%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%paper version 2003.2.14 
%JHEP class version No.1 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%% FOR JHEPcls 3.1.0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[preprint,12pt]{JHEP3} % 10pt is ignored!
%\documentclass[published]{JHEP3} % 10pt is ignored!

%\JHEP{00(2002)000}

%\JHEPspecialurl{http://jhep.sissa.it/JOURNAL/JHEP3.tar.gz}

\usepackage{epsfig,multicol}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% Options: preprint* published, (no)hyper*, paper, draft, %%%%%%%
%%%%%%%%%%%%          a4paper*, letterpaper, legalpaper, executivepaper,%%%%
%%%%%%%%%%%%          11pt, 12pt*, oneside*, twoside %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *=default %%%%%%%%
%%%%%%%%%%%% \title{...} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% \author{...\\...} %%%%%%%%%%%%%%%%%%%%%%%% \email{...} %%%%%%%%
%%%%%%%%%%%% \author{...\thanks{...}\\...} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% \abstract{...} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% \keywords{...} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% \preprint{...} %% or \received{...} \accepted{...} \JHEP{...} %
%%%%%%%%%%%% \dedicated{...} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% \aknowledgments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% -- No pagestyle formatting. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% -- No size formatting. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% Your definitions: %%%%%%%%%%% MINE :) %%%%%%%%%%%%%%%%%%%%%%%%%
%   ... 								   %
\newcommand{\ttbs}{\char'134}           % \backslash for \tt (Nucl.Phys. :)%
\newcommand\fverb{\setbox\pippobox=\hbox\bgroup\verb}
\newcommand\fverbdo{\egroup\medskip\noindent%
			\fbox{\unhbox\pippobox}\ }
\newcommand\fverbit{\egroup\item[\fbox{\unhbox\pippobox}]}
\newbox\pippobox
%   ...                                                                    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\EQ}{\begin{equation}}
\newcommand{\EN}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\ena}{\end{eqnarray}}
\newcommand{\bdis}{\begin{displaymath}}
\newcommand{\edis}{\end{displaymath}}
\newcommand{\vs}[1]{\vspace{#1 mm}}
\newcommand{\hs}[1]{\hspace{#1 mm}}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\c}{\gamma}
\renewcommand{\d}{\delta}
%\renewcommand{\v}{\Delta}
\renewcommand{\o}{\omega}
\renewcommand{\t}{\tau}
\newcommand{\ep}{\epsilon}
%\renewcommand{\th}{\theta}
%\newcommand{\bth}{{\bar \theta}}
%\newcommand{\shalf}{\frac{1}{2}}
\newcommand{\tq}{\tilde q}
\newcommand{\dslash}{D\!\!\!\! \slash}
%\newcommand{\caldslash}{{\cal D}\!\!\!\! \slash}
\newcommand{\pa}{\partial}
\newcommand{\dz}{\frac{dzd^2\t}{2\pi i}}
\newcommand{\nn}{\nonumber \\}




\title{ N=1 Super Yang-Mills Theory \nn
          in Ito Calculus }

\author{Naohito\ Nakazawa\\
     High Energy Accelerator Research Organization(KEK) \\
     Tsukuba, Ibaraki, 305-0801, Japan\\
	 E-mail: \email{naohito@post.kek.jp}}
\received{February 20, 2001} 		%%
\revised{May 1, 2001}
\accepted{November 27, 2001}		%% These are for published papers.


%\preprint{\hepth{9912999}}	% OR: \preprint{Aaaa/Mm/Yy\\Aaa-aa/Nnnnnn}
			  	% Use \hepth etc. also in bibliography.  

\abstract{Stochastic quantization method is applied to $N = 1$ super Yang-Mills theory especially in 4 and 10 dimensions. In 4 dimensional case, based on the It${\bar {\rm o}}$ calculus, the Langevin equation is derived in the superfield formalism. The stochastic process manifestly preserves both the local gauge symmetry and the global $N = 1$ supersymmetry in the sense of It${\bar {\rm o}}$. The corresponding Fokker-Planck equation reproduces the superfield $({\rm SYM})_{4}$ at the equilibrium limit. The component expression of the superfield Langevin equation is naturally extended to a Langevin equation for 10 dimensional case where the spinor field is Majorana-Weyl. By taking a naive zero volume limit of $({\rm SYM})_{10}$, IIB matirx model is discussed in this context. }

%\keywords{Super Yang-Mills Theory, Stochastic Quantization, Matrix Model}

%\dedicated{Dedicated to\ldots\\if you want.}




\begin{document} 

\newcommand{\caldslash}{{\cal D}\!\!\!\! \slash}


%\maketitle  IS IGNORED %%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Introduction}                   2003.2.10 version 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

$N=1$ Supersymmetric Yang-Mills theory in ten dimensions\cite{BSS} attracts special interests in relation to superstring theories not only because it describes a part of low energy effective theory of superstrings\cite{GSO} but also, its reduction to one dimension describes N D-particles system as the light-cone eleven dimensional M-theory\cite{IIA}. In a naive zero volume limit at large N, it provides a constructive definition of type IIB superstring\cite{IIB} where the supersymmetry is enhanced to $N=2$ inherited from the Green-Schwarz superstring action. The constructive approach is supported by the fact that, due to the $N=2$ supersymmetry, the light-cone IIB superstring field theory\cite{GSB} can be derived from the Schwinger-Dyson equation, or so called loop equation\cite{MM}, for Wilson loops in IIB matrix model\cite{IIB2}. Since there is no manifestly Lorentz invariant formulation of IIB superstring field theory, the light-cone setting is inevitable to check the equivalence. While if the large N matrix model provides a constructive definition of IIB superstring, we prefer to keep the manifest Lorentz covariance to obtain some hints for a manifestly Lorentz invariant description of superstring field theories. This motivates us to apply stochastic quantization method (SQM)\cite{PW}\cite{DH} to large N matirx models. As a remarkable advantage of SQM, it enables us to calculate the expectation values of gauge invariant observables such as the Wilson loop without gauge fixing procedure in terms of the Langevin equation which preserves the manifest Lorentz covariance. It is also possible to lift up the symmetry property of the $d$ dimensional system to $d$+1 dimensions by a geometric interpretation of the covariant nature of It${\bar {\rm o}}$ stochastic calculus\cite{Ito}\cite{Graham}\cite{ZZ-KOT}\cite{Nakazawa}. In this note, we consider stochastic quantization of N = 1 super Yang-Mills theory (SYM$_d$). A supersymmetric model, supersymmetric QED$_4$, was first discussed by Ishikawa in SQM\cite{Ishikawa} with superfield formalism in which supersymmetric extension of the Langevin system deduces a Langevin equation with a kernel for the fermion field\cite{BGZ}\cite{Sakita}. For SYM$_{10}$, the basic problem is to apply SQM to Majorana-Weyl fermion. In order to clarify the covariance of the Langevin equations both for the local gauge transformation and the global supersymmetry transformation, we firstly apply SQM to SYM$_{4}$ in superfield formalism. We also derive the corresponding Fokker-Planck equation written by superfield which ensures that the probability distribution reproduces the superfield action of SYM$_4$ at the equilibrium limit. It is shown that the superfield Langevin equation manifestly preserves the local gauge symmetry and the global $N=1$ supersymmetry in the sense of It${\bar {\rm o}}$ and its geometrical meaning is also clarified in superspace. Next, we extend it to ten dimensional case. Since there is no superfield formalism for SYM$_{10}$, The Langevin equations for SYM$_4$ written in component fields are naturally extended to those for SYM$_{10}$ by introducing a chiral projection for the two Majorana noise variables introduced for SYM$_4$. Then we discuss IIB matrix model by taking a naive zero volume limit of SYM$_{10}$. The Fokker-Planck hamiltonian is also derived motivated to construct a collective field theory of Wilson loops in this context\cite{EKN}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{(SYM)_4}                        2003.2.10 version fixed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{SYM$_4$ in SQM}

We apply SQM to (SYM)$_4$ in superfield formalism to clarify its symmetry property. A vector superfield $V = V^\dagger$ is given by 
%
\bea
\label{eq:superfield}
V (x,\ \theta,\ {\bar \theta})
& = & 
C + i\theta\chi - i{\bar \theta}{\bar \chi} 
+ \displaystyle{\frac{i}{2}}\theta^2 [ M + i N ] 
- \displaystyle{\frac{i}{2}}{\bar \theta}^2 [ M - i N ] 
- \theta\sigma^m {\bar \theta} v_m  \nn
& + & 
i \theta^2 {\bar \theta} [ {\bar \lambda} 
        + \displaystyle{\frac{i}{2}} {\bar \sigma}^m \pa_m \chi ] 
- i {\bar \theta}^2 \theta [ {\bar \lambda} 
        + \displaystyle{\frac{i}{2}} \sigma^m \pa_m {\bar \chi} ] 
+ \displaystyle{\frac{1}{2}}\theta^2 {\bar \theta}^2 [ D
        + \displaystyle{\frac{1}{2}} \pa^2 C]    \ ,
\ena
%
where superfield $V$ and the component fields are SU(N) algebra valued; $V = V^a t_a$. $t_a$'s are elements of the SU(N) algebra, $[ t_a, t_b] = if_{ab}^{\ \ c} t_c$, which satisfy ${\rm Tr}( t_a t_b ) = {\rm k}\delta_{ab}$. The complete set is define by 
$(t^A)_{ij} (t_A)_{kl} = {\rm k} \delta_{il}\delta_{kj}
$, 
where $t^A = t^a $ for $A=a= 1,...,N^2 -1$ 
 and 
$t^0 \equiv \sqrt{\displaystyle{\frac{{\rm k}}{N}}} {\bf 1}$. 
We use a closely related notation to Ref.\cite{WB}. The transformation property of the vector superfield under the local gauge transformation is given by 
%
\bea
\label{eq:local-gauge-tr.}
{\rm e}^{2V} 
& = & 
{\rm e}^{-i\Lambda^\dagger}{\rm e}^{2V}{\rm e}^{i\Lambda}     \ , \nn 
{\rm e}^{-2V} 
& = & 
{\rm e}^{-i\Lambda}{\rm e}^{- 2V}{\rm e}^{i\Lambda^\dagger}     \ ,
\ena
%
where $\Lambda$ and $\Lambda^\dagger$ are SU(N) algebra valued chiral superfields which satisfy 
%
$
{\bar D}_{\dot \alpha} \Lambda = D_\alpha \Lambda^\dagger = 0 \ . 
$
%
By introducing a local gauge covariant superfield $W_\alpha$ and its conjugation ${\bar W}_{\dot \alpha}$, 
%
\bea
\label{eq:covariant-fiel-strength}
W_\alpha 
& = & 
- \displaystyle{\frac{1}{8}} {\bar D}^2 {\rm e}^{-2V}D_\alpha {\rm e}^{2V} 
     \ , \nn 
{\bar W}_{\dot \alpha} 
& = & 
 \displaystyle{\frac{1}{8}} D^2 {\rm e}^{2V}{\bar D}_{\dot \alpha} {\rm e}^{-2V}            \ , 
\ena
%
the action of SYM$_4$ is given by 
%
\bea
\label{eq:4Daction}
S 
& = & 
\int d^4x d^2\theta d^2{\bar \theta} \displaystyle{\frac{1}{4{\rm k}g^2}} {\rm Tr}    \Big(
W^\alpha W_\alpha  + {\bar W}_{\dot \alpha}{\bar W}^{\dot \alpha}
\Big)           \  , \nn 
& = & 
\displaystyle{\frac{1}{{\rm k}g^2}}  \int d^4x {\rm Tr} \Big( 
- \displaystyle{\frac{1}{4}} v_{mn} v^{mn} 
- \displaystyle{\frac{i}{2}} \lambda \sigma^m {\cal D}_m {\bar \lambda} 
- \displaystyle{\frac{i}{2}} {\bar \lambda} {\bar \sigma}^m {\cal D}_m \lambda 
+ \displaystyle{\frac{1}{2}} D^2         
\Big) \ . 
\ena
%
Here the expression with component fields is written in Wess-Zumino gauge. 
%
$
v_{mn} = \pa_m v_n - \pa_n v_m + i \Big[ v_m,\ v_n \Big]  ,\
$
%
and 
%
$
{\cal D}_m {\lambda} = \pa_m {\lambda} + i \Big[ v_m,\ {\lambda} \Big]  . \ 
$
%
In the following, we do $not$ assume the Wess-Zumino gauge.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Langevin equation and Fokker-Planck equation for (SYM)_4 } 
% 2003.2.10 version fixed 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We define the time evolution of the vector superfield, 
$V( \t+\Delta \t ) \equiv V ( \t ) + \Delta V( \t )$ in terms of It${\bar {\rm o}}$ calculus\cite{Ito} with respect to the stochastic time $\t$. We use the discretized notation for clear understanding of the concept of It${\bar {\rm o}}$ calculus. The Langevin equation is defined from the Fokker-Planck equation which reproduces the probability distribution ${\rm e}^{-S}$ with $S$ in (\ref{eq:4Daction}) at the limit 
$\t \rightarrow \infty$. 
At first, we regard ${\rm e}^{2V}$ as a fundamental variable because of its simple transformation property. Then we derive the Langevin equation for the vector superfield $V$. Let us intoduce a derivative operator ${\hat E}_a$ which is invariant under the right multiplication to ${\rm e}^{2V}$. 
%
\bea
\label{eq:left-derivative}
\displaystyle{\frac{\pa}{\pa V^a}} {\rm e}^{2V} 
& = & 
\displaystyle{\frac{2}{{\rm k}}}\int^1_0 ds {\rm Tr}\Big( 
{\rm e}^{2sV}t_a {\rm e}^{-2sV} t^b 
\Big) t_b {\rm e}^{2V}        
\equiv
K_a^{\ b} t_b {\rm e}^{2V}             \ , \nn
{\hat E}_a  
& \equiv & 
L_a^{\ b} \displaystyle{\frac{\pa}{\pa V^b}}      \ , \quad
K_a^{\ c} L_c^{\ b} 
 =  
L_a^{\ c} K_c^{\ b} = \delta_a^{\ b}      \ .
\ena
%
By definition, we have 
$
{\hat E}_a {\rm e}^{2V} = t_a {\rm e}^{2V} \ 
$ 
and 
$
[ {\hat E}_a, {\hat E}_b] = -if_{ab}^{\ \ c} {\hat E}_c . 
$
%
The coefficients, $L_a^{\ b}$ and $K_a^{\ b} $, satisfy Maurer-Cartan equations.
%
\bea
\label{eq:Maurer-Cartan}
L_a^{\ c} \pa_c L_b^{\ d} - L_b^{\ c} \pa_c L_a^{\ d} 
& = & 
- i f_{ab}^{\ \ c}L_c^{\ d}       \ , \nn  
\pa_b K_c^{\ a} - \pa_c K_b^{\ a} 
& = & 
+ i f_{b'c'}^{\ \ a}K_b^{\ b'}K_c^{\ c'}       \ .
\ena
%

To derive the superfield Langevin equation, we assume the following form for the time evolution of ${\rm e}^{2V}$; \  
$
{\rm e}^{2V}( \t + \Delta \t ) 
=
 {\rm e}^{2V}( \t ) + \Delta {\rm e}^{2V}( \t )
$, 
%
\bea
\label{eq:4DLangevin-eq1}
\Delta  {\rm e}^{2V} 
& = & 
- \Delta \t X + ( \Delta w ) {\rm e}^{2V} \ ,   \nn 
< \Delta w_{ij} \Delta w_{kl} > 
& = & 
2{\rm k} \Delta \t \Big( 
\delta_{il}\delta_{jk} - \displaystyle{\frac{1}{N}}\delta_{ij}\delta_{kl} 
\Big) \delta^2 ( \theta - \theta' ) 
\delta^2 ( {\bar \theta} - {\bar \theta}' ) \delta^4 ( x - x' )     \ , 
\ena
% 
where $\Delta w$ is a noise superfield. Since ${\rm Tr}V = 0$, we require for the time evolution of ${\rm e}^{2V}$ to satisfy 
$
{\rm det}( {\rm e}^{2V}( \t ) + \Delta {\rm e}^{2V}( \t ) ) = 1 
$, 
i.e., 
$
{\rm Tr}{\rm e}^{-2V}( \t ) \Delta {\rm e}^{2V}( \t ) = 0 
$. 
This implies that $( \Delta {\rm e}^{2V}( \t ) ) {\rm e}^{2V}( \t )$ is also SU(N) algebra valued. We assume that, in (\ref{eq:4DLangevin-eq1}), $X$ and $\Delta w$ are SU(N) algebra valued. 
The corresponding Fokker-Planck equation is given by, 
%
\bea
\label{eq:4DFokker-Planck-eq1}
\displaystyle{\frac{\pa}{\pa \t}} P( \t ) 
= 
{\hat E}_a \Big( 
{\hat E}_a + \displaystyle{\frac{1}{{\rm k}}}{\rm Tr}(t_a X {\rm e}^{-2V})
\Big) P( \t ) 
\ena
%
If we chose 
$
X = ( {\hat E}S ){\rm e}^{2V} \ ,
$
with ${\hat E} \equiv t^a {\hat E}_a$, the probability distribution $P( \t )$ behaves 
$
\lim_{\t \rightarrow \infty} P( \t ) = {\rm e}^{-S}  .\ 
$
We thus obtain the Langevin equation for ${\rm e}^{2V}$
%
\bea
\label{eq:4DLangevin-eq2}
( \Delta {\rm e}^{2V} ){\rm e}^{-2V}
& = & 
- \Delta \t {\hat E}S + \Delta w  \ ,   \nn
& = & 
- \Delta \t \displaystyle{\frac{1}{4g^2}} {\rm e}^{2V} \Big( 
( D^\alpha W_\alpha + \Big\{ 
W_\alpha , {\rm e}^{-2V} D^\alpha {\rm e}^{2V}
\Big\} ) {\rm e}^{-2V}       \nn
& & 
\quad + {\rm e}^{-2V} ( {\bar D}_{\dot \alpha} {\bar W}^{\dot \alpha} + \Big\{ 
{\bar W}^{\dot \alpha} , {\rm e}^{2V} {\bar D}_{\dot \alpha} {\rm e}^{-2V} 
\Big\} ) 
\Big)  + \Delta w  \ . 
\ena 
%

Now the transformation property of the Langevin equation (\ref{eq:4DLangevin-eq2}) is in question. The l.h.s. and the first term in the r.h.s. of (\ref{eq:4DLangevin-eq2}) are transformed as 
$
( \Delta {\rm e}^{2V} ){\rm e}^{-2V} 
\rightarrow {\rm e}^{-i\Lambda\dagger} ( \Delta {\rm e}^{2V} ){\rm e}^{-2V} {\rm e}^{i\Lambda\dagger}   \ .
$
For the covariance of the Langevin equation, we must require for the noise superfield, 
%
\bea
\label{eq:noise-transformation}
\Delta w
\rightarrow {\rm e}^{-i\Lambda\dagger} \Delta w {\rm e}^{i\Lambda\dagger} . 
\ena
%
We notice that the noise correlation in (\ref{eq:4DLangevin-eq1}) is $invariant$ under the transformation (\ref{eq:noise-transformation}). Hence we conclude that the Langevin equation and the noise correlation manifestly preserve the global $N=1$ supersymmetry and the local gauge symmetry. It should be remarked that the transformation property of the noise superfield shows that it is not a vector superfield, i.e., $\Delta w \neq \Delta w^\dagger$. We also comment on the transformation property of the Fokker-Planck equation. It takes the form, 
%
\bea
\label{eq:4DFokker-Planck-eq2}
\displaystyle{\frac{\pa}{\pa \t}} P( \t ) 
= 
\displaystyle{\frac{1}{{\rm k}}}{\rm Tr} {\hat E} \Big( 
{\hat E} + ( {\hat E}S )
\Big) P( \t )           \ .
\ena
%
Since 
$
{\hat E}S 
= {\rm k} ( {\rm e}^{2V} )\displaystyle{\frac{\delta S}{\delta ({\rm e}^{2V})^t}} 
$ is transformed as 
%
$
( {\hat E}S )
\rightarrow {\rm e}^{-i\Lambda\dagger} ( {\hat E}S ) {\rm e}^{i\Lambda\dagger} ,\ 
$ 
%
the differential operator ${\hat E}$ must be transformed as 
%
$
{\hat E}
\rightarrow {\rm e}^{-i\Lambda\dagger} {\hat E} {\rm e}^{i\Lambda\dagger} 
$. 
%
Under the transformation, we have 
%
$
{\hat E}'_a {\rm e}^{2V'} = {\tilde t}_a {\rm e}^{2V'}  \ , 
$
%
where by definition 
%
$
{\hat E}'_a  =  {\hat E}_a  \ ,
$
%
and 
%
$
{\tilde t}_a  = {\rm e}^{-i\Lambda\dagger} t_a {\rm e}^{i\Lambda\dagger} \ .
$
%
The transformed elements of SU(N) algebra also satisfy 
%
$
[ {\tilde t}_a, {\tilde t}_b] = if_{ab}^{\ \ c} {\tilde t}_c ,\ ({\tilde t}^a)_{ij} ({\tilde t}_a)_{kl} = {\rm k} ( \delta_{il}\delta_{kj} 
- \displaystyle{\frac{1}{N}}\delta_{ij}\delta_{kl} ),    \
$
%
hence the Fokker-Planck equation is invariant. 

Now we derive the Langevin equation for the vector superfield $V$. To do this, we must carefully evaluate the r.h.s. of (\ref{eq:4DLangevin-eq2}) in terms of It${\bar {\rm o}}$ stochastic calculus. We define $g \equiv {\rm e}^{2V}$ and evaluate, 
%
\bea
\label{eq:contact-term-eq1}
( \Delta g ) g^{-1} 
& = & 
\Delta V^a (\pa_a g ) g^{-1} + {1\over 2}\Delta V^a \Delta V^b ( \pa_a \pa_b g ) g^{-1}  + O(\Delta\t^{3/2})          \ , \nn
& = & 
\Delta V^a (\pa_a g ) g^{-1} + \Delta \t L_c^{\ a} L_c^{\ b} ( \pa_a \pa_b g ) g^{-1}    + O(\Delta\t^{3/2})           \ .
\ena
%
Here the second line is derived by the help of (\ref{eq:4DLangevin-eq2}). From (\ref{eq:contact-term-eq1}) we obtain, 
%
\bea
\label{eq:contact-term-eq2}
\Delta V^a 
& = & 
 \Big( ( \Delta g ) g^{-1} \Big) ^b L_b^{\ a} - \Delta \t L_c^{\ b} L_c^{\ d} \Big( ( \pa_b \pa_d g ) g^{-1} \Big) ^e L_e^{\ a}          \ , \nn
& = &  
 \Big( - \Delta \t {\hat E}S + \Delta w   \Big) ^b L_b^{\ a} - \Delta \t L_c^{\ b} L_c^{\ d} \Big( ( \pa_b \pa_d g ) g^{-1} \Big) ^e L_e^{\ a}          \ .
\ena
%
The second term is evaluated to yield,  
%
$
+ \Delta \t ( \pa_b L_c^{\ a} ) L_c^{\ b}           .\ 
$
%
The contribution plays an essential role for the covariant nature of the Langevin equation of the vector superfield. The interpretation is the following. Let us introduce a metric, 
$ G^{ab} \equiv L_c^{\ a} L_c^{\ b}$, and 
$ G_{ab} \equiv K_a^{\ c} K_b^{\ c}$. 
% 
$G$ denotes ${\rm det}G_{ab}$. 
From the metric tensor, we can define a covariant form of $\Delta V^a$, 
%
\bea
\label{eq:covariant-derivative}
\Delta_{\rm cov} V^a \equiv \Delta V^a 
+ \Delta \t \Gamma^a_{\ bc}G^{bc} 
\ena
%
This yields 
$
\Delta \t \Gamma^a_{\ bc}G^{bc}  
=
- \Delta \t 
\displaystyle{\frac{1}{\sqrt{G}}} \pa_b ( \sqrt{G} G^{ab} ) 
= 
- \Delta \t ( \pa_b L_c^{\ a} ) L_c^{\ b}     \ .
$
 The covariance is more transparent if we write the Langevin equation (\ref{eq:contact-term-eq2}) and the noise correlation as follows.
%
\bea
\label{eq:4DLangevin-eq3}
\Delta V^a 
& = & 
-\Delta \t G^{ab} \displaystyle{\frac{\delta S}{\delta V^b}} 
-  \Delta \t \Gamma^a_{\ bc}G^{bc} 
+ \Delta_w V^a     \ , \nn
< \Delta_w V^a \Delta_w V^b > 
& = & 
2 \Delta \t G^{ab} \delta ( \theta - \theta' ) 
\delta ( {\bar \theta} - {\bar \theta}' ) \delta ( x - x' )   , 
\ena
%
where we have introdced a collective noise superfield 
$ \Delta_w V^a = \Delta w^b L_b^{\ a}$. The corresponding Fokker-Planck equation written by the superfield appears in a manifestly general coordinate invariant form in sperspace.
%
\bea 
\label{eq:4DFokker-Planck-eq3}
\displaystyle{\frac{\pa}{\pa \t}} P( \t ) 
= 
\displaystyle{\frac{1}{\sqrt{G}}} \pa_a \Big( 
\sqrt{G}G^{ab} ( \pa_b + 
 \displaystyle{\frac{\delta S}{\delta V^b}} )
 P( \t )  \Big)         \ ,
\ena
%
where $P(\t)$ is a scalar probability defined by 
$<{\cal O}(\t)> = \int\!\! {\cal O}P(\t)\sqrt{G}{\cal D}V $. 
Thus we conclude that the Langevin equation of vector superfield for SYM$_4$ in superspace formulation has a precise geometrical meaning which comes from the particular fact that $\Delta V^a$ is $not$ a covariant object in the sense of Ito calculus. The geometricaly covatriant nature of the Langevin equation, in the sense of Ito calculus, was first pointed out by Graham\cite{Graham}. The geometrical interpretation is also essential to lift up the symmetry property in $d$ dimensional system to $d$+1 dimension in SQM  by identifing the variation of the vector superfield $V^a$ under the local gauge transformation to the Killing vector in the \lq\lq superspace \rq\rq $\{ V^a,\ G_{ab}\}$\cite{Nakazawa}. As an application, by the Langevin equation, we can derive the supersymmetric Schwinger-Dyson equation ( or the supersymmetric loop equation ) written in superfield formalism which has been already  found in ref.\cite{IT}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Langevin equation of SYM$_4$ in Component Fields} 
% 2003.2.10 version
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the extension to ten dimensional case, we need an expression with component fields. We comment on some useful relations for a practical calculation in component fields in order to expand the non-polynomial form of the Langevin equation and the Fokker-Planck hamiltonian. Let us introduce an operator $L_V$ which is defined by $L_V X = [V, X]$\cite{GGRS}. If we define an inner product $(X\cdot Y) = {\rm Tr} (XY)$, then we have $(X\cdot L_V Y) = (Y\cdot (L_V)^t X) = - (Y\cdot L_V X)$. By introducing a notation $(L_V)_a^{\ b} \equiv \displaystyle{\frac{1}{{\rm k}}}\Big( t_a\cdot L_V t^b \Big)$, we define the operators 
$
K 
\equiv  \displaystyle{\frac{1}{L_V}}( 1 - {\rm e}^{-2L_V} )     \
$
and 
$
L 
\equiv  \displaystyle{\frac{L_V}{ 1 - {\rm e}^{-2L_V}}}    \ 
$
which corresponds to the coefficients $K_a^{\ b}$ and $L_a^{\ b}$ respectively. In this notation, quantities in the Langevin equation is expressed as, 
%
\bea
\label{eq:operation1}
G \displaystyle{\frac{\delta S}{\delta V^t}}     
& = & 
\Big( 
{\rm e}^{2L_V} - 1 \Big)^{-1} L_V \Big( 1 - {\rm e}^{-2L_V} 
\Big)^{-1} L_V \displaystyle{\frac{\delta S}{\delta V^t}}  \   , \nn 
\Delta_w V 
& = & 
\Big( {\rm e}^{2L_V} - 1 \Big)^{-1} L_V \Delta w     \ . 
\ena
%
By using these operations, we can evaluate the non-polynomial superfield Langevin equation perturbatively. In Wess-Zumino gauge, 
%
$ 
V 
= 
- \theta\sigma^m {\bar \theta} v_m  
+ i \theta^2 {\bar \theta} {\bar \lambda} 
- i {\bar \theta}^2 \theta \lambda 
+ \displaystyle{\frac{1}{2}}\theta^2 {\bar \theta}^2 D   \ , 
$ 
%
the \lq\lq equations of motion \rq\rq in the Langevin equation gives, 
%
\bea
\label{eq:operation2}
G \displaystyle{\frac{\delta S}{\delta V^t}}   
& = & 
\Big( 
{\rm e}^{2L_V} - 1 \Big)^{-1} L_V \Big( 1 - {\rm e}^{-2L_V} 
 \Big)^{-1} L_V \displaystyle{\frac{\delta S}{\delta V^t}}   \ ,   \nn 
& = & 
\Delta \t \displaystyle{\frac{1}{4g^2}}  \Big( 
( 1 - {\rm e}^{-2L_V} )^{-1} L_V( D^\alpha W_\alpha + \{ 
W_\alpha , {\rm e}^{-2V} D^\alpha {\rm e}^{2V} 
\} )      \nn
& & 
\quad + ({\rm e}^{2L_V} - 1 )^{-1} L_V ( {\bar D}_{\dot \alpha} {\bar W}^{\dot \alpha} + \{ 
{\bar W}^{\dot \alpha} , {\rm e}^{2V} {\bar D}_{\dot \alpha} {\rm e}^{-2V} 
\} ) 
\Big)  \ ,  \nn
& = & 
- D + \theta\sigma^m{\cal D}_m {\bar \lambda} - {\bar \theta}{\bar \sigma}^m{\cal D}_m \lambda  
+  \theta\sigma^m {\bar \theta} ( {\cal D}^n v_{mn}     
+ \lambda\sigma_m{\bar \lambda} + {\bar \lambda}{\bar \sigma}_m\lambda )   \nn
& - & 
 \displaystyle{\frac{i}{2}} \theta^2 {\bar \theta} 
{\bar \sigma}^m \sigma^n {\cal D}_m {\cal D}_n {\bar \lambda}
- i \theta^2 {\bar \theta} [ D,\ {\bar \lambda} ]       
 +  
 \displaystyle{\frac{i}{2}} {\bar \theta}^2 \theta 
\sigma^m {\bar \sigma}^n {\cal D}_m {\cal D}_n \lambda 
- i {\bar \theta}^2 \theta [ D,\  \lambda ]                    \nn
& + & 
\theta^2{\bar \theta}^2 (\ {\rm D\ dependent\ terms}\ )   \  . 
\ena 
%
The expression gives the correct equations of motion for component fields in Wess-Zumino gauge, while it is also clear from the expression that the Wess-Zumino gauge breaks the symmetry structure of the Langevin equation because in Wess-Zumino gauge there is no time evolution in $\Delta V$ corresponding $D$, $\lambda$ and ${\bar \lambda}$. In fact, we find that the time evolution of these component fields is governed by the terms such as $\sigma^m {\bar \sigma}^n {\cal D}_m {\cal D}_n \lambda$. This shows that, in general, the supersymmetric Langevin equations for component fields, for example $D$, $\lambda$ and ${\bar \lambda}$, include so called kernels. 
 The appearance of the kernel  
$
( \approx {\bar \sigma}^m {\cal D}_m )
$ 
is inebitable because the canonical dimension of the stochastic time is $[ \Delta \t ] = 2-d $ ( where $d$=4 in this case ). Without it, we would have to introduce a dimensionful parameter artificially on the Langevin equation of the fermion fields for the proper canonical dimension of the stochastic time. But the parameter would break the supersymmetry at finite stochastic time because it would only scale the time development of the fermion field. 

With an appropriate choice of noise variable $\Delta w$, we may write,  
%
\bea
\label{eq:4Dcomponent-Langevin}
\Delta v_m
& = & 
- \Delta \t \displaystyle{\frac{1}{g^2}}\Big( 
{\cal D}^n v_{mn} + ( \lambda\sigma^m{\bar \lambda} + {\bar \lambda}{\bar \sigma}^m\lambda ) 
\Big)       + \Delta w |_{\theta\sigma^m {\bar \theta}}   \ , \nn 
\Delta \lambda 
& = &  
- \Delta \t \displaystyle{\frac{1}{g^2}}
 \displaystyle{\frac{1}{2}}\sigma^m{\bar \sigma}^n 
{\cal D}_m{\cal D}_n \lambda 
        + \Delta w |_{{\bar \theta}^2\theta}  
        + \displaystyle{\frac{1}{2}}\sigma^m 
{\cal D}_m \Delta w |_{{\bar \theta}}  , \nn 
\Delta {\bar \lambda} 
& = &   
- \Delta \t \displaystyle{\frac{1}{g^2}}
 \displaystyle{\frac{1}{2}}{\bar \sigma}^m \sigma^n 
{\cal D}_m{\cal D}_n {\bar \lambda}  
+ \Delta w |_{\theta^2 {\bar \theta}}  
+ \displaystyle{\frac{1}{2}}{\bar \sigma}^m 
{\cal D}_m \Delta w |_{\theta}    \ , 
\ena
%
Here we have also set $D = 0$ for the auxiliary scalar field in the vector multiplet. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Langevin equation of SYM$_4$ in Component Fields II; D=0} 
% 2003.2.10 version
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the extension to SYM$_{10}$, we reconstruct the Langevin equation for SYM$_4$ with 4-dimensional Majorana spinor by starting from the action of SYM$_4$ in Minkowski space-time  
%
\bea
\label{eq:4Daction-component1}
S 
= 
\int\!\!\!d^4\!\! x \displaystyle{\frac{1}{g^2}} {\rm Tr}\big( 
- {1\over4} v_{mn}v^{mn} + {i\over 2} {\bar \psi}\caldslash \psi
\big)         \ .  
\ena
%
$\psi$ is a Majorana spinor 
$\psi^t \equiv ( (\lambda_\alpha )^t , ({\bar \lambda}^{\dot \alpha} )^t )$. $\gamma^m$'s are 4-dimensional $\gamma$-matrices in a Weyl representation which satisfy 
$\big\{ \gamma^m, \gamma^n \big\} = -2 \eta^{mn}$, with 
$\eta^{mn}= (-,+,+,+)$. 
$
\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3           
$. 
%
The charge conjugation matrix ${\cal C}$ satisfies,
%
$
( \gamma^m )^{t} 
= - {\cal C}^{-1}\gamma^m {\cal C}    \ , 
{\cal C}^{t} 
= - {\cal C}    \ ,
$
%
The Majorana fermion satisfies the relation, 
%
$
\psi^c \equiv {\cal C} {\bar \psi}^t = \psi
$ .
%
The model has the local gauge symmetry and the N=1 supersymmetry. 
%
\bea
\label{eq:4Dsupersymmetry-component}
\d A_\mu
& = & 
i{\bar \epsilon}\gamma_m \psi \,  \nn
\d \psi 
& = & 
-{1\over 2}v^{mn}\gamma_{mn}\epsilon  \, 
\ena
%

We fix the Wick rotation prescription for the Majorana fermion following Nicolai\cite{Nicolai}, though Majorana spinor does not exist in the Euclidean space, which keeps the explicit connection of the Euclidean theory to the Minkowski one. We introduce the independent Majorana spinors $\psi$ and ${\bar \psi}$ and require the constraint, 
%
\bea
\label{eq:4DMajorana-condition}
{\cal C} {\bar \psi}^t = \psi
\ena
%
We then perform the Wick rotation, 
$
x^0 = -i x^4 \ , \gamma^0 =-i \gamma^4 \ . 
$ 
The expression of the Euclidean action is given by 
%
$
iS \equiv - S_E \ ,
$
%
%
\bea
\label{eq:4Daction-component2}
S_E 
= 
\int\!\!\!d^4\!\! x \displaystyle{\frac{1}{g^2}} {\rm Tr}\big( 
{1\over4} v_{mn}^2 - {i\over 2} {\bar \psi}\caldslash \psi
\big)         \ .   
\ena
%
The Euclidean supersymmetry transformation is the same as that in Minkowski space-time. It is sufficient to prove the supersymmetry of the Euclidean action that the \lq\lq charge conjugation \rq\rq matrix satisfies the relation such as 
%
$
\gamma_m^{t} 
= - {\cal C}^{-1}\gamma_m {\cal C}    \ , m=1,2,3,4 \ .
$
%

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

The time evolution of the componet field in (\ref{eq:4Dcomponent-Langevin}), for $v_m ( \t+\Delta \t ) \equiv v_m ( \t ) + \Delta v_m ( \t )$, $\psi ( \t+\Delta \t ) \equiv \psi ( \t ) + \Delta \psi ( \t )$, is summarized in the following Langevin equation with the Majorana fermion. It reproduces the probability distribution ${\rm e}^{-S_E}$ with $S_E$ in (\ref{eq:4Daction-component2}) at $\t \rightarrow \infty$, 
%
%
\bea
\label{eq:4DLangevin-component}
{\Delta}v_m (\t)
& = & 
-\Delta\t \displaystyle{\frac{1}{g^2}} \big( 
{\cal D}_n v_{mn} - {\bar \psi}\gamma_m \psi 
\big) (\t) + \Delta\xi_m (\t)          \ , \nn
{\Delta}\psi (\t)
& = & 
- \Delta\t \displaystyle{\frac{1}{g^2}} \caldslash\ ^2 \psi (\t) 
+ \Delta \chi ( \t )          \ , \nn
\Delta \chi ( \t )
& = & 
\Delta\xi (\t) + i\caldslash \Delta\eta (\t)  \ .
\ena
%
Since the Majorana fermion fields, $\psi$ and ${\bar \psi}$, are not independent, the Langevin equation for ${\bar \psi}$ is given by \lq\lq charge conjugation \rq\rq;  
%
$
\Delta{\bar \psi} (\t)
 = 
 -\Delta\t {\bar \psi} \overleftarrow{\caldslash}\ ^2(\t) 
+ {\overline {\Delta\chi}} (\t) 
        \ 
$
%
with
$
{\overline {\Delta\chi}} (\t)         
 =  
 {\overline {\Delta\xi}} (\t) - i {\overline {\Delta\eta}} \overleftarrow{\caldslash}\ (\t)   \ ,
$
%
where we write 
$
{\bar \psi}\overleftarrow{{\cal D}_m}
\equiv {\cal D}_m{\bar \psi}  . 
$
We have indtroduced noise variables $\Delta\xi_\mu$, $\Delta\xi$ which are a 4-dimensional vector and a Majorana spinor, respectively. All these noise variables are SU(N) algebra valued. Their correlations are defined by 
%
\bea
\label{eq:4Dnoise-component}  
<\Delta\xi^a_m(\t,x) \Delta\xi^b_n(\t,y)>  
& = & 
2\Delta\t \d^{ab}\d_{mn}  \d^4(x-y)     \ , \nn  
<\Delta\xi^a_\a(\t,x) {\overline {\Delta\eta}}^b_\b(\t,y)>  
& = & 
- <{\overline {\Delta\eta}}^b_\b(\t,y) \Delta\xi^a_\a(\t,x) >     
= 
\Delta\t \d^{ab} \d^4(x-y)     .  
\ena
%
This deduces the correlation for the collective noise
%
\bea
\label{eq:4Dcollective-correlation}
<\Delta\chi^a_\a(\t,x) \Delta\chi^b_\b(\t,y)>  
= 
2\Delta\t ( \Gamma_\mu{\cal C} 
)_{\a\b} (-iD_\mu^{ab})_x  \d^4(x-y)          .  
\ena
%




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

The Fokker-Planck equation for the component fields is also derived from (\ref{eq:4DLangevin-component}). The time evolution of an observable, ${\cal O}( v_m, \psi_\alpha )$, is given by 
$
\Delta {\cal O} 
= 
- \Delta\t {\cal H}_{\rm FP} {\cal O}    \      
$
with
%
\bea
\label{eq:4DFP-hamiltonian}
{\cal H}_{\rm FP} 
& = & 
\int\!\!\! d^4x \Big\{ \displaystyle{\frac{1}{g^2}}\Big( 
 {\cal D}_n v_{mn} - {\bar \psi}\gamma_m \psi \Big)^a (x) - \displaystyle{\frac{\d\quad }{\d v^a_m(x)}} \Big\} \displaystyle{\frac{\d\quad }{\d v^a_m(x)}}             \nn
& + & 
\int\!\!\! d^4x ( 
\gamma_m{\cal C} 
) _{\a\b} (-i{\cal D}_m)^{ab}_x  \Big\{  \displaystyle{\frac{1}{g^2}}\Big( 
 i {\cal C}^{-1} \caldslash\ \psi (x) \Big) ^b_\b  
- \displaystyle{\frac{\d\quad }{\d \psi^b_\b(x)}}  \Big\} \displaystyle{\frac{\d\quad }{\d \psi^a_\a(x)}}                   .
\ena
%
We have used the left derivative for the fermionic variables. 
We also obtain the following Fokker-Planck equation for the probability distribution. 
%
\bea
\label{eq:4DFokker-Planck-eq-component}
\displaystyle{\frac{\pa P}{\pa \t}}
& = &  
\int\!\!\! d^4x\displaystyle{\frac{\d }{\d v^a_m}}\Big\{ \Big( \displaystyle{\frac{\d S_E}{\d v^a_m}} 
+ \displaystyle{\frac{\d }{\d v^a_m}} \Big) P \Big\}   \nn
& + & 
\int\!\!\! d^4x\displaystyle{\frac{\d }{\d \psi^a_\a}}\big( 
(-i\caldslash \ ) {\cal C} 
\big)^{ab}_{\a\b} \Big\{ \Big( \displaystyle{\frac{\d S_E}{\d \psi^b_\b}} + \displaystyle{\frac{\d }{\d \psi^b_\b}} \Big) P \Big\}  
\ena
%
Hence we obtain the action (\ref{eq:4Daction-component2}) at the equilibrium limit,  
$ 
\lim_{\t \rightarrow \infty} P = {\rm e}^{- S_E} 
$ . 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Langevin equation for (SYM)$_{10}$}      2003.2.10 version fixed 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{SYM$_{10}$ in SQM}
Now we consider ten dimensional case.  We begin with the Euclidean action of SU(N) SYM$_{10}$, 
%
\bea
\label{eq:10Daction}
S 
= 
\int\!\!\!d^{10}\!\! x \displaystyle{\frac{1}{g^2}} {\rm Tr}\big( 
{1\over4} F_{\mu\nu}^2 -{i\over 2} {\bar \Psi}\dslash \Psi
\big)         \ ,   
\ena
%
where $A_\mu$ and $\Psi$, a vector and a Majorana-Weyl spinor in ten dimensions respectively, are SU(N) algebra valued. 
%
$
F_{\mu\nu} = \pa_\mu A_\nu - \pa_\nu A_\mu - i\Big[ 
A_\mu, A_\nu \Big], 
$
%
and 
%
$
D_\mu \Psi = \pa_\mu \Psi - i \Big[ 
A_\mu, \Psi \Big] 
$. 
%
$\Gamma_\mu$'s are 10-dimensional $\gamma$-matrices which satisfy 
$\big\{ \Gamma_\mu, \Gamma_\nu \big\} = -2\delta_{\mu\nu}$. We use the convention, 
%
$
\Gamma_j = i\sigma_1\otimes\gamma_j\ $ for $j=1,...,8, \Gamma_9 = i\sigma_3\otimes {\bf 1}_{16} 
$
and 
$
\Gamma_{10} = i\sigma_2\otimes{\bf 1}_{16} 
$. With real symmetric $\gamma_i$'s which satisfy $\{ \gamma_i, \gamma_j \} = 2\d_{ij}$, all $\Gamma_\mu$'s are anti-hermitian and 
$
\Gamma_{11} = i\Gamma_1 ... \Gamma_{10} \ 
$
is real symmetric.
For later convenience, we fix the chirality of the spinor, $\Gamma_{11} \Psi = \Psi$. ${\bar \Psi}$ is defined by \lq\lq charge conjugation \rq\rq, 
%
$
{\bar \Psi} = - \Psi^t {\cal C} .
$
%
The charge conjugation matrix ${\cal C}$ satisfies,
%
$
\Gamma_\mu 
= 
- {\cal C}^{-1}\Gamma_\mu^{t} {\cal C}    \ , 
{\cal C}^{t} 
= 
- {\cal C}    \ ,
$
%
which ensures necessary relations to prove the supersymmetry of the Euclidean action such as 
${\bar \epsilon} \Gamma_{\mu_1}\cdots \Gamma_{\mu_M}\Psi 
= 
(-)^{M}{\bar \Psi} \Gamma_{\mu_M}\cdots \Gamma_{\mu_1}\epsilon \ .$ 
The model has the local gauge symmetry and the N=1 supersymmetry; 
%
$
\d A_\mu
= 
i{\bar \epsilon}\Gamma_\mu \Psi \ , 
\d \Psi 
= 
-{1\over 2}F_{\mu\nu}\Gamma_{\mu\nu}\epsilon  \ . 
$
%

The Langevin equations for SYM$_{10}$ are given by, 
%
\bea
\label{eq:10DLangevin}
{\Delta}A_\mu (\t)
& = & 
-\Delta\t \displaystyle{\frac{1}{g^2}} \big( 
D_\nu F_{\mu\nu} + {\bar \Psi}\Gamma_\mu \Psi 
\big) (\t) + \Delta\xi_\mu(\t)          \ , \nn
{\Delta}\Psi (\t)
& = & 
- \Delta\t \displaystyle{\frac{1}{g^2}} \dslash\ ^2 \Psi (\t) 
+ \Delta\xi (\t) + i\dslash \Delta\eta (\t)          \  . 
\ena
%
We have indtroduced noise variables $\Delta\xi_\mu$, $\Delta\xi$ and $\Delta\eta$, a 10-dimensional vector and two 10-dimensional Majorana-Weyl spinors, respectively. All these noise variables are SU(N) algebra valued. The main difference of this Langevin equations compared to those for SYM$_4$ is appeared in the following noise correlation which includes the chiral projection corresponding to the Majorana-Weyl condition on noise variables. 
Their correlations are given by 
%
\bea
\label{eq:10Dnoise}  
<\Delta\xi^a_\mu(\t,x) \Delta\xi^b_\nu(\t,y)>  
& = & 
2\Delta\t \d^{ab}\d_{\mu\nu}  \d^{10}(x-y)     \ , \nn  
<\Delta\xi^a_\a(\t,x) {\overline {\Delta\eta}}^b_\b(\t,y)>  
& = & 
- <{\overline {\Delta\eta}}^b_\b(\t,y) \Delta\xi^a_\a(\t,x) >     \nn  
& = & 
\Delta\t \d^{ab}{1\over 2}(1+\Gamma_{11})_{\a\b}  \d^{10}(x-y)     \ .  
\ena
%
Here we notice the chirality assignment of the fermionic (Majorana-Weyl) noise variables. The time development must preserve the chirality of the Majorana-Weyl fermion while the operator $\dslash\ $ flips it. Thus we define the chirality of the two independent Majorana noises, 
$\Gamma_{11}\Delta \xi = \Delta \xi$ and 
$\Gamma_{11}\Delta \eta = - \Delta \eta$. By the definition, $\Delta\eta\equiv {\cal C}^{-1}{\overline {\Delta\eta}}^t$,  (\ref{eq:10Dnoise}) deduces, 
%
\bea
\label{eq:10Dnoise2}
<\Delta\xi^a_\a(\t) \Delta\eta^b_\b(\t)>  
= 
- \Delta\t \d^{ab}\Big( {1\over 2}(1+\Gamma_{11}){\cal C}\ ^{-1} \Big)_{\a\b}  \d^{10}(x-y)       \ .  
\ena
%
By introducing a collective fermionic ( Majorana-Weyl ) noise variable,
$
\Delta\chi , 
$
the correlation is written in a compact form,  
%
\bea
\label{eq:10Dcollective-correlation} 
\Delta\chi 
& \equiv & 
 \Delta\xi + i\dslash \Delta\eta   \ ,  \nn
<\Delta\chi^a_\a(\t,x) \Delta\chi^b_\b(\t,y)>  
& = & 
2\Delta\t\Big( {1\over 2}(1+\Gamma_{11})\Gamma_\mu{\cal C}\ ^{-1} 
\Big)_{\a\b} (-iD_\mu^{ab})_x  \d^{10}(x-y)          \ .  
\ena
%
For the Majorana-Weyl fermion, $\Psi$ and ${\bar \Psi}$ are not independent each other and we have 
%
$
\Delta{\bar \Psi} (\t)
 = 
 -\Delta\t {\bar \Psi} \overleftarrow{\dslash}\ ^2(\t) 
+ {\overline {\Delta\chi}} (\t) 
        \ , 
$
%
where 
$
{\overline {\Delta\chi}} (\t)         
 =  
 {\overline {\Delta\xi}} (\t) 
 - i {\overline {\Delta\eta}} \overleftarrow{\dslash}\ (\t)   \ .
$
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The time development of an observable ${\cal O}(A_\mu, \Psi)$ is also defined by 
$
\Delta {\cal O} 
= 
- \Delta\t {\cal H}_{\rm FP} {\cal O}    \      
$
with the Fokker-Planck hamiltonian, 
%
\bea
\label{eq:10DFokker-Planck-hamiltonian}
& {} & {\cal H}_{\rm HP}    \nn
& = & 
\int\!\!\! d^{10}x \Big\{ \displaystyle{\frac{1}{g^2}} \Big( 
 D_\nu F_{\mu\nu} + {\bar \Psi}\Gamma_\mu \Psi \Big)^a (x) - \displaystyle{\frac{\d\quad }{\d A^a_\mu(x)}} \Big\} \displaystyle{\frac{\d\quad }{\d A^a_\mu(x)}}             \nn
& + & 
\int\!\!\! d^{10}x \Big( 
{1\over 2}(1+\Gamma_{11})( -i\dslash \ ){\cal C}\ ^{-1} 
\Big) ^{ab}_{\a\b}  \Big\{  \displaystyle{\frac{1}{g^2}} \Big( 
i {\cal C} \dslash\ \Psi (x) \Big) ^b_\b  
- \displaystyle{\frac{\d\quad }{\d \Psi^b_\b(x)}}  \Big\} \displaystyle{\frac{\d\quad }{\d \Psi^a_\a(x)}}                   .
\ena
%
The corresponding Fokker-Planck equation is given by,
%
\bea
\label{eq:10DFokker-Planck-eq1}
\displaystyle{\frac{\pa P}{\pa \t}}
& = &  
\int\!\!\! d^{10}x\displaystyle{\frac{\d }{\d A^a_\mu}}\Big\{ 
\Big( \displaystyle{\frac{\d S}{\d A^a_\mu}} + \displaystyle{\frac{\d }{\d A^a_\mu}} \Big) P \Big\}     \nn
& + & 
\int\!\!\! d^{10}x\displaystyle{\frac{\d }{\d \Psi^a_\a}}\big( 
\displaystyle{\frac{1}{2}}(1+ \Gamma_{11}) (-i\dslash \ ) {\cal C}\ ^{-1}
\big)^{ab}_{\a\b} \Big\{ \Big( \displaystyle{\frac{\d S}{\d \Psi^b_\b}} + \displaystyle{\frac{\d }{\d \Psi^b_\b}} \Big) P \Big\}   \ .
\ena
%
This ensures that the action (\ref{eq:10Daction}) is reproduced at the equilibrium,  
$ 
\lim_{\t \rightarrow \infty} P = {\rm e}^{- S} 
$ . 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{IIB matrix Model in SQM}        2003.2.10 version fixed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{IIB Matrix Model in SQM}

Now we discuss a zero volume limit of the Langevin equation (\ref{eq:10DLangevin}). One of the main interests is to apply it to IIB matrix model\cite{IIB} 
and construct a collective field theory of Wilson loops. 
For the bosonic part, the construction is illustrated in this context\cite{EKN}. The IIB matrix model is obtained as a naive zero volume limit of the SU(N) SYM$_{10}$ in (\ref{eq:10Daction}).
%
\bea
\label{eq:IIBaction}
S_{{\rm IIB}} 
= 
- \displaystyle{\frac{1}{g^2}}{\rm Tr}\Big( 
{1\over4} [ A_\mu, A_\nu ]^2 +{1\over 2} {\bar \Psi}\Gamma_\mu [ A_\mu, \Psi ] 
\Big)         .  
\ena
%
The model has the $ N = 2$ supersymmetry, one is defined by the zero volume limit of the supersymmetry of SYM$_{10}$ and the other is enhanced at the zero volume limit which is identified as the reminicent of Green-Schwarz IIB superstring action\cite{IIB}. 
%
\bea
\label{eq:IIBsupersymmetry1}
\d_1 A_\mu 
& = & 
i{\bar \epsilon}\Gamma_\mu \Psi \ ,  \nn
\d_1 \Psi    
& = & 
{i\over 2}[ A_\mu, A_\nu ]\Gamma_{\mu\nu}\epsilon  \, \nn
\d_2 A_\mu 
& = & 
0                     \ , \nn
\d_2 \Psi 
& = & 
\lambda{\bf 1}        \ ,
\ena
%
where the transformation parameter, $\epsilon$ and $\lambda$, are Majorana-Weyl spinors in ten dimensions and ${\bf 1}$, the $N\times N$ unit matrix, respectively. In the definition of IIB matrix model, the matrix variables are $N\times N$ hermitian matrices with non-zero trace parts. After the zero volume limit of SU(N) SYM$_{10}$, the model has been extended to U(N) algebra valued. Although a precise proof of the finiteness is given for the SU(N) algebra valued case\cite{AW}, the trace part is necessary for the realization of the N=2 supersummetry.  

The reduced version of the Langevin equation is defined by the zero volume limit of (\ref{eq:10DLangevin}) with an extension for fundametal variables and noises to be U(N) algebra valued.
%
\bea
\label{eq:IIBLangevin}
{\Delta}A_\mu (\t)
& = & 
-\Delta\t \displaystyle{\frac{1}{g^2}} \Big( 
[ A_\nu, [ A_\nu, A_\mu ] ](\t) + {\bar \Psi}\Gamma_\mu \Psi(\t) 
\Big) + \Delta\xi_\mu(\t)          \ , \nn
{\Delta}\Psi (\t)
& = & 
\Delta\t \displaystyle{\frac{1}{g^2}} \Gamma_\mu 
 [ A_\mu, \Gamma_\nu [A_\nu, \Psi ] ](\t) 
+ \Delta\chi (\t)           \ , \nn
\Delta\chi (\t)  
& \equiv & 
\Delta\xi (\t) + \Gamma_\mu[ A_\mu, \Delta\eta ](\t)   \ .
\ena
%
The correlation of noise variables is also obtained from the zero volume limit of (\ref{eq:10Dnoise}). It is given by, 
%
\bea
\label{eq:IIBnoise}  
<\Delta\xi_\mu(\t)_{ij} \Delta\xi_\nu(\t)_{kl}>  
& = & 
2\Delta\t \d_{il}\d_{kj}\d_{\mu\nu}      \ , \nn  
<\Delta\xi_\a(\t)_{ij} {\overline {\Delta\eta}}_\b(\t)_{kl}>  
& = & 
- <{\overline {\Delta\eta}}_\b(\t)_{kl} \Delta\xi_\a(\t)_{ij} >     
= 
\Delta\t \d_{il}\d_{kj}{1\over 2}(1+\Gamma_{11})_{\a\b}       .  
\ena
%
It deduces, 
%
\bea
\label{eq:IIB-correlation}
<\Delta\chi_\a(\t)_{ij} \Delta\chi_\b(\t)_{kl}>  
= 
2\Delta\t\Big( {1\over 2}(1+\Gamma_{11})\Gamma_\mu{\cal C}\ ^{-1} 
\Big)_{\a\b}\Big\{ \d_{il}(A_\mu)_{kj} - \d_{kj}(A_\mu)_{il} 
  \Big\}        .  
\ena
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{IIB Fokker-Planck Hamiltonian}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For practical calculations, it is suffcient to consider a generalized Langevin equation for Wilson loops,, 
$
W_M \equiv {\rm Tr}\prod_{n=1}^M U_n , \ 
U_n = {\rm e}^{ i\epsilon ( k_n^\mu A_\mu + {\bar \lambda}_n\Psi ) } 
$. While in the field theoretical interpretation of SQM, the hamiltonian operator is given by the Fokker-Planck hamiltonian\cite{EKN}. From the zero volume limit of (\ref{eq:10DFokker-Planck-hamiltonian}), we obtain the Fokker-Planck hamiltonian for IIB matrix model.
%
\bea
\label{eq:IIBF-P hamiltonian}
{\cal H}_{\rm IIB}   
& = & 
\Big\{ \displaystyle{\frac{1}{g^2}} \Big( 
 [ A_\nu, [ A_\nu, A_\mu ] ] + {\bar \Psi}\Gamma_\mu \Psi \Big)_{ij}  - \displaystyle{\frac{\d\quad }{\d ( A_\mu )_{ji}}} \Big\} \displaystyle{\frac{\d\quad }{\d (A_\mu)_{ij} }}             \nn
& - &  
\displaystyle{\frac{1}{g^2}} 
\Big( \Gamma_\mu [ A_\mu, \Gamma_\nu [ A_\nu, \Psi ]]
  \Big)_{\a\ ij}\displaystyle{\frac{\d\quad }{\d ( \Psi_\a )_{ij} }}    \nn
& - & 
\Big( 
{1\over 2}(1+\Gamma_{11})\Gamma_\mu{\cal C}\ ^{-1} 
\Big) _{\a\b} \Big\{ \d_{il} ( A_\mu )_{kj} - \d_{kj} ( A_\mu )_{il}  
\Big\} \displaystyle{\frac{\d^2 \quad }{\d ( \Psi_\b )_{kl} \d ( \Psi_\a )_{ij} }}                  \  . 
\ena 
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Conclusion}                                 2003.2.10 version 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}

In this paper, we have applied SQM to SYM in both four and ten dimensions. In 4-dimensional case, it has been shown in superfield formalism that the local gauge symmetry as well as the grobal $N=1$ supersymmetry are manifestly preserved in the sense of It${\bar {\rm o}}$ calculus. In superfield formalism for SYM$_4$, the Langevin equation and the corresponding Fokker-Planck equation are formulated in a manifestly general coordinate invariant form in superspace. It is a manifestation of the covariant nature of It${\bar {\rm o}}$ stochastic calculus. The consequence of it, in component fields, is the enevitable introduction of a kerneled Langevin equation for fermion fields in the vector multiplet. In ten dimensions, though there is no superfield formulation, the structure of the Langevin equation and the Fokker-Planck hamiltonian is similar to 4-dimensional case except that, correponding to the Majorana-Weyl fermion, the chiral projection is introduced for the noise correlation. The formulation may be useful for construction of collective field theories of Wilson loops in SYM both in four and ten dimensions. Especially in SYM$_4$, if we work in Wess-Zumino gauge which breaks the manifest supersymmetry and the commutator algebra closes up to the local gauge transformation. While the supersymmetry is restored by the local gauge transformation with $\Lambda$ and $\Lambda^\dagger$, the gauge fixing procedure by breaking the gauge invariance may also breaks the supersymmetry in vector multiplet. SQM approach in superfield formalism may provide a solution for this problem at least gauge invariant observables are concerned in the perturbative analysis. It may also provide a basis for numerical analysis of SYM. As for SYM$_{10}$, from the Ward-Takahashi identity for chiral gauge symmetry, there exists a chiral anomaly as expected which is not explained in this note. Application of our formulation to lower dimensional SYM ( i.e. $d=3,6$ ) is straightforward.  

We have also derived the Langevin equation and the Fokker-Planck hamiltonian for IIB matrix model by taking the zero volume limit of those for SYM$_{10}$. One of the main interests is to apply our formulation to construct a collective field theory of Wilson loops in IIB matirx model as illustrated for the bosonic part\cite{EKN} and we hope to obtain some hints for the manifestly Lorentz invariant formulation of superstring field theories. It seems to be also possible to evaluate the expectation values of Wilson loops perturbatively by using the Langevin equation. To do this, we have to clarify the structure of ground states of the colletive field theory of IIB matrix model in SQM approach. It may be a fantasy, however, to hope to describe the M-theory by lifting up the symmetry properties of IIB matrix model to eleven dimensions in the context of SQM. 

\bigskip


\noindent
{\bf Acknowledgements}

The author would like to thank S. Iso and T. Suyama for enlightening discussions, and all members in theory group at KEK for hospitality. 
The work is supported in the early stage by the Ministry of Education, Science and Culture of Japan, Grant-in-Aid for Scientific Research (B), No.13135216.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{References}                           2003.2.10 version 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{999}
%
\bibitem{BSS} L. Brink, J. Schwarz and J. Scherk, \npb{121}{1977}{77}.
%
\bibitem{GSO} F. Gliozzi, J. Scherk and D. Olive, \npb{122}{1977}{253}.
%
\bibitem{IIA} T. Banks, W. Fischler, S. Shenker and L. Susskind, \pr{D55}{1997}{5112}.
%
\bibitem{IIB} N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, \npb{498}{1997}{467}.
%
\bibitem{GSB} M. Green, J. Schwarz and L. Brink, \npb{219}{1983}{437}.
%
\bibitem{MM} Y. Makeenko and A. Migdal, \plb{88}{1979}{135}; \plb{89}{1980}{437}.
%
\bibitem{IIB2} M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, \npb{510}{1998}{158}.
%
\bibitem{PW} G. Parisi and Y. Wu, Sci. Sin. {\bf 24}(1981) 483.
%
\bibitem{DH} P. Damgaard and H. Huffel, \prep{152}{1987}{227}, and  
references therein.
%
\bibitem{Ito} K. Ito, Proc. Imp. Acad. {\bf 20}(1944) 519;\nn
              K. Ito and S. Watanabe, in \lq\lq Stochastic Differential Equations \rq\rq, ed. K. Ito ( Wiley, New Yorl, 1978 ).
%
\bibitem{Graham} R. Graham, \pla{109}{1985}{209}. 
%
\bibitem{ZZ-KOT} J. Zinn-Justin and D. Zwanziger, \npb{295}{1988}{297}, \\
                 A. Kappor, H. Ohba and S. Tanaka, \plb{221}{1989}{125}.
%
\bibitem{Nakazawa} N. Nakazawa, \npb{335}{1990}{546}.
%
\bibitem{Ishikawa} K. Ishikawa, \npb{241}{1984}{589}.
%
\bibitem{BGZ} J. Breit, S. Gupta and A. Zaks, \npb{233}{1984}{61}.
%
\bibitem{Sakita} B. Sakita, Proceedings of the 7th Johns Hopkins Workshop, ed. G. Domokos and S. Kovesi-Domkos, ( World Scientific, Singapore, 1983) 115.
%
\bibitem{EKN} D. Ennyu, H. Kawabe and N. Nakazawa, \jhep{01}{2003}{025}.
%
\bibitem{WB} J. Wess and J. Bagger, \lq\lq Supersymmetry and Supergravity \rq\rq ( Princeton Univ. Press, Princeton, 1982 ).
%
\bibitem{IT} H. Itoyama and H. Takashino, \plb{381}{1996}{163}; \ptp{97}{1997}{963}.
%
\bibitem{GGRS} S. Gates, M. Grisaru, M. Rocek and W. Siegel, \lq\lq Superspace \rq\rq ( Benjamin, Massachusetts,1983 ).  
%
\bibitem{Nicolai} H. Nicolai, \npb{140}{1978}{294}.
%
\bibitem{AW} P. Austing and J. Wheater, \jhep{04}{2001}{019}, \\
             T. Suyama and A. Tsuchiya, \ptp{99}{1998}{321}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\end{thebibliography}
\end{document}


