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ncindex15rev.tex
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\preprint{
SAGA-HE-223 \\ 
KEK-TH-1085 \\
%KEK-TH-1050 \\
UTHEP-514}
\title{
Probability distribution of the index\\
in gauge theory on 2d non-commutative geometry
%topological charge
%Topologically nontrivial gauge configurations
%in 2d non-commutative U(1) gauge theory
%geometry
%
%Suppression of topologically nontrivial sectors\\
%in gauge theory on 2d non-commutative geometry
%in gauge theory on non-commutative geometry \\
%--- a possible solution to the strong CP problem ---
%--- a Monte Carlo study in 2d U(1) case ---
%possible solution to strong CP problem 
%Probability distribution of the topological charge
%Topologically nontrivial gauge configurations
%in 2d non-commutative U(1) gauge theory
%geometry
%in non-commutative geometry
}

\author{
Hajime Aoki${}^{a}$,
Jun Nishimura${}^{bc}$ and
Yoshiaki Susaki${}^{bd}$\\ 
\llap{$^a$}Department of Physics, Saga University, 
Saga 840-8502, Japan \\
\llap{$^b$}High Energy Accelerator Research Organization (KEK), \\
Tsukuba, Ibaraki, 305-0801, Japan \\
\llap{$^c$}Department of Particle and Nuclear Physics,\\
%School of High Energy Accelerator Science,\\
Graduate University for Advanced Studies (SOKENDAI),\\
Tsukuba, Ibaraki 305-0801, Japan \\
\llap{$^d$}Graduate School of Pure and Applied Science,
University of Tsukuba,\\
Tsukuba, Ibaraki 305-8571, Japan\\
%\llap{$^d$}Institute of Physics, University of Tsukuba, 
%Tsukuba, Ibaraki 305-8571, Japan\\
%{}%Email:
\email{haoki@cc.saga-u.ac.jp,
jnishi@post.kek.jp,
susaki@post.kek.jp}} 

\abstract{
We investigate the effects of non-commutative geometry
on the topological aspects of gauge theory
using a non-perturbative formulation based on the twisted reduced
model.
The configuration space is decomposed into topological sectors
labeled by the index $\nu$ of the overlap Dirac operator 
satisfying the Ginsparg-Wilson relation.
We study the probability distribution of $\nu$ by Monte Carlo 
simulation
of the U(1) gauge theory on 2d non-commutative space with
periodic boundary conditions.
%in the 2d U(1) case.
%which was proved to have a sensible continuum limit
%and hence to be nonperturbatively renormalizable.
In general the distribution is asymmetric under 
$\nu \mapsto -\nu$,
%which is considered 
reflecting the parity violation due to non-commutative geometry.
%is possible due to the parity violation 
%by non-commutative geometry.
%should be generally the case in non-commutative geometry
%due to lack of parity invariance.
In the continuum and infinite-volume limits, however,
the distribution turns out to be dominated 
%by a single topological sector.
by the topologically trivial sector.
This conclusion is consistent with the
instanton calculus in the continuum theory.
However, it is in striking contrast to the known results in the
commutative case obtained from lattice simulation,
where
%In the latter case, 
%In that case, 
the distribution is Gaussian in a finite volume,
but the width diverges
%  and it becomes {\em flat}
in the infinite-volume limit.
%grows
%linearly with the physical extent of the 2d space.
We also calculate
the average action in each topological sector,
%as has been done by Gattringer, Hip and Lang in the commutative case,
and provide deeper understanding of the observed phenomenon.
%effects of non-commutative geometry.
%% Our conclusion is also supported by the
%% behavior of
%% the average action in each topological sector in the above limit,
%% and it is consistent with the instanton calculus
%% in the continuum theory.
%we obtain only topologically
%trivial configurations at least in the present model.
%
%% As we increase the system size, we find that the
%% probability of finding a topologically nontrivial 
%% configuration is suppressed exponentially,
%% and in the continuum limit we obtain only topologically
%% trivial configurations.
%
%% The observed situation is in striking contrast to 
%% the corresponding commutative case, 
%% %which was studied by Gattringer, Hip and Lang.
%% where the distribution becomes gaussian of finite width
%% with an appropriate scaling.
%
%We speculate that non-commutative geometry may provide
%a possible solution to the strong CP problem.
%
%on a possible solution to the strong CP problem.
%As an interesting consequence, the so-called $\theta$-vacuum
%there is no dependence
%on the CP violating theta term 
}

\keywords{Non-Commutative Geometry,
%Non-commutative geometry, %
Nonperturbative Effects,
Solitons Monopoles and Instantons}



\begin{document}

\section{Introduction}

Non-commutative (NC) geometry \cite{Sny,Connes}
has been studied for quite a long time
as a simple modification of our notion of space-time
at short distances possibly due to effects of 
quantum gravity \cite{gravity}.
It has attracted much attention since it was shown to 
appear naturally from matrix models \cite{CDS,AIIKKT}
and string theories \cite{String}.
In particular, field theory on NC geometry has
a peculiar property known as the UV/IR mixing \cite{rf:MRS},
which may cause a drastic change of the long-distance physics
through quantum effects.
This phenomenon has been first discovered in perturbation theory, 
but it was shown to appear also in a fully nonperturbative
setup \cite{AMNS}.
A typical example is the spontaneous breaking of the
translational symmetry in NC scalar field theory, 
which was first conjectured 
from a self-consistent one-loop analysis \cite{GuSo}
and confirmed later on by Monte Carlo simulation 
\cite{Procs,AC,Bietenholz:2004xs}. (See also \cite{ChenWu,CZ}.)


The appearance of a new type of IR divergence due to the UV/IR mixing
spoils the perturbative renormalizability in most cases 
\cite{Chepelev:1999tt},
%pauses a severe obstacle to the perturbative renormalization procedure
%\cite{Chepelev:1999tt}
and therefore, even the existence of a sensible field theory on a NC
geometry is {\em a priori} debatable.
%A nonperturbative approach based on the lattice regularization is 
In order to study such a nonperturbative issue,
% as well as other nonperturbative ones,
one has to define a regularized field theory on NC geometry.
This can be done by using matrix models.
In the case of NC torus, for instance, 
the so-called twisted reduced model \cite{EK,GAO}
is interpreted
as a lattice formulation of NC field theories \cite{AMNS},
in which finite $N$ matrices are mapped
one-to-one onto fields on a periodic lattice.
%the finite $N$ matrix model
%formulation of NC field theories can be interpreted
%as a lattice formulation of NC field theories \cite{AMNS}.
The existence of a sensible continuum limit
and hence the nonperturbative renormalizability
have been shown by Monte Carlo simulations
in NC U(1) gauge theory in 2d \cite{2dU1} and 4d
\cite{4dU1}
%{Bietenholz:2004wk,Bietenholz:2005iz,Volkholz:2005mp}
as well as in NC scalar field theory 
in 3d \cite{Bietenholz:2004xs,Bietenholz:2004as}.

In the case of fuzzy sphere \cite{Madore}, finite $N$ matrices
are mapped one-to-one onto functions on the sphere
with a specific cutoff on the angular momentum.
The fuzzy sphere (or fuzzy manifolds 
\cite{Dolan:2003kq,O'Connor:2003aj} in general)
preserves the continuous
symmetry of the base manifold, which makes it an interesting 
candidate for a novel regularization of {\em commutative} field theories
alternative to the lattice \cite{Grosse:1995ar}.
%It is also interesting to use fuzzy spheres in
%the coset space dimensional reduction \cite{Aschieri:2003vy}.
Stability of fuzzy manifolds in
matrix models with the Chern-Simons term \cite{Myers:1999ps,0101102}
has been studied by Monte Carlo simulations
\cite{Azuma:2004zq,fuzzy-MC}.
%Azuma:2004yg,
%Azuma:2005pm,Anagnostopoulos:2005cy,Azuma:2004ie,Azuma:2004qe}.


One of the interesting features of NC field theories
is the appearance of a new type of topological objects,
which are referred to as NC solitons \cite{GMS},
%,Harvey:2000jb},
NC monopoles, NC instantons, and fluxons \cite{fluxon}
in the literature.
They are constructed by using a projection operator,
and the matrices describing such configurations
are assumed to be infinite dimensional.
%Solitons have been constructed in the NC scalar field theory
%with a large NC parameter by using a projection operator
%\cite{GMS}.
In finite NC geometries topological objects have been constructed 
by using the algebraic K-theory and projective modules 
\cite{non-trivial_config,%non_chi,Valtancoli:2001gx,
%Steinacker:2003sd,Karabali:2001te,Carow-Watamura:2004ct,%
Balachandran:2003ay,AIN3}.

Dynamical aspects of these topological objects are
of particular importance
in the realization of a chiral gauge theory in our four-dimensional
world by compactifying a string theory
with a nontrivial index in the compactified dimensions.
Ultimately we hope to realize such a scenario {\em dynamically},
for instance, in the IIB matrix model \cite{9612115},
in which the dynamical generation of {\em four-dimensional} space-time
\cite{Aoki:1998vn,spaceiib,%
%{Aoki:1998vn,9811220,Ambjorn:2000bf,%
%Ambjorn:2000dx,NV,Burda:2000mn,%
%Ambjorn:2001xs,exact,sign,
gaussian}
%Nishimura:2001sx,KKKMS,
%Vernizzi:2002mu,%
%Kawai:2002ub}
%0307007,Nishimura:2004ts,Bal:2004ai}
as well as the gauge group \cite{Iso:1999xs,0504217}
has been studied intensively.
%In particular the emergence of  space-time
%has been suggested in refs.\ 
%\cite{Nishimura:2001sx,KKKMS,Kawai:2002ub}.
A crucial link in generating chiral fermions from a matrix model
is provided 
%In the commutative continuum space, the index theorem relates 
by the index theorem \cite{Atiyah:1971rm}, which relates 
the topological charge of an arbitrary gauge configuration
to the index of the Dirac operator on that background.
%with definite chirality.
The index theorem can be proved 
in noncommutative $\R ^ d$ in the same way as in the
commutative case \cite{Kim:2002qm}.


Extension of the index theorem to {\em finite} NC geometry
is a non-trivial
issue due to the doubling problem of the naive Dirac action.
In lattice gauge theory, an analogous problem was
solved by the use of the so-called overlap 
Dirac operator \cite{Neuberger,Narayanan:1994gw,Hasenfratzindex,Luscher},
which satisfies
the Ginsparg-Wilson relation \cite{GinspargWilson}.
The ideas developed in lattice gauge theory
have been successfully extended to NC geometry.
%The aim of this paper is to extend the index theorem 
%to NC geometry by generalizing the ideas 
%developed in lattice gauge theory.
In the case of NC torus, the overlap Dirac operator has been
introduced in ref.\ \cite{Nishimura:2001dq}, and 
it was used to define a NC chiral gauge theory
with manifest star-gauge invariance.
For general NC manifolds,
a prescription to define 
the Ginsparg-Wilson Dirac operator and its index
has been provided in ref.\ \cite{AIN2},
and the fuzzy sphere was considered as a concrete example
\footnote{The Ginsparg-Wilson Dirac operator for vanishing gauge field 
was constructed earlier in refs.\ \cite{balagovi}.}.
The Ginsparg-Wilson algebra
for the fuzzy sphere 
has been studied in detail in each topological 
sector \cite{Balachandran:2003ay}.
%Chiral anomaly on the fuzzy sphere was evaluated
%in refs.\ \cite{AIN2,AIN3,Balachandran:2003ay}.
%Ydri:2002nt,
%
%(See ref.\ \cite{Balachandran:2003ay} for a more detailed analysis.)
%In ref.\ \cite{Balachandran:2003ay} a more detailed analysis has
%been performed in the case of fuzzy sphere.
In ref.\ \cite{isonagao}
the overlap Dirac operator on the NC torus 
\cite{Nishimura:2001dq}
was derived also from this general prescription \cite{AIN2},
and the axial anomaly has been calculated in the continuum limit.
%Owing to the GW relation, an index theorem can be proved 
%even in finite NC geometries.
%A topological charge was defined, 
%and shown to take only integer values 
%and become the Chern character in the commutative limit.


In an attempt to construct a topologically nontrivial configuration
on the fuzzy sphere, an analogue of the 't Hooft-Polyakov monopole
was obtained \cite{Balachandran:2003ay,AIN3}.
Although the index defined through the Ginsparg-Wilson Dirac operator
vanishes for these configurations, one can 
make it non-zero
%define a non-zero topological charge 
by inserting a projection operator, which
picks up the unbroken U(1) component of the SU(2) gauge group.
%It has a correct commutative limit,
%and takes suitable integer values 
%for the TP monopole configurations. 
In fact the 't Hooft-Polyakov monopole configurations are precisely
the meta-stable states observed in Monte Carlo simulations \cite{Azuma:2004zq}
taking the two coincident fuzzy spheres as the initial configuration,
which eventually decays into a single fuzzy sphere.
In ref.\ \cite{AIMN}
this instability was studied analytically by
the one-loop calculation of free energy
around the 't Hooft-Polyakov monopole 
configurations,
%Since the two coincident fuzzy spheres
%correspond to the trivial U(2) vacuum, 
%the above instability 
and it was interpreted as
the dynamical generation of a nontrivial index,
which may be used for the realization of a chiral fermion in our space-time.
%in the context of string theories
%
%% Note, however, that the definition of the topological charge,
%% which includes the projection operator 
%% to assign nonzero indices to the 
%% 't Hooft-Polyakov monopoles,
%% cannot be extended to other gauge configurations.



%To our knowledge, the index theorem has not been studied yet 
%in NC geometry.
In our previous work \cite{Aoki:2006sb}, 
we have demonstrated the validity of 
the index theorem in finite NC geometry,
taking the 2d U(1) gauge theory on a discretized NC torus as a 
simple example,
which is studied extensively in the literature
both numerically \cite{2dU1}
and analytically \cite{Mafia,Paniak:2002fi,
%Torrielli:2003sf,%
%Paniak:2003gn,Paniak:2003yf,Paniak:2003xm,%
Griguolo:2003kq}.
%Dorn:2003wi,Ambjorn:2004ck,%
%Fatollahi:2005ri,Bassetto:2005hn,Cirafici:2005af}.
In particular, ref.\ \cite{Griguolo:2003kq} presents
general classical solutions carrying the topological charge.
We computed the index defined through the Ginsparg-Wilson Dirac operator
for these classical solutions 
%% \footnote{Unlike in previous works we do not introduce 
%% the projection operator,
%% and therefore the index can be defined for arbitrary configurations.
%% }
and compared the results with the topological charge. 
%to assign nonzero indices to the 't Hooft-Polyakov monopoles,
%(except for singular configurations, which are irrelevant in the
%path integral), 
The index theorem holds when the action is small,
but the index takes only multiple integer values 
of $N$, the size of the 2d lattice.
For non-zero indices, the action is finite in the large $N$ limit,
but it diverges when the bare coupling constant is tuned in the continuum
limit.
By interpolating the classical solutions, we constructed explicit 
configurations
for which the index is of order 1, but
the action becomes of order $N$.
These results 
%implies that it is difficult to construct 
%a configuration with non-zero indices which survive 
suggested that the probability of obtaining 
a non-zero index
vanishes
%is highly suppressed 
in the continuum limit. 


In this paper we confirm this statement 
{\em at the quantum level}
%from first principles
by performing Monte Carlo simulation of
the 2d U(1) gauge theory on a NC discretized torus.
Since the theory is known to have a sensible 
continuum limit \cite{2dU1},
we investigate the probability distribution of the index
in that limit.
Comparison with the known results in the corresponding commutative case
obtained from lattice simulation
\cite{GHL} allows us to reveal the striking effects of NC geometry.


%% This property is in striking contrast to the
%% corresponding commutative case,
%% where the probability of obtaining a nonzero index remains 
%% finite
%% \cite{GHL,Bardeen:1998eq,FukayaOnogi},
%% and provides a possibility
%% to solve the strong CP problem on account of
%% non-commutative geometry.

The rest of this paper is organized as follows.
In section \ref{model} 
we define the model and the index of the
overlap Dirac operator.
In section \ref{sec:prob-index} we show our results
for the probability distribution of the index.
In section \ref{sec:average-action} we discuss
the average action in each topological sector,
which provides qualitative understanding for
the behavior of the probability distribution.
Section \ref{summary} is devoted to a summary and discussions.

%\bigskip
%\noindent {\bf Note added.~}


\section{The model and the topological sectors}
\label{model}
% model
In this section we define the model and 
the topological sectors based on the matrix model formulation
of NC gauge theory.
%, and discuss how to take the continuum limit.
% the matrix model,
%which represents the non-perturbative formulation
%of gauge theory on  
For more details such as the interpretation
of matrices as fields on a NC torus,
we refer the reader to our previous paper \cite{Aoki:2006sb}.

 
The model we study in this paper is given by the action
%\footnote{This action differs from the one written
%in }
\beq
S = N^2 \beta \, \sum_{\mu \ne \nu} 
\left\{ 1 - \frac{1}{N}
{\cal  Z}_{\nu\mu} \, 
\tr \Bigl(V_\mu\,V_\nu\,V_\mu^\dag\,V_\nu^\dag\Bigr)
\right\}  \ ,
\label{TEK-action}
\eeq
%% \beqa
%% S &=& - N \beta \, \sum_{\mu \ne \nu} 
%% \tr ~\Bigl(\hat{U}_\mu\,\Gamma_\mu 
%% \hat{U}_\nu \Gamma_\mu^\dag\,
%% \Gamma_\nu \hat{U}_\mu^\dag \Gamma_\nu^\dag
%% \, \hat{U}_\nu^\dag \Bigr) \\
%% &=& - N \beta \, \sum_{\mu \ne \nu} 
%% {\cal  Z}_{\mu\nu}
%% \tr ~\Bigl(V_\mu\,V_\nu\,V_\mu^\dag\,V_\nu^\dag\Bigr) \ ,
%% \label{TEK-action}
%% \eeqa
where ${\cal Z}_{\mu\nu}={\cal  Z}_{\nu\mu}^*$ is a
phase factor given by \cite{Nishimura:2001dq}
\beq
%{\cal  Z}_{12}=\exp\left(\pi i \frac{N+1}{N} \right) \ . 
{\cal  Z}_{12}=\exp\left(\pi i \frac{N+1}{N} \right)
%  \ ,
%\quad \quad \quad  M = \frac{N+1}{2} 
\label{def-twist}
\eeq
with $N$ being an odd integer.
The NC tensor $\Theta_{\mu\nu}$, 
which characterizes NC geometry
$[x_\mu , x_\nu] = i \Theta_{\mu\nu}$, is given by
\beq
\Theta_{\mu\nu} = \vartheta \, \epsilon_{\mu\nu}  \ ,
\quad \quad \quad
\vartheta =\frac{1}{\pi} N a^2  \ .
\label{theta-def}
\eeq
%Note that the size of the torus $\ell = N a$ goes to $\infty$
%in the continuum limit $a \rightarrow 0$ fixing $\theta$.
%
%A finite torus can be obtained 
%by other constructions given in the first paper of refs.\ \cite{AMNS} 
%and ref.\ \cite{Griguolo:2003kq}, which are mutually equivalent.


%\subsection{the double scaling limit}

%% This is nothing but the twisted Eguchi-Kawai (TEK) model \cite{GAO}
%% except that the phase factor ${\cal  Z}_{\mu\nu}$,
%% which is called the ``twist'', should be taken to be of the 
%% specific form (\ref{tH-W-alg}).
%% Historically, the TEK model appeared as a matrix model which is 
%% equivalent to the large $N$ gauge theory.
%% \footnote{In that context, the ``twist'' does not have to be of
%% this particular form.
%% In fact in the present 2d case, one can even omit the twist; i.e., 
%% the original Eguchi-Kawai model is fine.}
%% By taking the large-$N$ limit with fixed $\beta$, one can calculate
%% Wilson loops in the SU($\infty$) lattice gauge theory, which
%% was obtained earlier by Gross and Witten \cite{GW}.
%% This large-$N$ limit is called the ``planar limit'' since only planar
%% diagrams survive in the perturbation theory.


Since the NC parameter $\vartheta$
is related to the lattice spacing by (\ref{theta-def}),
we have to take the large $N$ limit together with the continuum limit
$a \rightarrow 0$ in order to obtain
a continuum theory with finite $\vartheta$.
In that limit the physical extent of the torus $\ell = a N$
goes to infinity at the same time.
Whether one can obtain a sensible continuum limit
by tuning $\beta$ appropriately
is a non-trivial issue, which has been addressed in ref.\ \cite{2dU1}.
It turned out that $\beta$ should be sent to $\infty$ as
\beq
\beta \propto \frac{1}{a^2} \ .
\label{relation_a_beta}
\eeq
Combining this with (\ref{theta-def}), one finds that
the large $N$ limit should be taken together with 
$\beta \rightarrow \infty$ limit so that 
$\beta/N$ is fixed. This limit is
called the ``double scaling limit'', in which non-planar
diagrams survive.
If one takes the planar limit ($N \rightarrow \infty$ with fixed $\beta$)
instead, one obtains a gauge theory 
on a NC space with $\vartheta=\infty$.
In this limit the Wilson loops agree 
\footnote{
At finite $\vartheta$,
the agreement holds 
only when the physical area surrounded by the Wilson loop is much
smaller than $\vartheta$. 
%(See ref.\ \cite{Konagaya:2005nb}
%for a clarification of this fact in perturbation theory.)
As a consequence, the relation (\ref{relation_a_beta}) agrees
with the one required for the continuum limit of
the SU($\infty$) lattice gauge theory \cite{GW}.
}
with the SU($\infty$) lattice gauge theory \cite{GW}
due to the Eguchi-Kawai equivalence \cite{EK}.
%% We study the topological charge distribution
%% in NC gauge theory using the TEK model,
%% and discuss in particular its behavior in the double scaling limit.
In particular the expectation value of the action in this limit
is given by \cite{GW}
\beq
\langle S \rangle =\left\{
\begin{array}{ccc}
2 \beta N^2 (1-\beta) &~~ \mbox{for} & \beta  < \frac{1}{2}  \ , \\
 \frac{1}{2}N^2 &~~\mbox{for}& \beta \geq \frac{1}{2}  \ ,
\end{array}\right. 
\label{S-GW}
\eeq
which shows that the system undergoes a third order phase transition
at $\beta = \beta_{\rm cr}\equiv 1/2$.


%%%%%%%%
% definition of the topological charge
%\subsection{the Ginsparg-Wilson Dirac operator and its index}

The configuration space can be naturally 
decomposed into topological
sectors by the index of the overlap Dirac operator 
on the discretized NC torus \cite{Nishimura:2001dq,AIN2,isonagao}.
%for which the index is well defined.
Let us define the covariant forward (backward) difference operator
\beqa
\nabla_\mu \Psi&=&
\frac{1}{a}\left[V_\mu \Psi \Gamma_\mu
- \Psi  \right] \ , \nonumber \\
\nabla_\mu^* \Psi &=&
\frac{1}{a}\left[\Psi - V_\mu ^\dagger 
\Psi  \Gamma_\mu \right] \ ,
\label{def-cov-shift}
\eeqa
%instead of eq.\ (\ref{def-cov-shift}).
where the SU($N$) matrices $\Gamma_\mu$ ($\mu = 1, 2$)
satisfy the 't Hooft-Weyl algebra
\beq
\label{tH-W-alg}
\Gamma_\mu \Gamma_\nu = {\cal  Z}_{\mu\nu}
\Gamma_\nu \Gamma_\mu   \ .
\eeq
Given the covariant forward (backward) difference operator,
we can define the overlap Dirac operator in precisely the same
way as in the commutative case.


First the Wilson-Dirac operator can be defined as
\beq
D_{\rm W}=\frac{1}{2}\sum_{\mu=1}^2
\left\{\gamma_\mu\left(\nabla_\mu^* 
+\nabla_\mu \right) - a \nabla_\mu^* \nabla_\mu \right\} \ ,
\label{def-Wilson-Dirac}
\eeq
where $\gamma_\mu$ ($\mu = 1,2$) are the gamma matrices in 2d.
A crucial property of the overlap Dirac operator $D$
%In order to define the topological sectors for finite NC geometry,
%we need a Dirac operator $D$ satisfying 
is the Ginsparg-Wilson relation \cite{GinspargWilson}
\beq
\gamma_5 D + D\gamma_5 = a \, D\gamma_5 D \ ,
\label{GWrel}
\eeq
where $\gamma_5 = - i \gamma_1 \gamma_2$ is the chirality operator.
Assuming the $\gamma_5$-hermiticity $D^\dagger =\gamma_5 D\gamma_5$,
we can define a hermitean operator $\hat\gamma_5$ by
\beq
{\hat \gamma_5}=\gamma_5\left(1- a D\right) \ ,
\eeq
which may be solved for $D$ as 
$D=\frac{1}{a} (1 - \gamma_5 \hat{\gamma}_5)$.
Then the Ginsparg-Wilson relation (\ref{GWrel}) is equivalent to
requiring $\hat\gamma_5$ to be unitary.
%\beq
%(\hat \gamma_5)^2=1  \ ,
%\eeq
The overlap Dirac operator 
%from the overlap formalism \cite{Narayanan:1994gw}
corresponds to taking $\hat\gamma_5$ to be \cite{Neuberger}
\beqa
\hat\gamma_5 &=& \frac{H}{\sqrt{H^2}} \ , \\
H &=& \gamma_5 \left(1-aD_{\rm W}\right) \ ,
\label{H-def}
\eeqa
where $D_{\rm W}$ is the Wilson-Dirac operator.

One can define the index of $D$ unambiguously by
%othe index of the Dirac operator is defined as
$\nu \equiv
%{\rm Index} (D) 
n_+ - n_-$,
where $n_{\pm}$ is the number of zero modes
with the chirality $\pm 1$.
%$\hat{\gamma}_5 = \pm 1$
%Due to eq.\ (\ref{GWrelation}), 
It turns out that \cite{Narayanan:1994gw,Hasenfratzindex,Luscher}
\beq
\nu 
%{\rm Index} (D)
=\frac{1}{2} {\cal T}r ( \gamma_5 + \hat\gamma_5) 
= \frac{1}{2} {\cal T}r \frac{H}{\sqrt{H^2}} \ , 
\label{indexD-H}
\eeq
where ${\cal T}r$ represents a trace over the space of matrices
and over the spinor index.


We performed Monte Carlo simulation of the model (\ref{TEK-action})
% at odd $N$ 
using the heat bath algorithm as in ref.\ \cite{2dU1}.
For each configuration $V_\mu$ generated by simulation,
we calculate the index (\ref{indexD-H}).
We diagonalize the hermitean matrix $H$ defined by (\ref{H-def}), 
and count the number of positive and negative eigenvalues.
(Note that the lattice spacing $a$ which appears in the expressions 
(\ref{def-cov-shift}), (\ref{def-Wilson-Dirac}) and (\ref{H-def})
cancel each other, and the index does not depend explicitly
on $a$.)
The computational effort for calculating
%the topological charge and the action is of order $N^3$,
%while that for calculating 
the index is of order $N^6$,
since we have to diagonalize the $2N^2 \times 2N^2$
hermitean matrix $H$.
%This situation is similar to simulating
%supersymmetric matrix models using the hybrid Monte Carlo algorithm


%We define the index 
%as half of the difference between the positive and 
%the negative eigenvalues of the matrix,
%\beq
%H=\gamma_5 \left(1-aD_{\rm W}\right)
%\eeq
%where
%\beq
%D_{\rm W}=\frac{1}{2}\sum_{\mu=1}^D 
%\left\{\gamma_\mu\left(\nabla_\mu^* +\nabla_\mu -
%\nabla_\mu^* \nabla_\mu\right) \right\}.
%\eeq

%%%%%%%%
% definition of the topological charge
%\subsection{definition of the topological charge}



%\newpage

%\section{Results}
%\label{results}

% beta=0
\section{Probability distribution of the index}
\label{sec:prob-index}

In this section we present our results for 
the probability distribution of the index $\nu$
--- as computed 
by the definition (\ref{indexD-H}) ---
in the gauge theory on the NC torus.


    \FIGURE{
    \epsfig{file=eps_%
Ndistribution_indexoverN_planar_b0.00.eps,
%distribution_index_N15.eps,%
angle=270,width=7.4cm}
%    \epsfig{file=distribution_index_planar_b0.55.eps,%
%angle=270,width=7.4cm}
\caption{The probability
distribution of $\frac{\nu}{N}$ for various $N$ at $\beta=0$.}
\label{b0}
}

In figure \ref{b0}
we plot the probability distribution of $\nu$ 
for various $N$ at $\beta=0$.
This represents the distribution
in the configuration space
without taking account of the action.
%\footnote{Since $\beta = 0$ corresponds to the strong coupling limit, 
%the notion of the topological charge may not make sense.}
To our surprise, it turns out that the distribution 
of $\nu$ is asymmetric under $\nu \mapsto -\nu$.
This is in striking contrast to ordinary commutative theories,
in which the distribution of $\nu$ is symmetric 
due to parity invariance.
%(See, for instance, the result for
%the commutative 2d U(1) gauge theory \cite{GHL}.)
We also find that the distribution for the rescaled topological
charge $\nu/N$ at different $N$ lies on top of each other.
This behavior is analogous to what one obtains in the 
commutative continuum theory (See, for instance,
section 6 of ref.\ \cite{Aoki:2006sb}.).
%This behavior is analogous to the result
% (\ref{commutative})
%in the commutative case.
The plot also confirms 
the existence of $\nu \ne 0$ configurations
on the discretized NC torus. An example of such configurations
is found numerically in ref.\ \cite{Nagao:2005st},
and constructed analytically in 
section 5 of ref.\ \cite{Aoki:2006sb}.
The crucial question we address in what follows is 
whether such configurations survive in the continuum limit.


%% It means that the width of the distribution grows 
%% in order of $N$ as we increase $N$.
%% In other words, it implies that the contribution 
%% from the path integral measure to the probability 
%% of the distribution of the index should be
%% an asymmetric function of $\nu/N$.
%
%This is quite natural since the situation is 
%essentially the same as the random walk in the configuration space.




% various beta at N=15 --

    \FIGURE{
%    \epsfig{file=actoSwithFS.eps,width=7.4cm}
%    \epsfig{file=a-sqSwithFS.eps,width=7.4cm}
    \epsfig{file=eps_%
distribution_index_N15.eps,%
%log_distribution_index_N15.eps,%
%distribution_index_N15.eps,%
angle=270,width=7.4cm}
    \epsfig{file=eps_%
distribution_index_planar_b0.55.eps,%
angle=270,width=7.4cm}
\caption{The probability distribution of $\nu$ is plotted
for various $\beta$ at $N=15$ (left) and 
for various $N$ at $\beta=0.55$ (right).
In the right plot, the log scale is taken for the y-axis to
magnify the results at $\nu \neq 0$.
}
\label{distrib-N-beta}
}

Let us see how 
the probability distribution of $\nu$ changes as we switch on $\beta$.
In figure \ref{distrib-N-beta} (left) we 
plot the probability distribution $P(\nu)$
for various $\beta$ at $N=15$.
(Throughout this paper, we assume the normalization
$\sum_\nu P(\nu) = 1$.)
We find that the probability for $\nu \neq 0$ decreases rapidly,
and the probability for $\nu = 0$ approaches unity.
In figure \ref{distrib-N-beta} (right) we plot 
the probability distribution $P(\nu)$
for various $N$ at $\beta = 0.55$.
Note that the value of $\beta$ we have chosen
is above the critical point $\beta = \beta_{\rm cr}\equiv 1/2$
of the Gross-Witten phase transition.
We find that the distribution approaches
the Kronecker delta $\delta_{\nu 0}$
% instead of a gaussian distribution
not only for increasing $\beta$ but also for increasing $N$.
In figure \ref{logrho1overrho0}
we plot the ratio $P(\nu)/P(0)$ for $\nu=1,-1$
for various $\beta$ at $N=15$ (left) and 
for various $N$ at $\beta = 0.55$ (right).
In both cases we observe an exponentially decreasing behavior.

    \FIGURE{
%    \epsfig{file=actoSwithFS.eps,width=7.4cm}
%    \epsfig{file=a-sqSwithFS.eps,width=7.4cm}
%    \epsfig{file=logrho1overrho0_vs_N_b0.55_planar.eps,%
%angle=270,width=7.4cm}
    \epsfig{file=eps_%
rho1overrho0_vs_beta.eps,%
angle=270,width=7.4cm}
    \epsfig{file=eps_%
rho1overrho0_vs_N_b0.55_planar.eps,%
angle=270,width=7.4cm}
\caption{
The ratio $P(\nu)/P(0)$ for $\nu=1,-1$
is plotted in the log scale
as a function of $\beta$ at $N=15$ (left) and
as a function of $N$ at $\beta=0.55$ (right).
The straight lines represent a fit to an exponentially decreasing
behavior.
}
\label{logrho1overrho0}
}

As we mentioned in the previous section,
in order to take the continuum limit,
we have to send $N$ and $\beta$ to $\infty$
simultaneously fixing the ratio
$\beta/N$.
It is clear from the above results that
the distribution $P(\nu)$ approaches $\delta_{\nu 0}$
very rapidly in that limit.


\section{Average action in each topological sector}
\label{sec:average-action}

    \FIGURE{
%    \epsfig{file=actoSwithFS.eps,width=7.4cm}
%    \epsfig{file=a-sqSwithFS.eps,width=7.4cm}
    \epsfig{file=eps_%
distribution_action_b0.10_0.50.eps,%
angle=270,width=7.4cm}
\caption{The distribution of the action in the
%various topological sectors 
$\nu = 0$ and $\nu =-1$ topological sectors
is plotted for $\beta= 0.1$ and $\beta=0.5$ at $N=15$.
}
\label{dist_actionN15}
}



In this section 
we provide an explanation of our results in the previous section
by studying the action in each topological sector.
%A similar analysis was performed in the commutative case \cite{GHL}.
In figure \ref{dist_actionN15} we plot 
the distribution of the action $S$
for $\nu=0, -1$ at $\beta = 0.1$ and $\beta = 0.5$.
%where we see that there is a quantum fluctuation 
%around the average value of the action for each topological sector.
We find that at $\beta = 0.1$ the distribution for different
topological sector lies on top of each other, while
at $\beta = 0.5$ the distribution for $\nu =0$ differs much
from $\nu=-1$. We have also measured 
the distribution for $\nu =1,2,-2$, which turns out to be 
very close to the result for $\nu=-1$.



In figure \ref{actionN15}
we plot the average value of the action $\bar{S}(\nu)$ in 
each topological sector.
We find that the result is almost flat
except at $\nu = 0$.
%, and the overall scale grows as O($N^2$).
Note that the weighted sum of $\bar{S}(\nu)$ yields
%the expectation value of the action can be decomposed as
\beq
%\langle S \rangle = \sum_{\nu} \bar{S}(\nu)  P(\nu) \ .
\sum_{\nu} \bar{S}(\nu)  P(\nu) = 
\langle S \rangle \ ,
\eeq
where $\langle S \rangle$ is given by (\ref{S-GW})
in the planar limit.
When the $\nu=0$ sector dominates,
we have $\bar{S}(0) \simeq \langle S \rangle$.
This explains the behavior of $\bar{S}(0)$ in figure \ref{actionN15}.
%In the right plot, we indeed observe that 
%$\bar{S}(0) \simeq \frac{1}{2}N^2$.
%In the left plot, we find that
%the $\bar{S}(0) \simeq \frac{1}{2}N^2=112.5$ for $\beta \ge \frac{1}{2}$.


In both plots in figure \ref{actionN15},
we observe a dip at $\nu=0$.
%that the avarage action in the $\nu=0$ sector is substantially
%smaller than those in the $\nu \neq 0$ sectors.
In figure \ref{dipN15}
we plot the size of the dip defined by
\beq
\Delta S \equiv {\bar S}(-1) - {\bar S}(0) \ ,
\eeq
which shows that the dip grows linearly with both $\beta$ and $N$.
(From the left plot, we find that the linear behavior
sets in at $\beta \sim 0.5$, which is close to the critical point
of the Gross-Witten phase transition.)
This is consistent with the exponentially decreasing behavior
of the probability $P(\nu)/P(0)$
for $\nu \neq 0$ discussed in the previous section. 

In the commutative case \cite{GHL},
lattice simulation shows that 
the average action in each sector
increases quadratically with $\nu$, 
but the coefficient vanishes
in the infinite-volume limit.
Correspondingly the distribution of $\nu$
is Gaussian in a finite volume, but the 
width diverges in the infinite-volume limit.
Thus the situation in the NC case differs
drastically from the commutative case.


    \FIGURE{
%    \epsfig{file=actoSwithFS.eps,width=7.4cm}
%    \epsfig{file=a-sqSwithFS.eps,width=7.4cm}
    \epsfig{file=eps_%
action_vs_index_N15.eps,%
angle=270,width=7.4cm}
    \epsfig{file=eps_%
action_vs_index_planar_b0.55.eps,%
angle=270,width=7.4cm}
\caption{
The average value of the action
is plotted 
against the index $\nu$ for various $\beta$ at $N=15$ (left)
and for various $N$ at $\beta=0.55$ (right).
}
\label{actionN15}
}

    \FIGURE{
%    \epsfig{file=actoSwithFS.eps,width=7.4cm}
%    \epsfig{file=a-sqSwithFS.eps,width=7.4cm}
    \epsfig{file=eps_%
dip_of_nu0andnu-1_vs_beta_N15.eps,%
angle=270,width=7.4cm}
    \epsfig{file=eps_%
dip_vs_N_of_nu0andnu1_planar.eps,%
angle=270,width=7.4cm}
\caption{
The dip $\Delta S$ is plotted 
as a function of $\beta$ for $N=15$ (left)
and 
as a function of $N$ for various $\beta$ (right).
}
\label{dipN15}
}


\section{Summary and discussions}
\label{summary}

In this paper we have studied the effects
of NC geometry on the probability distribution 
of the index $\nu$ of the Dirac operator.
%of the topological sectors in gauge theory.
In the 2d U(1) gauge theory with periodic boundary conditions,
we found that
%the distribution is dominated by the trivial sector.
the probability for $\nu \neq 0$ is exponentially suppressed
in the continuum and infinite-volume limits.
% thermodynamic limits.
% for arbitrary $\vartheta$.
Our conclusion is consistent with our previous analysis at
the classical level \cite{Aoki:2006sb}
and with the instanton calculus
in the continuum theory \cite{Paniak:2002fi}.
In fact the topologically trivial sector includes
all the instanton configurations that contribute to 
the partition function.


In order to understand our conclusion intuitively, 
let us recall that in NC geometry,
the action involves the star product, 
which must have certain smoothening effects 
on the gauge field.
In the commutative case with periodic boundary
conditions,
%recall that in the commutative case with periodic boundary
a classical solution in a topologically non-trivial sector
has a constant field strength, but the vector potential
has a singularity. (See e.g., section 6 of 
ref.\ \cite{Aoki:2006sb}.)
It is therefore conceivable that such configurations 
cannot be realized in NC geometry.
Our results in section \ref{sec:average-action}
substantiate this argument.
% with finite 
%action, since the action involves the star product.


It follows from our conclusion that
special care must be taken when
one studies the $\theta$-vacuum in NC
geometry.\footnote{Strictly speaking, we need
to have the ordinary (commutative) time
in order to be able to speak about a ``vacuum''.
We may think of, for instance,
four-dimensional space-time
with non-commutativity introduced only
in two spatial directions \cite{4dU1}.
Let us also remind the readers that
the parameter $\theta$ should not be confused with
the non-commutativity parameter $\vartheta$.}
In general one has to sum over (topologically different) 
twisted boundary conditions labeled by $\nu$ 
with the phase factor $\ee^{i \nu \theta}$.
In the commutative case, however, one may equivalently
%simply 
add a $\theta$-term to the action and just integrate over
the lattice configuration with periodic 
boundary conditions, as is done e.g., 
in ref.\ \cite{FukayaOnogi}.
Our conclusion implies that this is no longer true
in NC geometry. In ref.\ \cite{Aoki:2006sb}
we speculated that NC geometry may 
provide a solution to the strong CP problem,
but this remains to be seen.
% whether NC geometry provides
%something to do with 
%a solution to the strong CP problem.


As another effect of NC geometry,
we found that in general the probability distribution 
of 
%the topological charge 
$\nu$ becomes 
asymmetric under $\nu \mapsto -\nu$,
reflecting the parity violation due to NC geometry.
This is interesting since it suggests
a possibility to obtain a non-zero vacuum expectation value
for the index $\nu$ in some NC model.
Alternatively, one can twist the boundary condition
to make a topologically non-trivial 
sector dominate in the continuum and infinite-volume 
limits \cite{ANS}.
%, in much the same way as we have seen in the
%present periodic  by 
%We have also checked 
%that we can make a topologically non-trivial 
%sector dominate in the continuum and infinite-volume 
%limits by 
%This will be reported in a forth-coming paper.
We expect that these unusual properties of NC geometry
%such a phenomenon 
may provide
%be closely related to 
a dynamical mechanism for realizing chiral fermions in
string theory compactifications, or a mechanism for
generating non-zero baryon number density in the universe.
See refs.\ \cite{AIMN,Aschieri:2003vy} for a related line of research
using fuzzy spheres in the extra dimensions.


From the motivations mentioned above, 
it would be interesting to extend the present analysis
to four dimensions.
Unlike the 2d case studied in this paper,
the perturbative vacuum is actually unstable 
due to the UV/IR mixing 
\cite{LLT,ruiz,Bassetto:2001vf,rf:MVR,rf:AL,Guralnik:2002ru}.
However, the system stabilizes after the condensation of the Wilson lines
and finds a stable nonperturbative vacuum
\cite{4dU1},
%{Bietenholz:2004wk,Bietenholz:2005iz,Volkholz:2005mp},
in which the translational invariance is spontaneously broken.
One can also stabilize the perturbative vacuum by keeping the
UV cutoff finite and regarding the model as a low-energy effective
theory. The situation may depend on which vacuum one chooses.
We hope to address such issues
%report on the results 
in future publications.
 

%% \begin{figure}[htb]
%% \begin{center}
%%      \resizebox{60mm}{!}{\includegraphics{figure_NCindex.eps}} 
%%      \caption{Configuration space in the non-commutaive space}
%%      \label{NCconfigspace}
%% \end{center}
%% \end{figure}

\acknowledgments

It is our pleasure to thank Hidenori Fukaya,
Satoshi Iso, Hikaru Kawai, Kenji Ogawa and Kentaroh Yoshida
for valuable discussions.


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\jhep{9912}{1999}{022}
[\hepth{9910053}].
%%CITATION = HEP-TH 9910053;%%

\bibitem{0101102}
S.~Iso, Y.~Kimura, K.~Tanaka and K.~Wakatsuki, \emph{Noncommutative
  gauge theory on fuzzy sphere from matrix model},
\npb{604}{2001}{121} [\hepth{0101102}].




%\cite{Azuma:2004zq}
\bibitem{Azuma:2004zq}
T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Nonperturbative studies of fuzzy spheres in a matrix model with the
Chern-Simons term},
\jhep{0405}{2004}{005} [\hepth{0401038}].
%%CITATION = HEP-TH 0401038;%%


%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{fuzzy-MC}
%\cite{Azuma:2004yg}
%\bibitem{Azuma:2004yg}
T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
{\it Absence of a fuzzy $S^4$ phase in the dimensionally reduced 5d
Yang-Mills-Chern-Simons model},
\jhep{0407}{2004}{066} [\hepth{0405096}];
%%CITATION = HEP-TH 0405096;%%
%\cite{Azuma:2005pm}
%\bibitem{Azuma:2005pm}
%T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Perturbative versus nonperturbative dynamics of the fuzzy $S^2 \times
S^2$},
\jhep{0509}{2005}{047} [\hepth{0506205}];
%%CITATION = HEP-TH 0506205;%%
%
%\cite{Azuma:2004qe}
%\bibitem{Azuma:2004qe}
%T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Dynamical aspects of the fuzzy CP(2) in the large N reduced model 
with a cubic term}, {\tt hep-th/0405277};
%%CITATION = HEP-TH 0405277;%%
%\cite{Anagnostopoulos:2005cy}
%\bibitem{Anagnostopoulos:2005cy}
K.~N.~Anagnostopoulos, T.~Azuma, K.~Nagao and J.~Nishimura,
\emph{Impact of supersymmetry 
on the nonperturbative dynamics of fuzzy spheres},
\jhep{0509}{2005}{046} [\hepth{0506062}];
%%CITATION = HEP-TH 0506062;%%
%
%\cite{Azuma:2004ie}
%\bibitem{Azuma:2004ie}
T.~Azuma, K.~Nagao and J.~Nishimura,
\emph{Perturbative dynamics of fuzzy spheres at large N},
\jhep{0506}{2005}{081} [\hepth{0410263}].
%%CITATION = HEP-TH 0410263;%%


%----------------NC solitons---------------
\bibitem{GMS}
R.~Gopakumar, S.~Minwalla and A.~Strominger,
\emph{Noncommutative solitons},
\jhep{0005}{2000}{020} [\hepth{0003160}];
%%CITATION = HEP-TH 0003160;%%
%%%%%%%%%%%%%
%\cite{Harvey:2000jb}
%\bibitem{Harvey:2000jb}
J.~A.~Harvey, P.~Kraus and F.~Larsen,
\emph{Exact noncommutative solitons},
\jhep{0012}{2000}{024} [\hepth{0010060}].
%%CITATION = HEP-TH 0010060;%%

%%%%%%%%

\bibitem{fluxon}
N.~Nekrasov and A.~Schwarz,
\emph{Instantons on noncommutative $R^4$ and 
(2,0) superconformal six  dimensional theory},
\cmp{198}{1998}{689} [\hepth{9802068}];
%%CITATION = HEP-TH 9802068;%%
%
A.~P.~Polychronakos,
\emph{Flux tube solutions in noncommutative gauge theories},
\plb{495}{2000}{407} [\hepth{0007043}];
%%CITATION = HEP-TH 0007043;%%
%
D.~J.~Gross and N.~A.~Nekrasov,
\emph{Dynamics of strings in noncommutative gauge theory},
\jhep{0010}{2000}{021} [\hepth{0007204}];
%%CITATION = HEP-TH 0007204;%%
%
M.~Aganagic, R.~Gopakumar, S.~Minwalla and A.~Strominger,
\emph{Unstable solitons in noncommutative gauge theory},
\jhep{0104}{2001}{001} [\hepth{0009142}];
%%CITATION = HEP-TH 0009142;%%
%
D.~Bak,
\emph{Exact multi-vortex solutions in noncommutative Abelian-Higgs theory},
\plb{495}{2000}{251} [\hepth{0008204}];
%%CITATION = HEP-TH 0008204;%%
%
D.~J.~Gross and N.~A.~Nekrasov,
\emph{Monopoles and strings in noncommutative gauge theory},
\jhep{0007}{2000}{034} [\hepth{0005204}];
%%CITATION = HEP-TH 0005204;%%

%-----------------------Non-trivial config.-----------------
\bibitem{non-trivial_config}
%\bibitem{23_5}
H.~Grosse, C.~Klimcik and P.~Presnajder,
\emph{Topologically nontrivial field configurations in noncommutative
geometry},
\cmp{178}{1996}{507} [\hepth{9510083}];
%%CITATION = HEP-TH 9510083;%%
%
S.~Baez, A.~P.~Balachandran, B.~Ydri and S.~Vaidya,
\emph{Monopoles and solitons in fuzzy physics},
\cmp{208}{2000}{787} [\hepth{9811169}];
% 
G.~Landi,
\emph{Projective modules of finite type and monopoles over S(2)},
{\em J.\ Geom.\ Phys.\  {\bf 37}} (2001) 47
[{\tt math-ph/9905014}];
%%CITATION = MATH-PH 9905014;%%
% 
%-----------non-trivial config.and chiral anomaly-------
%\bibitem{non_chi}
A.~P.~Balachandran and S.~Vaidya,
\emph{Instantons and chiral anomaly in fuzzy physics},
\ijmpa{16}{2001}{17} [\hepth{9910129}];
%%CITATION = HEP-TH 9910129;%%
%------------------------------------------------------------
% 
%--------------------------------------------------------------
%\cite{Valtancoli:2001gx}
%\bibitem{Valtancoli:2001gx}
P.~Valtancoli,
\emph{Projectors for the fuzzy sphere},
\mpla{16}{2001}{639}
%Mod.\ Phys.\ Lett.\ A {\bf 16}, 639 (2001)
[\hepth{0101189}];
%%CITATION = HEP-TH 0101189;%%
% 
%\cite{Steinacker:2003sd}
%\bibitem{Steinacker:2003sd}
H.~Steinacker,
\emph{Quantized gauge theory on the fuzzy sphere as random matrix model},
\npb{679}{2004}{66} [\hepth{0307075}];
%%CITATION = HEP-TH 0307075;%%
%
%
%\bibitem{Karabali:2001te}
D.~Karabali, V.~P.~Nair and A.~P.~Polychronakos,
\emph{Spectrum of Schroedinger field in a noncommutative magnetic monopole},
\npb{627}{2002}{565} [\hepth{0111249}];
%%CITATION = HEP-TH 0111249;%%
%
%\bibitem{Carow-Watamura:2004ct}
U.~Carow-Watamura, H.~Steinacker and S.~Watamura,
\emph{Monopole bundles over fuzzy complex projective spaces},
{\em J.\ Geom.\ Phys.\  {\bf 54}} (2005) 373 [\hepth{0404130}].
%%CITATION = HEP-TH 0404130;%%

\bibitem{Balachandran:2003ay}
A.~P.~Balachandran and G.~Immirzi,
{\it The fuzzy Ginsparg-Wilson algebra: A solution of the fermion doubling
problem},
\prd{68}{2003}{065023} [\hepth{0301242}].
%%CITATION = HEP-TH 0301242;%%


 
\bibitem{AIN3}
H.~Aoki, S.~Iso and K.~Nagao,
{\it Ginsparg-Wilson relation and 
't Hooft-Polyakov monopole on fuzzy 2-sphere},
\npb{684}{2004}{162} [\hepth{0312199}].
%%CITATION = HEP-TH 0312199;%% 
%



%--------- 4d space-time

\bibitem{9612115}
 N.~Ishibashi, H.~Kawai, Y.~Kitazawa and A.~Tsuchiya,
\emph{A large-N reduced model as superstring,}
\npb{498}{1997}{467}
[\hepth{9612115}].


\bibitem{Aoki:1998vn}
H.~Aoki, S.~Iso, H.~Kawai, Y.~Kitazawa and T.~Tada,
\emph{Space-time structures from IIB matrix model,}
\ptp{99}{1998}{713}
[\hepth{9802085}].
%%CITATION = HEP-TH 9802085;%%


\bibitem{spaceiib}
%
%\bibitem{9811220}
T.~Hotta, J.~Nishimura and A.~Tsuchiya,
\emph{Dynamical aspects of large N reduced models,}
\npb{545}{1999}{543}
[\hepth{9811220}];
%%CITATION = HEP-TH 9811220;%%
%
%\cite{Ambjorn:2000bf}
%\bibitem{Ambjorn:2000bf}
J.~Ambjorn, K.~N.~Anagnostopoulos, W.~Bietenholz, T.~Hotta and J.~Nishimura,
\emph{Large N dynamics of dimensionally reduced 4D SU(N) super Yang-Mills
theory,}
\jhep{0007}{2000}{013}
[\hepth{0003208}];
%%CITATION = HEP-TH 0003208;%%
%
%\cite{Ambjorn:2000dx}
%\bibitem{Ambjorn:2000dx}
J.~Ambj\o rn, K.~N.~Anagnostopoulos, W.~Bietenholz, T.~Hotta and J.~Nishimura,
\emph{Monte Carlo studies of the IIB matrix model at large N,}
\jhep{0007}{2000}{011}
[\hepth{0005147}];
%%CITATION = HEP-TH 0005147;%%
%
%\bibitem{NV}
J.\ Nishimura and G.\ Vernizzi,
\emph{Spontaneous breakdown of Lorentz invariance in IIB matrix model,}
\jhep{0004}{2000}{015}
[\hepth{0003223}];
\emph{Brane world generated dynamically from string type IIB matrices,}
\prl{85}{2000}{4664}
[\hepth{0007022}];
%
%\cite{Burda:2000mn}
%\bibitem{Burda:2000mn}
Z.~Burda, B.~Petersson and J.~Tabaczek,
\emph{Geometry of reduced supersymmetric 4D Yang-Mills integrals,}
\npb{602}{2001}{399}
[\heplat{0012001}];
%%CITATION = HEP-LAT 0012001;%%
%
%\cite{Ambjorn:2001xs}
%\bibitem{Ambjorn:2001xs}
J.~Ambj\o rn, K.~N.~Anagnostopoulos, W.~Bietenholz, 
F.~Hofheinz and J.~Nishimura,
\emph{On the spontaneous breakdown of Lorentz symmetry in matrix models of 
superstrings,}
\prd{65}{2002}{086001}
[\hepth{0104260}];
%%CITATION = HEP-TH 0104260;%%
%
%\bibitem{exact}
J.~Nishimura,
\emph{Exactly solvable matrix models for the dynamical generation of  
space-time in superstring theory,}
\prd{65}{2002}{105012}
[\hepth{0108070}];
%%CITATION = HEP-TH 0108070;%%
%
%\bibitem{sign}
K.~N.~Anagnostopoulos and J.~Nishimura,
\emph{New approach to the complex-action problem 
and its application to a nonperturbative study of superstring theory,}
\prd{66}{2002}{106008}
[\hepth{0108041}];
%
%\cite{Vernizzi:2002mu}
%\bibitem{Vernizzi:2002mu}
G.~Vernizzi and J.~F.~Wheater,
\emph{Rotational symmetry breaking in multi-matrix models,}
\prd{66}{2002}{085024}
[\hepth{0206226}];
%%CITATION = HEP-TH 0206226;%%
%
%\bibitem{0307007}
T.~Imai, Y.~Kitazawa, Y.~Takayama and D.~Tomino,
\emph{Effective actions of matrix models on homogeneous spaces,}
\npb{679}{2004}{143}
[\hepth{0307007}];
%%CITATION = HEP-TH 0307007;%%
%
%\cite{Nishimura:2004ts}
%\bibitem{Nishimura:2004ts}
J.~Nishimura, T.~Okubo and F.~Sugino,
\emph{Gaussian expansion analysis of a matrix model 
with the spontaneous breakdown of rotational symmetry,}
\ptp{114}{2005}{487} [\hepth{0412194}];
%%CITATION = HEP-TH 0412194;%%
%
%\cite{Bal:2004ai}
%\bibitem{Bal:2004ai}
S.~Bal, M.~Hanada, H.~Kawai and F.~Kubo,
\emph{Fuzzy torus in matrix model},
\npb{727}{2005}{196} [\hepth{0412303}];
%%CITATION = HEP-TH 0412303;%%
%
%\bibitem{0506033}
H.~Kaneko, Y.~Kitazawa and D.~Tomino,
\emph{Stability of fuzzy S$^2 \times$  S $^2 \times$ S$^2$ 
in IIB type matrix models},
\npb{725}{2005}{93} [\hepth{0506033}];
%%CITATION = HEP-TH 0506033;%%
%\cite{Kaneko:2005kp}
%\bibitem{Kaneko:2005kp}
%H.~Kaneko, Y.~Kitazawa and D.~Tomino,
\emph{Fuzzy spacetime with SU(3) isometry in IIB matrix model},
\prd{73}{2006}{066001} [\hepth{0510263}].
% Phys.\ Rev.\  D {\bf 73} (2006) 066001
%{\tt hep-th/0510263}.
%%CITATION = HEP-TH 0510263;%%


%%%%%%%%%%%%%

\bibitem{gaussian}
%
%\cite{Nishimura:2001sx}
%\bibitem{Nishimura:2001sx}
J.~Nishimura and F.~Sugino,
\emph{Dynamical generation of four-dimensional space-time 
in the IIB matrix model,}
\jhep{0205}{2002}{001} 
[\hepth{0111102}];
%%CITATION = HEP-TH 0111102;%%
%
%
%\bibitem{KKKMS}
H.~Kawai, S.~Kawamoto, T.~Kuroki, T.~Matsuo and S.~Shinohara,
\emph{Mean field approximation of IIB matrix model and 
 emergence of four dimensional space-time,}
\npb{647}{2002}{153}
[\hepth{0204240}];
%%CITATION = HEP-TH 0204240;%%
%
%\cite{Kawai:2002ub}
%\bibitem{Kawai:2002ub}
H.~Kawai, S.~Kawamoto, T.~Kuroki and S.~Shinohara,
\emph{Improved perturbation theory and four-dimensional 
space-time in IIB  matrix model,}
\ptp{109}{2003}{115}
[\hepth{0211272}];
%%CITATION = HEP-TH 0211272;%%
%
%\cite{Aoyama:2006di}
%\bibitem{Aoyama:2006di}
T.~Aoyama, H.~Kawai and Y.~Shibusa,
\emph{Stability of 4-dimensional space-time from IIB matrix model 
via improved mean field approximation},
\ptp{115}{2006}{1179} [\hepth{0602244}];
%Prog.\ Theor.\ Phys.\  {\bf 115} (2006) 1179
%\hepth{0602244};
  %%CITATION = HEP-TH 0602244;%%
%
%\cite{Aoyama:2006rk}
%\bibitem{Aoyama:2006rk}
  T.~Aoyama and H.~Kawai,
\emph{Higher order terms of improved mean field approximation 
for IIB matrix model and emergence of four-dimensional space-time},
\ptp{116}{2006}{405} [\hepth{0603146}].
% Prog.\ Theor.\ Phys.\  {\bf 116} (2006) 405
%\hepth{0603146}.
  %%CITATION = HEP-TH 0603146;%%


%\cite{Iso:1999xs}
\bibitem{Iso:1999xs}
S.~Iso and H.~Kawai,
\emph{Space-time and matter in IIB matrix model: Gauge symmetry and
diffeomorphism},
\ijmpa{15}{2000}{651}
[{\tt hep-th/9903217}].
%%CITATION = HEP-TH 9903217;%%

\bibitem{0504217}
  T.~Azuma, S.~Bal and J.~Nishimura,
  \emph{Dynamical generation of gauge groups 
in the massive Yang-Mills-Chern-Simons
  matrix model},
\prd{72}{2005}{066005} [\hepth{0504217}].
  %%CITATION = HEP-TH 0504217;%%

%-------------------------------------
%\cite{Atiyah:1971rm}
\bibitem{Atiyah:1971rm}
  M.~F.~Atiyah and I.~M.~Singer,
 \emph{The index of elliptic operators. 5},
  {\em Annals Math.\  {\bf 93}} (1971) 139.
  %%CITATION = ANMAA,93,139;%%

%\cite{Kim:2002qm}
\bibitem{Kim:2002qm}
  K.~Y.~Kim, B.~H.~Lee and H.~S.~Yang,
\emph{Zero-modes and Atiyah-Singer index in noncommutative instantons},
\prd{66}{2002}{025034} [\hepth{0205010}].
  %%CITATION = HEP-TH 0205010;%%
  %%Cited 17 times in SPIRES-HEP



%
\bibitem{Neuberger}H.~Neuberger,
\emph{Exactly massless quarks on the lattice},
\plb{417}{1998}{141}
[\heplat{9707022}];
%%CITATION = HEP-LAT 9707022;%%
% 
\emph{Vector like gauge theories with almost massless fermions on the
lattice},
\prd{57}{1998}{5417} [\heplat{9710089}];
%%CITATION = HEP-LAT 9710089;%%
% 
\emph{More about exactly massless quarks on the lattice},
\plb{427}{1998}{353}
[\heplat{9801031}].
%%CITATION = HEP-LAT 9801031;%% 
%--------------------------------------



%%%%%%%%%% lattice chirality %%%%%%%%%%%


%\cite{Narayanan:1994gw}
\bibitem{Narayanan:1994gw}
R.~Narayanan and H.~Neuberger,
\emph{A Construction of lattice chiral gauge theories},
\npb{443}{1995}{305} [\hepth{9411108}].
%%CITATION = HEP-TH 9411108;%%

\bibitem{Hasenfratzindex}P.~Hasenfratz,
\emph{Prospects for perfect actions},
{\em Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 63}} (1998) 53
[\heplat{9709110}];
%%CITATION = HEP-LAT 9709110;%%
% 
P.~Hasenfratz, V.~Laliena and F.~Niedermayer,
\emph{The index theorem in QCD with a finite cut-off},
\plb{427}{1998}{125}
[\heplat{9801021}].
%%CITATION = HEP-LAT 9801021;%% 

 %
\bibitem{Luscher}M.~L\"uscher,
\emph{Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation},
\plb{428}{1998}{342}
[\heplat{9802011}].
%%CITATION = HEP-LAT 9802011;%% 




%---GW---------------------------
\bibitem{GinspargWilson}P.~H.~Ginsparg and K.~G.~Wilson,
\emph{A remnant of chiral symmetry on the lattice},
\prd{25}{1982}{2649}.
%%CITATION = PHRVA,D25,2649;%% 


\bibitem{Nishimura:2001dq}
J.~Nishimura and M.~A.~Vazquez-Mozo,
{\it Noncommutative chiral gauge theories 
on the lattice with manifest star-gauge invariance},
\jhep{0108}{2001}{033} [\hepth{0107110}].
%%CITATION = HEP-TH 0107110;%% 


\bibitem{AIN2}
H.~Aoki, S.~Iso and K.~Nagao,
\emph{Ginsparg-Wilson relation, topological invariants and finite
noncommutative geometry},
\prd{67}{2003}{085005}
[\hepth{0209223}].
%%CITATION = HEP-TH 0209223;%% 
%


%---------------------------------------------------------------
\bibitem{balagovi}
A.~P.~Balachandran, T.~R.~Govindarajan and B.~Ydri,
\emph{The fermion doubling problem and noncommutative geometry},
\mpla{15}{2000}{1279}
[\hepth{9911087}];
%%CITATION = HEP-TH 9911087;%%
% 
%\bibitem{balaGW}
%A.~P.~Balachandran, T.~R.~Govindarajan and B.~Ydri,
\emph{The fermion doubling problem and noncommutative geometry. II},
\hepth{0006216}.
%%CITATION = HEP-TH 0006216;%% 


%% \bibitem{Ydri:2002nt}
%% B.~Ydri,
%% %``Noncommutative chiral anomaly and the Dirac-Ginsparg-Wilson operator,''
%%  {\bf 0308}, 046 (2003)
%% [arXiv:hep-th/0211209].
%% %%CITATION = HEP-TH 0211209;%%


%
\bibitem{isonagao}
S.~Iso and K.~Nagao,
\emph{Chiral anomaly and Ginsparg-Wilson relation 
on the noncommutative torus},
\ptp{109}{2003}{1017} [\hepth{0212284}].
%%CITATION = HEP-TH 0212284;%% 



\bibitem{AIMN}
H.~Aoki, S.~Iso, T.~Maeda and K.~Nagao,
\emph{Dynamical generation of a nontrivial index on the fuzzy 2-sphere},
\prd{71}{2005}{045017} [\hepth{0412052}],
Erratum~:~\prd{71}{2005}{069905}.
%{\em ibid.\ {\bf D71}} (2005) 069905.
%%CITATION = HEP-TH 0412052;%%

%\cite{Aoki:2006sb}
\bibitem{Aoki:2006sb}
  H.~Aoki, J.~Nishimura and Y.~Susaki,
\emph{The index of the overlap Dirac operator 
on a discretized 2d non-commutative torus},
\jhep{0702}{2007}{033} [\hepth{0602078}].
  %%CITATION = JHEPA,0702,033;%%


\bibitem{Mafia}
L.~Griguolo, D.~Seminara and P.~Valtancoli, \emph{Towards the solution
  of noncommutative $YM_2$: Morita equivalence and large-$N$-limit},
  \jhep{12}{2001}{024} [\hepth{0110293}];\\
A.~Bassetto, G.~Nardelli and A.~Torrielli, \emph{Perturbative Wilson
  loop in two-dimensional non-commutative Yang-Mills theory},
  \npb{617}{2001}{308} [\hepth{0107147}];
\emph{Scaling properties of the perturbative Wilson loop in
two-dimensional non-commutative Yang-Mills theory},
\prd{66}{2002}{085012} [\hepth{0205210}];
%
%\bibitem{other2dncym}
%\cite{Torrielli:2003sf}
%\bibitem{Torrielli:2003sf}
A.~Torrielli,
\emph{Noncommutative perturbative quantum field theory: Wilson loop in
two-dimensional Yang-Mills, and unitarity from string theory},
{\em Ph.D thesis, Padua University} (2003)
[\hepth{0301091}];
%%CITATION = HEP-TH 0301091;%%
%
%\cite{Paniak:2003gn}
%\bibitem{Paniak:2003gn}
L.~D.~Paniak and R.~J.~Szabo,
\emph{Open Wilson lines and group theory of noncommutative Yang-Mills theory
in two dimensions},
\jhep{0305}{2003}{029}
[\hepth{0302162}];
%%CITATION = HEP-TH 0302162;%%
%
%\cite{Paniak:2003yf}
%\bibitem{Paniak:2003yf}
%L.~D.~Paniak and R.~J.~Szabo,
%\emph{Lectures on two-dimensional noncommutative gauge theory. I:
%Classical aspects},
%\hepth{0302195};
%%CITATION = HEP-TH 0302195;%%
%
%\cite{Paniak:2003xm}
%\bibitem{Paniak:2003xm}
%L.~D.~Paniak and R.~J.~Szabo,
%\emph{Lectures on two-dimensional noncommutative gauge theory. 2:
%Quantization},
%\hepth{0304268};
%%CITATION = HEP-TH 0304268;%%
%
%\cite{Dorn:2003wi}
%\bibitem{Dorn:2003wi}
H.~Dorn and A.~Torrielli,
\emph{Loop equation in two-dimensional noncommutative Yang-Mills theory},
\jhep{0401}{2004}{026}
[\hepth{0312047}];
%%CITATION = HEP-TH 0312047;%%
%
%\cite{Ambjorn:2004ck}
%\bibitem{Ambjorn:2004ck}
J.~Ambjorn, A.~Dubin and Y.~Makeenko,
\emph{Wilson loops in 2D noncommutative Euclidean gauge theory. I:
Perturbative expansion},
\jhep{0407}{2004}{044}
[\hepth{0406187}];
%%CITATION = HEP-TH 0406187;%%
%
%\cite{Fatollahi:2005ri}
%\bibitem{Fatollahi:2005ri}
A.~H.~Fatollahi and A.~Jafari,
\emph{On the bound states of photons in noncommutative 
quantum electrodynamics},
\epjc{46}{2006}{235} [\hepth{0503078}];
%Eur.\ Phys.\ J.\  C {\bf 46} (2006) 235
%\hepth{0503078};
%%CITATION = HEP-TH 0503078;%%
%
%\cite{Bassetto:2005hn}
%\bibitem{Bassetto:2005hn}
A.~Bassetto, G.~De Pol, A.~Torrielli and F.~Vian,
\emph{On the invariance under area preserving diffeomorphisms 
of noncommutative Yang-Mills theory in two dimensions},
\jhep{0505}{2005}{061}
[\hepth{0503175}];
%%CITATION = HEP-TH 0503175;%%
%
%\cite{Cirafici:2005af}
%\bibitem{Cirafici:2005af}
M.~Cirafici, L.~Griguolo, D.~Seminara and R.~J.~Szabo,
\emph{Morita duality and noncommutative Wilson loops in two dimensions},
\jhep{0510}{2005}{030} [\hepth{0506016}].
%%CITATION = HEP-TH 0506016;%%



%\cite{Paniak:2002fi}
\bibitem{Paniak:2002fi}
L.~D.~Paniak and R.~J.~Szabo,
\emph{Instanton expansion of noncommutative gauge theory in two dimensions},
\cmp{243}{2003}{343} [\hepth{0203166}].
%%CITATION = HEP-TH 0203166;%%


%\cite{Griguolo:2003kq}
\bibitem{Griguolo:2003kq}
L.~Griguolo and D.~Seminara,
\emph{Classical solutions of the TEK model and 
noncommutative instantons in two dimensions},
\jhep{0403}{2004}{068} 
[\hepth{0311041}].
%%CITATION = HEP-TH 0311041;%%


\bibitem{GHL}
C.~R.~Gattringer, I.~Hip and C.~B.~Lang,
\emph{Quantum fluctuations versus topology: 
A study in U(1)$_2$ lattice gauge theory},
\plb{409}{1997}{371} [\heplat{9706010}].
%[arXiv:hep-lat/9706010].
%%CITATION = HEP-LAT 9706010;%%



%% \bibitem{Morita} A.\ Schwarz, {\it Morita equivalence and duality},
%% \npb{534}{1998}{720} [\hepth{9805034}];
%% B.\ Pioline and A.\ Schwarz, {\it Morita equivalence and T-duality 
%% (or B versus $\Theta$)}, \jhep{9908}{1999}{021} [\hepth{9908019}];
%% K.\ Saraikin, {\it Comments on the Morita equivalence},
%% {\it J.\ Exp.\ Theor.\ Phys.} {\bf 91} (2000) 653
%% [\hepth{0005138}]; L.\ Alvarez-Gaum\'{e} and J.L.F.\ Barb\'{o}n,
%% {\it Morita duality and large-$N$ limits}, \npb{623}{2002}{165}
%% [\hepth{0109176}].



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%\cite{Nagao:2005st}
\bibitem{Nagao:2005st}
  K.~Nagao,
\emph{An admissibility condition 
and nontrivial indices on a noncommutative
torus},
\prd{73}{2006}{065002} [\hepth{0509034}].
  %%CITATION = HEP-TH 0509034;%%

\bibitem{ANS}
H.~Aoki, J.~Nishimura and Y.~Susaki,
work in progress.


%\cite{Fukaya:2004kp}
\bibitem{FukayaOnogi}
H.~Fukaya and T.~Onogi,
\emph{$\theta$ vacuum effects on the chiral 
condensation and the $\eta'$ meson
correlators in the two-flavor massive QED$_2$ on the lattice},
\prd{70}{2004}{054508} [\heplat{0403024}];
%%CITATION = HEP-LAT 0403024;%%
%
%\cite{Fukaya:2003ph}
%\bibitem{Fukaya:2003ph}
%H.~Fukaya and T.~Onogi,
\emph{Lattice study of the massive Schwinger model with $\theta$ term under
Luescher's `admissibility' condition},
\prd{68}{2003}{074503} [\heplat{0305004}].
%%CITATION = HEP-LAT 0305004;%%


%\cite{Aschieri:2003vy}
\bibitem{Aschieri:2003vy}
  P.~Aschieri, J.~Madore, P.~Manousselis and G.~Zoupanos,
  \emph{Dimensional reduction over fuzzy coset spaces},
  \jhep{0404}{2004}{034} [\hepth{0310072}];
  %%CITATION = HEP-TH 0310072;%%
  %%Cited 12 times in SPIRES-HEP
%
%\cite{Aschieri:2006uw}
%\bibitem{Aschieri:2006uw}
  P.~Aschieri, T.~Grammatikopoulos, H.~Steinacker and G.~Zoupanos,
\emph{Dynamical generation of fuzzy extra dimensions, dimensional reduction and
symmetry breaking},
\jhep{0609}{2006}{026} [\hepth{0606021}];
%  JHEP {\bf 0609} (2006) 026
%  [arXiv:hep-th/0606021].
  %%CITATION = JHEPA,0609,026;%%
%
%\cite{Steinacker:2007ay}
%\bibitem{Steinacker:2007ay}
  H.~Steinacker and G.~Zoupanos,
\emph{Fermions on spontaneously generated spherical extra dimensions},
{\tt arXiv:0706.0398}.
%\hepth{0706.0398}.
%%CITATION = ARXIV:0706.0398;%%


\bibitem{LLT}
%\cite{Landsteiner:2000bw}
%\bibitem{Landsteiner:2000bw}
K.~Landsteiner, E.~Lopez and M.~H.~G.~Tytgat,
{\it Excitations in hot non-commutative theories},
\jhep{0009}{2000}{027} [\hepth{0006210}];
%%CITATION = HEP-TH 0006210;%%
%\cite{Landsteiner:2001ky}
%\bibitem{Landsteiner:2001ky}
%K.~Landsteiner, E.~Lopez and M.~H.~G.~Tytgat,
{\it Instability of non-commutative SYM theories 
at finite temperature},
\jhep{0106}{2001}{055} [\hepth{0104133}].
%%CITATION = HEP-TH 0104133;%%




\bibitem{ruiz}
%\cite{Martin:2000bk}
%\bibitem{Martin:2000bk}
C.~P.~Martin and F.~Ruiz Ruiz,
{\it Paramagnetic dominance, 
the sign of the beta function and UV/IR mixing in
non-commutative U(1)},''
\npb{597}{2001}{197} [\hepth{0007131}];
%%CITATION = HEP-TH 0007131;%%
%
%\cite{Ruiz:2000hu}
%\bibitem{Ruiz:2000hu}
F.~R.~Ruiz,
{\it Gauge-fixing independence of IR divergences in non-commutative U(1),
perturbative tachyonic instabilities and supersymmetry},
\plb{502}{2001}{274} [\hepth{0012171}].
%%CITATION = HEP-TH 0012171;%%



%\cite{Bassetto:2001vf}
\bibitem{Bassetto:2001vf}
A.~Bassetto, L.~Griguolo, G.~Nardelli and F.~Vian,
\emph{On the unitarity of quantum gauge theories on noncommutative spaces},
\jhep{0107}{2001}{008} [\hepth{0105257}].
%%CITATION = HEP-TH 0105257;%%


\bibitem{rf:MVR}
M.~Van Raamsdonk,
{\it The meaning of infrared singularities in noncommutative gauge theories},
\jhep{0111}{2001}{006} [\hepth{0110093}].


\bibitem{rf:AL}
A.~Armoni and E.~Lopez,
{\it UV/IR mixing via closed strings and tachyonic instabilities},
\npb{632}{2002}{240} [\hepth{0110113}].



%\cite{Guralnik:2002ru}
\bibitem{Guralnik:2002ru}
Z.~Guralnik, R.~C.~Helling, K.~Landsteiner and E.~Lopez,
{\it Perturbative instabilities on the non-commutative torus, 
Morita duality and twisted boundary conditions},
\jhep{0205}{2002}{025} [\hepth{0204037}].
%%CITATION = HEP-TH 0204037;%%



\end{thebibliography}

\end{document}

hep-th0604093.extracted_bib
\begin{thebibliography}{50}


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%\cite{Martin:2004un}
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%%CITATION = HEP-TH 0402230;%%


%\cite{Bietenholz:2004xs}
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W.~Bietenholz, F.~Hofheinz and J.~Nishimura,
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%%CITATION = HEP-TH 0404020;%%


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I.~Chepelev and R.~Roiban,
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Scalars},
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%{\bf 0005} (2000) 037
%[arXiv:hep-th/9911098].
%%CITATION = HEP-TH 9911098;%%


\bibitem{EK}
%\cite{Eguchi:1982nm}
%\bibitem{Eguchi:1982nm}
T.~Eguchi and H.~Kawai,
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in the large N gauge theory},
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%%CITATION = PRLTA,48,1063;%%

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%%CITATION = ,D27,2397;%%
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\bibitem{2dU1} W.\ Bietenholz, F.\ Hofheinz and J.\ Nishimura,
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plane}, \jhep{09}{2002}{009} [\hepth{0203151}].


\bibitem{4dU1}
%\cite{Bietenholz:2006cz}
%\bibitem{Bietenholz:2006cz}
W.~Bietenholz, J.~Nishimura, Y.~Susaki and J.~Volkholz,
\emph{A non-perturbative study of 4d U(1) 
non-commutative gauge theory: The fate of one-loop instability},
\jhep{10}{2006}{042} [\hepth{0608072}];
  %%CITATION = HEP-TH 0608072;%%
%
%\cite{Bietenholz:2004wk}
%\bibitem{Bietenholz:2004wk}
W.~Bietenholz, F.~Hofheinz, J.~Nishimura, Y.~Susaki and J.~Volkholz,
\emph{First simulation results for the photon in a non-commutative space},
{\em Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 140}} (2005) 772
[\heplat{0409059}];
%%CITATION = HEP-LAT 0409059;%%
%
%\cite{Bietenholz:2005iz}
%\bibitem{Bietenholz:2005iz}
W.~Bietenholz, A.~Bigarini, F.~Hofheinz, 
J.~Nishimura, Y.~Susaki and J.~Volkholz,
\emph{Numerical results for U(1) gauge theory on 2d and 4d non-commutative
spaces},
{\em Fortsch.\ Phys.\  {\bf 53}} (2005) 418
[\hepth{0501147}];
%%CITATION = HEP-TH 0501147;%%
%
%\cite{Volkholz:2005mp}
%\bibitem{Volkholz:2005mp}
J.~Volkholz, W.~Bietenholz, J.~Nishimura and Y.~Susaki,
\emph{The scaling of QED in a non-commutative space-time},
{\em Proc.\ Sci.\  {\bf LAT2005}} (2005) 264
[\heplat{0509146}].
%%CITATION = HEP-LAT 0509146;%%



%\cite{Bietenholz:2004as}
\bibitem{Bietenholz:2004as}
W.~Bietenholz, F.~Hofheinz and J.~Nishimura,
\emph{On the relation between non-commutative 
field theories at $\theta = \infty$
and large N matrix field theories},
\jhep{0405}{2004}{047} [\hepth{0404179}].
%%CITATION = HEP-TH 0404179;%%

%% \bibitem{KikNarNeu}
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%% %%CITATION = HEP-LAT 9705006;%%

\bibitem{Madore}
J.~Madore,
\emph{The fuzzy sphere,}
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%%CITATION = CQGRD,9,69;%%


%\cite{Dolan:2003kq}
\bibitem{Dolan:2003kq}
B.~P.~Dolan and D.~O'Connor,
\emph{A fuzzy three sphere and fuzzy tori,}
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%%CITATION = HEP-TH 0306231;%%

%\cite{O'Connor:2003aj}
\bibitem{O'Connor:2003aj}
D.~O'Connor,
\emph{Field theory on low dimensional fuzzy spaces,}
\mpla{18}{2003}{2423}.
%%CITATION = MPLAE,A18,2423;%%


%\cite{Grosse:1995ar}
\bibitem{Grosse:1995ar}
H.~Grosse, C.~Klimcik and P.~Presnajder,
\emph{Towards finite quantum field theory in noncommutative geometry,}
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%%CITATION = HEP-TH 9505175;%%


%\cite{Myers:1999ps}
\bibitem{Myers:1999ps}
R.~C.~Myers,
\emph{Dielectric-branes},
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%%CITATION = HEP-TH 9910053;%%

\bibitem{0101102}
S.~Iso, Y.~Kimura, K.~Tanaka and K.~Wakatsuki, \emph{Noncommutative
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%\cite{Azuma:2004zq}
\bibitem{Azuma:2004zq}
T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Nonperturbative studies of fuzzy spheres in a matrix model with the
Chern-Simons term},
\jhep{0405}{2004}{005} [\hepth{0401038}].
%%CITATION = HEP-TH 0401038;%%


%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{fuzzy-MC}
%\cite{Azuma:2004yg}
%\bibitem{Azuma:2004yg}
T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
{\it Absence of a fuzzy $S^4$ phase in the dimensionally reduced 5d
Yang-Mills-Chern-Simons model},
\jhep{0407}{2004}{066} [\hepth{0405096}];
%%CITATION = HEP-TH 0405096;%%
%\cite{Azuma:2005pm}
%\bibitem{Azuma:2005pm}
%T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Perturbative versus nonperturbative dynamics of the fuzzy $S^2 \times
S^2$},
\jhep{0509}{2005}{047} [\hepth{0506205}];
%%CITATION = HEP-TH 0506205;%%
%
%\cite{Azuma:2004qe}
%\bibitem{Azuma:2004qe}
%T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
\emph{Dynamical aspects of the fuzzy CP(2) in the large N reduced model 
with a cubic term}, {\tt hep-th/0405277};
%%CITATION = HEP-TH 0405277;%%
%\cite{Anagnostopoulos:2005cy}
%\bibitem{Anagnostopoulos:2005cy}
K.~N.~Anagnostopoulos, T.~Azuma, K.~Nagao and J.~Nishimura,
\emph{Impact of supersymmetry 
on the nonperturbative dynamics of fuzzy spheres},
\jhep{0509}{2005}{046} [\hepth{0506062}];
%%CITATION = HEP-TH 0506062;%%
%
%\cite{Azuma:2004ie}
%\bibitem{Azuma:2004ie}
T.~Azuma, K.~Nagao and J.~Nishimura,
\emph{Perturbative dynamics of fuzzy spheres at large N},
\jhep{0506}{2005}{081} [\hepth{0410263}].
%%CITATION = HEP-TH 0410263;%%


%----------------NC solitons---------------
\bibitem{GMS}
R.~Gopakumar, S.~Minwalla and A.~Strominger,
\emph{Noncommutative solitons},
\jhep{0005}{2000}{020} [\hepth{0003160}];
%%CITATION = HEP-TH 0003160;%%
%%%%%%%%%%%%%
%\cite{Harvey:2000jb}
%\bibitem{Harvey:2000jb}
J.~A.~Harvey, P.~Kraus and F.~Larsen,
\emph{Exact noncommutative solitons},
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%%CITATION = HEP-TH 0010060;%%

%%%%%%%%

\bibitem{fluxon}
N.~Nekrasov and A.~Schwarz,
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(2,0) superconformal six  dimensional theory},
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%%CITATION = HEP-TH 9802068;%%
%
A.~P.~Polychronakos,
\emph{Flux tube solutions in noncommutative gauge theories},
\plb{495}{2000}{407} [\hepth{0007043}];
%%CITATION = HEP-TH 0007043;%%
%
D.~J.~Gross and N.~A.~Nekrasov,
\emph{Dynamics of strings in noncommutative gauge theory},
\jhep{0010}{2000}{021} [\hepth{0007204}];
%%CITATION = HEP-TH 0007204;%%
%
M.~Aganagic, R.~Gopakumar, S.~Minwalla and A.~Strominger,
\emph{Unstable solitons in noncommutative gauge theory},
\jhep{0104}{2001}{001} [\hepth{0009142}];
%%CITATION = HEP-TH 0009142;%%
%
D.~Bak,
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%%CITATION = HEP-TH 0008204;%%
%
D.~J.~Gross and N.~A.~Nekrasov,
\emph{Monopoles and strings in noncommutative gauge theory},
\jhep{0007}{2000}{034} [\hepth{0005204}];
%%CITATION = HEP-TH 0005204;%%

%-----------------------Non-trivial config.-----------------
\bibitem{non-trivial_config}
%\bibitem{23_5}
H.~Grosse, C.~Klimcik and P.~Presnajder,
\emph{Topologically nontrivial field configurations in noncommutative
geometry},
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%%CITATION = HEP-TH 9510083;%%
%
S.~Baez, A.~P.~Balachandran, B.~Ydri and S.~Vaidya,
\emph{Monopoles and solitons in fuzzy physics},
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% 
G.~Landi,
\emph{Projective modules of finite type and monopoles over S(2)},
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[{\tt math-ph/9905014}];
%%CITATION = MATH-PH 9905014;%%
% 
%-----------non-trivial config.and chiral anomaly-------
%\bibitem{non_chi}
A.~P.~Balachandran and S.~Vaidya,
\emph{Instantons and chiral anomaly in fuzzy physics},
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%%CITATION = HEP-TH 9910129;%%
%------------------------------------------------------------
% 
%--------------------------------------------------------------
%\cite{Valtancoli:2001gx}
%\bibitem{Valtancoli:2001gx}
P.~Valtancoli,
\emph{Projectors for the fuzzy sphere},
\mpla{16}{2001}{639}
%Mod.\ Phys.\ Lett.\ A {\bf 16}, 639 (2001)
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%%CITATION = HEP-TH 0101189;%%
% 
%\cite{Steinacker:2003sd}
%\bibitem{Steinacker:2003sd}
H.~Steinacker,
\emph{Quantized gauge theory on the fuzzy sphere as random matrix model},
\npb{679}{2004}{66} [\hepth{0307075}];
%%CITATION = HEP-TH 0307075;%%
%
%
%\bibitem{Karabali:2001te}
D.~Karabali, V.~P.~Nair and A.~P.~Polychronakos,
\emph{Spectrum of Schroedinger field in a noncommutative magnetic monopole},
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%%CITATION = HEP-TH 0111249;%%
%
%\bibitem{Carow-Watamura:2004ct}
U.~Carow-Watamura, H.~Steinacker and S.~Watamura,
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%%CITATION = HEP-TH 0404130;%%

\bibitem{Balachandran:2003ay}
A.~P.~Balachandran and G.~Immirzi,
{\it The fuzzy Ginsparg-Wilson algebra: A solution of the fermion doubling
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%%CITATION = HEP-TH 0301242;%%


 
\bibitem{AIN3}
H.~Aoki, S.~Iso and K.~Nagao,
{\it Ginsparg-Wilson relation and 
't Hooft-Polyakov monopole on fuzzy 2-sphere},
\npb{684}{2004}{162} [\hepth{0312199}].
%%CITATION = HEP-TH 0312199;%% 
%



%--------- 4d space-time

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H.~Aoki, S.~Iso, H.~Kawai, Y.~Kitazawa and T.~Tada,
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[\hepth{9802085}].
%%CITATION = HEP-TH 9802085;%%


\bibitem{spaceiib}
%
%\bibitem{9811220}
T.~Hotta, J.~Nishimura and A.~Tsuchiya,
\emph{Dynamical aspects of large N reduced models,}
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[\hepth{9811220}];
%%CITATION = HEP-TH 9811220;%%
%
%\cite{Ambjorn:2000bf}
%\bibitem{Ambjorn:2000bf}
J.~Ambjorn, K.~N.~Anagnostopoulos, W.~Bietenholz, T.~Hotta and J.~Nishimura,
\emph{Large N dynamics of dimensionally reduced 4D SU(N) super Yang-Mills
theory,}
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[\hepth{0003208}];
%%CITATION = HEP-TH 0003208;%%
%
%\cite{Ambjorn:2000dx}
%\bibitem{Ambjorn:2000dx}
J.~Ambj\o rn, K.~N.~Anagnostopoulos, W.~Bietenholz, T.~Hotta and J.~Nishimura,
\emph{Monte Carlo studies of the IIB matrix model at large N,}
\jhep{0007}{2000}{011}
[\hepth{0005147}];
%%CITATION = HEP-TH 0005147;%%
%
%\bibitem{NV}
J.\ Nishimura and G.\ Vernizzi,
\emph{Spontaneous breakdown of Lorentz invariance in IIB matrix model,}
\jhep{0004}{2000}{015}
[\hepth{0003223}];
\emph{Brane world generated dynamically from string type IIB matrices,}
\prl{85}{2000}{4664}
[\hepth{0007022}];
%
%\cite{Burda:2000mn}
%\bibitem{Burda:2000mn}
Z.~Burda, B.~Petersson and J.~Tabaczek,
\emph{Geometry of reduced supersymmetric 4D Yang-Mills integrals,}
\npb{602}{2001}{399}
[\heplat{0012001}];
%%CITATION = HEP-LAT 0012001;%%
%
%\cite{Ambjorn:2001xs}
%\bibitem{Ambjorn:2001xs}
J.~Ambj\o rn, K.~N.~Anagnostopoulos, W.~Bietenholz, 
F.~Hofheinz and J.~Nishimura,
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superstrings,}
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%%CITATION = HEP-TH 0104260;%%
%
%\bibitem{exact}
J.~Nishimura,
\emph{Exactly solvable matrix models for the dynamical generation of  
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%%CITATION = HEP-TH 0108070;%%
%
%\bibitem{sign}
K.~N.~Anagnostopoulos and J.~Nishimura,
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%
%\cite{Vernizzi:2002mu}
%\bibitem{Vernizzi:2002mu}
G.~Vernizzi and J.~F.~Wheater,
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%
%\bibitem{0307007}
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%%CITATION = HEP-TH 0307007;%%
%
%\cite{Nishimura:2004ts}
%\bibitem{Nishimura:2004ts}
J.~Nishimura, T.~Okubo and F.~Sugino,
\emph{Gaussian expansion analysis of a matrix model 
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%%CITATION = HEP-TH 0412194;%%
%
%\cite{Bal:2004ai}
%\bibitem{Bal:2004ai}
S.~Bal, M.~Hanada, H.~Kawai and F.~Kubo,
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%%CITATION = HEP-TH 0412303;%%
%
%\bibitem{0506033}
H.~Kaneko, Y.~Kitazawa and D.~Tomino,
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%%CITATION = HEP-TH 0506033;%%
%\cite{Kaneko:2005kp}
%\bibitem{Kaneko:2005kp}
%H.~Kaneko, Y.~Kitazawa and D.~Tomino,
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% Phys.\ Rev.\  D {\bf 73} (2006) 066001
%{\tt hep-th/0510263}.
%%CITATION = HEP-TH 0510263;%%


%%%%%%%%%%%%%

\bibitem{gaussian}
%
%\cite{Nishimura:2001sx}
%\bibitem{Nishimura:2001sx}
J.~Nishimura and F.~Sugino,
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in the IIB matrix model,}
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%%CITATION = HEP-TH 0111102;%%
%
%
%\bibitem{KKKMS}
H.~Kawai, S.~Kawamoto, T.~Kuroki, T.~Matsuo and S.~Shinohara,
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 emergence of four dimensional space-time,}
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%
%\cite{Kawai:2002ub}
%\bibitem{Kawai:2002ub}
H.~Kawai, S.~Kawamoto, T.~Kuroki and S.~Shinohara,
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%
%\cite{Aoyama:2006di}
%\bibitem{Aoyama:2006di}
T.~Aoyama, H.~Kawai and Y.~Shibusa,
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%Prog.\ Theor.\ Phys.\  {\bf 115} (2006) 1179
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  %%CITATION = HEP-TH 0602244;%%
%
%\cite{Aoyama:2006rk}
%\bibitem{Aoyama:2006rk}
  T.~Aoyama and H.~Kawai,
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% Prog.\ Theor.\ Phys.\  {\bf 116} (2006) 405
%\hepth{0603146}.
  %%CITATION = HEP-TH 0603146;%%


%\cite{Iso:1999xs}
\bibitem{Iso:1999xs}
S.~Iso and H.~Kawai,
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%%CITATION = HEP-TH 9903217;%%

\bibitem{0504217}
  T.~Azuma, S.~Bal and J.~Nishimura,
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  %%CITATION = HEP-TH 0504217;%%

%-------------------------------------
%\cite{Atiyah:1971rm}
\bibitem{Atiyah:1971rm}
  M.~F.~Atiyah and I.~M.~Singer,
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%\cite{Kim:2002qm}
\bibitem{Kim:2002qm}
  K.~Y.~Kim, B.~H.~Lee and H.~S.~Yang,
\emph{Zero-modes and Atiyah-Singer index in noncommutative instantons},
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  %%Cited 17 times in SPIRES-HEP



%
\bibitem{Neuberger}H.~Neuberger,
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%%CITATION = HEP-LAT 9707022;%%
% 
\emph{Vector like gauge theories with almost massless fermions on the
lattice},
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%%CITATION = HEP-LAT 9710089;%%
% 
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%%CITATION = HEP-LAT 9801031;%% 
%--------------------------------------



%%%%%%%%%% lattice chirality %%%%%%%%%%%


%\cite{Narayanan:1994gw}
\bibitem{Narayanan:1994gw}
R.~Narayanan and H.~Neuberger,
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\bibitem{Hasenfratzindex}P.~Hasenfratz,
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% 
P.~Hasenfratz, V.~Laliena and F.~Niedermayer,
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%%CITATION = HEP-LAT 9801021;%% 

 %
\bibitem{Luscher}M.~L\"uscher,
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%%CITATION = HEP-LAT 9802011;%% 




%---GW---------------------------
\bibitem{GinspargWilson}P.~H.~Ginsparg and K.~G.~Wilson,
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%%CITATION = PHRVA,D25,2649;%% 


\bibitem{Nishimura:2001dq}
J.~Nishimura and M.~A.~Vazquez-Mozo,
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%%CITATION = HEP-TH 0107110;%% 


\bibitem{AIN2}
H.~Aoki, S.~Iso and K.~Nagao,
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%%CITATION = HEP-TH 0209223;%% 
%


%---------------------------------------------------------------
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A.~P.~Balachandran, T.~R.~Govindarajan and B.~Ydri,
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%%CITATION = HEP-TH 9911087;%%
% 
%\bibitem{balaGW}
%A.~P.~Balachandran, T.~R.~Govindarajan and B.~Ydri,
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\hepth{0006216}.
%%CITATION = HEP-TH 0006216;%% 


%% \bibitem{Ydri:2002nt}
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%% %%CITATION = HEP-TH 0211209;%%


%
\bibitem{isonagao}
S.~Iso and K.~Nagao,
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on the noncommutative torus},
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%%CITATION = HEP-TH 0212284;%% 



\bibitem{AIMN}
H.~Aoki, S.~Iso, T.~Maeda and K.~Nagao,
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Erratum~:~\prd{71}{2005}{069905}.
%{\em ibid.\ {\bf D71}} (2005) 069905.
%%CITATION = HEP-TH 0412052;%%

%\cite{Aoki:2006sb}
\bibitem{Aoki:2006sb}
  H.~Aoki, J.~Nishimura and Y.~Susaki,
\emph{The index of the overlap Dirac operator 
on a discretized 2d non-commutative torus},
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  %%CITATION = JHEPA,0702,033;%%


\bibitem{Mafia}
L.~Griguolo, D.~Seminara and P.~Valtancoli, \emph{Towards the solution
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%
%\bibitem{other2dncym}
%\cite{Torrielli:2003sf}
%\bibitem{Torrielli:2003sf}
A.~Torrielli,
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two-dimensional Yang-Mills, and unitarity from string theory},
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%%CITATION = HEP-TH 0301091;%%
%
%\cite{Paniak:2003gn}
%\bibitem{Paniak:2003gn}
L.~D.~Paniak and R.~J.~Szabo,
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in two dimensions},
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%%CITATION = HEP-TH 0302162;%%
%
%\cite{Paniak:2003yf}
%\bibitem{Paniak:2003yf}
%L.~D.~Paniak and R.~J.~Szabo,
%\emph{Lectures on two-dimensional noncommutative gauge theory. I:
%Classical aspects},
%\hepth{0302195};
%%CITATION = HEP-TH 0302195;%%
%
%\cite{Paniak:2003xm}
%\bibitem{Paniak:2003xm}
%L.~D.~Paniak and R.~J.~Szabo,
%\emph{Lectures on two-dimensional noncommutative gauge theory. 2:
%Quantization},
%\hepth{0304268};
%%CITATION = HEP-TH 0304268;%%
%
%\cite{Dorn:2003wi}
%\bibitem{Dorn:2003wi}
H.~Dorn and A.~Torrielli,
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%%CITATION = HEP-TH 0312047;%%
%
%\cite{Ambjorn:2004ck}
%\bibitem{Ambjorn:2004ck}
J.~Ambjorn, A.~Dubin and Y.~Makeenko,
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Perturbative expansion},
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%%CITATION = HEP-TH 0406187;%%
%
%\cite{Fatollahi:2005ri}
%\bibitem{Fatollahi:2005ri}
A.~H.~Fatollahi and A.~Jafari,
\emph{On the bound states of photons in noncommutative 
quantum electrodynamics},
\epjc{46}{2006}{235} [\hepth{0503078}];
%Eur.\ Phys.\ J.\  C {\bf 46} (2006) 235
%\hepth{0503078};
%%CITATION = HEP-TH 0503078;%%
%
%\cite{Bassetto:2005hn}
%\bibitem{Bassetto:2005hn}
A.~Bassetto, G.~De Pol, A.~Torrielli and F.~Vian,
\emph{On the invariance under area preserving diffeomorphisms 
of noncommutative Yang-Mills theory in two dimensions},
\jhep{0505}{2005}{061}
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%%CITATION = HEP-TH 0503175;%%
%
%\cite{Cirafici:2005af}
%\bibitem{Cirafici:2005af}
M.~Cirafici, L.~Griguolo, D.~Seminara and R.~J.~Szabo,
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%%CITATION = HEP-TH 0506016;%%



%\cite{Paniak:2002fi}
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L.~D.~Paniak and R.~J.~Szabo,
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%%CITATION = HEP-TH 0203166;%%


%\cite{Griguolo:2003kq}
\bibitem{Griguolo:2003kq}
L.~Griguolo and D.~Seminara,
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noncommutative instantons in two dimensions},
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%%CITATION = HEP-TH 0311041;%%


\bibitem{GHL}
C.~R.~Gattringer, I.~Hip and C.~B.~Lang,
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%%CITATION = HEP-LAT 9706010;%%



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%\cite{Nagao:2005st}
\bibitem{Nagao:2005st}
  K.~Nagao,
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and nontrivial indices on a noncommutative
torus},
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  %%CITATION = HEP-TH 0509034;%%

\bibitem{ANS}
H.~Aoki, J.~Nishimura and Y.~Susaki,
work in progress.


%\cite{Fukaya:2004kp}
\bibitem{FukayaOnogi}
H.~Fukaya and T.~Onogi,
\emph{$\theta$ vacuum effects on the chiral 
condensation and the $\eta'$ meson
correlators in the two-flavor massive QED$_2$ on the lattice},
\prd{70}{2004}{054508} [\heplat{0403024}];
%%CITATION = HEP-LAT 0403024;%%
%
%\cite{Fukaya:2003ph}
%\bibitem{Fukaya:2003ph}
%H.~Fukaya and T.~Onogi,
\emph{Lattice study of the massive Schwinger model with $\theta$ term under
Luescher's `admissibility' condition},
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%%CITATION = HEP-LAT 0305004;%%


%\cite{Aschieri:2003vy}
\bibitem{Aschieri:2003vy}
  P.~Aschieri, J.~Madore, P.~Manousselis and G.~Zoupanos,
  \emph{Dimensional reduction over fuzzy coset spaces},
  \jhep{0404}{2004}{034} [\hepth{0310072}];
  %%CITATION = HEP-TH 0310072;%%
  %%Cited 12 times in SPIRES-HEP
%
%\cite{Aschieri:2006uw}
%\bibitem{Aschieri:2006uw}
  P.~Aschieri, T.~Grammatikopoulos, H.~Steinacker and G.~Zoupanos,
\emph{Dynamical generation of fuzzy extra dimensions, dimensional reduction and
symmetry breaking},
\jhep{0609}{2006}{026} [\hepth{0606021}];
%  JHEP {\bf 0609} (2006) 026
%  [arXiv:hep-th/0606021].
  %%CITATION = JHEPA,0609,026;%%
%
%\cite{Steinacker:2007ay}
%\bibitem{Steinacker:2007ay}
  H.~Steinacker and G.~Zoupanos,
\emph{Fermions on spontaneously generated spherical extra dimensions},
{\tt arXiv:0706.0398}.
%\hepth{0706.0398}.
%%CITATION = ARXIV:0706.0398;%%


\bibitem{LLT}
%\cite{Landsteiner:2000bw}
%\bibitem{Landsteiner:2000bw}
K.~Landsteiner, E.~Lopez and M.~H.~G.~Tytgat,
{\it Excitations in hot non-commutative theories},
\jhep{0009}{2000}{027} [\hepth{0006210}];
%%CITATION = HEP-TH 0006210;%%
%\cite{Landsteiner:2001ky}
%\bibitem{Landsteiner:2001ky}
%K.~Landsteiner, E.~Lopez and M.~H.~G.~Tytgat,
{\it Instability of non-commutative SYM theories 
at finite temperature},
\jhep{0106}{2001}{055} [\hepth{0104133}].
%%CITATION = HEP-TH 0104133;%%




\bibitem{ruiz}
%\cite{Martin:2000bk}
%\bibitem{Martin:2000bk}
C.~P.~Martin and F.~Ruiz Ruiz,
{\it Paramagnetic dominance, 
the sign of the beta function and UV/IR mixing in
non-commutative U(1)},''
\npb{597}{2001}{197} [\hepth{0007131}];
%%CITATION = HEP-TH 0007131;%%
%
%\cite{Ruiz:2000hu}
%\bibitem{Ruiz:2000hu}
F.~R.~Ruiz,
{\it Gauge-fixing independence of IR divergences in non-commutative U(1),
perturbative tachyonic instabilities and supersymmetry},
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%%CITATION = HEP-TH 0012171;%%



%\cite{Bassetto:2001vf}
\bibitem{Bassetto:2001vf}
A.~Bassetto, L.~Griguolo, G.~Nardelli and F.~Vian,
\emph{On the unitarity of quantum gauge theories on noncommutative spaces},
\jhep{0107}{2001}{008} [\hepth{0105257}].
%%CITATION = HEP-TH 0105257;%%


\bibitem{rf:MVR}
M.~Van Raamsdonk,
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\bibitem{rf:AL}
A.~Armoni and E.~Lopez,
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%\cite{Guralnik:2002ru}
\bibitem{Guralnik:2002ru}
Z.~Guralnik, R.~C.~Helling, K.~Landsteiner and E.~Lopez,
{\it Perturbative instabilities on the non-commutative torus, 
Morita duality and twisted boundary conditions},
\jhep{0205}{2002}{025} [\hepth{0204037}].
%%CITATION = HEP-TH 0204037;%%



\end{thebibliography}