\noindent %%%version including plus minus problem We study the IIB matrix model, which is conjectured to be a nonperturbative definition of superstring theory, % is hard to study by standard Monte Carlo %simulation due to the notorious sign problem. by introducing an integer deformation parameter $\nu$ which couples to the imaginary part of the effective action induced by fermions. The deformed IIB matrix model continues to be well-defined for arbitrary $\nu$, and it preserves %The deformation preserves the well-definedness of the IIB matrix model, %and is compatible with gauge invariance, Lorentz invariance, and the cluster property. We study the model at $\nu=\infty$ using a saddle-point analysis, and show that ten-dimensional Lorentz invariance is spontaneously broken at least down to an eight-dimensional one. We argue that it is likely that the remaining eight-dimensional Lorentz invariance is further broken, which can be checked by integrating over the saddle-point configurations using standard Monte Carlo simulation. %without any sign problem. %We introduce an integer deformation parameter $\nu$ in the IIB matrix model, %which couples to the %imaginary part of the effective action induced by fermions. %The deformation preserves the well-definedness of the IIB matrix model, %and is compatible with gauge invariance, Lorentz invariance, %and the cluster property. %While the deformed model reduces to the IIB matrix model %at $\nu=1$, we study the model at $\nu=\infty$ using a saddle-point analysis, %and show that ten-dimensional Lorentz invariance is spontaneously broken %at least down to an eight-dimensional one. %By studying the hessian around saddle-point configurations, %we argue that it is likely that %the remaining eight-dimensional Lorentz invariance is %further broken, which can be checked in principle, %by integrating over the saddle-point configurations %using standard Monte Carlo simulations without any sign problem.
hep-th0003223.tex
\documentclass[12pt]{article}

\usepackage{amssymb}
\usepackage[dvips]{graphicx}

\setlength{\textwidth}{16.5cm}
\setlength{\textheight}{23cm}
%\setlength{\topmargin}{-2.5cm}
\setlength{\topmargin}{-0.5cm}

\def\appendix#1{
  \addtocounter{section}{1}
  \setcounter{equation}{0}
  \renewcommand{\thesection}{\Alph{section}}
  \section*{Appendix \thesection\protect\indent \parbox[t]{11.715cm} {#1}}
  \addcontentsline{toc}{section}{Appendix \thesection\ \ \ #1}
  }

\def\l{\Big(}
\def\r{\Big)}
\def\E{{\cal E}}
\newcommand {\bd}{\begin{displaymath}}
\newcommand {\ed}{\end{displaymath}}
\newcommand {\eq}{\begin{equation}}
\newcommand {\beq}{\begin{equation}}
\newcommand {\eeq}{\end{equation}}
\newcommand {\beqa}{\begin{eqnarray}}
\newcommand {\eeqa}{\end{eqnarray}}
\newcommand {\n}{\nonumber \\}
\newcommand {\tr}{{\rm tr\,}}
\newcommand {\Tr}{\mbox{Tr\,}}
\newcommand {\eff}{\mbox{\scriptsize eff}}
\newcommand {\Det}{\mbox{Det}}
\newcommand {\Pf}{\mbox{Pf}}
\newcommand {\Imag}{\mbox{Im}}
\newcommand {\trs}{\mbox{\scriptsize tr}}
\newcommand {\trees}{\mbox{\scriptsize tree}}
\newcommand {\ee}{\mbox{e}}
\newcommand {\ees}{\mbox{\scriptsize e}}
\newcommand {\dd}{\mbox{d}}
\newcommand {\dlangle}{\langle \! \langle}
\newcommand {\drangle}{\rangle \! \rangle}
\newcommand {\dds}{\mbox{\scriptsize d}}
\newcommand {\del}{\partial}
\newcommand {\fac}{\mbox{\scriptsize !}}
\newcommand {\defeq}{\stackrel{\rm def}{=}}
%\newcommand {\lsim}{\raisebox{-.6ex}{$\stackrel{<}{\sim}$}}
%\newcommand {\gsim}{\raisebox{-.6ex}{$\stackrel{>}{\sim}$}}
\newcommand {\lead}{\stackrel{\rm leading }{\simeq}}

%
\font\mybbs=msbm10 at 9pt
\def\bbs#1{\hbox{\mybbs#1}}
\font\mybb=msbm10 at 12pt
\def\bb#1{\hbox{\mybb#1}}

\def\IT{{\bb T}}
\def\ITs{{\bbs T}}
\def\IC{{\bb C}}
\def\IR{{\bb R}}
\def\IZ{{\bb Z}}
\def\ICs{{\bbs C}}
\def\IRs{{\bbs R}}
\def\IZs{{\bbs Z}}
\newcommand{\vl}{\ell}
\newcommand{\K}{s}
%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\id}{{1\!\!1}} %% identity operator
\def\wt{\widetilde}
\def\ep{\epsilon}
\def\th{\theta}
\def\la{\lambda}
\def\da{\dot{a}}
\def\pa{\partial}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\theequation}{\arabic{equation}}
\def\rref#1{(\ref{#1})}

\begin{document}

\setlength{\oddsidemargin}{0cm}
\setlength{\baselineskip}{7mm}


\begin{titlepage}

\baselineskip=14pt
%\baselineskip=12pt

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{normalsize}
\begin{flushright}
\begin{tabular}{l}
NBI-HE-00-15\\
hep-th/0003223\\
\hfill{ }\\
March 2000
\end{tabular}
\end{flushright}
  \end{normalsize}

%{}~~\\

\vskip 2 cm

\vspace{1cm}

\vspace*{0cm}
    \begin{Large}
       \begin{center}
{Spontaneous Breakdown of Lorentz Invariance in IIB Matrix Model}\\
%\vspace{1cm}


       \end{center}
    \end{Large}
\vspace{1cm}

\begin{center}
           Jun N{\sc ishimura}\footnote{
Permanent address : Department of Physics, Nagoya University,
Nagoya 464-8602, Japan,\\
e-mail address : nisimura@nbi.dk}
\setcounter{footnote}{2}
%a\footnote{a}
%b\footnote{b}
%c\footnote{c}
%d\footnote{d}
%e\footnote{e}
%f\footnote{f}
%g\footnote{g}
           {\sc and}
           Graziano V{\sc ernizzi}\footnote{
e-mail address :
vernizzi@nbi.dk}\\
      \vspace{1cm}
         {\it The Niels Bohr Institute\\ Blegdamsvej 17, DK-2100
                 Copenhagen \O, Denmark}\\[4mm]
\end{center}

\vskip 2 cm

\hspace{5cm}


\begin{abstract}
\noindent
%%%version including plus minus problem
We study the IIB matrix model, 
which is conjectured to be a nonperturbative definition
of superstring theory,
% is hard to study by standard Monte Carlo 
%simulation due to the notorious sign problem.
by introducing an integer deformation parameter $\nu$
which couples to the 
imaginary part of the effective action induced by fermions.
The deformed IIB matrix model continues to 
be well-defined for arbitrary $\nu$,
and it preserves
%The deformation preserves the well-definedness of the IIB matrix model,
%and is compatible with 
gauge invariance, Lorentz invariance,
and the cluster property.
We study the model at $\nu=\infty$ using a saddle-point analysis, 
and show that ten-dimensional Lorentz invariance is spontaneously broken
at least down to an eight-dimensional one.
We argue that it is likely that 
the remaining eight-dimensional Lorentz invariance is 
further broken, which can be checked
by integrating over the saddle-point configurations 
using standard Monte Carlo simulation.
%without any sign problem.
%We introduce an integer deformation parameter $\nu$ in the IIB matrix model,
%which couples to the 
%imaginary part of the effective action induced by fermions.
%The deformation preserves the well-definedness of the IIB matrix model,
%and is compatible with gauge invariance, Lorentz invariance,
%and the cluster property.
%While the deformed model reduces to the IIB matrix model
%at $\nu=1$, we study the model at $\nu=\infty$ using a saddle-point analysis, 
%and show that ten-dimensional Lorentz invariance is spontaneously broken
%at least down to an eight-dimensional one.
%By studying the hessian around saddle-point configurations,
%we argue that it is likely that 
%the remaining eight-dimensional Lorentz invariance is 
%further broken, which can be checked in principle,
%by integrating over the saddle-point configurations 
%using standard Monte Carlo simulations without any sign problem.
\end{abstract}
\vfill
\end{titlepage}
\vfil\eject
%\setcounter{page}{2}
\setcounter{footnote}{0}

\setcounter{equation}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

\baselineskip=18pt
%\baselineskip=15pt

\setcounter{footnote}{0}
%\section{Introduction}
%\setcounter{equation}{0}

\renewcommand{\thefootnote}{\arabic{footnote}}
Recent developments in string dualities have suggested that the known
five types of superstring theories in ten dimensions and 
M-theory in eleven dimensions are just different microscopic descriptions
of the same underlying physics.
%,which gives rise to the same low energy dynamics.
This means that if any of the descriptions can be defined nonperturbatively,
%as one does for QCD with lattice gauge theory,
it is as good as any other
to study the 
%low-energy 
dynamics of 
the universality class of string/M theories.
A celebrated first step towards this end has been made by 
Ref. \cite{BFSS}, where the so-called Matrix theory has been proposed as
a nonperturbative definition of M-theory in the infinite momentum frame.
%The precise definition of infinite momentum frame has been clarified in 
%\cite{SeibergSen}.
Soon after, 
the IIB matrix model \cite{IKKT} has been proposed as a nonperturbative
definition of type IIB superstring theory.
%An advantage of the latter proposal is that 
%it preserves manifest Lorentz invariance.
Although there are many evidences that support these conjectures,
there is no direct proof that these models reproduce the string/M theories
perturbatively\footnote{See, however, Ref. \cite{FKKT}
as an attempt in this direction.}.
%Evidences supporting these conjectures
%are concerned with the resemblance of the action 
%or the Hamiltonian of the theory
%to the corresponding one
If these conjectures are true, 
there is a hope 
%one should be able in principle 
to understand all the fundamental questions 
about the Standard Model, 
%all the ingredients of the Standard Model, 
%dynamical origin
including the space-time dimensionality, gauge group, matter contents, 
the hierarchy problem, 
the cosmological constant problem, and so on,
in terms of the dynamics of these models.
%On the other hand, if it turns out that the vacuum of the theories are given
%by a world which is different from ours, then either the conjecture is wrong
%or the string/M theories simply do not describe our world.



%A remarkable fact of large-$N$ gauge theory in the 't Hooft's large-$N$ limit
%is that the theory does not depend on the volume as far as the nonwinding 
%sector
%of the theory is concerned \cite{EK}.
%%This means that the space-time degrees of freedom is absorbed into the 
%%infinitely many internal degrees of freedom.
%Large $N$ reduced model \cite{EK} was originally developed
%as a model which is equivalent to large $N$ gauge theory in the 
%'t Hooft limit.
%Although the model is completely well defined for finite $N$,
%and one can in principle perform Monte Carlo simulation as
%has been done in Ref. \cite{KNS} for $N=3$, 

%So far, the understanding of the vacuum of these models
%is quite limited.
%In this Letter, we attempt to extract information about
%%an important piece of information of the vacuum
%%dynamical property namely
%the dimensionality of the space-time which
%is dynamically generated in the IIB matrix model.


So far, the understanding of the vacuum of these models
is quite limited.
In this Letter, we attempt to extract information about
%an important piece of information of the vacuum
%dynamical property namely
the dimensionality of the space-time which
is dynamically generated in the IIB matrix model.
The IIB matrix model is a supersymmetric matrix model obtained
formally by taking a zero-volume limit \cite{EK} of ten-dimensional
SU($N$) super Yang-Mills theory.
Unlike in field theories,
%where the space-time dimensionality is an input paraeter,
the space-time is treated in this model as a dynamical object,
which is represented by ten bosonic $N \times N$ hermitian matrices, 
where $N$ should be sent to infinity eventually.
%Therefore, one may hope to understand the dynamical origin of the 
%dimensionality of our space-time by studying the large $N$ dynamics
%of this model.
Therefore, the model has a potential 
to explain the dynamical origin of the 
dimensionality of our space-time,
which, in the Standard Model, has to be given as an input parameter. 
%field theories.
%
% by studying the large $N$ dynamics
% of this model.
If we are ever going to explain 
our four-dimensional space-time in this way,
the vacuum 
%dominant configurations 
of the IIB matrix model should be
dominated by configurations 
which have only four-dimensional extent\footnote{This is similar
in spirit to a description of
our four-dimensional space-time as a ``brane'' in a higher-dimensional
{\em non-compact} space-time, which is proposed \cite{RS}
as an alternative to a more conventional
compactification approach.}.
This means, in particular, that the ten-dimensional Lorentz invariance
of the model should be spontaneously broken to a four-dimensional 
one.\footnote{The possibility of a non-trivial vacuum structure in 
string theory has been considered also in \cite{AK}.}

%one may hope to address a most fascinating issue
%the model has a potential to explain why our space-time
%is four-dimentional.

Such 
an
%a fascinating 
issue can 
{\em in principle} be addressed by performing Monte Carlo simulation,
since the model is completely well defined for arbitrary $N$
%larger than 2 
without any cutoff \cite{SuyamaTsuchiya,AIKKT,KNS}.
%Note first that the model is completely well defined for finite $N$ larger 
%than 2 without any cutoff \cite{SuyamaTsuchiya,AIKKT,KNS},
%%as has been done in Ref. \cite{} for $N=3$.
%and one can in principle perform Monte Carlo simulation to address
%such an issue.
Indeed the large-$N$ dynamics of 
the four-dimensional version of the IIB matrix model, 
which can be obtained by a  zero-volume limit
of four-dimensional super Yang-Mills theory, 
has been understood to a considerable extent through Monte Carlo simulation 
\cite{AABHN}.
Comparison of the results with those for the corresponding 
bosonic theory \cite{HNT}, 
%which can be obtained by simply dropping the fermions,
revealed that supersymmetry indeed plays an 
important role in the dynamics of large-$N$ reduced models
as a nonperturbative definition of string theories.
Unfortunately, such a direct approach does not work in the IIB matrix model
as it stands,
due to the fact that the effective action induced by fermions
%the fermion determinant 
is generically complex,
whereas in the above mentioned four-dimensional version, it is real.
In general, when the action of a theory has a non-zero imaginary part,
the number of configurations needed to extract any information
increases as exponential of the system size.
%which is O($N^2$) in the present case.
This is the notorious
%well-known 
sign problem, which occurs also
in many other interesting systems related to particle physics, 
such as theories with a chiral fermion (as in the present case),
theories with a $\theta$ vacuum, Chern-Simons theories
and theories with a finite baryon number density.
We emphasize, however, that the problem is a purely technical one,
and indeed its complete solution in some particular class
of systems has been obtained recently \cite{sign}.
%A complete solution to this problem in some particular class
%of systems has been obtained recently \cite{sign},
%but the technique does not seem to be directly 
%applicable to the present case.


%One thing one can try in this kind of situation is to
%study the model by dropping the imaginary part by hand.
%
%In order to 
%In this Letter, 
In order to search for a way out of this difficulty,
let us consider a deformation of the IIB matrix model by
%Let us 
%we 
introducing
an integer parameter $\nu$, which couples
to the 
%phase of the fermion determinant.
imaginary part of the effective action induced by fermions.
The deformed model, which reduces to the IIB matrix model
at $\nu=1$, 
is well-defined for arbitrary $\nu$.
It preserves both gauge invariance and Lorentz invariance,
and 
%
%The deformed model 
%and $N$
% \ge 2$
%and possesses both gauge invariance and Lorentz invariance.
moreover, it preserves the cluster property, which is an important 
consequence of the supersymmetry of the original model.
At $\nu=0$, the sign problem disappears and standard Monte Carlo
techniques become applicable.
Monte Carlo studies up to $N=512$ show, however, that
the ten-dimensional Lorentz invariance is 
{\em not} spontaneously broken \cite{oneloop},
which suggests that if Lorentz invariance is spontaneously
broken in the original model,
% ($\nu = 1$),
the imaginary part of the effective action must play a 
crucial role.
This motivated us to consider the opposite extreme,
namely $\nu = \infty$.
%In fact, we will argue that it is likely that
%the original model $\nu = 1$ belongs to the same phase 
%as $\nu = \infty$.
%is qualitatively
%closer to $\nu = \infty$ rather than to $\nu = 0$.

At $\nu = \infty$, 
the integration over the bosonic matrices
%$A_\mu$ 
is dominated by the
saddle-points of the imaginary part of the effective action.
We find that the saddle-points are
given by configurations 
%$\{A_\mu\}$ 
which have only eight-dimensional extent.
This implies
%, first of all, 
that 
the ten-dimensional Lorentz invariance {\em is} 
spontaneously broken 
%in this limit 
at least to an eight-dimensional one.
%Secondly, 
Since we still have to integrate over 
the saddle-point configurations,
the question arises whether 
the remaining eight-dimensional Lorentz invariance further breaks down, 
say, to a four-dimensional one.
%We will show that the effective action for these saddle-point
%configurations is real, which means that
%the sign problem, which is expected naively to become maximally severe
%at $\nu = \infty$, actually disappears in this limit.
%This means that standard Monte Carlo simulations can be used 
%to perform the integration over the saddle point configurations.
%extract the information of the vacuum at $\nu = \infty$.
To this end, we study the hessian at configurations
which have only $d$-dimensional extent ($d\le 8$)
and find that it is zero of order $(8-d)\, \{2\, (N^2-1)-16\}$.
In other words, the imaginary part of the
effective action becomes {\em more} stationary for smaller $d$.
This gives a huge enhancement
to configurations with smaller $d$,
%This gives an enhancement to configurations with smaller $d$,
which is found to cancel exactly
the entropical barrier against having such configurations.
%exactly the $N$-dependent suppression
%factor coming from the integration measure.
%
%and this effect cancels completely 
%$smaller $d$.
Considering the effect of the real part of the effective action,
% \cite{AIKKT},
which is studied in Ref. \cite{AIKKT} using perturbation theory, 
%argued to induce a collapse of eigenvalue distribution of
%$A_\mu$ in Ref. \cite{AIKKT}, 
%a spontaneous breakdown of Lorentz invariance,
it is plausible that the vacuum of the model at $\nu = \infty$
is actually given by $d<8$.
%We also show that a lower bound on $d$ can be given as $d > 2$.
We also find a lower bound on $d$ as $d > 2$.
Further information about the vacuum can be obtained
by performing explicitly
the integration over the 
saddle-point configurations by Monte Carlo simulation.
Naively, one might consider that the sign problem, which already 
exists in the original model ($\nu = 1$), 
becomes maximally severe at $\nu = \infty$.
%Surprisingly, this is not the case, as we will see.
This is not the case, as we will see.
%We hope that Monte Carlo studies of the model we derive at 
%$\nu = \infty$ will shed light to the dynamics of the IIB matrix model
%in near future.
%
%saddle-point configurations can be studied
%by standard Monte Carlo simulation without any sign problem.

%Finally we derive a model which describes the integration over the
%Finally, we find that the limiting theory 
%in the $\nu \rightarrow \infty$ limit
%actually depends on the parity of $\nu$.
%For the even $\nu$ sequence, the model
%which describes the integration over the 
%saddle-point configurations can be studied
%by standard Monte Carlo simulation without any sign problem.


%We put a bound on the space-time dimensionality $d$, which is obtained 
%dynamically from the eigenvalue distribution of the bosonic matrices, 
%as $3\le d \le 8$.
%We also propose a 
%direct Monte Carlo simulation without sign problem,
%with which we can determine $d$ explicitly.

%While the deformed model reduces to the IIB matrix model
%at $\nu=1$, we solve the model at $\nu = \infty$, and show that
%the vacuum of the theory is given by flat four-dimensional space-time.

 
\vspace*{1cm}


%\section{The model and properties of fermion integral}
%\label{fermiondet}
%\setcounter{equation}{0}

The IIB matrix model is formally a zero-volume limit of
ten-dimensional pure ${\cal N}=1$ supersymmetric Yang-Mills theory.
The action, therefore, is given by
\beqa
S &=& S_b + S_f \ ,\n
S_b &=& -\frac{1}{4} \, \tr \left( [A_{\mu},A_{\nu}]^2 \right) \ , \n
S_f  &=& - \frac{1}{2} \,
\tr \left( \psi _\alpha (\, {\cal C} \,  \Gamma_{\mu})_{\alpha\beta} 
[A_{\mu},\psi _\beta] \right)  \ .
\label{action}
\eeqa
$A_\mu$ ($\mu = 1,\cdots,10$) and 
$\psi_\alpha$ ($\alpha = 1,\cdots , 16$) are
$N \times N$ traceless hermitian matrices,
which can be expanded in terms of the generators 
$t^a$ of SU($N$) as
\beq
(A_{\mu })_{ij} = \sum_{a=1}^{N^2-1} 
A_{\mu}^{ a} \, (t^{a})_{ij} ~~~~~;~~~~~  
(\psi_\alpha ) _{ij} = \sum_{a=1}^{N^2-1} 
\psi_{\alpha}^{ a} \, (t^{a})_{ij} \ , 
%(\psi _{ij}^{\alpha})^* &=& \sum_{a=1}^{N^2-1} 
%(\psi_{\alpha}^{~ a})^* ~ (t^{a})_{ij} \ , 
\eeq
where $A_\mu ^{a}$ is a real variable and 
$\psi_{\alpha}^{a}$ is a real Grassmann variable.
We have made a Wick rotation in the action (\ref{action}),
so that the metric has Euclidean signature.
The $16 \times 16$ matrices $\Gamma _\mu$ are 
ten-dimensional gamma matrices after Weyl projection,
and the unitary matrix ${\cal C}$ is a 
charge conjugation matrix satisfying
\beq
%(\, {\cal C} \, \Gamma _\mu)^\top = {\cal C} \, \Gamma _\mu ~~~~~;~~~~~
%{\cal C} ^\top = {\cal C}  \ .
{\cal C} \, \Gamma _\mu \, {\cal C}^\dag
= (\Gamma _\mu)^\top ~~~~~;~~~~~
{\cal C} ^\top = {\cal C}  \ .
\label{chargeconj}
\eeq
The model has a manifest ten-dimensional 
Lorentz invariance,
by which we actually mean an SO(10) invariance.
$A_{\mu}$ transforms as a 
%10-dimensional 
vector and
$\psi _\alpha$ transforms as a Majorana-Weyl spinor.
%

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

Pure ${\cal N}=1$ supersymmetric Yang-Mills theory
can be also defined in 3D, 4D and 6D, as well as in 10D.
Hence, by taking a zero-volume limit of these theories,
we can define supersymmetric large-$N$ reduced models,
which are $D=3,4,6$ versions of the IIB matrix model.
A nontrivial question then concerns
whether the integration over the bosonic matrices 
is convergent, since 
the integration domain for hermitian matrices is non-compact.
A potential danger of divergence, 
even for finite $N$, exists when 
the eigenvalues of $A_\mu$ become large. 
%Since the eigenvalues of $A_\mu$ are interpreted as 
%space-time coordinates in IIB matrix model, 
%this divergence can be regarded as the infrared divergence.
This issue has been addressed in Ref. \cite{AIKKT}
using a one-loop perturbative argument.
When all the eigenvalues are well separated from each other,
one can expand the matrices $A_\mu$ and $\psi _\alpha$ 
around diagonal matrices as
\beq
(A_\mu)_{ij} = x_{i \mu} \, \delta _{ij} + a_{\mu ij} ~~~~~;~~~~~
(\psi_\alpha)_{ij}  = \xi_{i \alpha} \, \delta _{ij} + \varphi_{\alpha ij}  \ ,
\label{perturbation}
\eeq
where $a_\mu$ and $\varphi_\alpha$ are matrices containing only off-diagonal
elements.
One can then integrate over the off-diagonal elements
$a_\mu$ and $\varphi_\alpha$ up to one-loop,
by fixing the gauge properly and including the Faddeev-Popov ghosts.
The integration over the bosonic off-diagonal elements $a_\mu$ 
and the Faddeev-Popov ghosts, 
gives a logarithmic attractive potential between
all the pairs of $x_i$ \cite{BHN}.
This potential, however, is exactly cancelled by the contribution of the
fermionic off-diagonal elements $\varphi_\alpha$ due to supersymmetry.
This cancellation is responsible for the cluster property of the model
\cite{IKKT}, which is important for the interpretation of the model
as a string theory.
In order to calculate the effective potential for $x_{i \mu}$,
one still has to integrate 
over the fermionic diagonal elements $\xi_{i \alpha}$, 
which is nontrivial.
In Ref. \cite{AIKKT}, it has been shown that
%regarding $x_{i \mu}$ as 
%$D$-dimensional coordinate of $i$-th point,
the effective potential for $x_{i\mu}$ can be given by
a branched-polymer like attractive interaction among $N$ points in
$D$-dimensional space-time represented by $x_{i\mu}$ ($i=1,\cdots ,N$).
Now counting the power of $x_i$ in the partition function, 
the integration measure for $x_{i \mu}$
gives $D\, (N-1)$ and 
the integration over $\xi _{i \alpha}$ gives
$- \frac{3}{2}\, p \, (N-1)$, where $p=2\, (D-2)$ is the number of
real components of the spinor considered for each $D$.
Therefore, naively one concludes that when 
\beq
D\, (N-1) - \frac{3}{2} \, p \, (N-1) = 2 \, (3-D) \, (N-1) \ge 0  \ , 
\eeq
the integration over $x_{i \mu}$ is divergent
and otherwise convergent.
This means that the integral is divergent for $D=3$ and 
convergent for $D=4,6,10$, irrespectively of $N$.
This conclusion is in agreement with
an exact result available for $N=2$ \cite{SuyamaTsuchiya}
and a numerical result obtained for $N=3$ \cite{KNS}.
%as well as the bosonic case \cite{KS}.
For $D=4$, 
Monte Carlo simulations show that the conclusion extends
to the large-$N$ limit \cite{AABHN}.
%up to $N=48$ 
%confirms further this conclusion.
Therefore, 
%From these observations,
it is conceivable
%reasonable to expect 
that the above conclusion obtained by the 
one-loop argument holds in general, 
although it is not a rigorous proof.
%for arbitrary $N$.


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

Going back to the definition of the model (\ref{action}),
let us first integrate over the fermionic matrices $\psi _ \alpha$,
which induces an effective action for $A_\mu$.
When fermions are real Grassmann variables as in the present $D=10$ case,
the fermion integral yields
%induced effective action for $A_\mu$ 
%is represented as 
a pfaffian of a $p \, (N^2 -1) \times p \, (N^2 -1)$ matrix.
On the other hand, when fermions are complex Grassmann variables 
as in the $D=4,6$ case,
the fermion integral yields
%the induced effective action for $A_\mu$ 
%is represented as 
a determinant of a 
$p \,(N^2 -1)/2 \times p \, (N^2 -1)/2$ matrix.
%or pfaffian (when fermions are of Majorana type) of 
%a fermion determinant.
% (or ``pfaffian'' when the fermions are Majorana, 
%to be precise).
For $D=4$, the fermion determinant is real positive, which allows
a direct Monte Carlo study of the model \cite{AABHN}.
For $D=6,10$, the fermion determinant ($D=6$) or pfaffian ($D=10$)
is complex in general, since fermions are essentially chiral in these
cases. This causes the notorious sign problem.
There are some exceptions\footnote{This is because,
when $(N^2 -1)$ is smaller than $D$, one can always make a 
$D$-dimensional rotation to set $A_{\mu} = 0$ for
$\mu = 1, \cdots , D-(N^2 -1)$,
since one can regard $A_{\mu}^a$ as $(N^2 -1)$, $D$-dimensional 
real vectors.
The statements in $D=10$ for $N=2$ and $N=3$, for example, follow from
(\ref{omega6}) and (\ref{omega9}), respectively.} 
when $N$ is small \cite{SuyamaTsuchiya,KNS}.
For $D=6$, the fermion determinant is real positive for $N=2$.
For $D=10$, the pfaffian is real positive for $N=2$ and
real (but not positive definite) for $N=3$.
Although we consider only the $D=10$ case
in what follows, 
the analysis can be readily applied to the $D=6$ case as well.
We also restrict ourselves to $N \ge 4$ so that 
the pfaffian is complex generically.
%We collect the results for $D=6$ in Appendix.

In $D=10$, the fermion integral yields
%In $D=10$, the fermion determinant\footnote{To be precise,
%it is not a determinant but a pfaffian. We, however, keep on using 
%this terminology.} can be given by
\beq
Z_f [A] = 
 \int \dd \psi   ~ \ee ^{-S_f } = \Pf \, {\cal M}  \ ,
\eeq
where
\beq
{\cal M}_{a \alpha ,  b \beta} = 
-  i \, f_{abc} (\,{\cal C} \, \Gamma _\mu)_{\alpha \beta} A_\mu ^c  
\eeq
is a $16 \,  (N^2-1) \times 16 \, (N^2-1)$ anti-symmetric matrix, regarding
each of $(a\alpha)$ and $(b\beta)$ as a single index.
The real totally-antisymmetric tensor $f_{abc}$ gives 
the structure constants of SU($N$).
In what follows, it proves convenient to work with an explicit 
representation of the gamma matrices given by
\beqa
%&~&\Gamma ^1 = i \sigma_2  \otimes \sigma_2  
%              \otimes \sigma_2  \otimes \sigma_2 ~~;~~
%\Gamma ^2 = i \sigma_2  \otimes \sigma_2
%              \otimes {\bf 1} \otimes \sigma_1 ~~;~~
%\Gamma ^3 = i \sigma_2  \otimes \sigma_2 
%              \otimes {\bf 1} \otimes \sigma_3 ~~;~~\n
%&~&\Gamma ^4 = i \sigma_2  \otimes \sigma_1
%              \otimes \sigma_2 \otimes {\bf 1} ~~;~~
%\Gamma ^5 = i \sigma_2  \otimes \sigma_3
%             \otimes \sigma_2 \otimes {\bf 1} ~~;~~
%\Gamma ^6 = i \sigma_2  \otimes {\bf 1}
%              \otimes \sigma_1 \otimes \sigma_2  ~~;~~ \n
%&~&\Gamma ^7 = i \sigma_2  \otimes {\bf 1}
%             \otimes \sigma_3 \otimes \sigma_2 ~~;~~
%\Gamma ^8 = i \sigma_1  \otimes {\bf 1}
%              \otimes {\bf 1} \otimes {\bf 1} ~~;~~
%\Gamma ^9 = i \sigma_3  \otimes {\bf 1}
%              \otimes {\bf 1} \otimes {\bf 1} ~~;~~\n
%&~&\Gamma ^{10} = {\bf 1}  \otimes {\bf 1} \otimes {\bf 1} \otimes {\bf 1} ,
&~&\Gamma _1 = i \, \sigma_2  \otimes \sigma_2  
              \otimes \sigma_2  \otimes \sigma_2 ~;~
\Gamma _2 = i \, \sigma_2  \otimes \sigma_2
              \otimes {\bf 1} \otimes \sigma_1 ~;~
\Gamma _3 = i \, \sigma_2  \otimes \sigma_2 
              \otimes {\bf 1} \otimes \sigma_3 ~;~\n
&~&\Gamma _4 = i \, \sigma_2  \otimes \sigma_1
              \otimes \sigma_2 \otimes {\bf 1} ~;~
\Gamma _5 = i \, \sigma_2  \otimes \sigma_3
             \otimes \sigma_2 \otimes {\bf 1} ~;~
\Gamma _6 = i \, \sigma_2  \otimes {\bf 1}
              \otimes \sigma_1 \otimes \sigma_2  ~;~ \n
&~&\Gamma _7 = i \, \sigma_2  \otimes {\bf 1}
             \otimes \sigma_3 \otimes \sigma_2 ~;~
\Gamma _8 = i \, \sigma_1  \otimes {\bf 1}
              \otimes {\bf 1} \otimes {\bf 1} ~;~
\Gamma _9 = i \, \sigma_3  \otimes {\bf 1}
              \otimes {\bf 1} \otimes {\bf 1} ~;~\n
&~&\Gamma _{10} = {\bf 1}  \otimes {\bf 1} \otimes {\bf 1} \otimes {\bf 1} \ ,
%~~~&~&~~~ 
\label{Gamma}
\eeqa
for which the charge conjugation matrix ${\cal C}$ becomes a unit matrix.

Some important properties of the pfaffian $\Pf \, {\cal M}(A)$
are in order. First of all, it is invariant under a ten-dimensional 
Lorentz transformation,
$A'_\mu = \Lambda _{\mu \nu} A_\nu$, 
where $\Lambda$ is an SO(10) matrix.
It is also invariant under a gauge transformation
$A'_\mu = g   A_\nu g ^\dag$,
where $g$ is an SU($N$) matrix.
Under a parity transformation,
\beq
A^P _{10}  = - A_{10} ~~~~~;~~~~~
A^P_i  =   A_i ~~~~~(\mbox{for}~~~i=1,\cdots, 9) \ ,
\label{parity}
\eeq
%\beqa
%A^P_i  &=&   A_i ~~~~~~~~~~(\mbox{for}~~~i=1,\cdots, 9) \\
%A^P _{10}  &=& - A_{10} ,
%\eeqa
the pfaffian becomes complex conjugate,
\beq
\Pf \, {\cal M}(A^P)  = \{ \Pf \, {\cal M}(A)\} ^*  \ ,
\label{paritytransf}
\eeq
since ${\cal M} (A^P) = {\cal M} (A) ^*$.
As a consequence, the pfaffian $\Pf \, {\cal M}(A)$ is real,
when $A_{10}=0$.
This also means that 
$\det {\cal M} = (\Pf \, {\cal M})^2$ is real positive,
when $A_{10}=0$.

%The fact that $\Pf \, {\cal M}(A_{10}=0)$ is real 
%can be understood also from a property of eigenvalues
%of the matrix ${\cal M}$.
%Due to the fact that ${\cal M}$ is anti-symmetric,
%if $\lambda$ is an eigenvalue of ${\cal M}$,
%so is $-\lambda$.
%When $A_{10}=0$, ${\cal M}$ becomes anti-hermitian,
%which means that all the eigenvalues are pure imaginary.
%Thus one concludes that $\det {\cal M}(A_{10}=0)$ is real positive,
%which is equivalent to $\Pf \, {\cal M}(A_{10}=0)$ being real,
%due to $\det {\cal M} = (\Pf \, {\cal M})^2$.

We next consider the case with $A_9 =A_{10} = 0$.
In this case, the pfaffian is actually equal to
the fermion determinant 
of eight-dimensional Weyl fermion
which we denote as $\det {\cal M}^{(8)}$.
${\cal M}^{(8)}$ is an $8 \,(N^2-1)\times 8\, (N^2-1)$ matrix 
defined as ${\cal M}^{(8)}_{a \alpha ,  b \beta} = 
-  i \, f_{abc} (\tilde{\Gamma} _\mu)_{\alpha \beta} A_\mu ^c $,
where the eight-dimensional gamma-matrices $\tilde{\Gamma}_\mu$
after Weyl projection are given, for example, by
\beqa
&~& \tilde{\Gamma} _1 = i \, \sigma_2 \otimes \sigma_2  \otimes \sigma_2 ~;~
\tilde{\Gamma} _2 = i \, \sigma_2 \otimes {\bf 1} \otimes \sigma_1 ~;~
\tilde{\Gamma} _3 = i \, \sigma_2 \otimes {\bf 1} \otimes \sigma_3 ~;~
\tilde{\Gamma} _4 = i \,  \sigma_1 \otimes \sigma_2 \otimes {\bf 1} ~;~\n
&~&\tilde{\Gamma} _5 = i \, \sigma_3  \otimes \sigma_2 \otimes {\bf 1} ~;~
\tilde{\Gamma} _6 = i \,  {\bf 1} \otimes \sigma_1 \otimes \sigma_2  ~;~
\tilde{\Gamma} _7 = i \,  {\bf 1}  \otimes \sigma_3 \otimes \sigma_2 ~;~
\tilde{\Gamma} _8 =   {\bf 1} \otimes {\bf 1} \otimes {\bf 1} \ .
\label{Gamma8}
\eeqa
With this representation, one finds that ${\cal M}^{(8)}$
is a real matrix, which means that
$\det {\cal M}^{(8)} $ is real.
However, it is not positive definite, as we have checked
numerically.
In fact, this is the case
%The eight-dimensional determinant $\det {\cal M}^{(8)}$
%is not necessarily positive definite
even for $A_8 = 0$.
%A_9 =A_{10} = 0$.
However, if we take $A_7 = A_8 = 0$,
%A_9 =A_{10} = 0$, 
it becomes positive definite,
since in this case, we have
$\det {\cal M}^{(8)} = | \det {\cal M}^{(6)}| ^2  $,
where $\det {\cal M}^{(6)}$ is the fermion determinant 
of six-dimensional Weyl fermion \cite{KNS}.


Finally, when $A_3 =A_4 = \cdots = A_{10} = 0$,
one finds that $\Pf \, {\cal M} (A)=0$ \cite{KNS}.
This can be proved in the following way.
First we note that 
$(\Pf \, {\cal M})^2  =\det {\cal M} = |\det U|^{16}$, where
$U$ is an $(N^2 -1)\times (N^2 -1)$ matrix defined as
$U_{ab}=f_{abc} X^c$, where $X^c = A_1^c + i \, A_2^c$.
Since $U_{ab}X^b=0$, the matrix $U$ has a zero-eigenvalue,
and therefore $\det  U = 0$.
This completes the proof. 

Using Lorentz invariance of the pfaffian $\Pf \, {\cal M}(A)$,
we summarize some of its important properties as follows.
Let us first define sets of ``degenerate'' 
configurations $\Omega _d$ as
\beq
\Omega _d = \{ ~~\{ A_\mu\}~~ ;~~ n_\mu ^{(i)} A_\mu = 0 ~~\mbox{for}~~
\exists \, n_\mu ^{(i)} (i=1,\cdots , 10-d)~~\mbox{linearly independent}\} \ .
\eeq
Obviously, $\Omega _1 \subset \Omega _2 \subset 
\cdots \subset \Omega _9$.
Then the statements are
\beqa
&~&\mbox{(a)~When}~\{A_\mu \}\in \Omega _9 \ , ~~~\Pf \, {\cal M}(A)\in \IR \ .
% is real (but not positive definite)} \ .  
%;$\det {\cal M}$ is real positive
\label{omega9}
\\
&~&
\mbox{(b)~When}~\{A_\mu \}\in \Omega _6 \ , ~~~\Pf \, {\cal M}(A) \ge 0 \ .
% is real positive}.
\label{omega6}
\\
&~& \mbox{(c)~When}~\{A_\mu \}\in \Omega _2 \ , ~~~\Pf \, {\cal M}(A) = 0 \ .
\label{omega2}
\eeqa
Here we recall that in the IIB matrix model, 
the space-time is treated as a
dynamical object represented by the bosonic matrices $A_\mu$.
In this regard,
generic configurations in $\Omega _d$ 
describe $d$-dimensional space-time.
%Since the bosonic matrices $A_\mu$ represent the
%space-time in IIB matrix model,
%generic configurations in $\Omega _d$ describe $d$-dimensional space-time.
It is very suggestive
that the phase of the pfaffian is sensitive to the
dimensionality $d$ of the space-time.
%configuration. 
%
%already suggests that
%the phase may play an important role in the dimensionality
%of the space time.
Indeed, the above results will play a crucial role
in the following analysis.
%These results will be crucial in the following analysis.

\vspace*{1cm}

%\section{Generalized model}

Let us write the effective action induced by fermion integral
as $\Gamma_{\eff}
= - \ln (\Pf \, {\cal M})$, which we decompose into the real part
$\Gamma ^{(r)}$ and the imaginary part $\Gamma ^{(i)}$ as
$\Gamma _{\eff} =\Gamma ^{(r)} + i \, \Gamma ^{(i)}$.
Then we generalize the theory by introducing an integer parameter $\nu$,
which couples only to $\Gamma ^{(i)}$, the imaginary part of the 
effective action, as 
\beq
Z_{\nu} = 
\int \dd A  ~ \ee ^{-S_b[A] - \Gamma ^{(r)} [A] } 
~ \ee ^{- i \, \nu \, \Gamma^{(i)} [A]} \ .
\label{fullPTnu}
\eeq
The parameter $\nu$ has to be 
an integer since $\Gamma^{(i)}$ is defined only up to
modulo $2 \pi$.
For $\nu =1$, the model reduces to the original model.
Since $\Gamma^{(i)}$ flips its sign under a parity transformation
due to (\ref{paritytransf}),
the models with $\nu$ and $-\nu$ are nothing but parity partners.
%For $N=2,3$, it is known that $\Gamma^{(i)}$ is zero for any 
%configurations \cite{SuyamaTsuchiya,KNS}.
%In what follows, we restrict ourselves to $N \ge 4$.
%in order to make the deformation meaningful.
%Since $\Gamma^{(i)}$ plays no role
Since $\Gamma ^{(r)}$ and $\Gamma ^{(i)}$ are both Lorentz invariant
and gauge invariant separately, the generalized model is also
both Lorentz invariant and gauge invariant.
%using
According to 
the one-loop perturbative argument given below (\ref{perturbation}),
the deformed IIB matrix model continues to be
well-defined for arbitrary $\nu$, and moreover, it preserves the
cluster property, which is an important consequence of the
supersymmetry of the original model.
%Since the fact that the IIB matrix model
%is well-defined is 
%due entirely to 
%the real part of the effecitive action $\Gamma ^{(r)}$
%as can be seen from the previous arguments,
%the deformation to $\nu \ne 1$ preserves these important properties.

Another important point about the generalization we consider is that
$\langle  S_b
%- \Tr [A_\mu , A_\nu]^2 
\rangle$ can be calculated
exactly and the result is independent of the parameter $\nu$.
For this, we recall that
in the original supersymmetric large-$N$ reduced models ($D=4,6,10$),
one obtains
% for $\langle \Tr F^2 \rangle$,
\beq
\langle  
%- \Tr [A_\mu , A_\nu]^2 
S_b \rangle
= \frac{1}{4} \, \left(D + \frac{p}{2}\right) (N^2 -1 ) 
= \frac{1}{2} \,  (D-1) \, (N^2 -1 )  \ ,
%\left( D + \frac{p}{2} \right) 
\label{trF2exact}
\eeq
using a scaling argument similar to \cite{HNT}.
% for arbitrary $\nu$
Due to the fact that $\Gamma ^ {(i)} [A]$ is invariant under
a scale transformation of $A_\mu$,
one can easily find that the above result is unaltered by
the generalization to arbitrary $\nu$ for $D=6,10$.
In particular, this holds true even in the $\nu \rightarrow \infty$ limit.
%which we discuss below.
%
%to which we now proceed.
%Note, however, that this does not necessarily suggest
%that all the vacuum expectation values of the generalized model
%are independent of $\nu$.
Note, however, that this does not necessarily imply
%$\nu \rightarrow \infty$ limit leads to a 
that all the vacuum expectation values of the generalized model
are independent of $\nu$.
Even their convergence in the $\nu \rightarrow \infty$ limit 
is nontrivial.
%We will answer to this question later.
This point shall be clarified later.
%clarify this point later.
%In fact, we will show below that the two sequences of $\nu$,
%$(I):\nu = 0,2,4,\cdots, \ 2\infty$ and
%$(II):\nu = 1,3,5,\cdots, \ (2\infty +1)$ lead
%to two distinctive limiting theories.

We prove a property of $\Gamma ^{(i)}$, which
turns out to be essential in the analysis of 
the $\nu \rightarrow \infty$ limit.
We first recall that
the pfaffian $\Pf \, {\cal M}(A)$ is
a polynomial of $A_\mu ^{a}$ of order $8 \, (N^2 -1)$.
Hence, $\Gamma ^{(i)}$ is
infinitely differentiable at a configuration,
for which the matrix ${\cal M}$ is invertible.
%Then, 
When the configuration belongs to $\Omega_d$ ($d=1,\cdots , 8$),
%$\{ A_\mu \} \in \Omega_d$,
we find that 
%
\beq
\frac{\del ^n \, \Gamma ^{(i)}}
{\del A_{\mu_1} ^{a_1} \del A_{\mu_2} ^{a_2}
\cdots \del A_{\mu_n} ^{a_n} } 
=
%
%\left.
- \frac{1}{2} \, 
\frac{\del ^n }{
\del A_{\mu_1} ^{a_1} \del A_{\mu_2} ^{a_2}
\cdots \del A_{\mu_n} ^{a_n}}
\,  \Imag \, \ln \det {\cal M } 
= 0 ~~~~~\mbox{for}~n=1,\cdots , (9-d) \ .
\label{derivatives}
\eeq
%due to (\ref{omega9}).
This is because, 
up to $(9-d)$-th order of perturbations, the configuration stays
within $\Omega _9$, and therefore $\det {\cal M}$ 
remains to be real positive due to (\ref{omega9}).


In the $\nu \rightarrow \infty$ limit,
the integration over $A_\mu$ is dominated by the configurations
which satisfy the saddle-point equation given by
\beq
0 = \frac{\del \, \Gamma ^{(i)}}{\del A_\mu ^a} =
- \frac{1}{2} \,
\frac{\del}{\del A_\mu ^a} \, \Imag \, \ln \det {\cal M}  
= - \frac{1}{2} \, \Imag \, \Tr \left({\cal M}^{-1} {\cal B} ^{(\mu a)} 
\right)  \ ,
\label{stationaryphase}
\eeq
where ${\cal B}^{(\mu a)} $ 
is an $A$-independent $16 \, (N^2-1) \times 16 \, (N^2-1)$ matrix defined by 
\beq
\left({\cal B}^{(\mu a)}\right)_{b \beta , c \gamma} 
= \frac{\del \, {\cal M}_{b\beta , c\gamma}}{\del A_\mu ^a}
= -i \, f_{abc} (\, {\cal C} \,  \Gamma_\mu)_{\beta\gamma}  \ .
\eeq
%This equation is invariant under Lorentz transformation 
%and gauge transformation.
%As a consequence, if $A_\mu ^a$ is a solution to these equations,
%so is $\Omega A_\mu ^a$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Taking the derivative explicitly, Eq. (\ref{stationaryphase}) can be
%rewritten as
%\beq
%\Imag \ \Tr \left({\cal M}^{-1} {\cal B} ^{(\mu a)} \right)
% = 0 ,
%\label{stationaryphase2}
%\eeq
Since the equation (\ref{stationaryphase})
gives $10 \, (N^2 -1)$ constraints among
$10 \, (N^2 -1)$ real variables,
naively one would expect that 
only a very small portion of the configuration space survives.
This is not the case, however.
%Since ${\cal M}^{-1} $ is proportional to the cofactor matrix of 
%${\cal M}$, 
%the equations are polynomial of the equal order $10 (N^2 -1 ) -1$.
%Counting the number of equations (\ref{stationaryphase}),
%generically the number of configurations which give the stationary point
%must be finite.
In fact, we find that
all the configurations that belong to $\Omega _8$
satisfy the saddle-point equation (\ref{stationaryphase})
due to (\ref{derivatives}).
%This can be shown as follows. 
%
%It is hard to prove the inverse, namely 
%that there are no solutions to (\ref{stationaryphase}) 
%other than of this type,
%but even if there exists, it is reasonable to consider
%that the configurations in $\Omega _8$ dominate
%in the sense of Lebesgue measure.
The existence of solutions to (\ref{stationaryphase}) 
other than of this type, cannot be excluded.
It is reasonable, however, to consider
that the configurations in $\Omega _8$ dominate
in the sense of Lebesgue measure.
%we should note then that the fermion determinant is real positive for 
%these configurations as we discussed already.
%Assuming that this is the case,
%we are lead to the conclusion that
Hence,
%these configurations are expected to dominate and 
at $\nu = \infty$
the full ten-dimensional Lorentz invariance is broken down at least to
an eight-dimensional one.
In order to examine whether the remaining eight-dimensional 
Lorentz invariance further breaks down,
we still have to integrate over the saddle-point configurations.
An important point here is that the imaginary part of the effective
action $\Gamma ^{(i)}$ for the saddle-point configurations
takes only 0 or $\pi$ due to (\ref{omega9}).
This means that there are actually two sequences of $\nu$,
(I)~$\nu = 0,2,4,\cdots,  2\infty$ and 
(II)~$\nu = 1,3,5,\cdots,  (2\infty +1)$,
which give two {\em a priori} different limiting theories.
%which means that the sign problem,
%which is expected naively to become maximally severe
%at $\nu = \infty$, actually disappears in this limit.
%
%We will show that the effective action for these saddle-point
%configurations is real, which means that
%the sign problem, which is expected naively to become maximally severe
%at $\nu = \infty$, actually disappears in this limit.
%This means that standard Monte Carlo simulations can be used 
%to perform the integration over the saddle point configurations.
%extract the information of the vacuum at $\nu = \infty$.
%
%We emphasize that this is not at all a trivial feature
%and is quite peculiar to the present system.


In order to formulate the integration over
the saddle-point configurations,
we first rotate them
%saddle-point configurations to
so that they satisfy $A_9 = A_{10}=0$
and then integrate over $A_1,A_2,\cdots, A_8$.
We define the Hesse matrix for $\Gamma ^{(i)}[A]$ 
within the directions transverse to the integration domain as
\beq
%J = \Det _{ia,jb} 
H_{ja,kb} =
%
\frac{\del ^2 \, \Gamma ^{(i)}}{\del A_j ^a \del A_k ^b}  =
%
%\left.
- \frac{1}{2} \, \frac{\del ^2 }{\del A_j ^a
\del A_k ^b} \,  \Imag \, \ln \det {\cal M } 
%\right|_{A_9 = A_{10}=0} 
=  
%\left.
\frac{1}{2} \,  \Imag \, \Tr \left({\cal M}^{-1} {\cal B} ^{(j a)}
{\cal M}^{-1} {\cal B} ^{(k b)}  \right)  \ ,
%\right|_{A_9 = A_{10}=0} ,
\eeq
where $j,k = 9,10$.
%Because of these properties,
%if $(\delta A_9 ^a , \delta A_{10} ^b)= (\varphi ^a, \psi ^b)$ 
%belongs to an eigenvalue $\lambda$,
%so does $(\delta A_9 ^a , \delta A_{10} ^b)= (\varphi ^a, - \psi ^b)$
%and moreover 
%$(\delta A_9 ^a , \delta A_{10} ^b)=(\varphi ^a, \pm \psi ^b)$ 
%belong to an eigenvalue $ - \lambda$.
We find that the Hesse matrix $H_{ja,kb}$ 
has $16$ zero-eigenvalues with eigenvectors corresponding to
perturbations 
\beq
(\delta A_9 ^a ,\delta A_{10}^a ) =
(A_1 ^a , 0),(A_2 ^a , 0), \cdots , (A_8 ^a , 0), 
 (0 , A_1 ^a ),(0 , A_2 ^a ), \cdots , (0 , A_8 ^a)    \ .
\label{zeromodes}
\eeq
These zero-modes are a reflection of the fact that
the configurations after these perturbations still stay within
$\Omega _8$, and thus satisfy
the saddle-point equation (\ref{stationaryphase}).
When $N$ is even, the Hesse matrix $H_{ja,kb}$ 
has actually two more zero-eigenvalues.
To see this,
we first note that $H_{9a,9b}=H_{10a,10b}=0$ and 
$H_{9a,10b}=H_{10b,9a}$.
Note also that $H_{9a,10b}=-H_{9b,10a}$
due to (\ref{paritytransf}),
which means that
an $(N^2 -1)\times (N^2 -1)$ matrix
$K_{ab}$ defined by $K_{ab} = H_{9a,10b}$
is antisymmetric.
% due to $H_{9a,10b}=-H_{9b,10a}$.
Due to a general property of an antisymmetric matrix of odd size,
$K_{ab}$ for even $N$ should have a zero-eigenvalue, 
whose corresponding eigenvector we denote as $\chi ^a$.
Then one finds that 
$(\delta A_9 ^a ,\delta A_{10}^a ) = (\chi ^a , 0), (0,  \chi ^a )$
are eigenvectors of the Hesse matrix $H_{ja,kb}$ with zero-eigenvalues.
Unlike the 16 zero-modes in (\ref{zeromodes}),
these two additional zero-modes, which exist only for even $N$,
have nothing to do with the symmetry of the space of solutions
to the saddle-point equation (\ref{stationaryphase}).
Hence, they should be considered as accidental zero-modes.
Since the saddle-point analysis in such a case becomes more complicated,
we restrict ourselves to the odd $N$ case in what follows.

In order to deal with the 16 zero-modes,
which actually correspond 
to the Lorentz transformation in the $(i,9)$ and 
$(i,10)$ planes ($i=1,2,\cdots , 8$),
we have to take into account the phase volume analogous to 
the Faddeev-Popov determinant.
In the present case, it can be given by
the 16-dimensional volume 
spanned by the 16 vectors (\ref{zeromodes}) in the configuration space.
We denote this phase volume as ${\cal V}^{(16)}$.
Thus we arrive at the models, which describe 
the two limiting theories corresponding to
the even/odd $\nu$ sequences,
% as
\beqa
Z_{\nu = 2 \infty}
&=& \int \left(\prod _{\mu = 1} ^8 \dd A_{\mu}\right) 
\, {\cal V}^{(16)} \, |J| ^{-1/2} \, 
\ee ^{-S_b} \,  \left| \det {\cal M}^{(8)} \right|  \  ,
\label{ZnuinftyEVEN}
\\
Z_{\nu = 2 \infty+1}
&=& \int \left(\prod _{\mu = 1} ^8 \dd A_{\mu} \right)
\,  {\cal V}^{(16)} \, |J| ^{-1/2} \, 
\ee ^{-S_b} \, \det {\cal M}^{(8)}  \ ,
%|\det {\cal M}|.
\label{ZnuinftyODD}
\eeqa
where $J$ denotes the determinant of
the Hesse matrix $H_{ja,kb}$ after removing the zero-modes.
We recall that $\det {\cal M}^{(8)}$ is real but not
positive definite, which makes the two models 
{\em a priori} different. 
One can see that the exact result (\ref{trF2exact}) for 
$\langle S_b \rangle$ can be reproduced from (\ref{ZnuinftyEVEN})
and (\ref{ZnuinftyODD})
as it should, by noting that
${\cal V}^{(16)}\mapsto \lambda ^{16} {\cal V}^{(16)}$ and
$ J \mapsto \lambda ^{- 2\{2(N^2 -1) - 16\}}J  $
under a scale transformation $A_\mu \mapsto \lambda A_\mu$.

Let us consider the question whether the remaining eight-dimensional
Lorentz invariance 
of the models (\ref{ZnuinftyEVEN}) and (\ref{ZnuinftyODD})
is further broken.
For this we show that the hessian $J$ becomes zero for
configurations in $\Omega _7$ and
%, one finds that $J=0$.
that the order of zero increases for configurations
in $\Omega _d$ with smaller $d$.
To quantify this statement, let us consider a configuration
with $A_1, \cdots , A_d$ being generic and $A_{d+1}, \cdots , A_8$ being
of order $\epsilon$.
We first note that each element of $H_{ja,kb}$ for such a configuration
becomes of order $\epsilon ^{(8-d)}$
due to (\ref{derivatives}).
Therefore, the non-zero eigenvalues of $H_{ja,kb}$ 
become of order $\epsilon ^{(8-d)}$.
We have checked numerically that they are not of order higher than
$\epsilon ^{(8-d)}$ generically. 
%We have checked this numerically for $N=5,7,9$.
This means that 
$J$ is of order $\epsilon ^{(8-d)\, \{ 2\,(N^2 -1 ) - 16 \} }$.
On the other hand, 
degenerate configurations with smaller $d$ is suppressed by the 
entropy factor $\epsilon ^{(8-d)\, (N^2 -1)}$
coming from the integration measure.
The phase volume factor ${\cal V}^{(16)}$ gives also a suppression
of order $\epsilon ^{2 \, (8-d)}$.
Collecting all the powers of $\epsilon$, we find a
suppression of order $\epsilon ^{10 \, (8-d)}$.
This means, first of all, 
that the integrals (\ref{ZnuinftyEVEN}) and (\ref{ZnuinftyODD})
are non-singular at the degenerate configurations.
%If it were that that the power of $\epsilon$ is
%$(8-d)$ times a negative number, then eight-dimensional Lorentz invariance
%would break further down to three-dimensional one.
%In this regard, we recall that for configurations in $\Omega _2$, 
%the pfaffian $\Pf \, {\cal M}(A)$ is zero, as we already discussed.
Secondly, we note that, as far as the $N$-dependent part is concerned, 
the suppression for smaller $d$ coming the entropy factor
$\epsilon ^{(8-d)\, (N^2 -1)}$ is exactly cancelled by the 
enhancement coming from $|J|^{-1/2}$.
This means that there is essentially no entropical barrier against
having smaller $d$.
%Considering the effect of $S_b$ and $\Gamma ^{(r)}$, which
%is argued to induce a collapse of eigenvalues distribution of $A_\mu$
%%seems to favour degenerate configurations 
%\cite{AIKKT}, 
%it is very plausible that 
%the remaining eight-dimensional Lorentz invariance of 
%the models (\ref{ZnuinftyEVEN}) and (\ref{ZnuinftyODD})
%is further broken down,
%and that the vacuum is given by degenerate configurations with $d < 8$.

So far, we have been focusing on the effect of $\Gamma ^{(i)}$ on 
the dynamics of the IIB matrix model.
The effect of $S_b$ and $\Gamma ^{(r)}$, on the other hand, has been studied
in Ref. \cite{AIKKT} by using the low-energy effective theory
obtained by the one-loop perturbation theory (\ref{perturbation}).
There, it was argued that the complicated branched-polymer interaction
among the diagonal elements $x_{i \mu}$ of the bosonic matrices $A_\mu$
might induce a collapse of the distribution of $x_{i \mu}$.
Monte Carlo simulation of the low-energy effective theory
at $\nu = 0$ \cite{oneloop} shows that the effect of 
$S_b$ and $\Gamma ^{(r)}$ is not sufficient to induce
a spontaneous breakdown of Lorentz invariance.
However, after taking into account
the effect of $\Gamma ^{(i)}$ by sending $\nu$ to infinity,
it is very plausible that the remaining eight-dimensional Lorentz invariance 
of the models (\ref{ZnuinftyEVEN}) and (\ref{ZnuinftyODD})
is further broken by the effect of $S_b$ and $\Gamma ^{(r)}$,
because there is no entropical barrier against having degenerate
configurations any more.
If this is the case, 
the vacuum at $\nu = \infty$
is given by degenerate configurations with $d < 8$.
In this regard, we recall also that 
the pfaffian $\Pf \, {\cal M}(A)$ is zero
when $\{ A_\mu \} \in \Omega _2$, 
as stated in (\ref{omega2}).
Therefore, the dimensionality $d$ of the vacuum configurations 
%is restricted to be larger than 2.
must be $d > 2$.

In order to check the above statements and to determine the 
dimensionality $d$ of the vacuum configurations at $\nu = \infty$,
one has to carry out the integration over the saddle-point
configurations described by
(\ref{ZnuinftyEVEN}) and (\ref{ZnuinftyODD}),
for example, by Monte Carlo simulation.
Note that the model (\ref{ZnuinftyEVEN}) is not plagued by
the sign problem any more.
%, which is quite contrary to what one expects
%naively to happen in the $\nu \rightarrow \infty$ limit.
%
%Note that 
%the sign problem, which is naively expected to become
%maximally severe at $\nu = \infty$, actually disappears for
%the model (\ref{ZnuinftyEVEN}).
%
The model (\ref{ZnuinftyODD}), on the other hand, 
can be studied using the configurations
generated with the model (\ref{ZnuinftyEVEN}) as
\beq
\left\langle {\cal O} \right\rangle _{\nu = 2\infty +1}
=
\frac{\left\langle {\cal O} \, \mbox{sgn}(\det {\cal M}^{(8)})
\right\rangle _{\nu = 2\infty }}
{\left\langle \mbox{sgn}(\det {\cal M}^{(8)}) 
\right\rangle _{\nu = 2\infty}}  \ .
\label{EVENtoODD}
\eeq
If the fermion determinant $\det {\cal M}^{(8)}$ 
does not have a definite sign
for dominant configurations, the denominator as well as the numerator 
of the r.h.s. in (\ref{EVENtoODD}) becomes very small, exhibiting
%manifesting
the sign problem.
However, if it turns out that the vacuum of the model (\ref{ZnuinftyEVEN}) 
is given by configurations with $d \le 6$, 
%which is not very unlikely to happen as we discussed, 
then due to (\ref{omega6}), we have 
$\mbox{sgn}(\det {\cal M}^{(8)})=1$ for dominant configurations.
In this case, the sign problem, which {\em a priori} exists in 
the model (\ref{ZnuinftyODD}), is solved in a sense {\em dynamically},
and the model (\ref{ZnuinftyODD}) is actually equivalent to
the model (\ref{ZnuinftyEVEN}),
since $\left\langle {\cal O} \right\rangle _{\nu = 2\infty +1}
= \left\langle {\cal O} \right\rangle _{\nu = 2\infty}$
for any observables ${\cal O}$.

%This can be done by considering the quantity
%\beq
%T_{\mu \nu} = \sum _{ij}   (A_\mu)_{ij} (A_\nu)_{ji} ,
%\eeq
%which is real symmetric 6 $\times$ 6 matrix with real positive 
%eigenvalues.
%We diagonalize it and would like to see that there are two eigenvalues
%that are very close to zero for each configuration $A_\mu$.
%The corresponding eigenvectors gives the  
%$n_\mu ^{(1)}$ and $n_\mu ^{(2)}$ in (\ref{SSB}).
%This clearly shows that spontaneous breakdown of the Lorentz invariance.


%Within the one-loop approximated model, 
%we can think of the corresponding
%equation of motion as
%\beq
%\frac{\delta}{\delta A_\mu ^a} \Imag \, \ln \det D = 0 ,
%\label{eqm_oneloop}
%\eeq
%where $A_\mu$ is given by (\ref{Aoneloop}).
%In order to obtain the solutions to the classical equation of motions,
%we simulate the system given by
%\beq
%Z^{cl} = \int \dd x \dd b 
%~ \ee ^{-S_g} \ee^{-S_{const}}  ,
%\eeq
%where
%\beq
%S_{cl} = \frac{1}{\epsilon ^2}
%\left(\frac{\delta}{\delta A_{\mu }^a} \Imag \, \ln \det D \right)^2
%\eeq
%Since the integrand is invariant under
%\beqa
%x_{i \mu} &\rightarrow &  x_{i \mu} + c_{\mu} \\
%x_{i \mu} &\rightarrow &  \lambda x_{i \mu}  ,
%\eeqa
%where $c_\mu$ is an arbitrary 6D vector and $\lambda$ is an
%arbitrary real number,
%we put the following constraint on the integration over $x$.
%\beqa
%\sum _i x_{i \mu} &=&  0 \\
%\frac{1}{N}\sum _i (x_{i \mu})^2  &= & R ^2  .
%\eeqa
%We generate ensemble for this system for small enough $\epsilon$.
%For each configuration, we would like to see if
%\beq
%n_\mu ^{(i)}  A_\mu  = 0  ~~~~~~~~~(i=1,2)
%\label{SSB}
%\eeq
%holds for the 6D vectors $n_\mu ^{(1)}$ and $n_\mu ^{(2)}$,
%which are linearly independent to each other.
%This can be done by considering the quantity
%\beq
%T_{\mu \nu} = \sum _{ij}   (A_\mu)_{ij} (A_\nu)_{ji} ,
%\eeq
%which is real symmetric 6 $\times$ 6 matrix with real positive 
%eigenvalues.
%We diagonalize it and would like to see that there are two eigenvalues
%that are very close to zero for each configuration $A_\mu$.
%The corresponding eigenvectors gives the  
%$n_\mu ^{(1)}$ and $n_\mu ^{(2)}$ in (\ref{SSB}).
%This clearly shows that spontaneous breakdown of the Lorentz invariance.


%On the other hand, one can easily see that
%(\ref{eqm_oneloop}) is indeed satisfied for those configurations
%which satisfies (\ref{SSB}).


\vspace*{1cm}


%\section{Summary and Discussion}
%\label{Summary}
%\setcounter{equation}{0}

To summarize, 
%In this Letter, 
we considered a deformation of the IIB matrix model
by introducing an integer parameter $\nu$ which couples to
$\Gamma ^{(i)}$,
the imaginary part of the effective action induced by fermions.
We studied the deformed model at $\nu = \infty$,
where the integration over the bosonic matrices is
dominated by the configurations for which $\Gamma ^{(i)}$ is stationary.
First of all,
%A first surprise is that 
there is still a huge configuration space
left as the saddle-point configurations which we have to integrate over.
Secondly, these saddle-point configurations have more than
two shrunken directions.
Thirdly, the more shrunken directions the configuration has, 
the more stationary $\Gamma ^{(i)}$ becomes.
This gives rise to an enhancement for configurations
with more shrunken directions,
and the effect was shown to cancel exactly the $N$-dependent entropical
barrier against having such configurations.
An intriguing feature of this enhancement is that it occurs 
exactly when the configuration becomes a lower-dimensional
hyperplane. This may be responsible for generating a 
{\em flat} space-time instead of a curved one
or a fractal one, as a result of the dynamics of the IIB matrix model.
We argued that the dimensionality $d$ of the space-time generated 
dynamically in the deformed IIB matrix model at $\nu = \infty$
is $2 < d \le 8$ and most likely $d<8$.
%There are actually two limiting theories for 
%$\nu \rightarrow \infty$ depending
%on the parity of $\nu$.
% {\em a priori}.
We derived the models which describe the integration
over the saddle-point configurations for the two {\em a priori} different
limiting theories
corresponding to the even/odd $\nu$ sequences.
Remarkably, the model with the even $\nu$ sequence 
is not plagued by the sign problem.
One can therefore study the model
by standard Monte Carlo simulation to extract the
dimensionality $d$ of the vacuum configurations.
If this turns out to be $d \le 6$,
%By doing this, one can investigate, for example,
%whether the remaining eight-dimensional Lorentz invariance further breaks
%down and determine the dimensionality of the space-time dynamically
%generated.
the model with the odd $\nu$ sequence
is actually equivalent to the model with the even $\nu$ sequence.
% due to the 
%indefinite sign of the eight-dimensional fermion determinant.
%However, we argued a possibility that the two limiting
%theories are actually equivalent.
%However, we argued that if the dimensionality of the space-time
%dynamically generated 
%for the model with the even $\nu$ sequence,
%turns out to be less than 6,
%then the model with the odd $\nu$ sequence must be equivalent to
%the model with the even $\nu$ sequence.
%However, we argued that if the dimensionality of the space-time
%dynamically generated turns out to be 
%less than 6 for the model with the even $\nu$ sequence,
%then the model with the odd $\nu$ sequence must be equivalent to
%the model with the even $\nu$ sequence.

%limit, the model shows
%SSB of full ten-dimensional Lorentz invariance down to eight-dimensional one.
%Hybrid R algorithm can be applied in a straightforward
%manner \cite{AABHN}.

Whether the original model ($\nu = 1$) belongs to the same phase
as $\nu = \infty$
% or not
%there are any phase transition in between 
is a nontrivial question,
which is not accessible through standard
Monte Carlo simulation due to the sign problem.
We quote, however, an example from history, where
an analogous approach was successful.
It is the strong coupling limit in the lattice formulation
of nonabelian gauge theories.
Although one should send the bare coupling constant to zero in the 
continuum limit, confinement in these theories has been clearly
demonstrated in the strong coupling limit \cite{Wilson}.
The limit also provides a qualitative 
%(or even semi-quantative)
understanding of
the spontaneous breakdown of chiral symmetry \cite{KawamotoSmit}.
%(see, e.g., \cite{KawamotoSmit}).
%allows an analytical understanding of
%
%a semi-quantitative understanding of 
%the hadron-mass spectrum and the
%spontaneous breakdown of chiral symmetry \cite{KawamotoSmit}.
% has also been understood
%in this limit \cite{KawamotoSmit}.
% and indeed quite a good hadron mass formula has been obtained.
The existence of such an approach
% using the notion of ``phase'' 
is indeed one of the advantages of having a nonperturbative formulation, 
which should also apply to the case of string theory.
Note, however, that
the above successes in the case of nonabelian gauge theories
rely crucially on the fact that
there is no phase transition between the strong coupling regime
and the weak coupling regime \cite{Creutz}.
An illustrative example clarifying this point is that 
confinement holds in the strong coupling limit even for
abelian gauge theories, for which there is actually a phase transition
to a deconfining phase at an intermediate coupling constant.

%We feel that the fact that even in the $\nu = \infty$ limit,
%the model has a huge configuration space to integrate over,
%which is quite peculiar to this system,
%suggests that the original model ($\nu =1$) is in fact quite close
%to the $\nu = \infty$ limit.
%Although it is hard to justify this statement rigorously,
%it would be certainly worthwhile to explore further the dynamics of 
%the deformed IIB matrix model in this limit.
%
%We feel that 
The fact that 
the model at $\nu = \infty$
still has a huge configuration space to integrate over,
which is quite peculiar to this system,
may suggest that it is in fact quite close
to the original model ($\nu =1$).
Although it is hard to justify this statement rigorously,
it would be certainly worthwhile to explore further the dynamics of 
the deformed IIB matrix model in this limit.


\vspace*{1cm}

%\section*{Acknowledgment}
J.\ N.\ would like to thank K.N.\ Anagnostopoulos, T.\ Hotta, T.\ Izubuchi
and A.\ Tsuchiya for collaborations at the earlier stage of this
work.
We are also grateful to J.\ Ambj\o rn, W.\ Bietenholz, 
D.\ B\"odeker, F.R.\ Klinkhamer and
N.\ Obers for valuable comments and discussions.
J.\ N.\ is supported by the Japan Society for the Promotion of
Science as a Postdoctoral Fellow for Research Abroad.  
The work of G.\ V.\ is supported by MURST (Italy) within the project
``Fisica Teorica delle Interazioni Fondamentali''.

%\vspace*{1cm}

\newpage

\begin{thebibliography}{99}

%\baselineskip=12pt
%\baselineskip=14pt
%\baselineskip=12pt

\bibitem{BFSS} T.\ Banks, W.\ Fischler, S.H.\ Shenker and L.\ Susskind,
{\em M Theory as a Matrix Model: A Conjecture,
Phys.\ Rev.} {\bf D~55} (1997) 5112 [{\tt hep-th/9610043}].

\bibitem{IKKT}N.\ Ishibashi, H.\ Kawai, Y.\ Kitazawa and A.\ Tsuchiya,
{\em A Large-$N$ Reduced Model as Superstring,
Nucl.\ Phys.} {\bf B~498} (1997) 467 [{\tt hep-th/9612115}].

\bibitem{FKKT}M.\ Fukuma, H.\ Kawai, Y.\ Kitazawa and A.\ Tsuchiya,
{\em String Field Theory from IIB Matrix Model,
Nucl.\ Phys.} {\bf B~510} (1998) 158 [{\tt hep-th/9705128}].

\bibitem{EK}
T.\ Eguchi and H.\ Kawai,
{\em Reduction of Dynamical Degrees of Freedom 
in the Large-$N$ Gauge Theory,
Phys.\ Rev.\ Lett.} {\bf 48} (1982) 1063.

\bibitem{RS}
L.\ Randall and R.\ Sundrum,
{\em An Alternative to Compactification,
Phys.\ Rev.\ Lett.} {\bf 83} (1999) 4690 [{\tt hep-th/9906064}].

\bibitem{AK}
J.\ Greensite and F.R.\ Klinkhamer,
{\em Superstring Amplitudes and Contact Interactions, Nucl.\ Phys.} {\bf B~304} (1988) 108; 
V.A.\ Kosteleck\'{y} and S.\ Samuel, {\em Spontaneous Breaking of Lorentz Symmetry in
 String Theory,  Phys.\ Rev.} {\bf D~39} (1989) 683; {\em Gravitational Phenomenology in Higher-dimensional Theories 
and Strings, Phys.\ Rev.} {\bf D~40} (1989) 1886; 
V.A.\ Kosteleck\'{y} and R.\ Potting, {\em CPT and Strings,  Nucl.\ Phys.} {\bf B~359} (1991) 545;
{\em Expectation Values, Lorentz Invariance, and CPT in the Open Bosonic String, Phys.\ Lett.} {\bf B~381} (1996) 89. V.A.\ Kosteleck\'{y}, M.J.\ Perry and R.\ Potting, {\em Off-shell Structure of the String Sigma
Model}, [{\tt hep-th/9912243}].


\bibitem{SuyamaTsuchiya}T.\ Suyama and A.\ Tsuchiya,
{\em Exact Results in $N_c =2$ IIB Matrix Model,
Prog.\ Theor.\ Phys.} {\bf 99} (1998) 321 [{\tt hep-th/9711073}].

\bibitem{AIKKT}H.\ Aoki, S.\ Iso, H.\ Kawai, Y.\ Kitazawa and T.\ Tada,
{\em Space-time Structures from IIB Matrix Model,
Prog.\ Theor.\ Phys.} {\bf 99} (1998) 713 [{\tt hep-th/9802085}].

\bibitem{KNS}W.\ Krauth, H.\ Nicolai and M.\ Staudacher,
{\em Monte Carlo Approach to M-Theory,
Phys.\ Lett.} {\bf B~431} (1998) 31 [{\tt hep-th/9803117}].

\bibitem{AABHN}J.\ Ambj\o rn, K.N.\ Anagnostopoulos, W.\ Bietenholz,
T.\ Hotta and J.\ Nishimura,
{\em Large-$N$ Dynamics of Dimensionally Reduced 4D SU($N$) Super Yang-Mills 
Theory}, [{\tt hep-th/0003208}].

\bibitem{HNT} T.\ Hotta, J.\ Nishimura and A.\ Tsuchiya, 
{\em Dynamical Aspects of Large-$N$ Reduced Models,
Nucl.\ Phys.} {\bf B~545} (1999) 543 [{\tt hep-th/9811220}].

\bibitem{sign}
W.\ Bietenholz, A.\ Pochinsky and U.-J.\ Wiese
{\em Meron-Cluster Simulation of the $\theta$ Vacuum in the 2D
O(3) Model,
Phys.\ Rev.\ Lett.} {\bf 75} (1995) 4524 [{\tt hep-lat/9505019}];\\
S.\ Chandrasekharan and U.-J.\ Wiese,
{\em Meron-Cluster Solution of a Fermion Sign Problem,
Phys.\ Rev.\ Lett.} {\bf 83} (1999) 3116 [{\tt cond-mat/9902128}].

\bibitem{oneloop}J.\ Ambj\o rn, K.N.\ Anagnostopoulos, W.\ Bietenholz,
T.\ Hotta and J.\ Nishimura,
%``Monte Carlo Studies of IIB Matrix Model'', 
in preparation.

\bibitem{BHN}
G.\ Bhanot, U.M.\ Heller and H.\ Neuberger, 
{\em The Quenched Eguchi-Kawai Model,
Phys.\ Lett.} {\bf 113~B} (1982) 47.


\bibitem{Wilson} K.G.\ Wilson, 
{\em Confinement of Quarks,
Phys. Rev.} {\bf D~10} (1974) 2445.

\bibitem{KawamotoSmit}
H.\ Kluberg-Stern, A.\ Morel, O.\ Napoly and B.\ Petersson,
{\em Spontaneous Chiral Symmetry Breaking for a U($N$) Gauge Theory 
on a Lattice, Nucl.\ Phys.} {\bf B~190} (1981) 504;\\
N.\ Kawamoto and J.\ Smit,
{\em Effective Lagrangian and Dynamical Symmetry Breaking in Strongly 
Coupled Lattice QCD,
Nucl.\ Phys.} {\bf B~192} (1981) 100.
%J.\ Hoek, N.\ Kawamoto and J.\ Smit
%``Baryons in the Effective Lagrangian of Strongly Coupled Lattice QCD'',
%%By Jaap Hoek, Noboru Kawamoto, Jan Smit (Amsterdam U.). 
%Nucl. Phys. {\bf B199} (1982) 495.

\bibitem{Creutz} M.\ Creutz, 
{\em Monte Carlo Study of Quantized SU(2) Gauge Theory,
Phys.\ Rev.} {\bf D~21} (1980) 2308.

\end{thebibliography}

\end{document}

hep-th0003223.extracted_bib
\begin{thebibliography}{99}

%\baselineskip=12pt
%\baselineskip=14pt
%\baselineskip=12pt

\bibitem{BFSS} T.\ Banks, W.\ Fischler, S.H.\ Shenker and L.\ Susskind,
{\em M Theory as a Matrix Model: A Conjecture,
Phys.\ Rev.} {\bf D~55} (1997) 5112 [{\tt hep-th/9610043}].

\bibitem{IKKT}N.\ Ishibashi, H.\ Kawai, Y.\ Kitazawa and A.\ Tsuchiya,
{\em A Large-$N$ Reduced Model as Superstring,
Nucl.\ Phys.} {\bf B~498} (1997) 467 [{\tt hep-th/9612115}].

\bibitem{FKKT}M.\ Fukuma, H.\ Kawai, Y.\ Kitazawa and A.\ Tsuchiya,
{\em String Field Theory from IIB Matrix Model,
Nucl.\ Phys.} {\bf B~510} (1998) 158 [{\tt hep-th/9705128}].

\bibitem{EK}
T.\ Eguchi and H.\ Kawai,
{\em Reduction of Dynamical Degrees of Freedom 
in the Large-$N$ Gauge Theory,
Phys.\ Rev.\ Lett.} {\bf 48} (1982) 1063.

\bibitem{RS}
L.\ Randall and R.\ Sundrum,
{\em An Alternative to Compactification,
Phys.\ Rev.\ Lett.} {\bf 83} (1999) 4690 [{\tt hep-th/9906064}].

\bibitem{AK}
J.\ Greensite and F.R.\ Klinkhamer,
{\em Superstring Amplitudes and Contact Interactions, Nucl.\ Phys.} {\bf B~304} (1988) 108; 
V.A.\ Kosteleck\'{y} and S.\ Samuel, {\em Spontaneous Breaking of Lorentz Symmetry in
 String Theory,  Phys.\ Rev.} {\bf D~39} (1989) 683; {\em Gravitational Phenomenology in Higher-dimensional Theories 
and Strings, Phys.\ Rev.} {\bf D~40} (1989) 1886; 
V.A.\ Kosteleck\'{y} and R.\ Potting, {\em CPT and Strings,  Nucl.\ Phys.} {\bf B~359} (1991) 545;
{\em Expectation Values, Lorentz Invariance, and CPT in the Open Bosonic String, Phys.\ Lett.} {\bf B~381} (1996) 89. V.A.\ Kosteleck\'{y}, M.J.\ Perry and R.\ Potting, {\em Off-shell Structure of the String Sigma
Model}, [{\tt hep-th/9912243}].


\bibitem{SuyamaTsuchiya}T.\ Suyama and A.\ Tsuchiya,
{\em Exact Results in $N_c =2$ IIB Matrix Model,
Prog.\ Theor.\ Phys.} {\bf 99} (1998) 321 [{\tt hep-th/9711073}].

\bibitem{AIKKT}H.\ Aoki, S.\ Iso, H.\ Kawai, Y.\ Kitazawa and T.\ Tada,
{\em Space-time Structures from IIB Matrix Model,
Prog.\ Theor.\ Phys.} {\bf 99} (1998) 713 [{\tt hep-th/9802085}].

\bibitem{KNS}W.\ Krauth, H.\ Nicolai and M.\ Staudacher,
{\em Monte Carlo Approach to M-Theory,
Phys.\ Lett.} {\bf B~431} (1998) 31 [{\tt hep-th/9803117}].

\bibitem{AABHN}J.\ Ambj\o rn, K.N.\ Anagnostopoulos, W.\ Bietenholz,
T.\ Hotta and J.\ Nishimura,
{\em Large-$N$ Dynamics of Dimensionally Reduced 4D SU($N$) Super Yang-Mills 
Theory}, [{\tt hep-th/0003208}].

\bibitem{HNT} T.\ Hotta, J.\ Nishimura and A.\ Tsuchiya, 
{\em Dynamical Aspects of Large-$N$ Reduced Models,
Nucl.\ Phys.} {\bf B~545} (1999) 543 [{\tt hep-th/9811220}].

\bibitem{sign}
W.\ Bietenholz, A.\ Pochinsky and U.-J.\ Wiese
{\em Meron-Cluster Simulation of the $\theta$ Vacuum in the 2D
O(3) Model,
Phys.\ Rev.\ Lett.} {\bf 75} (1995) 4524 [{\tt hep-lat/9505019}];\\
S.\ Chandrasekharan and U.-J.\ Wiese,
{\em Meron-Cluster Solution of a Fermion Sign Problem,
Phys.\ Rev.\ Lett.} {\bf 83} (1999) 3116 [{\tt cond-mat/9902128}].

\bibitem{oneloop}J.\ Ambj\o rn, K.N.\ Anagnostopoulos, W.\ Bietenholz,
T.\ Hotta and J.\ Nishimura,
%``Monte Carlo Studies of IIB Matrix Model'', 
in preparation.

\bibitem{BHN}
G.\ Bhanot, U.M.\ Heller and H.\ Neuberger, 
{\em The Quenched Eguchi-Kawai Model,
Phys.\ Lett.} {\bf 113~B} (1982) 47.


\bibitem{Wilson} K.G.\ Wilson, 
{\em Confinement of Quarks,
Phys. Rev.} {\bf D~10} (1974) 2445.

\bibitem{KawamotoSmit}
H.\ Kluberg-Stern, A.\ Morel, O.\ Napoly and B.\ Petersson,
{\em Spontaneous Chiral Symmetry Breaking for a U($N$) Gauge Theory 
on a Lattice, Nucl.\ Phys.} {\bf B~190} (1981) 504;\\
N.\ Kawamoto and J.\ Smit,
{\em Effective Lagrangian and Dynamical Symmetry Breaking in Strongly 
Coupled Lattice QCD,
Nucl.\ Phys.} {\bf B~192} (1981) 100.
%J.\ Hoek, N.\ Kawamoto and J.\ Smit
%``Baryons in the Effective Lagrangian of Strongly Coupled Lattice QCD'',
%%By Jaap Hoek, Noboru Kawamoto, Jan Smit (Amsterdam U.). 
%Nucl. Phys. {\bf B199} (1982) 495.

\bibitem{Creutz} M.\ Creutz, 
{\em Monte Carlo Study of Quantized SU(2) Gauge Theory,
Phys.\ Rev.} {\bf D~21} (1980) 2308.

\end{thebibliography}