[math-ph]

#  $\star$-Cohomology, Connes-Chern Characters, and Anomalies in
General Translation-Invariant Noncommutative Yang-Mills

Amir Abbass Varshovi
ab.varshovi@sci.ui.ac.ir/amirabbassv<EMAIL_ADDRESS>Department of Mathematics, University of Isfahan, Isfahan, IRAN.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
Tehran, IRAN.

###### Abstract

Abstract: Topological structure of translation-invariant noncommutative Yang-
Mills theoris are studied by means of a cohomology theory, so called
$\star$-cohomology, which plays an intermediate role between de Rham and
cyclic (co)homology theory for noncommutative algebras and gives rise to a
cohomological formulation comparable to Seiberg-Witten map.

Keywords: Translation-Invariant Star Product, Noncommutative Yang-Mills,
Spectral Triple, Chern Character, Connes-Chern Character, Family Index Theory,
Topological Anomaly, BRST.

##  I Introduction

Noncommutative geometry is one of the most prominent topics in theoretical
physics. Through with last three decades it was extensively believed that the
fundamental forces of the nature could be interpreted with more success via
the machinery of noncommutative geometry and its different viewpoints maeda ;
connes-book2 ; connes-ch ; lizzi . Moyal noncommutative fields, inspired by
fascinating formulations of strings, were proposed as the emergence of this
idea in order to ramify the singular behaviors of quantum field theories,
especially the quantum gravity.111See seiberg-witten  for a proper overview to
this topic. Appearing $\mathrm{UV/IR}$ mixing as a pathological feature of the
Moyal quantum fields led to a concrete generalization of Moyal product as
general translation-invariant noncommutative star products.222See lizzi-
galluccio  and the references therein for a complete list of references and a
short history on this topic.

However, the topology and the geometry of noncommutative field theories with
general translation-invariant star products have not been studied thoroughly
yet. Actually, on the one hand it is a problem in noncommutative geometry, and
on the other hand it is a physical behavior correlated to the topology and the
geometry of the underlying spacetime and corresponding fiber bundles. But, in
contrast to commutative algebras the correlation of noncommutative geometric
machineries, such as cyclic (co)homology, and those of the ordinary
differential structures of commutative geometry, such as the underlying
spacetime, is not clear for noncommutative algebras. The former demands a
point-free formulation, while the later one is based on ordinary topology and
differentiation.

In this article we try to fill this gap by introducing a new cohomology
theory, so called $\star$-cohomology, which can play an intermediate role
between noncommutative and commutative differential geometry for
noncommutative algebras. In the following we give a brief introduction to
basic algebraic structures of $\star$-cohomology and in next sections we will
develop it to study and formulate the topology of general translation-
invariant noncommutative Yang-Mills theories.

Suppose that $X=Y\times Z$ is a $2n$-dimensional closed spin manifold with
$\text{dim}Y=D$ and $Z=\mathbb{T}^{2m}$. Assume that there is a dense
contractible open set, say $U\subset X$, which defines a chart with coordinate
functions $x^{\mu}$, $\mu=0,\cdots,2n-1$. Let $x^{\mu}$s split $X$, that is
$x^{\mu}$ belongs to $Y$ for $\mu=0,\cdots,D-1$, and $x^{i}$ is a torus
canonical coordinate for $D\leq i\leq 2n-1$. Here $x^{0}=t$ is the time
parameter which varies from $-\infty$ to $+\infty$ on $U$. We also put a
metric on $X$ unit diagonal for $x^{\mu}$s. Now consider a general
translation-invariant noncommutative star product on $Z$, say $\star$, defined
for 2-cocycle $\alpha$ as lizzi-galluccio ; galluccio ;

$(f\star
g)(x)=\sum_{p,q\in\mathbb{Z}^{2m}}\tilde{f}(p)\tilde{g}(q)~{}e^{\alpha(p+q,p)}~{}e^{i(p+q).x/R}~{},~{}~{}~{}~{}~{}f,g\in
C^{\infty}(Z)~{},$ (I.1)

for Fourier transformation

$\tilde{f}(p)=\frac{{1}}{{(2\pi
R)^{2m}}}\int_{Z}f(x)~{}e_{-p}(x)~{},~{}~{}~{}~{}~{}f(x)=\sum_{p\in\mathbb{Z}^{2m}}\tilde{f}(p)~{}e_{p}(x)~{},$
(I.2)

with the Fourier basis $e_{p}(x)=e^{ip.x/R}$, $p\in\mathbb{Z}^{2m}$. The
integration in (I.2) is taken for the Riemannian volume form of $Z$. Due to
Hodge decomposition theorem in $\alpha^{*}$-cohomology varshovi1 ; varshovi2
one readily finds $\alpha=\alpha_{M}+\partial\beta$, where $\alpha_{M}$ is a
Moyal 2-cocycle providing a Moyal star product $\star_{M}$, and $\beta$ is a
1-cocycle ($\beta(0)=0$ and $\overline{\beta}(p)=\beta(-p)$) with
$\partial\beta(p,q)=\beta(p-q)-\beta(p)+\beta(q)$. In fact,
$f^{\prime}\star_{M}g^{\prime}=(f\star g)^{\prime}$ where

$f^{\prime}(x)=\sum_{p\in\mathbb{Z}^{2m}}\tilde{f}(p)~{}e^{\beta(p)}~{}e_{p}(x)~{},~{}~{}~{}~{}~{}f\in
C^{\infty}(Z)~{}.$ (I.3)

Assume $H$ is a separable Hilbert space with coordinate function with
orthonormal basis $\\{\left|p\right\rangle\\}_{p\in\mathbb{Z}^{2m}}$. Then, it
can be checked that $\pi:C^{\infty}(Z)\to\mathcal{L}(H)$ with

$\pi(f):=\hat{f}:=\sum_{p\in\mathbb{Z}^{2m}}\tilde{f}^{\prime}(p)~{}\hat{e}_{p}~{},~{}~{}~{}~{}~{}(\hat{e}_{p})_{r,s}=e^{\alpha_{M}(r,s)}~{}\delta_{p,r-s}~{},$
(I.4)

gives rise to a representation of $C^{\infty}_{\star}(Z)$ on $H$, i.e.
$\hat{f}.\hat{g}=\widehat{f\star g}$, $f,g\in C^{\infty}(Z)$. This
representation can be similarly extended to smooth functions on $X$. In this
case for any $f\in C^{\infty}(X)$ the mapped element $\hat{f}$ is an operator
valued function on $Y$. Let $\mathfrak{C}_{0}$ be the algebra generated with
$\hat{f}$, $f\in C^{\infty}(X)$. Then $\mathfrak{C}_{0}$ is isomorphic to
$C^{\infty}_{\star}(X)$ via $\pi$. This leads to definition of noncommutative
polynomial. If $P(x_{1},\cdots,x_{k})$ is a polynomial of probably
noncommutative variables $x_{1},\cdots,x_{k}$, the noncommutative version of
$P$, denoted by $P_{\star}$ is defined as:

$P_{\star}(f_{1},\cdots,f_{k})=\pi^{-1}(P(\hat{f}_{1},\cdots,\hat{f}_{k}))~{}.$
(I.5)

In fact, $\mathfrak{C}_{0}$ is a unital $*$-algebra for $1=\hat{1}$ and
$\hat{f}^{*}=\hat{f}^{{\dagger}}=\hat{\overline{f}}$. The domain of $\pi$ can
be simply extended to vector and matrix valued smooth functions on $X$ with
$v=(v_{i})\to\hat{v}=(\hat{v}_{i})$ and $g=(g_{ij})\to\hat{g}=(\hat{g}_{ij})$.
Moreover, if $\\{t^{a}\\}$ is a basis for Lie algebra $\mathfrak{g}$, then
$\mathfrak{g}$-valued smooth function $f=f^{a}t^{a}$ is mapped to
$\hat{f}=\hat{f}^{a}t^{a}$ via $\pi$.

##  II Star product, $\star$-Cohomology and Chern Characters

The representation map $\pi$ can be simply defined for differential forms on
$X$. Let us denote the image of $k$-forms by $\mathfrak{C}_{k}$ and set
$\mathfrak{C}=\oplus_{k=0}^{2n}\mathfrak{C}_{k}$. Actually, $\mathfrak{C}_{k}$
is the space of operator valued $k$-forms on $Y$ with extra dimensions on $Z$.
Moreover, $\mathfrak{C}$ is obviously a graded algebra with operator and wedge
product. We define an exterior derivative operator on $\mathfrak{C}_{0}$ as;

$d\hat{f}=\widehat{\partial_{\mu}f}~{}dx^{\mu}~{},$ (II.1)

and extend it to $\mathfrak{C}$. Then, $(\mathfrak{C},d)$ is a differential
graded algebra. The corresponding cohomology of $(\mathfrak{C},d)$ is referred
to as $\star$-cohomology and we denote its groups with
$H_{\star}^{k}(X,\mathbb{C})$, $0\leq k\leq 2n$. Since $\pi$ is an isomorphism
and $C^{\infty}(X)=C^{\infty}_{\star}(X)$ as sets we readily find that
$\pi:\Omega_{k}(X)\to\mathfrak{C}_{k}$ is a bijective map for any $k\geq 0$,
where here $\Omega_{k}(X)$ is the space of differential $k$-forms on $X$. Thus
any element of $\mathfrak{C}$ is simply shown with an overal symbol of
$\hat{~}{}$. Moreover, we readily see that $\pi\circ d_{X}=d\circ\pi$ for
$d_{X}$ the exterior differential operator on $X$. Therefore, the following
diagram commutes

$\begin{array}[]{*{20}{c}}0&\to&\Omega_{0}(X)&\to&\Omega_{1}(X)&\to&\cdots&\to&\Omega_{k}(X)&\to&\Omega_{k+1}(X)&\to&\cdots\\\
{}\hfil&{}\hfil&\pi\downarrow&{}\hfil&\pi\downarrow&{}\hfil&\cdots&{}\hfil&\pi\downarrow&{}\hfil&\pi\downarrow&{}\hfil&\cdots\\\
0&\to&\mathfrak{C}_{0}&\to&\mathfrak{C}_{1}&\to&\cdots&\to&\mathfrak{C}_{k}&\to&\mathfrak{C}_{k+1}&\to&\cdots\\\
\end{array}~{},$ (II.2)

wherein the upper arrows stand for $d_{X}$ and the lower ones indicate $d$.
This provides isomorphisms $\pi_{*}:H_{\mathrm{dR}}^{k}(X,\mathbb{C})\to
H_{\star}^{k}(X,\mathbb{C})$. We are mostly interested in integral group
$H^{2n}_{\star}(X,\mathbb{Z})$. To define it an integration structure is put
on $\mathfrak{C}$ as

$\int\hat{\omega}=\left\\{{\begin{array}[]{*{20}{c}}\int_{X}f~{},~{}~{}~{}~{}~{}\hat{\omega}=\hat{f}~{}d^{2n}x\in\mathfrak{C}_{2n}\\\
~{}~{}~{}0~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{other
wise}{}{}{}{}{}\\\ \end{array}}\right.$ (II.3)

where the integral is taken for Riemann volume form of $X$. It is easily seen
that (II.3) defines a graded closed trace over $(\mathfrak{C},d)$ for $\int
d\hat{\omega}=0$ and
$\int\hat{\omega}_{1}.\hat{\omega}_{2}=(-1)^{|\hat{\omega}_{1}||\hat{\omega}_{2}|}\int\hat{\omega}_{2}.\hat{\omega}_{1}$.
The space of classes $[\hat{\omega}]\in H_{\mathrm{dR}}^{2n}(X,\mathbb{C})$
with $\int\hat{\omega}\in\mathbb{Z}$ is referred to as $2n^{th}$ integral
$\star$-cohomology group and is denoted with $H^{2n}_{\star}(X,\mathbb{Z})$.
It can be seen that $\pi_{*}:H_{\mathrm{dR}}^{2n}(X,\mathbb{Z})\to
H_{\star}^{2n}(X,\mathbb{Z})$ is also an isomorphism. Therefore, due to Chern-
Weil theory $H_{\star}^{2n}(X,\mathbb{Z})$ is generated with the $\pi$ image
of $n^{th}$ Chern character $\mathrm{ch}_{n}(E)$ for vector bundles $E\to X$.

The noncommutativity via $\pi$ lets us to define three types of Chern
characters. The first type Chern character is simply
$\hat{\mathrm{ch}}_{n}(E)=\pi(\mathrm{ch}_{n}(E))$, which as mentioned above
represents integral class in $H_{\star}^{2n}(X,\mathbb{Z})$. Let us denote it
with $\hat{\mathrm{ch}}^{(1)}_{n}(E)$ to stress on its type. Second type Chern
character is333From now on the notation $\mathrm{Tr}$ is used for trace on
vector bundle dimensions or gauge group colors.

$\hat{\mathrm{ch}}_{n}^{(2)}(E)=\frac{{1}}{{(4\pi)^{n}n!}}~{}\mathrm{Tr}\left\\{\hat{F}^{n}\right\\}$
(II.4)

wherein $\nabla^{2}=-\frac{{i}}{{2}}F$ is the curvature of connection $\nabla$
on $\mathbb{C}^{k}\to E\to X$ and $\mathrm{Tr}$ is over $\mathbb{C}^{k}$.
Hence;

$\hat{\mathrm{ch}}_{n}^{(2)}(E)=\frac{{1}}{{(4\pi)^{n}n!}}~{}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\pi(\mathrm{Tr}\\{F_{\mu_{1}\nu_{1}}\star\cdots\star
F_{\mu_{n}\nu_{n}}\\})~{}d^{2n}x~{}.$

It is obvious that if $n=1$ then $\hat{\mathrm{ch}}_{n}^{(2)}(E)$ and
$\hat{\mathrm{ch}}_{n}^{(1)}(E)$ coincide and hence the cohomology class of
$\hat{\mathrm{ch}}_{n}^{(2)}(E)$ is independent of connection $\nabla$. But,
however, it is not the case in general;444Similar results for Moyal product
are worked out in varshovi-dR-moyal .

Theorem 1; _Suppose $\star$ is a general translation-invariant noncommutative
star product on $X$ with $n\geq 2$. Then, the cohomology class of
$\hat{\mathrm{ch}}^{(2)}_{n}(E)$ is independent of connection $\nabla$ if and
only if $\star$ is a Moyal star product. Hence after;_

a) _$\hat{\mathrm{ch}}^{(2)}_{n}(E)$ defines an integral cohomology class in
$H_{\star}^{2n}(X,\mathbb{Z})$._

b) _$\hat{\mathrm{ch}}^{(2)}_{n}(E)$ and $\hat{\mathrm{ch}}^{(1)}_{n}(E)$ are
cohomologous. That is;_

$\int\hat{\mathrm{ch}}^{(2)}_{n}(E)=\int\hat{\mathrm{ch}}^{(1)}_{n}(E)=\int_{X}\mathrm{ch}_{n}(E)\in\mathbb{Z}~{}.$
(II.5)

Hint to the Proof; Actually, according to Chern-Weil theory it is enough to
prove equality (II.5). The integral on the far left consists of the following
integration according to (I.3);

$\int_{X}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\mathrm{Tr}\left\\{F^{\prime}_{\mu_{1}\nu_{1}}\star_{M}\cdots\star_{M}F^{\prime}_{\mu_{n}\nu_{n}}\right\\}~{},$

where $\star_{M}$ is the Moyal product cohomologous to $\star$ via Hodge
decomposition varshovi1 . It is also equal to

$\int_{X}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\mathrm{Tr}\left\\{F^{\prime}_{\mu_{1}\nu_{1}}\cdots
F^{\prime}_{\mu_{n}\nu_{n}}\right\\}~{},$

since Moyal product can be replaced by the ordinary product for integration of
symmetric polynomials varshovi-dR-moyal . This integral is independent of
connection if and only if there exists some $C\in\mathbb{C}$ so that;

$\int_{X}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\mathrm{Tr}\left\\{F^{\prime}_{\mu_{1}\nu_{1}}\cdots
F^{\prime}_{\mu_{n}\nu_{n}}\right\\}=C\int_{X}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\mathrm{Tr}\left\\{F_{\mu_{1}\nu_{1}}\cdots
F_{\mu_{n}\nu_{n}}\right\\}~{}.$

But it is easily seen that the above equation holds if and only if 1-cocycle
$\beta$ is linear and $C=1$. That is $\star=\star_{M}$. The rest of the proof
is due to Chern-Weil theory. This finishes the theorem. Q.E.D

To define the third type Chern character we represent vector bundle
$\mathbb{C}^{k}\to E\to X$ by $\pi$. Set

$\hat{E}=\\{\hat{V};~{}V\in C^{\infty}(E)\\}~{}.$ (II.6)

Then, the connection $\nabla=d_{X}+A$, for
$A\in\Omega_{1}(X)\otimes\mathbb{M}_{k}(\mathbb{C})$, is mapped to
$\hat{\nabla}=d+\hat{A}$ on $\hat{E}$. Then, its curvature is an element of
$\mathfrak{C}_{2}\otimes\mathbb{M}_{k}(\mathbb{C})$ and is equal to
$\hat{\nabla}^{2}=d\hat{A}+\hat{A}^{2}=-\frac{{i}}{{2}}\hat{F}_{\star}$. The
third type Chern character is then defined with $\hat{\nabla}^{2}$ as;

$\hat{\mathrm{ch}}_{n}^{(3)}(E)=\frac{{1}}{{(4\pi)^{n}n!}}~{}\mathrm{Tr}\left\\{\hat{F}_{\star}^{n}\right\\}=\frac{{1}}{{(4\pi)^{n}n!}}~{}\epsilon^{\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}~{}\pi\left(\mathrm{Tr}\left\\{F_{\star\mu_{1}\nu_{1}}\star\cdots\star
F_{\star\mu_{n}\nu_{n}}\right\\}\right)~{}d^{2n}x~{}.$ (II.7)

We show $\hat{\mathrm{ch}}^{(3)}_{n}(E)$ with $\check{\mathrm{ch}}_{n}(E)$ for
simplicity. The next theorem is significant to our formalism.

Theorem 2; _For any general translation-invariant noncommutative star product
$\star$ on $X$ we have;_

a) _The cohomology class of $\check{\mathrm{ch}}_{n}(E)$ in
$H^{2n}_{\star}(X,\mathbb{C})$ is independent of connection $\nabla$
($\hat{\nabla}$)._

b) _$\check{\mathrm{ch}}_{n}(E)$ represents an integral cohomology class in
$H_{\star}^{2n}(X,\mathbb{Z})$._

c) _$\check{\mathrm{ch}}_{n}(E)$ and $\hat{\mathrm{ch}}^{(1)}_{n}(E)$ are
cohomologous. That is;_

$\int\check{\mathrm{ch}}_{n}(E)=\int\hat{\mathrm{ch}}^{(1)}_{n}(E)=\int_{X}\mathrm{ch}_{n}(E)\in\mathbb{Z}~{}.$
(II.8)

Hint to the Proof; For connection $\nabla=d_{X}+A$ we obtain;

$\int\check{\mathrm{ch}}_{n}(E,A;\star)=\int\check{\mathrm{ch}}_{n}(E,A^{\prime};\star_{M})=\int_{X}\mathrm{ch}_{n}(E,A^{\prime})=\int_{X}\mathrm{ch}_{n}(E)~{},$

where the first equality is due to (I.3), the second one is the replacement of
$\star_{M}$ with the ordinary product for integration of symmetric
polynomials, and the last equation is an immediate consequence of Chern-Weil
theory. This proves (II.8) and thus the theorem follows. Q.E.D

##  III Index Theorem in $\star$-Cohomology and Abelian Anomaly

Assume vector bundle $\mathbb{C}^{N}\to E\to X$ with structure group
$\mathrm{U}(N)$ for some $N$, in fundamental representation. The generators of
$\mathfrak{u}(N)$, say $t^{a}$s, are supposed to be Hermitian with totally
anti-symmetric structure group $if^{abc}$, anti-commutator
$\\{t^{a},t^{b}\\}=-c^{abc}t^{c}$ and normalization condition
$\mathrm{Tr}(t^{a}t^{b})=\frac{{1}}{{2}}\delta^{ab}$. We show the space of
$\mathfrak{g}$-valued differential forms on $X$ by
$\tilde{\Omega}(X)=\oplus_{k=0}^{2n}~{}\tilde{\Omega}_{k}(X)$, where
$\mathfrak{g}=\mathfrak{u}(N)$. Similarly the
$\mathfrak{C}_{0}\otimes\mathfrak{g}$-valued differential forms on $Y$ with
extra dimensions on $Z$ are denoted as $\tilde{\mathfrak{C}}$. We have the
natural grading for $\tilde{\mathfrak{C}}$ as
$\tilde{\mathfrak{C}}=\oplus_{k=0}^{2n}\tilde{\mathfrak{C}}$ which for $d$ is
defined accordingly. Thus, $(\tilde{\mathfrak{C}},d)$ is a differential graded
algebra on which $\mathrm{Tr}\otimes\int$ defines an integration structure or
a closed graded trace. It is easy to see that $\tilde{\mathfrak{C}}_{0}$ is a
unital $*$-algebra with $1=\hat{1}\mathbb{I}$ and
$(\hat{f}^{a}t^{a})^{*}=\hat{f^{a}}^{{\dagger}}{t^{a}}^{{\dagger}}=\hat{\overline{f^{a}}}t^{a}$.
We refer to it with $\cal{A}$.

The involution $*$ is naturally extended to $\tilde{\mathfrak{C}}_{k}$ for any
$k\geq 0$. Conventionally, $\sigma\in\tilde{\mathfrak{C}}$ is said to be
Hermitian if $\sigma^{*}=\sigma$, and it is anti-Hermitian when
$\sigma^{*}=-\sigma$. The set of anti-Hermitian elements of $\mathcal{A}$,
denoted by $\tilde{\mathfrak{g}}$, is in fact a Lie algebra. Each element of
$\tilde{\mathfrak{g}}$ also is known as infinitesimal gauge transformation.
The Lie group generated by exponential of elements of $\tilde{\mathfrak{g}}$,
indicated with symbol $\tilde{G}$, is the gauge transformation group.
Accordingly, the space of anti-Hermitian elements in
$\tilde{\mathfrak{C}}_{1}$, shown with $\Gamma$, is called the connection
space. Any element of $\Gamma$, say
$\hat{A}=-i\hat{A}^{a}_{\mu}t^{a}~{}dx^{\mu}$, for real functions
$A^{a}_{\mu}\in C^{\infty}(X)$, is a connection form on mapped vector bundle
$\hat{E}$ for connection $\hat{\nabla}=d+\hat{A}$ due to (II.6).

The gauge transformation group $\tilde{G}$ (resp. $\tilde{\mathfrak{g}}$) acts
on $\Gamma$ form right:

$A\triangleleft
g=g^{-1}dg+g^{-1}Ag~{},~{}~{}~{}A\in\Gamma,~{}g\in\tilde{G}~{}~{}(\text{resp.~{}}A\triangleleft\alpha=d\alpha+[A,\alpha]~{},~{}~{}~{}\alpha\in\tilde{\mathfrak{g}}~{})~{}.$
(III.1)

The Yang-Mills theory of vector bundle $SE:=S(X)\otimes E\to X$ is mapped to
that of $\widehat{SE}$ which by definition is equipped with connection
$\hat{\nabla}=d-i\hat{A}^{a}_{\mu}t^{a}~{}dx^{\mu}$ and curvature
$\hat{\nabla}^{2}=-\frac{{i}}{{2}}\hat{F}_{\star\mu\nu}^{a}t^{a}~{}dx^{\mu}dx^{\nu}$.
The Lagrangian and the action of the $\mathrm{U}(N)$-Yang-Mills theory is then
given by;555For $\mathrm{U}(1)$ gauge theory one needs an overal factor of
$\frac{{1}}{{2}}$. This is mandatory to compensate the normalization condition
$\mathrm{Tr}(t^{a}t^{b})=\frac{{1}}{{2}}\delta^{ab}$ for $N>1$.

$L_{\mathrm{Y-M}}(\widehat{SE},\hat{\nabla})=\mathrm{Tr}\left\\{\hat{\nabla}^{2}.*\hat{\nabla}^{2}\right\\}\in\mathfrak{C}_{2n}~{},~{}~{}~{}~{}~{}S_{\mathrm{Y-M}}=\int
L_{\mathrm{Y-M}}(\widehat{SE},\hat{\nabla})~{},$ (III.2)

for $*$ the Hodge star. Vector bundle $\widehat{SE}$, the space of spinors, is
subject to Dirac operator $\mathcal{D}_{0}$ as

$\mathcal{D}_{0}\hat{\psi}=i\gamma^{\mu}\partial_{\mu}\hat{\psi}~{},$ (III.3)

on $U$ for Dirac matrices $\gamma^{\mu}$. Dirac operator $\mathcal{D}_{0}$ is
usually perturbed to $\mathcal{D}_{A}$, for
$A=-i\hat{A}^{a}_{\mu}t^{a}~{}dx^{\mu}\in\Gamma$;

$\mathcal{D}_{A}\hat{\psi}=i\gamma^{\mu}\partial_{\mu}\hat{\psi}+\gamma^{\mu}t^{a}\hat{A}^{a}_{\mu}\hat{\psi}~{}.$
(III.4)

The relevant Lagrangian and action are respectively

$L_{\mathcal{D}_{0}}(\widehat{SE})=\overline{\hat{\psi}}.\mathcal{D}_{0}\hat{\psi}~{}d^{2n}x=\hat{\psi}^{{\dagger}}\gamma^{0}\mathcal{D}_{0}\hat{\psi}~{}d^{2n}x\in\mathfrak{C}_{2n}~{},~{}~{}~{}~{}~{}S_{\mathcal{D}_{0}}=\int
L_{\mathcal{D}_{0}}(\widehat{SE})~{},$ (III.5)
$L_{\mathrm{int}}(\widehat{SE},\hat{\nabla})=\overline{\hat{\psi}}\gamma^{\mu}\hat{A}^{a}_{\mu}t^{a}\hat{\psi}~{}d^{2n}x\in\mathfrak{C}_{2n}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{\mathrm{int}}=\int
L_{\mathrm{int}}(\widehat{SE},\hat{\nabla})~{}.$ (III.6)

The total action then defines the well-known noncommutative
$\mathrm{U}(N)$-Yang-Mills theory;

$S_{\mathrm{total}}=S_{\mathrm{Y-M}}+S_{\mathcal{D}_{0}}+S_{\mathrm{int}}=-\frac{{1}}{{4}}\int_{X}F_{\star\mu\nu}F_{\star}^{\mu\nu}+i\int_{X}\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi+\int_{X}j^{a}_{\star}A^{a}_{\mu}~{},$
(III.7)

for
$j^{a\mu}_{\star}=\psi_{\beta,j}\star\overline{\psi}_{\alpha,i}~{}\gamma^{\mu}_{ij}~{}t^{a}_{\alpha\beta}$.
Replacing $t^{a}$ and $\gamma^{\mu}$ respectively with $\mathbb{I}$ and
$\gamma^{\mu}P_{\pm}$, for $P_{\pm}=\frac{{1\pm\gamma}}{{2}}$ and

$\gamma=\gamma_{2n+1}:=i^{n-1}\gamma^{0}\cdots\gamma^{2n-1}~{},$ (III.8)

leads to chiral singlet currents $j^{\mu}_{\star\pm}$, which obey the
classical equation

$\partial_{\mu}j^{\mu}_{\star\pm}+i[A^{a}_{\mu},j^{a\mu}_{\star\pm}]_{\star}=0~{}.$
(III.9)

Let this equation be anomalously broken at quantum levels with appearing
$\mathcal{A}_{\star\pm}$ on the right hand side. Then, since any translation-
invariant noncommutative star product $\star$ is cyclic under integration the
corresponding charges $Q_{\star\pm}(t)$ receive variations from $t=-\infty$ to
$t=+\infty$ as;

$\Delta
Q_{\star\pm}=Q_{\star\pm}(+\infty)-Q_{\star\pm}(-\infty)=\int_{-\infty}^{+\infty}\frac{{d}}{{dt}}Q_{\star\pm}(t)=\int_{X}\mathcal{A}_{\star\pm}=\int\hat{\mathcal{A}}_{\star\pm}~{}.$
(III.10)

Now let us make an _ansatz_ here. By this ansatz, so called _physical
consistency_ , we assume that despite to appearence of consistent anomaly
$\mathcal{A}_{\star\pm}$ the charge variation $\Delta Q_{\star\pm}$ respects
the theory so that it is equal to an integer times the unit charge of it.
Actually, the results achieved through with various methods of anomaly
derivation in Moyal noncommutative gauge theories, such as noncommutative
calculus langmann , perturbative loop calculations ardalan ; bonora ; bonora2
, Seiberg-Witten map brandt  and matrix model varshovi-consistent ,666We
should emphasize that the results of varshovi-consistent  is in fact for
general translation-invariant star products. See also varshovi-dR-moyal  and
the references therein for a more complete list of such papers. confirm the
reasonability of the ansatz for the Moyal case. However, the idea for
generalizing the results derived for Moyal product to general translation-
invariant star products comes in principal from quantum equivalence theorem
introduced in varshovi1 ; varshovi2  which asserts that the whole quantum
behaviors of a noncommutative quantum field theory with an arbitrary
translation-invariant star product coincide precisely with those of its Moyal
product case of the same $\alpha^{*}$-cohomology class.

Thus, according to physical consistency ansatz, we demand the results of
integrals in (III.10) to be integers for any given translation-invariant
noncommutative star product $\star$. Hence, for a homotopy of 2-cocyles as
$s\alpha$ with $s\in[0,1]$ and correponding star product $\star_{s}$ we obtain
$\int_{X}\mathcal{A}_{\star_{s}\pm}\in\mathbb{Z}$, which leads to
$\frac{{d}}{{ds}}\int\int_{X}\mathcal{A}_{\star_{s}\pm}=0$ due to continuity.
Furthermore, for the commutative fields, i.e. $s=0$, we have;
$\mathcal{A}_{\pm}=\mp\mathrm{ch}_{n}(SE)$. Therefore,
$\hat{\mathcal{A}}_{\star\pm}$ and $\mp\hat{\mathrm{ch}}_{n}(SE)$ must be
cohomologous in $H^{2n}_{\star}(X,\mathbb{Z})$. On the other hand, the left
hand side of (III.9) transforms covariantly under infinitesimal gauge
transformations via (III.1) and
$\hat{\psi}\to\alpha\triangleright\hat{\psi}:=-i\hat{\alpha}^{a}.t^{a}\psi$
for $\alpha=-i\hat{\alpha}^{a}~{}t^{a}\in\tilde{\mathfrak{g}}$. Hence,
$\hat{\mathcal{A}}_{\star\pm}$ is an equivariant form. That is,
$\hat{\omega}=\hat{\mathcal{A}}_{\star\pm}\pm\check{\mathrm{ch}}_{n}(SE)$ is
an equivariant form and thus it vanishes on $U$ for triviality of $E$ over it.
Since $\hat{\omega}$ represents the null integral cohomology class in
$H^{2n}_{\star}(X,\mathbb{Z})$ we readily conclude $\hat{\omega}=0$ . Thus, we
have already established the following theoerm.

Theorem 3; _The chiral Abelian anomaly in noncommutative Yang-Mills theories
with general translation-invariant noncommutative star product $\star$ is
given by $\mathcal{A}_{\star\pm}=\mp\check{\mathrm{ch}}_{n\star}(SE)$._

It is well-known that the Dirac operator in the ordinary commutative case,
i.e. $D_{A}=i\gamma^{\mu}\nabla_{\mu}$, $\nabla=d_{X}+A$, via
$\mathbb{Z}_{2}$-grading of $SE\to X$ due to $\gamma$ of (III.8), is given as

$\gamma=\left({\begin{array}[]{*{20}{c}}1&0\\\ 0&-1\\\
\end{array}}\right)~{},~{}~{}~{}~{}~{}D_{A}=\left({\begin{array}[]{*{20}{c}}0&D_{A}^{-}\\\
D_{A}^{+}&0\\\ \end{array}}\right)~{},~{}~{}~{}~{}~{}\gamma
D_{A}=-D_{A}\gamma~{},~{}~{}~{}~{}~{}{D_{A}^{+}}^{{\dagger}}=D_{A}^{-}~{}.$
(III.11)

Following the main approach of Atiyah-Singer index theorem atiyah-dirac 777See
also the approach of getzler . for $\star$-cohomology and also employing
Theorem 3 we readily find an index formula for any translation-invariant
noncommutative anomalies via the machinery of $\star$-cohomology. Show
$\pi(D_{A}^{(\mp)})$ with $\mathcal{D}_{A}^{(\mp)}$.

Theorem 4; _For any noncommutative $\mathrm{U}(N)$-Yang-Mills theory with
general translation-invariant noncommutative star product $\star$ the
topological index of $\mathcal{D}_{A}^{\mp}$ is given by
$\check{\mathrm{ch}}_{n}(SE)$ as;_

$\int_{X}\mathcal{A}_{\star\pm}=\mp\int\check{\mathrm{ch}}_{n}(SE)=\emph{\emph{Index}}(\mathcal{D}_{A}^{\mp})~{}.$
(III.12)

The topological index is given for de Rham integral class in
$H_{\mathrm{dR}}^{2n}(X,\mathbb{Z})$ with $\check{\mathrm{ch}}_{n\star}(SE)$
via (I.5);

$\emph{\emph{Index}}(\mathcal{D}_{A}^{\mp})=\int_{X}\mathcal{A}_{\star\pm}=\mp\int_{X}\check{\mathrm{ch}}_{n\star}(SE)~{}.$

##  IV Anomalies, $\star$-Cohomology and the Connes-Chern Characters

In previous section we established an intimate correlation between $\star$\-
and de Rham cohomology to figure out the topological structure in the
background of a translation-invariant noncommutative Yang-Mills theory. In
this section we try to demonstrate a similar relation for cyclic (co)homology
and the corresponding Connes-Chern character.888See varshovi-dR-moyal  for the
special case of Moyal star product. One should remember that since
$\mathcal{A}$ (or $C^{\infty}_{\star}(X)$) is a noncommutative algebra there
is no definite coincidence for de Rham and cyclic (co)homology (in the sense
of connes-book  via Hochschild-Kostant-Rosenberg formula999See also connes-
differential ; khalkhali .). Therefore, topological interpretation of
noncommutative anomalies need some breakthrough between cyclic (co)homology of
$\mathcal{A}$ on the one hand and de Rham cohomology of the topologically
commutative underlying spacetime $X$ on the other hand. In this section we
proved that this subjective is properly achieved by applying the machinery of
$\star$-cohomology.

Consider a trivial vector bundle over $X$, say $\mathbb{C}^{N^{\prime}}\to
E^{\prime}=\mathbb{C}^{N^{\prime}}\times X\to X$, for some large enough
$N^{\prime}$, so that $E\to X$ is embedded in via the image of an idemponent
$e\in\mathbb{M}_{N^{\prime}}(C^{\infty}(X))$, i.e.;

$C^{\infty}(E)=\\{\sigma\in C^{\infty}(E^{\prime});~{}e.\sigma=\sigma\\}~{}.$

Then, ${SE}^{\prime}:=S(X)\otimes E^{\prime}\to X$ is subject to Dirac
operator $D=i\gamma^{\mu}\partial_{\mu}$ on $U$. Actually,
$\widehat{SE}^{\prime}$ could be completed to a Hilbert space, $\cal H$, with
ordinary inner product
$\left\langle{\rho(\psi_{1})}\mathrel{\left|{\vphantom{\rho(\psi_{1})\rho(\psi_{2})}}\right.\kern-1.2pt}{\rho(\psi_{2})}\right\rangle=\int_{X}\overline{\psi}_{1}\psi_{2}$.
We see that $\pi$ and the Dirac operator $D$ commute, i.e. $\pi\circ
D=\mathcal{D}\circ\pi$ for $\mathcal{D}=i\gamma^{\mu}\partial_{\mu}$, and
therefore, the spectrum of $\mathcal{D}$ coincides with that of $D$. Hence,
Dirac operator $\mathcal{D}$ is a densely defined unbounded Hermitian operator
on $\mathcal{H}$. Thus $(\mathcal{A},\mathcal{H},\mathcal{D})$ is a spectral
triple. The action of unital $*$-algebra $\cal{A}$ on $\cal H$ is also given
as
$(\hat{\alpha}^{a}t^{a})\triangleright\hat{\psi}=\hat{\alpha}^{a}.\hat{t}^{a}\psi$.
In addition, we have a $\mathbb{Z}_{2}$-grading of $\mathcal{H}$ as
$\mathcal{H}=\mathcal{H}^{+}\oplus\mathcal{H}^{-}$ with;

$\mathcal{D}=\left({\begin{array}[]{*{20}{c}}0&\mathcal{D}^{-}\\\
\mathcal{D}^{+}&0\\\
\end{array}}\right)~{},~{}~{}~{}~{}~{}\gamma\mathcal{D}=-\mathcal{D}\gamma~{},~{}~{}~{}~{}~{}{\mathcal{D}^{+}}^{{\dagger}}=\mathcal{D}^{-}~{},$
(IV.1)

for $\gamma$ of (III.11). It is also clear that $\gamma\alpha=\alpha\gamma$,
$\alpha\in\mathcal{A}$. Now we employ the homotopy of last section for star
rpoducts $\star_{s}$. Then, it can be seen that for any
$\hat{a}\in\mathcal{A}$, the operator $[\mathcal{D},\hat{a}]$ is homotopic to
$[D,a]$ and thus is a densely defined operator which could be extended to a
bounded operator on $\mathcal{H}$. Also for any $p>2n$ we have
$(1+\mathcal{D}^{2})^{-1}\in\mathcal{L}^{p/2}(\mathcal{H})$ since
$\emph{\emph{spec}}(\mathcal{D})=\emph{\emph{spec}}(D)$ gilky . Set
$F=\mathcal{D}/|\mathcal{D}|$. Actually, $F$ is a bounded operator with
$\gamma F=-F\gamma$ and $F^{2}=1$. Hence, $(\mathcal{H},F,\gamma)$ is an even
$p$-summable Fredholm module over $\mathcal{A}$, and an element of
$K$-homology group $K^{0}(\mathcal{A})$.

The vector bundle $\widehat{SE}$ is in fact a dual element with respect to
$(\mathcal{H},F,\gamma)$ in $K$-theory group $K_{0}(\mathcal{A})$. Assume an
idemponent $e\in\mathbb{M}_{N^{\prime}}(\mathcal{A})$ so that $\widehat{SE}$
embeds in $\widehat{SE}^{\prime}$ as
$\widehat{SE}=\\{\hat{\psi}\in\widehat{SE}^{\prime};~{}e\hat{\psi}=\hat{\psi}\\}$.
The connection on $\widehat{SE}$ is canonically defined with
$\hat{\nabla}\hat{\psi}=e.d\hat{\psi}$. However, it is seen that if for a
local basis over $U$ we define
$\hat{\nabla}=d+A=d-i\hat{A}^{a}_{\mu}t^{a}~{}dx^{\mu}$, then;
$\emph{\emph{Index}}(F^{+}_{e})=\emph{\emph{Index}}(\mathcal{D}_{A}^{+})$, in
which

$F=\left({\begin{array}[]{*{20}{c}}0&F^{-}\\\ F^{+}&0\\\
\end{array}}\right)~{},$

for the $\mathbb{Z}_{2}$-grading of $\gamma$ and
$F^{+}_{e}=eF^{+}e:e\mathcal{H}^{+}\to e\mathcal{H}^{-}$. Hence, according to
noncommutative index theorem and Theorem 4 we find;

Theorem 5; _The integral cohomology class of the third type Chern character
$\check{\mathrm{ch}}_{n}(SE)$ in $H_{\star}^{2n}(X,\mathbb{Z})$ (and the
corresponding topological index of chiral Abelian anomaly) is given by the
pairing of Connes-Chern characters of $2n^{th}$ cyclic (co)homology groups due
to $K$-theory and $K$-cohomology. That is;_101010The notation $\mathrm{Trace}$
is used for trace of operators on the corresponding Hilbert space of Dirac
operator.

$\int\check{\mathrm{ch}}_{n}(SE)=\left\langle\emph{\emph{Ch}}^{2n}(\mathcal{H},F,\gamma),\emph{\emph{Ch}}_{2n}[e]\right\rangle=\frac{{(-1)^{n}}}{{2}}\mathrm{Trace}\\{\gamma
F[F,e][F,e]\cdots[F,e]\\}~{},$ (IV.2)

_for $2n+1$ copies of $e$ and for $\mathrm{Trace}$ the trace of operators._

We remember that $\emph{\emph{Ch}}^{2n}(\mathcal{H},F,\gamma)$ in (IV.2) is
the Connes-Chern character of $(\mathcal{H},F,\gamma)\in K^{0}(\mathcal{A})$
as

$\emph{\emph{Ch}}^{2n}(\mathcal{H},\mathcal{D},\gamma)(a_{0},a_{1},\cdots,a_{2n})=\left(-1\right)^{n}\left(\frac{{n!}}{{2}}\right)\mathrm{Trace}\left\\{\gamma
F[F,a_{0}][F,a_{1}]\cdots[F,a_{2n}]\right\\}~{},$ (IV.3)

for $a_{0},a_{1},\cdots,a_{2n}\in\mathcal{A}$, and

$\emph{\emph{Ch}}_{2n}[e]=\sum_{k=0}^{n}(-1)^{k}\frac{{(2k)!}}{{k!}}~{}\mathrm{tr}\left\\{\left(e-\frac{{1}}{{2}}\right)\otimes
e^{2k\otimes}\right\\}~{},$ (IV.4)

wherein $e\in\mathbb{M}_{N^{\prime}}(\mathcal{A})$ represents the $K$-theory
class $[e]$ in $K_{0}(\mathcal{A})$ and $\mathrm{tr}$ is the trace of
$\mathbb{C}^{N^{\prime}}$.

As we mentioned above $\star$-cohomology plays an intermediate role between de
Rham and cyclic (co)homologies for any general translation-invariant
noncommutative star product $\star$. In previous sections we established an
intimate correlation of $\star$\- and de Rham cohomology theories. Now, by
employing the special abilities of $\star$-cohomology, due to its partly
commutative geometric structures, we can also prove a geometric correspondence
between $\star$\- and cyclic (co)homology. We emphasize that this relation
must be implemented in the setting we just apply for demonstrating that of de
Rham and cyclic (co)homologies for commutative algebras. By means of
$\star$-cohomology we find the same arguments even for noncommutative algebra
$\mathcal{A}$ (or $\mathfrak{C}_{0}$).

This correlation is easy to see within familiar concepts of noncommutative
geometry. Actually, the well-known Hochschild-Kostant-Rosenberg map
$\alpha:HH_{k}(\mathcal{A})\to\Omega_{k}(X)$, for $HH_{*}(\mathcal{A})$ the
Hochschild homology group, leads to an isomorphism, similarly denoted by
$\alpha$, between cyclic homology group $HC_{2n}(\mathcal{A})$ and
$\oplus_{j=0}^{n}H_{\star}^{2j}(X,\mathbb{C})$ due to commutativity of
$d\circ\pi=\pi\circ d_{X}$. It is not hard to see that $\alpha$ produces the
third type Chern chracter $\check{\mathrm{ch}}_{n}(SE)$ from Connes-Chern
character $\mathrm{Ch}_{2n}[e]\in HC_{2n}(\mathcal{A})$. In fact, the
composition of isomorphism
$\alpha:HC_{2n}(\mathcal{A})\to\oplus_{j=0}^{n}H_{\star}^{2j}(X,\mathbb{C})$
and the canonical projection
$\pi_{2n}:\oplus_{j=0}^{n}H^{2j}_{\star}(X,\mathbb{C})\to
H^{2n}_{\star}(X,\mathbb{C})$ leads to the following result;

$\Xi=\left(\frac{{1}}{{2i\pi}}\right)^{n}\frac{{1}}{{(2n)!}}~{}\pi_{2n}\circ\alpha:HC_{2n}(\mathcal{A})\to
H^{2n}_{\star}(X,\mathbb{C})~{},~{}~{}~{}~{}~{}\Xi([\mathrm{Ch}_{2n}[e]])=[\check{\mathrm{ch}}_{n}(SE)]~{},$
(IV.5)

for corresponding cohomology classes $[\mathrm{Ch}_{2n}[e]]$ and
$[\check{\mathrm{ch}}_{n}(SE)]$. The homomorphism $\Xi$ can be derived with
more detailed formalism. To see this we note that for connection
$\hat{\nabla}=e.d=d-i\hat{A}^{a}_{\mu}t^{a}~{}dx^{\mu}$ the curvature
$\hat{\nabla}^{2}$ is actually
$e.de.de=-\frac{{i}}{{2}}\hat{F}_{\star\mu\nu}~{}dx^{\mu}dx^{\nu}$. However,
with $(e.de.de)^{n}=e.(de)^{2n}$ we readily find
$\alpha\left(\mathrm{Ch}_{2n}[e]\right)=\left(2i\pi\right)^{n}(2n)!~{}\check{\mathrm{ch}}_{n}(SE)+\Delta$,
where $\Delta$ is a direct summation of an exact $2n$-form and closed forms of
lower even orders in $\mathfrak{C}$. Now let
$\Omega^{*}(\mathcal{A})=\oplus_{k\geq 0}\mathcal{A}^{(k+1)\otimes}$ and
consider $\xi:\Omega^{*}(\mathcal{A})\to\mathfrak{C}$, with $\xi=0$ on
$\mathcal{A}^{(k+1)\otimes}$ for $k>2n$, and

$\xi\left(a_{0}\otimes\cdots\otimes
a_{k}\right)=-\left(\frac{{-1}}{{2\pi}}\right)^{n}\frac{{i}}{{2^{n}(2n)!}}~{}{\mathrm{Tr}}\\{\gamma
a_{0}[\mathcal{D},a_{1}]\cdots[\mathcal{D},a_{k}]\\}~{}d^{2n}x~{},$ (IV.6)

for $k\leq 2n$. Here $\mathrm{Tr}$ is the trace on Dirac matrices and on the
colors. Thereby, we readily find that
$\xi(\emph{\emph{Ch}}_{2n}[e])=\check{\mathrm{ch}}_{n}(SE)+d\phi$ for some
$\phi\in\mathfrak{C}_{2n-1}$. We have the following lemma.

Lemma 1; _The linear map $\xi$ of (IV.6) leads to a surjection from cyclic
cohomology group $HC_{2n}(\mathcal{A})$ to $H^{2n}_{\star}(X,\mathbb{C})$,
i.e. $\xi_{*}:HC_{2n}(\mathcal{A})\to H^{2n}_{\star}(X,\mathbb{C})$ for any
general translation-invariant noncommutative star product $\star$. Moreover,
$\xi_{*}$ is in fact a redefinition for Hochschild-Kostant-Rosenberg map due
to (IV.5) with $\xi_{*}=\Xi$._

Proof; First note that $[\mathcal{D},a]=i\gamma^{\mu}\partial_{\mu}a$,
$a\in\mathcal{A}$. Then since

$\mathrm{Tr}\left(\gamma\gamma^{\mu_{1}}\cdots\gamma^{\mu_{2n}}\right)=(-i)^{n-1}2^{n}~{}\epsilon^{\mu_{1}\cdots\mu_{2n}}~{},~{}~{}~{}~{}~{}\mathrm{Tr}\left(\gamma\gamma^{\mu_{1}}\cdots\gamma^{\mu_{k}}\right)=0~{},~{}~{}~{}~{}~{}k<2n~{},$

it is seen that $\xi$ vanishes on $\mathcal{A}^{(k+1)\otimes}$ for all $k\neq
2n$. On the other hand, $\phi:=\int\circ\xi$ is a closed graded trace on
$\Omega(\mathcal{A})$ with support in $\mathcal{A}^{(2n+1)\otimes}$. Thus, it
defines a cyclic cohomology class in $HC_{2n}(\mathcal{A})$. This proves that
the Connes’ $(b,B)$-bicomplex is compatible with the de Rham complex via $\xi$
so that $\xi\circ b=0$ and $\xi\circ B\in d\mathfrak{C}$. Thus $\xi$ is
reduced to a well-defined map $\xi_{*}:HC_{2n}(\mathcal{A})\to
H^{2n}_{\star}(X,\mathbb{C})$. Direct calculation shows that $\xi_{*}=\Xi$.
This finishes the lemma. Q.E.D

Therefore, we have already established the following theorem.

Theorem 6; _For any general translation-invariant noncommutative star product
$\star$ on $X$ the integral cohomology class of the third type Chern character
$\check{\mathrm{ch}}_{n}(SE)$ in $H_{\star}^{2n}(X,\mathbb{Z})$ is the image
of that of $\emph{Ch}_{2n}[e]$ in cyclic cohomology group
$HC_{2n}(\mathcal{A})$ via $\xi$ due to (IV.6)._

Corollary 1; _The topological index of chiral Abelian anomaly for any general
translation-invariant noncommutative $\mathrm{U}(N)$-Yang-Mills theory is
given by Connes-Chern character of the corresponding vector bundle
$\widehat{SE}$. That is;_

$\text{Index}(\mathcal{D}_{A}^{\mp})=\int\hat{\mathcal{A}}_{\star\pm}=\mp\int\xi\left(\emph{\emph{Ch}}_{2n}[e]\right)~{}.$

##  V Family Index and Homotopy Class of Topological Anomaly

Topological or consistent anomalies in noncommutative field theories have been
considered by several authors langmann ; bonora ; brandt ; varshovi-consistent
,111111In langmann ; bonora ; brandt  the authors considered the Moyal product
and in varshovi-consistent  general translation-invariant noncommutative star
product has been considered via some matrix model. but however, the
topological/geometric meaning of the solutions and of the corresponding
formulations remained unclear especially for the case of general translation-
invariant noncommutative star products. Thus, in this section the subjective
is to studying topological anomalies of translation-invariant noncommutative
Yang-Mills theories via homotopy classes due to Bismut-Freed determinant
bundle and the family index approach bismut-freed1 ; bismut-freed2 ; varshovi-
cyclic-moyal .

Let us assume the action of $\tilde{G}$ on $\Gamma$ of (III.1) is free so that
$\Gamma\to\Gamma/\tilde{G}$ provides a principal $\tilde{G}$-bundle. We also
suppose that $\tilde{X}:=\Gamma/\tilde{G}$ is a smooth manifold. Remember that
for any $A\in\Gamma$, the Dirac operator is perturbed to unbounded Hermitian
operator $\mathcal{D}_{A}:=\mathcal{D}+A$. Thus,
$(\mathcal{H},\mathcal{D}_{A},\gamma)$ is also regarded as an even
$p$-summable Fredholm module homotopic to $(\mathcal{H},\mathcal{D},\gamma)$.
Also

$\mathcal{D}_{A}=\left({\begin{array}[]{*{20}{c}}0&\mathcal{D}_{A}^{-}\\\
\mathcal{D}_{A}^{+}&0\\\ \end{array}}\right)~{},$

according to $\mathbb{Z}_{2}$-grading of $\gamma$. Thus,
$\mathrm{Ker}(\mathcal{D}_{A}^{+})\subset\mathcal{H}^{+}$ and
$\mathrm{Ker}(\mathcal{D}_{A}^{-})\subset\mathcal{H}^{-}$ are finite
dimensional. Consider the Bismut-Freed determinant line bundle bismut-freed1 ;
bismut-freed2

$\emph{\emph{Det}}:=\emph{\emph{Det}}(\mathcal{H},\mathcal{D},\gamma):=\emph{\emph{det}}(\mathrm{Ker}(\mathcal{D}_{A}^{+}))\otimes\emph{\emph{det}}(\mathrm{Ker}(\mathcal{D}_{A}^{-}))~{},$
(V.1)

which has a natural metric and unitary connection, say
$\nabla^{\emph{\emph{Det}}}=d_{\Gamma}-2\pi i\Pi(A)$, where $\Pi(A)$ is a one-
form on $\Gamma$, and $A\in\Gamma$. Here, $d_{\Gamma}$ is the exterior
differential operator on $\Gamma$. Actually $\Pi(A)$ is a closed form when it
is restricted to orbits of $\tilde{G}$ bismut-freed1 . Therefore, the
connection of $\nabla^{\emph{\emph{Det}}}$ is flat. That is;

$\displaystyle 2\pi i~{}\Pi(A)=\delta
W(A)~{},~{}~{}~{}~{}~{}\delta\Pi(A)=0~{},$ (V.2)

where $\delta$ is the exterior differential operator on $\tilde{G}$, the BRST
operator. Due to Bismut-Freed results $\Pi$ represents a non-trivial
cohomology class in $H^{1}_{\mathrm{dR}}(\tilde{G},\mathbb{C})$, since in fact
$W(A)$, the quantum action, is not in general a smooth function on $\Gamma$.
The BRST closedness of $\Pi$, the Wess-Zumino consistency condition, is then
an immediate consequence of (V.2). Actually, $\mathfrak{G}(A):=2\pi
i~{}\Pi(A)$ is the topological anomaly. The flatness of
$\nabla^{\emph{\emph{Det}}}$ on orbits of $\tilde{G}$ implies that the
parallelism structure on Det provides a covering space for Lie group
$\tilde{G}$. Thus, if $\mathfrak{i}:S^{1}\to\tilde{G}$ defines a smooth map
then;121212Actually, for commutative $\mathrm{SU}(N)$-Yang-Mills theories
$\mathbb{Q}$ can be replaced with $\mathbb{Z}$. But for the noncommutative
case the gauge group $\mathrm{SU}(N)$ has to be replaced with $\mathrm{U}(N)$,
since $F_{\star}$ and the Lagrangian contain anti-commutators of the
corresponding Lie algebra elements. This, leads to a non-trivial fundamental
group for the gauge transformation group $\tilde{G}$ hatcher . Thus, despite
to ordinary Yang-Mills theories, for those with noncommutative star products
we should deal with rational cohomology
$H^{*}_{\mathrm{dR}}(\tilde{G},\mathbb{Q})$ instead of
$H_{\mathrm{dR}}^{*}(\tilde{G},\mathbb{Z})$ for topological anomalies. This is
in fact due to $\mathrm{U}(1)$ component in
$\mathrm{U}(N)=\mathrm{U}(1)\times\mathrm{SU}(N)$. The only finite covering of
$\mathrm{U}(1)$ is $\mathrm{U}(1)$ itself. Therefore, the result of (V.3)
belongs to $\frac{{1}}{{l}}\mathbb{Z}$ for $l$-covering. For more details see
bertlman ; varshovi-cyclic-moyal ; griffith .

$\int_{S^{1}}\mathfrak{i}^{*}(\Pi)\in\mathbb{Q}~{}.$ (V.3)

Therefore, we readily find that $\Pi\in
H^{1}_{\mathrm{dR}}(\tilde{G},\mathbb{Q})$. In principal, the equality of
(V.3) should be invariant with respect to homotopy of 2-cocycles $s\alpha$ for
$s\in[0,1]$ and its corresponding translation-invariant noncommutative star
producs $\star_{s}$, with $\star_{1}=\star$ and $\star_{0}$ the ordinary
product. To see this let $\tilde{G}_{s}$ be the gauge transformation group
defined with $\star_{s}$. That is $\tilde{G}=\tilde{G}_{1}$ and
$\tilde{G}_{0}$ is the gauge transformation group in commutative
$\mathrm{U}(N)$-Yang-Mills theory. Then $M=\\{\tilde{G}_{s};~{}s\in
I=[0,1]\\}$ is topologically a cylindrical space with ends $\tilde{G}$ and
$\tilde{G}_{0}$. Actually, $M$ is topologically equivalent to
$I\times\tilde{G}$ and therefore it deformation retracts on both $\tilde{G}$
and $\tilde{G}_{0}$ hatcher ; varshovi-cyclic-moyal . Put a smooth structure
on $M$ with $d_{M}=\delta+ds\otimes\partial/\partial s$. Note that here
$\delta$ is the exterior derivative operator on $\tilde{G}_{s}$ for any
$s\in[0,1]$. Now, $\Pi$ can be assumed as a one-form on $M$. By homotopy
invariance of (V.3) we see that, there is some $\Phi\in C^{\infty}(M)$ so that
$\frac{{d}}{{ds}}\Pi=\delta\Phi$. In other words, if $\Pi_{0}$ be the Bismut-
Freed connection of commutative Yang-Mills theory;

$\Pi-\Pi_{0}=\delta\left(\int_{0}^{1}\Phi~{}ds\right)~{}.$ (V.4)

Moreover, $\tilde{G}$ and $\tilde{G}_{0}$ are homotopic equivalent. Let
$f:\tilde{G}\to\tilde{G}_{0}$ defines this equivalence relation. Then (V.4)
leads to the following theorem.

Theorem 7; _Let $\star$ be a general translation-invariant noncommutative star
product on $X$. Also suppose that $[\mathfrak{G}_{0}(A)]$ (resp.
$[\mathfrak{G}(A)]$) is the cohomology class of topological anomaly of
commutative (resp. $\star$-noncommutative) $\mathrm{U}(N)$-Yang-Mills theory
in $H^{1}_{\mathrm{dR}}(\tilde{G}_{0},\mathbb{C})$ (resp.
$H^{1}_{\mathrm{dR}}(\tilde{G},\mathbb{C})$). Then, we have;
$f^{*}([\mathfrak{G}_{0}(A)])=[\mathfrak{G}(A)]$. Moreover,
$\mathfrak{G}_{0}(A)/2\pi i$ (resp. $\mathfrak{G}(A)/2\pi i$) defines a
rational cohomology class in $H^{1}_{\mathrm{dR}}(\tilde{G}_{0},\mathbb{Q})$
(resp. $H^{1}_{\mathrm{dR}}(\tilde{G},\mathbb{Q})$)._

We should explain $f$ with more details. Let $\tilde{\mathfrak{g}}_{s}$ be the
infinitesimal gauge transformation group as the Lie algebra of
$\tilde{G}_{s}$. Moreover, let $\pi_{s}$ be the corresponding representation
map for star product $\star_{s}$. Therefore, $\\{\pi_{s}(e_{p})t^{a}\\}$ for
$p\in\mathbb{Z}^{2m}$ and $a=0,\cdots,N^{2}-1$ provides a basis for
$\tilde{\mathfrak{g}}_{s}$. Set
$df:\tilde{\mathfrak{g}}\to\tilde{\mathfrak{g}}_{0}$ with
$df(\pi(e_{p})t^{a})=\pi_{0}(e_{p})t^{a}$. Then, it is seen that its integral
provides a group isomorphism $f:\tilde{G}\to\tilde{G}_{0}$.131313For more
details of this proof see hall . In varshovi-cyclic-moyal  we also proved
similar results for the special case of Moyal product. Actually, $f^{*}$
replaces the ordinary product with translation-invariant noncommutative star
product $\star$ within differential forms. This leads us to the following
corollary.

Corollary 2; _Assume that the topological anomaly of commutative
$\mathrm{U}(N)$-Yang-Mills gauge theory, $\mathfrak{G}_{0}(A)$, is given by
integration of polynomial $P(A,C)$ over spacetime $X$ as;
$\mathfrak{G}_{0}(A)=\int_{X}P(A,C)$, for $C$ the ghost field. Then, the
topological anomaly of noncommutative $\mathrm{U}(N)$-Yang-Mills theory for
any translation-invariant noncommutative star product $\star$ is;_

$\mathfrak{G}(A)=\int_{X}P_{\star}(A,C)~{}.$ (V.5)

_wherein $P_{\star}$ is the noncommutative polynomial of $P$ due to (I.5)._

Note that according to Theorem 7 we know that

$\mathfrak{G}(A)=\int_{X}P_{\star}(A,C)+\delta\Phi(A)~{}$

for BRST exact term $\delta\Phi(A)$. Adding $-\Phi(A)$ to the quantum action
$W(A)$ as a counter term, i.e. $W^{\prime}(A)=W(A)-\Phi(A)$, to renormalize
the theory accordingly, then we obtain the topological anomaly
$\mathfrak{G}(A)$ as described in (V.5). In principal, the topological anomaly
of noncommutative $\mathrm{U}(N)$-Yang-Mills gauge theory on $4$-dimensional
spacetime $X$ and for any translation-invariant noncommutative star product
$\star$ is given for Weyl fermions as;141414Here the notation $\mathrm{tr}$ is
used for trace on both group colors and matrix representation due to $\pi$.

$\mathfrak{G}(A)=\frac{{1}}{{24\pi^{2}}}\int\mathrm{tr}\\{C\star d(A\star
dA+\frac{{1}}{{2}}A^{3})\\}~{}.$ (V.6)

Up to now we studied the topological anomaly within homotopy classes in
$H^{1}_{\mathrm{dR}}(\tilde{G},\mathbb{Q})$ via family index theory due to
Bismut-Freed determinant line bundle. However, it can also be regarded as a
noncommutative geometric problem through with the machinery of cyclic
(co)homology perrot ; perrot2 ; perrot1 .

To see this we note that the inclusion $\check{g}_{A}:\tilde{G}\to\Gamma$ as
an orbit which passes $A\in\Gamma$ is an invertible element of
$C^{\infty}(\Gamma)\otimes\mathcal{A}$, and hence defines a class in
$K_{1}^{\emph{\emph{alg}}}(C^{\infty}(\Gamma)\otimes\mathcal{A})$. Its Chern
character in odd periodic cyclic homology
$HP_{\mathrm{odd}}(C^{\infty}(\tilde{G}\otimes\mathcal{A})$ is in fact an
element of differential graded algebra $(\Lambda^{\prime},\hat{d})$ with
$\Lambda^{\prime}:=\oplus_{k\geq
0}\Lambda^{\prime}_{k}(C^{\infty}(\tilde{G})\otimes\mathcal{A})$ and
$\hat{d}=\delta+d$. For the Maurer-Cartan form
$\omega_{A}=\check{g}_{A}^{-1}\hat{d}\check{g}_{A}$ we read

$\emph{\emph{Ch}}^{1}_{*}[\check{g}_{A}]=\sum_{k\geq
0}~{}(-1)^{k}\frac{{k!}}{{(2k+1)!}}\omega_{A}^{2k+1}~{}\in
HP_{\mathrm{odd}}(C^{\infty}(\tilde{G})\otimes\mathcal{A})~{}.$ (V.7)

for $\emph{\emph{Ch}}^{1}_{*}[\check{g}_{A}]$ the corresponding Connes-Chern
character in odd periodic cyclic homology for $\check{g}_{A}$. An important
formula is perrot ; connes-moscovici-cs ;

$CS(\mathcal{H},\mathcal{D},\gamma)=\left\langle{\emph{\emph{Ch}}_{*}^{1}[\check{g}_{A}],\emph{\emph{Ch}}_{1}^{*}(\mathcal{H},\mathcal{D},\gamma)}\right\rangle~{},$
(V.8)

wherein $\emph{\emph{Ch}}_{1}^{*}(\mathcal{H},\mathcal{D},\gamma)$ is the
class of Connes-Chern character in odd periodic cyclic cohomology
$HP^{\mathrm{odd}}(C^{\infty}(\tilde{G})\otimes\mathcal{A})$ via (IV.3) and
$CS(\mathcal{H},\mathcal{D},\gamma)$ is the corresponding Chern-Simons form.
The pairing (V.8) is in fact taken place via cup product

$\cup:H_{k}(\tilde{G},\mathbb{C})\otimes HC^{j}(\mathcal{A})\to
HC^{k+j}(C^{\infty}(\tilde{G})\otimes\mathcal{A})~{},$

and hence, it represents de Rham cohomology classes in
$H_{\mathrm{dR}}^{\mathrm{odd}}(\tilde{G},\mathbb{C})$. The main result is
that the component in $H_{\mathrm{dR}}^{1}(\tilde{G},\mathbb{C})$ of pairing
(V.8) coincides with $\Pi(A)$ up to some exact form on $\tilde{G}$. Therefore,
if $i:S^{1}\to\tilde{G}$ is a smooth map, then we find;

$\int_{S^{1}}i^{*}(\mathfrak{G}(A))=2\pi
i\left\langle{\emph{\emph{Ch}}_{*}^{1}[\check{g}_{A}],[S^{1}]\cup\emph{\emph{Ch}}_{2}^{*}(\mathcal{H},\mathcal{D},\gamma)}\right\rangle\in
2\pi i~{}\mathbb{Q}~{},$ (V.9)

where $i:S^{1}\to\tilde{G}$ is considered to represent a class in
$H_{1}(\tilde{G},\mathbb{C})$, say $[S^{1}]$. Calculating (V.9) is actually
accomplished by using Connes-Moscovici local index formula as a residue of
some zeta function.151515See perrot  for more details.

##  VI Summary and Conclusions

Through this paper we introduced a cohomology theory, so called
$\star$-cohomology, with cohomology groups $H_{\star}^{k}(X,\mathbb{C})$, on
spacetime manifold $X$, to describe general translation-invariant
noncommutative quantum field theories by means of both commutative and
noncommutative geometric structures. In fact, $\star$-cohomology plays an
intermediate role between de Rham and cyclic (co)homology theories for
noncommutative algebras. It provides a breakthrough between commutative and
noncommutative quantum field theories and thus is comparable to the Seiberg-
Witten map.

Employing this framework for Chern-Weil theory we introduced three types of
Chern characters so that the third type, $\check{\mathrm{ch}}_{*}$, is shown
to belong to $H_{\star}^{*}(X,\mathbb{Z})$ and has intimate correlation to
Connes-Chern characters in cyclic (co)homology groups. On the other hand,
$\check{\emph{\emph{ch}}}_{*}$ induces integral classes in
$H_{\mathrm{dR}}^{*}(X,\mathbb{Z})$ which include significant information
about topology of translation-invariant noncommutative Yang-Mills theories
over $X$. Therefore, the topology of Abelian and topological anomalies of
translation-invariant noncommutative Yang-Mills theories were studied
thoroughly with correlation to Connes-Chern characters in cyclic (co)homology
groups and the machinery of noncommutative geometry.

##  VII Acknowledgments

The author says his gratitude to S. Ziaee who was the main reason for
appearing this article. Moreover, the author wishes to dedicate this work to
Mohammad Reza Shajarian for all he has done to Iranian art and culture along
last fifty years. Finally, my special thanks and highest regards go to the
esteemed referee and the respectable editor of ROMP for all their honest
considerations.

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