
also defined with;

$\hat{A}_{\star}(E)=\oplus_{k=0}^{[n/2]}\hat{A}_{\star
k}(E)=\frac{1}{K!}\sum_{\sigma\in
S_{K}}\left(\frac{x_{\star\sigma(1)}/2}{\sinh({x_{\star\sigma(1)}/2)}}\right)\cdots\left(\frac{x_{\star\sigma(K)}/2}{\sinh({x_{\star\sigma(K)}/2)}}\right).$
(V.16)

Each $\hat{A}_{\star k}(E)$ is referred to as the _$k$ -th noncommutative
Dirac genus_. Similar to the Hirzebruch polynomial, it can be seen that the
noncommutative Dirac genus also consists of $4k$-forms for $0\leq k\leq n/2$,
i.e. $\hat{A}_{\star k}(E)$. Hence, once again, to obtain a topological
invariant volume element, one has to assume that $n=2p$ for some integer $p$.
Then, $\hat{A}_{\star p}$, the _top noncommutative Dirac genus_ exists and
could be integrated over $X$.

Corollary 6; _For each vector bundle $\mathbb{F}^{K}\to E\to X$, the $k$-th
noncommutative Dirac genus $\hat{A}_{\star k}(E)$ is a topological invariant
and belongs to the integral cohomology class of $\hat{A}_{k}(E)$ in
$H^{4k}_{\text{dR}}(X,\mathbb{Q})$. Specially, if $n=2p$, then the top
noncommutative Dirac genus of $E$ is a topological invariant and_

$\int_{X}\hat{A}_{\star p}(E)=\int_{X}\hat{A}_{p}(E)\in\mathbb{Q}.$ (V.17)

_Moreover, according to the Borel-Hirzebruch theorembh ; bh2  if $X$ is a spin
manifold $A_{\star k}$ belongs to integral cohomology group
$H^{4k}_{\text{dR}}(X,\mathbb{Z})$, and if $p$ is odd;_

$\int_{X}\hat{A}_{\star p}(TX)\in 2\mathbb{Z},$ (V.18)

_while if the above result is non-zero, then due to the Lichnerowicz
obstruction theoremlic , $X$ would not admit a metric $g_{\mu\nu}$ with
positive scalar curvature._

If in addition the vector bundle $\mathbb{R}^{K}\to E\to X$ is orientable and
endowed with a smooth metric on the fibers, say $\mathfrak{g}_{pq}$, and a
metric compatible connection $\nabla$ (i.e. $\nabla\mathfrak{g}_{pq}=0$), then
the _noncommutative total Pontrjagin class_ is defined by;

$p_{\star}(E)={\det}_{\star}\left(1+\frac{F_{\star}}{2\pi}\right):=\oplus_{k=0}^{n}p_{\star
k/2}(E),$ (V.19)

wherein $p_{\star k/2}(E_{\star})$, the so called _$\frac{k}{2}$ -th
noncommutative Pontrjagin class_ is a $2k$-form. Also, similar to the Chern
class, the noncommutative Pontrjagin class would not provide a volume element
unless $K\geq n$. Hence, we may simply assume that $K\geq n$. One should note
that, just similar to the commutative case, the half integer noncommutative
Pontrjagin classes vanish identically; $p_{\star k/2}(E)=p_{k/2}(E)=0$ for
$k=1$ mod $2$. Specially, the _top noncommutative Pontrjagin class_ $p_{\star
n/2}(E_{\star})$ vanishes for odd $n$. In fact, for real vector bundle
$\mathbb{R}^{K}\to E\to X$ and the structure group $SO(K)$, both $F$ and
$F_{\star}$, are $\mathfrak{so}(K)$-valued two forms, hence traceless. For
instance if $K=2k$ is even, then we can block-diagonalize $\frac{F}{2\pi}$ and
$\frac{F_{\star}}{2\pi}$ as:

$F=\left[{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}0&x_{1}\\\
{-x_{1}}&0\\\ \end{array}}&{}\hfil&{}\hfil\\\ {}\hfil&\ddots&{}\hfil\\\
{}\hfil&{}\hfil&{\begin{array}[]{*{20}{c}}0&x_{k}\\\ {-x_{k}}&0\\\
\end{array}}\\\
\end{array}}\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{\star}=\left[{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}0&x_{\star
1}\\\ {-x_{\star 1}}&0\\\ \end{array}}&{}\hfil&{}\hfil\\\
{}\hfil&\ddots&{}\hfil\\\ {}\hfil&{}\hfil&{\begin{array}[]{*{20}{c}}0&x_{\star
k}\\\ {-x_{\star k}}&0\\\ \end{array}}\\\ \end{array}}\right].$ (V.20)

In fact, the generating function of $p_{\star}(E)$ is given by;

$p_{\star}(E)=\prod_{i=1}^{[K/2]}(1+x_{\star
i}^{2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{\star
k}=\sum x_{\star i_{1}}^{2}x_{\star i_{2}}^{2}\cdots x_{\star i_{k}}^{2},$
(V.21)

wherein the summation in the second equality is over all ${1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq[K/2]}$ and $x_{\star i}$s are two-form
eigenvalues of $\frac{F_{\star}}{2\pi}$ introduced in (V.20).

Corollary 7; _For each Riemannian real vector bundle $\mathbb{R}^{K}\to E\to
X$, the $\frac{k}{2}$-th noncommutative Pontrjagin class $p_{\star k/2}(E)$ is
a topological invariant and is cohomologous to $p_{k/2}(E)$ in the integral
cohomology group of $H^{2k}_{\text{dR}}(X,\mathbb{Z})$. Specially, if $n=2p$
and $K\geq n$, then the top noncommutative Pontrjagin class $p_{\star p}(E)$
is cohomologous to $p_{p}(E)$ in $H^{2n}_{\text{dR}}(X,\mathbb{Z})$ and_

$\int_{X}p_{\star n/2}(E_{\star})=\int_{X}p_{n/2}(E)\in\mathbb{Z}.$ (V.22)

_For this situation $p_{\star p}(E)$, $(-1)^{p}c_{\star n}(E^{\mathbb{C}})$
and $(-1)^{p}c_{n}(E^{\mathbb{C}})$ all are cohomologous, wherein
$E^{\mathbb{C}}$ is the complexification of $E$ milnor . Moreover, if $K=2l$
or $2l+1$, then for each set of integers $\\{a_{i}\\}_{i=1}^{l}$ with weighted
sum $a_{1}+2a_{2}+\cdots+ka_{k}=p$, the integral of_

$\int_{X}p_{\star 1}^{a_{1}}(E)p_{\star 2}^{a_{2}}(E)\cdots p_{\star
l}^{a_{l}}(E)$ (V.23)

_coincides with the Pontrjagin number of $E$ for $l$-plet
$(a_{1},\cdots,a_{l})$ tu ._

The _noncommutative Euler class_ is similarly defined for the real vector
bundle $\mathbb{R}^{K}\to E\to X$ as;

$e_{\star}(E)=\text{Pf}\left(\frac{F_{\star}}{2\pi}\right)=\frac{(-1)^{K}}{2^{K}K!}\epsilon^{a_{1}\cdots
a_{2K}}\left(\frac{F_{\star}}{2\pi}\right)_{a_{1}a_{2}}\cdots\left(\frac{F_{\star}}{2\pi}\right)_{a_{2K-1}a_{2K}},$
(V.24)

wherein $a_{i}$s are the indices of
$\frac{F_{\star}}{2\pi}\in\mathbb{M}_{K\times K}(\mathbb{R})$. Obviously,
$e_{\star^{\prime}}(E_{\star})$ would be a volume element only for $n=K$. For
instance $e_{\star}(TX)=x_{\star 1}\cdots x_{\star n}$.

Corollary 8; _Assume that $\mathbb{R}^{K}\to E\to X$ is a real vector bundle.
Then, the noncommutative Euler class $e_{\star}(E)$ is a topological invariant
and belongs to the integral cohomology class of $e(E)$ in
$H^{2K}_{\emph{dR}}(X,\mathbb{Z})$. Specially, if $K=n$, then;_

$\int_{X}e_{\star}(E)=\int_{X}e(E)\in\mathbb{Z}.$ (V.25)

##  VI Associative Vector Bundles, Noncommutative Elliptic Operators and
Noncommutative Topological Index Theorem

Following the statements of the last section leads us naturally to obtain a
noncommutative version of the celebrated Atiyah-Singer index theorem.161616As
mentioned before, through this section all vector bundles are assumed to be
associative. For instance, the Corollary 8 naturally provides the
noncommutative Gauss-Bonnet theorem:

Corollary 9; _Assume that $e_{\star}(TX)$ is the noncommutative Euler class of
the tangent bundle $TX$. Then;_

$\int_{X}e_{\star}(TX)=\chi(X)\in\mathbb{Z},$ (VI.1)

_wherein $\chi(X)$ is the Euler characteristic of $X$. Moreover, if
$\mathcal{X}$ is a smooth vector field on $X$ with finitely many zeros and
index sum $j(\mathcal{X})$, then curv-conn ;_

$j(\mathcal{X})=\int_{X}e_{\star}(TX_{\star}).$ (VI.2)

Actually, since the star product $\star$ never touches the CW-complex
structure of $X$, or the differential setting of the de Rham cohomology
classes,171717It appears only as a deformation of the cup product of the de
Rham cohomology classes. we see that the noncommutative Euler character
$\chi_{\star}(X)$, obtained by pairing $e_{\star}(TX)$ and the fundamental
homology class of $X$, coincides with the ordinary Euler character $\chi(X)$
via;

$\chi_{\star}(X)=\chi(X)=\sum_{k=0}^{2n}(-1)^{k}\\#(\text{\emph{k}-dimensional
cells})=\sum_{k=0}^{2n}(-1)^{k}\dim H^{k}_{\text{dR}}(X,\mathbb{R}),$

hence, Corollary 9 could be understood as important evidence for existing a
pure formulation of noncommutative topological index theorem.181818Actually,
what we are looking for is a topological noncommutative index theorem which,
of course, needs its own construction independent of the algebraic
noncommutative index theorem introduced in noncommutative geometry connes .
See the viewpoints of singer ; tellman  for more discussions. Moreover, as we
stated above, the noncommutative Hirzebruch signature theorem, reported in
Corollary 5, provides another clue for such an existence. To work out a
noncommutative topological index theorem we need some more preliminaries.

Assume that $\mathbb{C}^{K}\to E\to X$ is a complex vector bundle endowed with
a Hermitian inner product. This inner product will induce an inner product on
the vector space of smooth sections of $E$, i.e. $\Gamma(E)$. Let us show it
with
$\left\langle{.}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{.}\right\rangle$;

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta^{\prime}}\right\rangle=\int_{X}*\left(\overline{\zeta_{p}}\zeta^{\prime
p}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta,\zeta^{\prime}\in\Gamma(E).$
(VI.3)

This, will put a noncommutative structure on $E$, say
$\left\langle{.}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{.}\right\rangle_{\star}$,
by employing (III.3);

$\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta^{\prime}}\right\rangle_{\star}:=\int_{X}*\left(\overline{\zeta_{p}}\star\zeta^{\prime
p}\right)=\int_{X}*\left(\overline{\zeta_{p}}\zeta^{\prime
p}\right)=\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta^{\prime}}\right\rangle,$
(VI.4)

hence,
$\left\langle{.}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{.}\right\rangle_{\star}=\left\langle{.}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{.}\right\rangle$.
Thus, the ordinary and the noncommutative norms coincide, i.e.;

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|\zeta|_{\star}=\sqrt{\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta}\right\rangle_{\star}}=\sqrt{\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta}\right\rangle}=|\zeta|,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta\in\Gamma(E).$
(VI.5)

Now, suppose that $\mathbb{C}^{K}\to E\to X$ and $\mathbb{C}^{K^{\prime}}\to
E^{\prime}\to X$ are two complex vector bundles over $X$ and
$D:\Gamma(E)\to\Gamma(E^{\prime})$ be a linear pseudo-differential operator
with noncommutative version of $D_{\star}$, defined with (IV.4). The
noncommutative norm of $D$ is defined as the norm of $D_{\star}$ i.e.;

$|D_{\star}|=\text{Sup}_{\zeta\in S(E)}|D_{\star}\zeta|=\text{Sup}_{\zeta\in
S(E)}|D_{\star}\zeta|,$ (VI.6)

wherein $S(E)=\\{\zeta\in\Gamma(E)||\zeta|=1\\}$ is the unit sphere of
$\Gamma(E)$. The following lemma shows that $D_{\star}$ inherits the most
significant properties of $D$.

Lemma 4; _The linear pseudo-Differential operator
$D_{\star}:\Gamma(E)\to\Gamma(E^{\prime})$ is a bounded linear map, if and
only if $D$ is bounded too. Moreover, $|D_{\star}\zeta|=|D\zeta|$ for each
$\zeta\in\Gamma(E)$, hence; $|D_{\star}|=|D|$._

Proof; Initially, we note that upon to (VI.4) the noncommutative setting is
furnished by the Cauchy-Schwarz inequality;

$|\left\langle{\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{\zeta^{\prime}}\right\rangle_{\star}|\leq|\zeta||\zeta^{\prime}|.$
(VI.7)

Let $\zeta\in\Gamma(E)$ be an arbitrary smooth section. We have;

$|D_{\star}\zeta|^{2}=\sqrt{\left\langle{D_{\star}\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{D_{\star}\zeta}\right\rangle}=\sqrt{\left\langle{D_{\star}\zeta}\mathrel{\left|{\vphantom{..}}\right.\kern-1.2pt}{D\zeta}\right\rangle}_{\star}\leq|D_{\star}\zeta||D\zeta|,$
(VI.8)

wherein in the second equality use Lemma 1 has been made. Hence, we obtain;

$|D_{\star}\zeta|\leq|D\zeta|.$ (VI.9)

Obviously, (VI.9) could be proved for $\star^{-1}$ which is defined by
$-\theta^{ij}$. Hence, if we replace $D$ and $\star$ respectively by
$D_{\star}$ and $\star^{-1}$, we would find:

$|D\zeta|\leq|D_{\star}\zeta|.$ (VI.10)

Consequently, upon (VI.9) and (VI.10) for each $\zeta\in\Gamma(E)$ we have:

$|D_{\star}\zeta|=|D\zeta|,$ (VI.11)

which yields: $|D_{\star}|=|D|$. This finishes the lemma. Q.E.D.

Now, we are ready to work out our main objective. The following theorem is the
noncommutative version of the celebrated Atiyah-Singer index theorem.

Theorem 2; _Let $\mathbb{F}^{K}\to E\to X$ and $\mathbb{F}^{K}\to
E^{\prime}\to X$, be two smooth vector bundles over $X$ and
$D:\Gamma(E)\to\Gamma(E^{\prime})$ be an elliptic pseudo-differential
operator. Then, $D_{\star}$ is also elliptic and;_

$\text{Index}(D_{\star})=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(E)-\text{ch}_{\star}(E)\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}},$
(VI.12)

_wherein;_

$\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}=\oplus_{k=0}^{n}\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)_{k},$

_is the noncommutative total Todd class modulo the noncommutative Euler class,
as is defined in the ordinary differential geometryatiyah68-2 ._

Proof; First of all we have to prove the following two statements:

a) _If $D$ is elliptic, then
$D_{\star}:\Gamma(E_{\star})\to\Gamma(E^{\prime}_{\star})$ is also an elliptic
operator._

b) _If $D:\Gamma(E)\to\Gamma(E^{\prime})$ is elliptic, then we have;_

$\text{Index}(D_{\star})=\text{Index}(D).$ (VI.13)

We prove the above claims simultaneously. According to Lemma 4 we find:

$\begin{split}Ker(D_{\star})=\\{\zeta\in\Gamma(E)|D_{\star}\zeta=0\\}=\\{\zeta\in\Gamma(E)||D_{\star}\zeta|=0\\}\\\
=\\{\zeta\in\Gamma(E)||D\zeta|=0\\}=\\{\zeta\in\Gamma(E)|D\zeta=0\\}=Ker(D).\end{split}$
(VI.14)

On the other hand, due to (III.3) we readily obtain:
${D_{\star}}^{\dagger}=\left(D^{\dagger}\right)_{\star}$. Therefore, similar
to the reasoning of (VI.14) we can prove:

$Ker({D_{\star}}^{\dagger})=Ker(D^{\dagger}).$ (VI.15)

Thus;

$\dim Ker(D_{\star})=\dim Ker(D)<\infty,~{}~{}~{}~{}~{}~{}~{}\dim
Ker({D_{\star}}^{\dagger})=\dim Ker(D^{\dagger})<\infty.$ (VI.16)

Now we establish that $ImD_{\star}$ is closed. Assume that
$\\{\xi_{i}\\}_{i=1}^{\infty}$ is a convergent sequence in $ImD_{\star}$.
Therefore, there exists a sequence in $\Gamma(E)$, say
$\\{\zeta_{i}\\}_{i=1}^{\infty}$, so that $D_{\star}\zeta_{i}=\xi_{i}$. On the
other hand, since $|D(\zeta_{m}-\zeta_{n})|=|D_{\star}(\zeta_{m}-\zeta_{n})|$,
then the image of $\\{\zeta_{i}\\}$ under the action of $D$ is a Cauchy
sequence, and convergent to some point in $ImD$, say $D\zeta$. We claim that
$\xi_{i}\to D_{\star}\zeta$, hence $ImD_{\star}$ is closed. In fact, because
of the identity

$|D_{\star}(\zeta-\zeta_{i})|=|D(\zeta-\zeta_{i})|$

the Cauchy sequence $D_{\star}\zeta_{i}$ must converge to $D_{\star}\zeta$,
which proves the closedness of $ImD_{\star}$. Actually, the same reasoning
holds for ${D_{\star}}^{\dagger}$, hence $ImD_{\star}$ is closed too. Thus,
according to (VI.16) and the closedness of $ImD_{\star}$ and
$Im{D_{\star}}^{\dagger}$ we conclude that
$D_{\star}\in\text{Hom}(\Gamma(E),\Gamma(E^{\prime}))$ is a Fredholm operator,
hence elliptic gilkey . Moreover, from (VI.14) and (VI.15) we read:

$\text{Index}(D_{\star})=\text{Index}(D),$

which proves (a) and (b).

In principle, one can demonstrate the identity (VI.13) by considering the
homotopy invariance of index of elliptic operators. Through this way we do not
refer to (VI.14) and (VI.15), but instead we only insist on the ellipticity of
$D_{\star}\in\text{Hom}(\Gamma(E),\Gamma(E^{\prime}))$. Let us define a
homotopy of operators in $\text{Hom}(\Gamma(E),\Gamma(E^{\prime}))$ with
replacing $\theta^{ij}\to s\theta^{ij}$ and $\star\to\star_{s}$ for
$s\in[0,1]$. Then, since the ellipticity does not depend on the values of
$\theta^{ij}$ we conclude that
$D_{\star_{s}}\in\text{Fred}(\Gamma(E),\Gamma(E^{\prime}))$ for all
$s\in[0.1]$. Therefore,
$\gamma(s):=D_{\star_{s}}:[0,1]\to\text{Fred}(\Gamma(E_{\star}),\Gamma(E^{\prime}_{\star}))$
is a smooth curve which initiates at $D$ and terminates in $D_{\star}$. Since,
the index of Fredholm operators is invariant via homotopic relations gilkey ,
then (VI.13) is obtained.

Now we are ready to prove (VI.12). Actually, according to (VI.13) and Theorem
1, and upon the Atiyah-Singer index theorem atiyah1 ; atiyah68-2  we readily
obtain the following equalities;

$\begin{gathered}\text{Index}(D_{\star})=\text{Index}(D)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=(-1)^{n}\int_{X}\left(\text{ch}(E)-\text{ch}(E^{\prime})\right)\left(\frac{\text{Td}(TX^{\mathbb{C}})}{e(TX)}\right)\mid_{\text{vol}}~{}~{}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(E)-\text{ch}_{\star}(E^{\prime})\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}}.~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(VI.17)

Hence, the theorem follows. Q.E.D.

Actually, the noncommutative topological index theorem must be generalized to
elliptic complexes on $X$. In principle, if

$\cdots\xrightarrow[\text{~{}}]{D_{i-2}}\Gamma(E_{i-1})\xrightarrow[\text{~{}}]{D_{i-1}}\Gamma(E_{i})\xrightarrow[\text{~{}}]{D_{i}}\Gamma(E_{i+1})\xrightarrow[\text{~{}}]{D_{i+1}}\cdots$
(VI.18)

is an elliptic complex, then its noncommutative version is defined as;

$\cdots\xrightarrow[\text{~{}}]{D_{i-2\star}}\Gamma(E_{i-1})\xrightarrow[\text{~{}}]{D_{i-1\star}}\Gamma(E_{i})\xrightarrow[\text{~{}}]{D_{i\star}}\Gamma(E_{i+1})\xrightarrow[\text{~{}}]{D_{i+1\star}}\cdots.$
(VI.19)

Actually, according to Lemma 4 and the proof of Theorem 1 we can see that;
$ImD=ImD_{\star}$. Thus, for each $\zeta\in\Gamma(E_{i})$ there exists some
$\zeta^{\prime}\in\Gamma(E_{i})$ so that;
$D_{i}\zeta^{\prime}=D_{i\star}\zeta$. Moreover, upon Lemma 4 we know that
$\xi\in KerD_{i+1}$ if and only if it belongs to $KerD_{i+1\star}$. All in
all, we conclude:

$\left(D_{i+1\star}\circ
D_{i\star}\right)\zeta=D_{i+1\star}(D_{i}\zeta^{\prime})=0,$ (VI.20)

for each $\zeta\in\Gamma(E_{i})$. Therefore, if (VI.18) is an elliptic
complex, then (VI.19) is elliptic too. Altogether, we obtain:

$H^{i}_{\star}(E,D)=KerD_{i\star}/ImD_{i-1\star}=KerD_{i}/ImD_{i-1}=H^{i}(E,D)$
(VI.21)

Consequently;

$\text{Index}(E_{i},D_{i\star})=\sum_{i}(-1)^{i}\dim
H^{i}_{\star}(E,D)=\sum_{i}(-1)^{i}\dim H^{i}(E,D)=\text{Index}(E_{i},D_{i}),$
(VI.22)

which is an equivalent formulation of the noncommutative index theorem for
elliptic complexes. For instance, the noncommutative version of the de Rham
complex

$\cdots\xrightarrow[\text{~{}}]{d_{i-2\star}}\Omega^{i-1}(X)\xrightarrow[\text{~{}}]{d_{i-1\star}}\Omega^{i}(X)\xrightarrow[\text{~{}}]{d_{i\star}}\Omega^{i+1}(X)\xrightarrow[\text{~{}}]{d_{i+1\star}}\cdots$
(VI.23)

fulfills the nilpotency condition $d_{i\star}\star d_{i-1\star}=0$, and
$H^{i}_{dR\star}(X,\mathbb{F})=H^{i}_{dR}(X,\mathbb{F})$. Thus, according to
the above statements we read:

$\text{Index}(\Omega^{i}(X),\text{d}_{i\star})=\sum_{i}(-1)^{i}\dim
H^{i}_{dR\star}(X,\mathbb{F})=\sum_{i}(-1)^{i}\dim
H^{i}_{dR}(X,\mathbb{F})=\text{Index}(\Omega^{i}(X),\text{d}_{i}),$ (VI.24)

which coincides with $\chi(X)$ in agreement with Corollary 9. However, with
employing the rolling up trick to sum the even and the odd bundles, once again
we are encountered with an elliptic operator $D$ and its noncommutative
version $D_{\star}$, i.e.;

$0\hookrightarrow\Gamma(E^{\text{even}}_{\star})\xrightarrow[\text{~{}}]{D_{\star}}\Gamma(E^{\text{odd}}_{\star})\xrightarrow[\text{~{}}]{0}0,$
(VI.25)

wherein $E^{\text{even}}=\oplus_{i}E_{2i}$,
$E^{\text{odd}}=\oplus_{i}E_{2i+1}$, and
$D=\oplus_{i}\left(D_{2i}+D^{\dagger}_{2i-1}\right)$. For this rolling up
bundle we can always use the noncommutative topological index theorem via the
following equality;

$\text{Index}(D_{\star})=\text{Index}(D)=\text{Index}(E_{i},D_{i}).$ (VI.26)

Hence, we have the following corollary.

Corollary 10; _The index of the noncommutative version of the de Rham operator
$\text{d}_{X}$, considered for the rolled up vector bundles
$\wedge^{\text{even}}T^{*}X$ and $\wedge^{\text{odd}}T^{*}X$ is given by;_

$\begin{gathered}\text{Index}(\Omega^{i}(X),\text{d}_{i\star})\\\
=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(\wedge^{\text{even}}T^{*}X)-\text{ch}_{\star}(\wedge^{\text{odd}}T^{*}X)\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}}=\int_{X}e_{\star}(TX).\end{gathered}$
(VI.27)

_In addition, according to Corollary 9 this index coincides with the Euler
character of $X$, i.e.;_

$\text{Index}(\text{d}_{\star})=\int_{X}e_{\star}(TX)=\chi(X)=\sum_{k}(-1)^{k}b^{k},$
(VI.28)

_wherein $b^{k}=\dim H^{k}_{\text{dR}}(X,\mathbb{R})$, $0\leq k\leq 2n$, are
the Betti numbers._

Now let us have a short review on the preliminaries of the Hirzebruch
signature theorem and fix the notations due. Let
$\Omega^{*}(X)=\oplus_{k=0}^{2n}\Omega^{k}(X)$, be the graded algebra of
differential forms on $X$, and define $\eta:\Omega^{*}(X)\to\Omega^{*}(X)$
with $\eta\mid_{\Omega^{k}(X)}:\Omega^{k}(X)\to\Omega^{2n-k}(X)$, as;
$\eta=i^{k(k-1)+n}~{}\textbf{*}$, which fulfills $\eta^{2}=1$ and
$[D,\eta]=0$, for $D=\text{d}+\text{d}^{\dagger}$. The operator $\eta$ has two
eigenvalues $\pm 1$, which the corresponding eigenspaces are respectively
denoted by $\Omega^{+}(X)$ and $\Omega^{-}(X)$. In fact, $D^{+}$, the
restriction of $D$ to $\Omega^{+}(X)$, is an elliptic operator from the rolled
up bundle $\Omega^{+}(X)$ to $\Omega^{-}(X)$. Thus, the noncommutative
operator $D^{+}_{\star}:\Omega^{+}(X)\to\Omega^{-}(X)$ is an elliptic operator
too and $\text{Index}(D^{+}_{\star})=\text{Index}(D^{+})$. The next corollary
is the noncommutative version of the Hirzebruch signature theorem.

Corollary 11; _The index of $D^{+}_{\star}$ is given with the following
formula;_

$\begin{gathered}\text{Index}(D^{+}_{\star})\\\
=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(\wedge^{+}T^{*}X)-\text{ch}_{\star}(\wedge^{-}T^{*}X)\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}}=\int_{X}L_{\star}(TX)\mid_{\text{vol}}.\end{gathered}$
(VI.29)

_More precisely, according to Corollary 5 this index is equal to the
Hirzebruch signature of $X$, i.e.;_

$\tau(X)=b_{+}-b_{-}=\text{Index}(D^{+}_{\star})=\int_{X}L_{\star}(TX)\mid_{\text{vol}},$
(VI.30)

_wherein $b_{+}$ (resp. $b_{-}$) is the number of positive (resp. negative)
eigenvalues of the noncommutative inner product on
$H^{n}_{\text{dR}}(X,\mathbb{R})$, introduced in (V.15 ). Moreover, if $X$ is
a $4p$-dimensional manifold, then upon the equation of
$\tau(X)=\chi(X)~{}(\text{mod}~{}2)$, we also find;_

$\int_{X}L_{\star p}(TX)=\int_{X}e_{\star}(TX)~{}\text{mod}~{}2.$ (VI.31)

One can work out the index of an elliptic complex for the corresponding
complex manifolds.

Corollary 12; _Assume that $X$ is also a complex manifold with the Dolbeault
operator $\overline{\partial}$. Then, the index of the noncommutative version
of the Dolbeault operator for the rolled up vector bundles
$\wedge^{(0,\text{even})}T^{*}X$ and $\wedge^{(0,\text{odd})}T^{*}X$ is given
as;_

$\begin{gathered}\text{Index}(\overline{\partial}_{\star})\\\
=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(\wedge^{\text{even}}TX^{-})-\text{ch}_{\star}(\wedge^{\text{odd}}TX^{-})\right)\star\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX_{\star})}\right)\mid_{\text{vol}}=\int_{X}\text{Td}_{\star
n}(TX^{+}).\end{gathered}$ (VI.32)

_In addition, upon to Corollary 4, this index coincides with the arithmetic
genus of $X$ as;_

$\text{Index}(\overline{\partial}_{\star})=\int_{X}\text{Td}_{\star
n}(TX^{+})=\sum_{k}(-1)^{k}b^{0,k},$ (VI.33)

_wherein $b^{0,k}=\dim_{\mathbb{C}}H^{0,k}(X,\mathbb{C})$, $0\leq k\leq n$,
are the Hodge numbers._

The following corollary is in fact, the noncommutative version of the
celebrated Hirzebruch-Riemann-Roch theorem.

Corollary 13; _Assume that $X$ is a complex manifold and $\mathbb{C}^{K}\to
E\to X$ is a holomorphic vector bundle. Then, the index of
$\overline{\partial}^{E}_{\star}$ for the rolled up vector bundles
$\mathfrak{E}^{even}=\wedge^{(0,\text{even})}TX^{\mathbb{C}}\otimes E$ and
$\mathfrak{E}^{odd}=\wedge^{(0,\text{odd})}TX^{\mathbb{C}}\otimes E$ is given
as;_

$\begin{gathered}\text{Index}(\overline{\partial}^{E}_{\star})=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(\mathfrak{E}^{even})-\text{ch}_{\star}(\mathfrak{E}^{odd})\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}}\\\
=\int_{X}\text{ch}_{\star}(E)\text{Td}_{\star}(TX^{+})\mid_{\text{vol}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(VI.34)

_Moreover, as a topological invariant, the holomorphic Euler character of $E$
is given as;_

$\chi(X,E)=\sum_{k=0}^{2n}(-1)^{k}\dim_{\mathbb{C}}H^{k}(X,E)=\text{Index}(\overline{\partial}^{E}_{\star})=\int_{X}\text{ch}_{\star}(E)\text{Td}_{\star}(TX^{+})\mid_{\text{vol}},$
(VI.35)

_where $H^{k}(X,E)$, $0\leq k\leq 2n$, are the sheaf cohomology groups of
$E\to X$._

Suppose that $X$ is a spin manifold and $S(X)=S_{(}X)+\oplus S_{-}(X)\to X$ is
its spinor bundle, wherein $S_{+}(X)$ (resp. $S_{-}(X)$) is the corresponding
subbundle with positive (resp. negative) chirality poor ; nakahara . Let
$D=D_{+}+D_{-}$ be the Dirac operator for
$D_{+}:\Gamma(S_{+}(X))\to\Gamma(S_{-}(X))$, and $D_{-}={D_{+}}^{\dagger}$.
Hence, we readily have a noncommutative elliptic complex as;

$0\xrightarrow[\text{~{}}]{0}\Gamma(S_{+}(X))\xrightarrow[\text{~{}}]{D_{+\star}}\Gamma(S_{-}(X))\xrightarrow[\text{~{}}]{0}0.$
(VI.36)

Moreover, if $\mathbb{C}^{K}\to E\to X$ is a complex vector bundle, then the
noncommutative Dirac operator is generalized to twisted spinor bundles as
$D_{E\star}=D_{+\star}\otimes 1$ and provides the following noncommutative
elliptic complex;

$0\xrightarrow[\text{~{}}]{0}\Gamma(S_{+}(X)\otimes
E)\xrightarrow[\text{~{}}]{D_{E\star}}\Gamma(S_{-}(X)\otimes
E)\xrightarrow[\text{~{}}]{0}0,$ (VI.37)

The next corollary has important applications in noncommutative Einstein-Yang-
Mills theory, which in fact, extracts the total amount of axial anomaly for a
noncommutative $SU(N)$-gauge theory in presence of gravity.191919See varshovi1
; varshovi2 ; varshovi3  for more discussions on flat spacetime.

Corollary 14; _If $X$ is a spin manifold, then the index of $D_{+}\star$, the
so called noncommutative Dirac index, is given by noncommutative
$\hat{A}_{\star}(TM)$ genus as atiyah84 ;_

$\begin{gathered}\text{Index}(D_{+\star})=(-1)^{n}\int_{X}\left(\text{ch}_{\star}(S_{+}(X))-\text{ch}_{\star}(S_{-}(X))\right)\left(\frac{\text{Td}_{\star}(TX^{\mathbb{C}})}{e_{\star}(TX)}\right)\mid_{\text{vol}}\\\
=\int_{X}\hat{A}_{\star}(TM)\mid_{\text{vol}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(VI.38)

_which obviously vanishes for odd $n$. Moreover, if $E\to X$ is a complex
vector bundle then the index of $D_{E\star}$ is calculated as getzler ;
nakahara :_

$\text{Index}(D_{E\star})=\int_{X}\text{ch}_{\star}(E)\hat{A}_{\star}(TM)\mid_{\text{vol}}.$
(VI.39)

##  VII Summary and Conclusions

In this paper we constructed a covariant star product on a semi-conformally
flat noncommutative product manifold $X=Y\times Z$, wherein $Y$ is a closed
and orientable even dimensional manifold and $Z$ is a flat Calaby-Yau
$m$-fold. We generalized the definition of the covariant star product $\star$
to any vector bundle and extracted a well-defined gauge covariant
noncommutative curvature $F_{\star}$ due. Actually, the noncommutativity
matrix in $F_{\star}$ is rescaled by the ordinary curvature elements and this
provides a consistent framework for studying noncommutative field theories
including that of Yang-Mills and of gravity, within a dynamical
noncommutativity matrix, as was pointed out in steinacker .

Then, it was shown that by inserting $F_{\star}$ into symmetric invariant
polynomials one could provide many fruitful noncommutative counterparts of
characteristic classes. Next, a noncommutative version of Chern-Weil theorem,
Theorem 1, was also established for covariant star product $\star$ which
states: a) The de Rham cohomology class of noncommutative characteristic
classes are independent of the corresponding connection, and; b) The
noncommutative characteristic classes are cohomologous to their ordinary
(commutative) versions. The correspondence of our formulation and the Seiberg-
Witten map is discussed and it was used for providing a second proof for
Theorem 1.

Afterwards, the noncommutative of elliptic operators was studied and a
noncommutative statement of the Atiyah-Singer index theorem on compact
manifold atiyah1 , Theorem 2, was stablished on $X$. This, noncommutative
index theorem could be compared with the algebraic Conne’s index theorem
connes  and the Fedosov’s formula of index theorem on noncommutative
symplectic manifolds fedosov . The former was studied thoroughly in varshovi2
; varshovi3 , but the latter one could be compared with a generalization of
our noncommutative machinery for non-associative vector bundles. However,
since the covariant star product will not touch the topological/algebraic
structures of vector bundles we readily conclude that a $K$-theoretic
viewpoint could be employed to our noncommutative index theorem, via a similar
approach used in atiyah68-1 .

##  VIII Acknowledgments

The author says his gratitude to S. Ziaee who was the main reason for
appearing this article. Also, it should be noted that this research was in
part supported by a grant from IPM (No.1400810418).

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