hep-th0108154.tex
%	 10	   20	     30        40	 50	   60	    70
\documentstyle[12pt]{article}
\def\doit#1#2{\ifcase#1\or#2\fi}

\doit0{
% Horizontal alignment of accents: bars, hats, tildes, etc.
\skewchar\fivmi='177 \skewchar\sixmi='177 \skewchar\sevmi='177
\skewchar\egtmi='177 \skewchar\ninmi='177 \skewchar\tenmi='177
\skewchar\elvmi='177 \skewchar\twlmi='177 \skewchar\frtnmi='177
\skewchar\svtnmi='177 \skewchar\twtymi='177
\def\@magscale#1{ scaled \magstep #1}
}

% For framing, put 1 in \doit below as well as in \framing{1}.	   
\doit{0}{
% Framing 
\def\framingfonts#1{
\doit{#1}{\font\twfvmi	= ammi10   \@magscale5 % math italic
\skewchar\twfvmi='177 \skewchar\fivsy='60 \skewchar\sixsy='60
\skewchar\sevsy='60 \skewchar\egtsy='60 \skewchar\ninsy='60
\skewchar\tensy='60 \skewchar\elvsy='60 \skewchar\twlsy='60
\skewchar\frtnsy='60 \skewchar\svtnsy='60 \skewchar\twtysy='60
\font\twfvsy  = amsy10	 \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018			% Gothic
\font\sc=font005			% script
\def\Go#1{{\hbox{\go #1}}}	% Gothic for single characters 
%  in equations
\def\Sc#1{{\hbox{\sc #1}}}	% script for single characters 
%  in equations
\def\Sf#1{{\hbox{\sf #1}}}	% sans serif for single characters 
%  in eqs
\font\oo=circlew10	  % thick circles (hollow: ` and a-n , 
%  solid: p-z)
\font\ooo=circle10			% thin circles	 
\font\ro=manfnt 			% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for 
%  left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

% Underline for text or math
\catcode`@=11
%\def\un#1{\relax\ifmmode\@@underline#1\else
%	$\@@underline{\hbox{#1}}$\relax\fi}
\catcode`@=12

% Accents and foreign (in text):
\let\under=\b			% bar-under (but see \un above)
\let\ced=\c			% cedilla
\let\du=\d			% dot-under
\let\um=\H			% Hungarian umlaut
\let\sll=\l			% slashed (suppressed) l (Polish)
\let\Sll=\L			% " L
\let\slo=\o			% slashed o (Scandinavian)
\let\Slo=\O			% " O
\let\tie=\t			% tie-after (semicircle 
%   connecting two letters)
\let\br=\u			% breve


% Abbreviations for Greek letters
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

% Use the following definitions of \pmb, \bfalpha, etc, 
% for `boldfaced' Greek characters. (03/05/99) 

\def\pmb#1{\setbox0=\hbox{${#1}$}%
   \kern-.025em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern.05em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern-.025em\box0 }

\def\bfalpha{\pmb{\alpha}} \def\bfbeta{\pmb{\beta}} 
\def\bfgamma{\pmb{\gamma}}
\def\bfdelta{\pmb{\delta}} \def\bfepsilon{\pmb{\epsilon}}
\def\bfzeta{\pmb{\zeta}}  \def\bfeta{\pmb{\eta}} 
\def\bftheta{\pmb{\theta}}
\def\bfiota{\pmb{\iota}}  \def\bfkappa{\pmb{\kappa}}
\def\bflambda{\pmb{\lambda}} \def\bfmu{\pmb{\mu}}  
\def\bfnu{\pmb{\nu}}
\def\bfxi{\pmb{\xi}} \def\bfomicron{\pmb{o}}  \def\bfpi{\pmb{\pi}}
\def\bfroh{\pmb{\rho}} \def\bfsigma{\pmb{\sigma}}  
\def\bftau{\pmb{\tau}}
\def\bfupsilon{\pmb{\upsilon}} \def\bfphi{\pmb{\phi}}  
\def\bfchi{\pmb{\chi}}
\def\bfpsi{\pmb{\psi}} \def\bfomega{\pmb{\omega}}

\def\bfAlpha{\pmb{A}} \def\bfBeta{\pmb{B}} \def\bfGamma{\pmb{\Gamma}}
\def\bfDelta{\pmb{\Delta}} \def\bfEpsilon{\pmb{E}} \def\bfZeta{\pmb{Z}} 
\def\bfEta{\pmb{H}} \def\bfTheta{\pmb{\Theta}} \def\bfIota{\pmb{I}} 
\def\bfKappa{\pmb{K}} \def\bfLambda{\pmb{\Lambda}} \def\bfMu{\pmb{M}} 
\def\bfNu{\pmb{N}} \def\bfXi{\pmb{\Xi}} \def\bfOmicron{\pmb{O}} 
\def\bfPi{\pmb{\Pi}} \def\bfRoh{\pmb{P}} \def\bfSigma{\pmb{\Sigma}} 
\def\bfTau{\pmb{T}} \def\bfUpsilon{\pmb{\Upsilon}} 
\def\bfPhi{\pmb{\Phi}} 
\def\bfChi{\pmb{X}} \def\bfPsi{\pmb{\Psi}} \def\bfOmega{\pmb{\Omega}}

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} 
\raisebox{-.45ex}{$\hskip .48ex { |}$}} 
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} 
\raisebox{.45ex}{$\hskip .48ex { 
|}$}}	
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~} % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}} % D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}} % D'Alembertian
\def\cbo{\Sc [} 			   % curly "
\def\pa{\partial}			   % curly d
\def\dell{\bigtriangledown}		  % hi ho the dairy-o

\def\pr{\prod}				  % product
\def\iff{\leftrightarrow}		 % <-->
\def\conj{{\hbox{\large *}}}		 % complex conjugate

\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}} %< or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}} %> or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu %
\llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu %
\llap {$\ddot
	\smile$}}}			     % happy face
\def\dg{\sp\dagger}			    % hermitian 
% conjugate
\def\ddg{\sp\ddagger}			   % double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}		   % superscript (unaligned)
\def\sb#1{{}_{#1}}				% sub"
\def\oldsl#1{\rlap/#1}				% poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1} % good slash for lower case

\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}	% " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}	% " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}	% 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}	% 

%\def\Tilde#1{{\widetilde{#1}}\hskip 0.03in}			
\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}	 % big tilde
\def\Hat#1{\widehat{#1}}			% big hat
\def\Bar#1{\overline{#1}}			% big bar
\def\bra#1{\left\langle #1\right|}		% < |
\def\ket#1{\left| #1\right\rangle}		% | >
\def\VEV#1{\left\langle #1\right\rangle}	% < >
\def\abs#1{\left| #1\right|}			% | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow 
 \mkern-6mu
	\cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
	\mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
	\leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
	$\hfil\displaystyle{#1}\hfil$\crcr}}}		% <--> accent
\def\dt#1{{\buildrel {\hbox{\LARGE .}} \over {#1}}}% dot-over 
% for sp/sb
\def\dtt#1{{\buildrel \bullet \over {#1}}}		% alternate "
\def\der#1{{\pa \over \pa {#1}}}	      % partial derivative
\def\fder#1{{\d \over \d {#1}}} 	      % functional derivative

% Math stuff with more than one argument
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
	\hbox{$\scriptstyle{#2}$}}}}}	% fraction
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
	\vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}}
       %alternate fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
	\vphantom1\smash{\raise.3ex\hbox{$#2$}}}}	% "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}} % "
\def\partder#1#2{{\partial #1\over\partial #2}} % partial derivative 
 % of
\def\parvar#1#2{{\d #1\over \d #2}}		  % variation of
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}	%second
 %  "
\def\on#1#2{\mathop{\null#2}\limits^{#1}}	% arbitrary accent
\def\bvec#1{\on\leftarrow{#1}}		      % backward vector accent
\def\oover#1{\on\circ{#1}}			% circle accent

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
	\ialign{\strut \hfil$\displaystyle{##}$&$
	\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
	\everycr{\noalign{\ifdtup \global\dtupfalse
	\vskip-\lineskiplimit \vskip\normallineskiplimit
	\else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous      % eqalignno
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}

% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
}

% Text style parameters
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\headheight=0in 			% height of heading (LaTeX)
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%\footheight=3ex			   % height of foot (LaTeX)
\footheight=15ex			   % height of foot (LaTeX)
\footskip=4ex	 % distance between bottoms of body & foot (LaTeX)
\textwidth=6in				% width of body (LaTeX)
\hsize=6in				% " (TeX)
\parskip=\medskipamount 	 % space between paragraphs (LaTeX)
\lineskip=0pt				% minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em	%space above equation 
%   (either)
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\abovedisplayshortskip=.5em plus.2em minus.4em	%"above 
%   when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em	%"below
\def\baselinestretch{1.2}	% magnification for line spacing 
%   (LaTeX)
\thicklines			    % thick straight lines for pictures 
%   (LaTeX)

% Title page
\thispagestyle{empty}		    % no heading or foot on title page 
%   (LaTeX)
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	\vskip.2in}
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	\setlength{\unitlength}{1mm}
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	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
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	\repeat
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	\put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
	\put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
	\end{picture}
	\par\vskip-8mm}
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	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
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	\put(\xco,\yco){$\ktr$}
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	\indent
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\def\endtitle{\end{quotation}\newpage}	% end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\nablasl{{{\nabla\!\!\!\!\!{\hskip 1.0pt}/}}}
\def\caldsl{{\calD\!\!\! /}}	
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}
\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}
\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}
\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}
\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}
\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}
\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\rma{{\rm a}} \def\rmb{{\rm b}} \def\rmc{{\rm c}} 
\def\rmd{{\rm d}} 
\def\rme{{\rm e}} \def\rmf{{\rm f}} \def\rmg{{\rm g}} 
\def\Mpl{M_{\rm Pl}}
\def\mw{m\low{\rm W}}
\def\Mgut{M\low{\rm GUT}}
\def\Re{{\rm Re}\,\,}  \def\Im{{\rm Im}\,\,} 
\def\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}
{\raise-0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}
{\raise-0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\jhep#1#2#3{Jour.~High Energy Phys.~{\bf {#1}} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} }
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}}
\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
\def\hepph#1{hep-ph/{#1}} 
\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
%\def\uln{{\underline n}} 
\def\uln#1{\underline{#1}}
\def\ulp{{\underline p}} \def\ulq{{\underline q}} 
\def\ulr{{\underline r}}
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\hatm{\hat m}\def\hatn{\hat n}\def\hatr{\hat r}\def\hats{\hat s}
\def\hatt{\hat t}
\def\<<{<\!\!<} \def\>>{>\!\!>} 
\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
\def\cdm{{\Sc D}_{--}} \def\cdp{{\Sc D}_{++}}
\def\Pisl{{\Pi\!\!\!\! /}\,}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}
%\def\fracmm#1#2{{{#1}\over{#2}}}
%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
	\hbox{$\scriptstyle{#2}$}}}}}	% fraction
%\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A 
{\buildrel\leftrightarrow\over D}_{#1} B\right)} 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}

% There are more than one line for the definition of \footnotew below:
%\def\footnotew#1{\footnote{{\hsize=7.0in {\def\baselinestrentch{0.7}
% \noindent #1}}}
% \baselineskip 16pt\oddsidemargin=0.03in 
% \evensidemargin=0.01in\hsize=6.5in\textwidth=6.5in} 

\def\footnotew#1{\footnote{\hsize=6.5in {#1}}} 

\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
\def\ip{{=\!\!\! \mid}}
%\def\Dot#1{\buildrel{\hskip2.5pt_{\scriptscriptstyle\bullet}}
% \over{#1}}
%\def\dt#1{\Dot{#1}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}\,} \def\Tr{\,\,{\rm Tr}\,}
\def\det{{\rm det}\,} 
\def\Norm{|\hskip -2.0pt|}
\def\Sgn{\hbox{\rm Sgn}\,}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} 
\def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
			      \scriptscriptfont0\fivebf
	    \textfont1\tenmib \scriptfont1\sevenmib
			       \scriptscriptfont1\fivemib
	    \textfont2\tenbsy \scriptfont2\sevenbsy
			       \scriptscriptfont2\fivebsy}
 %
\def\rlx{\relax\leavevmode}		     
 % Guess what this is for...
\def\BM#1{\rlx\ifmmode\mathchoice
		      {\hbox{$\BMfont#1$}}
		      {\hbox{$\BMfont#1$}}
		      {\hbox{$\scriptstyle\BMfont#1$}}
		      {\hbox{$\scriptscriptstyle\BMfont#1$}}
		 \else{$\BMfont#1$}\fi}

\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
			      \scriptscriptfont0\fivebf
	    \textfont1\tenmib \scriptfont1\sevenmib
			       \scriptscriptfont1\fivemib
	    \textfont2\tenbsy \scriptfont2\sevenbsy
			       \scriptscriptfont2\fivebsy}
 %
\def\BM#1{\rlx\ifmmode\mathchoice
		      {\hbox{$\BMfont#1$}}
		      {\hbox{$\BMfont#1$}}
		      {\hbox{$\scriptstyle\BMfont#1$}}
		      {\hbox{$\scriptscriptstyle\BMfont#1$}}
		 \else{$\BMfont#1$}\fi}

\def\inbar{\vrule height1.5ex width.4pt depth0pt}
\def\sinbar{\vrule height1ex width.35pt depth0pt}
\def\ssinbar{\vrule height.7ex width.3pt depth0pt}
\font\cmss=cmss10
\font\cmsss=cmss10 at 7pt
\def\ZZ{\rlx\leavevmode
	     \ifmmode\mathchoice
		    {\hbox{\cmss Z\kern-.4em Z}}
		    {\hbox{\cmss Z\kern-.4em Z}}
		    {\lower.9pt\hbox{\cmsss Z\kern-.36em Z}}
		    {\lower1.2pt\hbox{\cmsss Z\kern-.36em Z}}
	       \else{\cmss Z\kern-.4em Z}\fi}
\def\Ik{\rlx{\rm I\kern-.18em k}}  % Yes, I know. This ain't capital.
\def\IC{\rlx\leavevmode
	     \ifmmode\mathchoice
		    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
		    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
		    {\hbox{\kern.28em\sinbar\kern-.25em{\rm C}}}
		    {\hbox{\kern.25em\ssinbar\kern-.22em{\rm C}}}
	     \else{\hbox{\kern.3em\inbar\kern-.3em{\rm C}}}\fi}
\def\IP{\rlx{\rm I\kern-.18em P}}
\def\IR{\rlx{\rm I\kern-.18em R}}
\def\IN{\rlx{\rm I\kern-.20em N}}
\def\Ione{\rlx{\rm 1\kern-2.7pt l}}

%
%%% apple lw
\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.59truein}
\def\speclscape{\special{ps: landscape}}

\newbox\leftpage \newdimen\fullhsize \newdimen\hstitle\newdimen\hsbody
\tolerance=1000\hfuzz=2pt\def\fontflag{cm}
%
\catcode`\@=11 % This allows us to modify PLAIN macros.
% We need next two \doit commands to avoid the repeated questions of 
% reduction. 
\doit0
{
\def\bigans{b }
\message{ big or little (b/l)? }\read-1 to\answ
%
\ifx\answ\bigans\message{(This will come out unreduced.}
}
%\magnification=1200\unredoffs\baselineskip=16pt plus 2pt minus 1pt
\hsbody=\hsize \hstitle=\hsize %take default values for 
%   unreduced format
%
\doit0{
\else\message{(This will be reduced.} \let\l@r=L
%\magnification=1000\baselineskip=16pt plus 2pt minus 1pt 
\vsize=7truein
\redoffs \hstitle=8truein\hsbody=4.75truein\fullhsize=10truein%
\hsize=\hsbody
%
\output={\ifnum\pageno=0 %%% This is the HUTP version
  \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
    \hbox to \fullhsize{\hfill\pagebody\hfill}}\advancepageno
  \else
  \almostshipout{\leftline{\vbox{\pagebody\makefootline}}}%
   \advancepageno
  \fi}
}
\def\almostshipout#1{\if L\l@r \count1=1 \message{[\the\count0.%
\the\count1]}
      \global\setbox\leftpage=#1 \global\let\l@r=R
 \else \count1=2
  \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
      \hbox to\fullhsize{\box\leftpage\hfil#1}}  \global\let\l@r=L\fi}
\fi

% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string%
\string##1}}}
\nolabels
%
% tagged sec numbers
\global\newcount\secno \global\secno=0
\global\newcount\meqno \global\meqno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi %if desired
\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1
\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1%
\message{(\secsym\the\subsecno.%
 #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
%
\def\appendix#1#2{\global\meqno=1\global\subsecno=0%
\xdef\secsym{\hbox{#1.}}
\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
%
%	\eqn\label{a+b=c}	gives displayed equation, numbered
%				consecutively within sections.
%     \eqnn and \eqna define labels in advance (of eqalign?)
%
\def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1\wrlabeL#1}
\def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
\writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
\global\advance\meqno by1\wrlabeL{#1$\{\}$}}
\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}
%
%			 footnotes
\newskip\footskip\footskip8pt plus 1pt minus 1pt 
% \footskip sets footnote baselineskip 
\def\footnotefont{\ninepoint}
\def\f@t#1{\footnotefont #1\@foot}
\def\f@@t{\baselineskip\footskip\bgroup\footnotefont\aftergroup%
\@foot\let\next}
\setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt} %
\global\newcount\ftno \global\ftno=0
\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}
%
%say \footend to put footnotes at end
%will cause problems if \ref used inside \foot, %
%instead use \nref before
\newwrite\ftfile
\def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
$^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
\immediate\write\ftfile{\noexpand\smallskip%
\noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
\def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
\centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}
\def\footatend{}
%
%     \ref\label{text}
% generates a number, assigns it to \label, generates an entry.
% To list the refs on a separate page,	\listrefs
%
\global\newcount\refno \global\refno=1
\newwrite\rfile
%% We have tampered after #1 in \items which was originally %
% \item and also 
%% the argument of \xdef without [ ].  Also \\ after \items{#1}.
%
% We have to be careful about \ref, when using \label and \eq commands.
\def\ref{[\the\refno]\nref}%
\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi%
\global\advance\refno by1\chardef\wfile=\rfile\immediate%
\write\rfile{\noexpand\Item{#1}\reflabeL{#1\hskip.31in}\pctsign}%
\findarg\hskip10.0pt}%	
%	horrible hack to sidestep tex \write limitation
\def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
\gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
\edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup%
\else\ifx\next\empty%
\else\immediate\write\wfile{\the\toks0}%
\fi\let\next=\writ@line\fi\next\relax}}
\def\striprel@x#1{} \def\em@rk{\hbox{}}
%
\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now 
% unnecessary
%
\def\listrefs{\footatend\vfill\supereject\immediate\closeout%
\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp%
\vfill\eject}\nonfrenchspacing}
%
% The following is the revision of \listrefs to put the list in 
% the same page.
\def\listrefsr{\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}%
\nonfrenchspacing}
% The following is the revision of \listrefs to put the list %
% in the same page
% with the smaller fonts.
\def\listrefsrsmall{\immediate\closeout\rfile\writestoppt
\baselineskip=11pt\centerline{{\bf References}}
\font\smallreffonts=cmr9 \font\it=cmti9 \font\bf=cmbx9%
\bigskip{ {\smallreffonts% 
\parindent=15pt\escapechar=` \input refs.tmp\vfill\eject}}}
%
\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel %
\string#1 is undefined.}%
\nref#1{need to supply reference \string#1.}\fi%
\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
\else#1\count255=1\fi\let\next=\r@fs\fi\next}
%
\def\figures{\centerline{{\bf Figure Captions}}%
\medskip\parindent=40pt%
\def\fig##1##2{\medskip\item{Fig.~##1.	}##2}}
%
% this is ugly, but moore insists
\newwrite\ffile\global\newcount\figno \global\figno=1
%
\def\fig{fig.~\the\figno\nfig}
\def\nfig#1{\xdef#1{fig.~\the\figno}%
\writedef{#1\leftbracket fig.\noexpand~\the\figno}%
\ifnum\figno=1\immediate\openout\ffile=figs.tmp%
\fi\chardef\wfile=\ffile%
\immediate\write\ffile{\noexpand\medskip\noexpand%
\item{Fig.\ \the\figno. }
\reflabeL{#1\hskip.55in}\pctsign}\global\advance\figno by1\findarg}
%
\def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
\baselineskip14pt\centerline{{\bf Figure Captions}}\nobreak\medskip
\escapechar=` \input figs.tmp\vfill\eject}}
%
\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
\def\figs#1{figs.~\f@gs #1{\hbox{}}}
\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else
\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
%
\newwrite\lfile
{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
\def\writedefs{\immediate\openout\lfile=labeldefs.tmp %
\def\writedef##1{%
\immediate\write\lfile{\string\def\string##1\rightbracket}}}
%
\def\writestop{\def\writestoppt%
{\immediate\write\lfile{\string\pageno%
\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
\string\secno\the\secno\string\meqno\the\meqno}% 
\immediate\closeout\lfile}}
%
\def\writestoppt{}\def\writedef#1{}
%
\def\seclab#1{\xdef #1{\the\secno}\writedef{#1\leftbracket#1}%
\wrlabeL{#1=#1}}
\def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
%
\newwrite\tfile \def\writetoca#1{}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
%	use this to write file with table of contents
\def\writetoc{\immediate\openout\tfile=toc.tmp
   \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
   \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}
%	and this lists table of contents on second pass
\def\listtoc{\centerline{\bf Contents}\nobreak%
 \medskip{\baselineskip=12pt
 \parskip=0pt\catcode`\@=11 \input toc.tex \catcode`\@=12 %
 \bigbreak\bigskip}}
%
\catcode`\@=12 % at signs are no longer letters
%

\doit0{
\def\footnote#1{\let\@sf=\empty 
 \ifhmode\edef\@sf{\spacefactor=\the\spacefactor}\/\fi
 #1\@sf\vfootnote{#1}}
\def\vfootnote#1{\insert\footins\bgroup
 \interlinepenalty=\interfootnotelinepenalty
 \splittopskip=\ht\strutbox
 \splitmaxdepth=\dp\strutbox \floatingpenalty=20000
 \leftskip=0pt \rightskip=0pt \spaceskip=0pt \xspaceskip=0pt
 \textindent{#1}\footstrut\futurelet\next\fo@t}
\def\fo@t{\ifcat\bgroup\noexpand\next \let\next\f@@t
 \else\let\next\f@t\fi \next}
\def\f@@t{\bgroup\aftergroup\@foot\let\next}
\def\f@t#1{#1\@foot}
\def\@foot{\strut\egroup}
\def\footstrut{\vbox to\splittopskip{}}
\skip\footins=\bigskipamount 
\count\footins=1000
\dimen\footins=8in 
This is a test for footnote.\footnote*{This is a footnote.}
\end{document} 
}

%\def\eightpoint{\def\rm{\fam0\eightrm}% 
%  \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
%  \textfont1=\eighti  \scriptfont1=\sixi  \scriptscriptfont1=\fivei
%  \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
%  \textfont3=\tenex   \scriptfont3=\tenex \scriptscriptfont3=\tenex
%  \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}%
%  \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}%
%		\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}%
%  \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
%   \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}%
%   \tt \ttglue=.5em plus.25em minus.15em
%    \normalbaselineskip=9pt 
%    \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
%    \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm}
%\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf
%   \insert\footins\bgroup %\eightpoint
%\interlinepenalty100 \let\par=\endgraf
%     \leftskip=0pt \rightskip=0pt
%     \splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000
%     \smallskip\Item{#1}\bgroup\strut\aftergroup\@foot\let\next}
%\skip\footins=12pt plus 2pt minus 4pt % space added 
% when footnote exists
%\dimen\footins=30pc % maximim footnotes per page 
%This is a test for footnote.{\footnote{1}This is a footnote.}

% The following is to lift the bottom of the body from page number:
\countdef\pageno=0 \pageno=1
\newtoks\headline \headline={\hfil} 
\newtoks\footline 
 \footline={\bigskip\hss\tenrm\folio\hss}
 %\footline={\hss\tenrm\folio\hss}
\def\folio{\ifnum\pageno<0 \romannumeral-\pageno \else\number\pageno 
 \fi} 
\def\nopagenumbers{\footline={\hfil}} 
\def\advancepageno{\ifnum\pageno<0 \global\advance\pageno by -1 
 \else\global\advance\pageno by 1 \fi} 
\newif\ifraggedbottom
\def\raggedbottom{\topskip10pt plus60pt \raggedbottomtrue}
\def\normalbottom{\topskip10pt \raggedbottomfalse} 

\def\on#1#2{{\buildrel{\mkern2.5mu#1\mkern-2.5mu}\over{#2}}}
\def\dt#1{\on{\hbox{\bf .}}{#1}}		% (big) dot over
\def\Dot#1{\dt{#1}}

\count100=1 

\font\smallcmr=cmr6 scaled \magstep2 
\font\largetitle=cmr17 scaled \magstep1 

\font\LargeLarge=cmr17 scaled \magstep5 
 
\def\alephnull{~$\large{\aleph_0}\,$~} 
\def\alephnulllarge{${\displaystyle\aleph_0}$~} 
\def\Bo{\bo{\hskip 0.03in}} 
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) } 
\def\.{.$\,$} 
\def\alpr{\a{\hskip 1.2pt}'} 
\def\dim#1{\hbox{dim}\,{#1}} 
\font\goth = eufm7 scaled \magstep3 
\font\gothsmall = eufm5 scaled \magstep3 
\def\Real{\raise-1pt\hbox{\goth R}\!\raise-1pt\hbox{\goth e}\,} 
\def\Imaginary{\raise-1pt\hbox{\goth I}\!\raise-1pt\hbox{\goth m}\,} 
\def\Realsmall{\raise-3pt\hbox{\goth R}\!\raise-3pt\hbox{\gothsmall e}\,} 
\def\Imaginarysmall{\raise-3pt\hbox{\goth I}\!\raise-3pt% 
\hbox{\gothsmall m}\,} 
\def\leftarrowoverdel{{\buildrel\leftarrow\over\partial}}
\def\rightarrowoverdel{{\buildrel\rightarrow\over\partial}}
\def\Dsl{{}D \!\!\!\! /{}}

\def\eppp#1{\big( \Bar\e \star \g_{#1} \psi_1 \big) 
     \star \big(\Bar\psi_2 \star \g^{#1} \psi_3 \big)} 
\def\ppep#1{\big( \Bar\psi_1 \star \g_{#1} \psi_2 \big) 
     \star \big(\Bar\e \star \g^{#1} \psi_3 \big)}  
\def\epppo{\big( \Bar\e \star \psi_1 \big) 
     \star \big(\Bar\psi_2 \star \psi_3 \big)} 
\def\ppepo{\big( \Bar\psi_1 \star \psi_2 \big) 
     \star \big(\Bar\e \star \psi_3 \big)}  
\def\rpppp#1{\big( \Bar\psi_1 \star \g_{#1} \psi_4 \big) 
     \star \big(\Bar\psi_3 \star \g^{#1} \psi_2 \big)} 
\def\lppppo{\big( \Bar\psi_1 \star \psi_2 \big) 
     \star \big(\Bar\psi_3 \star \psi_4 \big)} 
\def\rppppo{\big( \Bar\psi_1 \star \psi_4 \big) 
     \star \big(\Bar\psi_3 \star \psi_2 \big)}	
\def\starproduct#1#2#3#4{{#1} \star {#2} \star {#3} \star {#4}}
\def\grass#1{(-1)^{#1}}

\def\ZZ{Z\!\!\! Z} 

\def\atmp#1#2#3{Adv.~Theor.~Math.~Phys.~{\bf{#1}} (19{#2}) {#3}} 

\def\bea{\begin{eqnarray}} 
\def\eea{\end{eqnarray}} 

\def\Christoffel#1#2#3{\left\{ {\buildrel #1 
\over{\scst #2 #3}} \right\} } 

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\begin{center} 
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{\large\bf Noncommutative Supersymmetric Yang-Mills Theory} 
\\[0.026in] 
{\large\bf in Ten-Dimensions with Higher-Derivative Terms}
\\[.1in]  

\baselineskip 9pt 

\vskip 0.26in 

\doit0{
Hitoshi ~N{\smallcmr ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
\\[.16in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland} \\ [.015in] 
{\it College Park, MD 20742-4111} \\[.20in]   
and \\[0.2in] 
}

Hitoshi ~N{\smallcmr ISHINO}\footnote{E-Mail: 
hnishino@csulb.edu} 
~and~ Subhash ~R{\smallcmr AJPOOT}\footnote{E-Mail: rajpoot@csulb.edu} 
\\[.16in] 
{\it Department of Physics \& Astronomy} \\ [.015in] 
{\it California State University} \\ [.015in] 
{\it Long Beach, CA 90840} \\[0.02in]	


\vskip 2.3in 

{\bf Abstract}\\[.1in]	
\end{center} 

\baselineskip 14.5pt 

~~~We present an action for noncommutative
supersymmetric Yang-Mills theory in ten-dimensions, and confirm its
invariance under  supersymmetry.  We next add higher-order derivative
terms to such a noncommutative supersymmetric action.  
These terms contain fields as high as the quartic order.
This resulting action can be regarded as
supersymmetric generalization of noncommutative non-Abelian
Dirac-Born-Infeld action.  Some ambiguities related to field
redefinitions are also clarified.   


\vskip 0.8in

\leftline{\small PACS: ~02.40.Gh, ~03.50.-z, ~03.50.De, ~11.10.E,
~11.15.-q, ~11.30.Pb, ~12.60.Jv} 
\vskip -0.05in
\leftline{\small Key Words: Non-Commutativity, Yang-Mills Theory, 
Supersymmetry,}
\vskip -0.05in
\leftline{\small {\hskip 0.8in} Dirac-Born-Infeld Action}

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\leftline{\bf 1.~~Introduction} 

The idea that the space-time coordinates should be noncommutative 
%%% 
\ref\noncomm{A.~Connes, M.R.~Douglas and A.~Schwarz, JHEP {\bf 9802} 
(1998) 003; 
Y.K.E.~Cheung and M.~Krogh, Nucl.~Phys.~{\bf B528} (1998) 185;
C.-S.Chu and P.-M.~Ho, Nucl.~Phys.~{\bf B528} (1999) 151; 
V.~Schomerus, JHEP {\bf 9906} (1999) 030; 
F.~Ardalan, H.~Arfaei and M.M.~Sheikh-Jabbari, JHEP {\bf 9902} (1999) 016;
J.~Hoppe, Phys.~Lett.~{\bf B250} (1990) 44;
D.B.~Fairlie, P.~Fletcher and C.K.~Zachos, Phys.~Lett.~{\bf B218} 
(1989) 203; {\it For reviews, see, e.g.,} M.R.~Douglas and N.A.~Nekrasov, 
{\it `Noncommutative Field Theory'}, ITEP-TH-31/01, IHES/P/01/27,
RUNHETC-2001-18, \hepth{0106048}, {\it and references therein}.}    
%%% 
has been motivated by the development of open strings or D-branes, leading
to the constant background antisymmetric field.  Accordingly, the 
low energy effective theory of such open strings attached to
noncommutative branes becomes a noncommutative gauge theory
%%% 
\ref\dhsj{M.R.~Douglas and C.~Hull, JHEP {\bf 9802} (1998) 008, 
\hepth{9711165};
M.M.~Sheikh-Jabbari, \pl{450}{99}{119}, \hepth{9810179}.}.   
%%%
It has been also proven 
%%% 
\ref\sw{N.~Seiberg and E.~Witten, JHEP {\bf 9909} (1999) 032, 
\hepth{9908142}.} 
%%%
that the noncommutative Dirac-Born-Infeld (DBI) theory is equivalent to 
the ordinary DBI theory under what is called Seiberg-Witten map \sw%
%%%
\ref\ot{Y.~Okawa, Nucl.~Phys.~B566 (2000) 348, 
hep-th/9909132; S.~Terashima, JHEP {\bf 02} (2000) 029, 
hep-th/0001111.}.  
%%%  
The supersymmetrization of DBI theory with non-vanishing
$~B_{\m\n}\-$field was studied in 
%%%
\ref\fl{S.~Ferrara and Lled\`o, {\it `Some Aspects of Deformations of
Supersymmetric Field Theories'}, \hepth{0002084}.},  
%%%
and the conclusion was that it leads to the noncommutative
supersymmetric gauge theory in a certain limit.  A supersymmetric 
noncommutative DBI theory in 4D has been studied in 
%%%
\ref\gps{N.~Grandi, R.L.~Pakman and F.A.~Schaposnik,
Nucl.~Phys.~{\bf B588} (2000) 508, hep-th/0004104.}   
%%%
both for Abelian and non-Abelian gauge groups, based on superfield
formulation.  As in the non-supersymmetric case, the possible 
non-Abelian gauge group for noncommutative DBI 
theory is to be $~U(N)$.  However, this restriction on Yang-Mills gauge
groups has been recently overcome by the use of anti-automorphism of
$~\star$~ matrix algebra, consistently restricting the $~u_\star(N)$~
algebra to be $~o_\star (N)$~ or $~usp_\star(N)$~ algebras 
%%% 
\ref\bsjv{L.~Bonora, M.~Schnabl, M.M.~Shaikh-Jabbari and
A.~Tormasiello, Nucl.~Phys.~{\bf B589} (2000) 461, 
\hepth{0006091}; I.~Bars, M.M.~Sheikh-Jabbari and M.A.~Vasiliev, {\it
`Noncommutative $~o_\star(N)$~ and $~usp_\star(N)$~ Algebras and the
Corresponding Gauge Field Theories'}, \hepth{0103209}.}.  
%%%  

In this present Letter, we will present the noncommutative version of
supersymmetric Yang-Mills theory in ten-dimensions (10D) with the gauge
group $~U(N)$, first for the covariant kinetic terms for a non-Abelian
Yang-Mills multiplet, and next with its generalization with
higher-derivative quartic terms that can be added to the kinetic terms
containing the next-leading terms in the DBI action, up to quintic terms
in fields.  To put it differently, we will study the supersymmetrization
%%% 
\ref\brs{E.~Bergshoeff, M.~Rakowski and E.~Sezgin, \pl{185}{87}{371}.}%
%%% 
\ref\aps{M.~Aganagic, C.~Popescu, and J.H.~Schwarz,
\pl{393}{97}{311}; Nucl.~Phys.~\hfill
\newline{\bf B495} (1997) 99.}% 
%%%
\ref\cederwall{M.~Cederwall, B.E.W. Nilsson, D.~Tsimpis, {\it `$D=10$
Super-Yang-Mills at $O(alpha'^2)$'}, Goteborg-ITP-preprint 2001, 
\hepth{0104236}.}
%%% 
of noncommutative DBI theory \sw\ot\ in 10D.  As the guiding principle, we
follow the result in \brs\cederwall\ for commutative supersymmetric
Yang-Mills theory with higher-order derivatives.   We will show that by
introducing a total symmetrization operator  as in \gps, the whole
computation is drastically simplified,	by avoiding the potentially
dangerous ordering problem with
$~\star$~ products  at quartic order.  



\bigskip\bigskip\bigskip

% \newpage


\leftline{\bf 2.~~Noncommutative Supersymmetric Yang-Mills 
Theory in 10D}	  

We start with the covariant kinetic terms of noncommutative supersymmetric
Yang-Mills theory in 10D.  Here we do not include higher-derivative
terms, but consider only the kinetic terms.  We first fix the total
action $~I_{\rm NCSYM}$~ to be 
$$ \li{ I_{\rm NCSYM} = & \int d^{10} x \, \tr 
      \Big[ \, + \frac14 F_{\m\n} \star F^{\m\n}  
     + 2 \Bar\l \star \g^\m D_\m \l  \, \Big] \cr 
\equiv & \int d^{10} x \, 
     \Big[ \, - \frac 14 F\du{\m\n} I \star F^{\m\n\, I} 
      - 2 \Bar\l{}^I \star \g^\m D_\m \l^I \, \Big]  ~~, 
&(2.1) \cr } $$
where the $~\tr\-$operator acts like $~\tr\big( T^I T^J \big) = - \d^{I
J}$~ for anti-hermitian generators $~T^I~{\scst (I,~J,~\cdots ~=~1, ~2,
~\cdots,~N^2)}$~ of the gauge group $~U(N)$~ for the fields $~A_\m 
\equiv A\du\m I T^I$~ and $~\l\equiv \l^I T^I$.  The $~\star$~ product is
the noncommutative product defined by 
$$ \li{ f \star g \equiv & f \, 
     \exp \big( i \leftarrowoverdel_\m\theta^{\m\n}
     \rightarrowoverdel_\n\big) \, g   \cr 
\equiv & \sum_{n=0}^\infty \fracm {i^n}{n!} \theta^{\m_1\n_1}
     \cdots \theta^{\m_n\n_n} 
     \big( \partial_{\m_1}\cdots\partial_{\m_n} f \big) 
     \big( \partial_{\n_1}\cdots\partial_{\n_n} g \big) ~~.  
&(2.2) \cr } $$
The field strength $~F_{\m\n}{}^I$~ and the covariant derivative
$~D_\m\l^I$~  are defined by 
$$ \li{ F_{\m\n}{}^I \equiv & \partial_\m A_\n{}^I  
     - \partial_\n A_\m{}^I 
     + \big( A_\m \star A_\n - A_\n \star A_\m \big)^I	
     \equiv \partial_\m A\du\n I - \partial_\n A\du\m I 
      + \[ A_\m , A_\n \]_\star^I {~~, ~~~~~ ~~~~~}  
&(2.3\rma) \cr 
D_\m \l^I \equiv & \partial_\m\l^I + \big( A_\m\star \l \big)^I 
      - \big(\l \star A_\m\big)^I  
      \equiv  \partial_\m \l^I	+ \[ A_\m , \l\]_\star^I ~~,   
&(2.3\rmb) \cr } $$
as has been given by many authors \noncomm\sw\gps.  The reason we put 
the factor $~2$~ in the $~\l\-$kinetic term is in order to comply with 
the notation in \brs.  Relevantly, under $~U(N)$~ the fields transform
as 
$$ \li{ \d_\a A\du\m I = & \partial_\m \a^I + \[ A_\m, \a \]_\star^I ~~, 
&(2.4\rma) \cr 
\d_\a \l^I = & - \[ \a, \l \]_\star^I ~~. 
&(2.4\rmb) \cr } $$
To save space, we sometimes omit the indices $~{\scst I,~J,~\cdots}$, 
using also the $~\star$~ commutator $~\[ A, B\]_\star\equiv A\star B -
B\star A$.  We skip the details for the confirmation of gauge invariance
of our action
$~I_{\rm NCSYM}$, due to its common feature shared with other 4D cases
\noncomm\sw.  

Our action $~I_{\rm NCSYM}$~ in (2.1) is also invariant under
supersymmetry
$$ \li{ \d_Q A\du\m I = & - \big( \Bar\e \g_\m \l^I \big) ~~, 
&(2.5\rma) \cr 
\d_Q \l^I = & + \frac 1 8 \g^{\m\n} F\du{\m\n} I ~~. 
&(2.5\rmb) \cr } $$ 
The superinvariance of $~I_{\rm NCSYM}$~ is confirmed 
by the frequent use of basic relations, such as 
$$\li {& \int d^{10} x \, f \star g = 
	  \int d^{10} x \, g \star f = 
	 \int d^{10} x \, f g ~~, 
&(2.6) \cr } $$
namely, the $~\star$~ product of two fields does not matter 
under the 10D integral $~\int d^{10} x$, because the difference is only
a total divergence.  In the variation of the kinetic term of $~A_\m$, 
a convenient lemma is 
$$ \li{ & \int d^{10} x\, \Big( A \star \[ B, C\]_\star 
     - \[ A, B \]_\star \star C \Big) \equiv 0 ~~, 
&(2.7) \cr } $$
for arbitrary fields $~A, ~B$~ and $~C$, formulated as a corollary
of  (2.6).  Another useful corollary is about the partial integration 
for the covariant derivative $~D_\m$: 
$$ \li{ & \int d^{10} x\, B^I \star D_\m C^I  
     = -  \int d^{10} x\, \big( D_\m B^I \big) \star C^I  ~~,  
&(2.8) \cr } $$ 
which is used in the variation of the $~\l\-$kinetic term.  Needless 
to say, the Bianchi identity 
$$\li{ & D_{\[ \m} F_{\n\r\]}{}^I \equiv 0~~, 
&(2.9) \cr } $$
plays an important role in the confirmation of superinvariance.  

As in the commutative case, after the cancellation of all the quadratic
terms, we are left with the cubic term of the
$~\l\-$field:	
$$ \li{ & \d_Q I_{\rm NCSYM} 
      = \int d^{10} x\, \Big[ \, - 2 f^{I J K} 
     \big( \Bar\e \g_\m \l^I \big) \star\big( \Bar\l^J 
     \star \g^\m \l^K \big) \, \Big]  ~~,  
&(2.10) \cr } $$
where $~f^{I J K}$~ is the totally antisymmetric structure constant 
for $~U(N)$.  As in the commutative case, we need the Fierz identity 
$$ \li{ \big(\Bar\e\star \psi_1\big) \star & \! \big(
\Bar\psi_2\star \psi_3\big)  \cr 
     = & - \frac1{16} \ppepo
	 - \frac1{16} \ppep{\m}   \cr 
& + \frac1{32} \ppep{\m\n}    
	 + \frac1{96} \ppep{\m\n\r}  \cr 
& - \frac1{384} \ppep{\m\n\r\s}   
	 - \frac1{1920} \ppep{\m\n\r\s\t} {~~,	~~~~~ ~~~~~}  
&(2.11) \cr } $$ 
for arbitrary Majorana-Weyl spinors $~\e,~\psi_1,~\psi_2$~ and
$~\psi_3$.   Depending on their chiralities, some terms above vanish,
{\it e.g.,}
$~\big( \Bar\psi_1 \star\psi_2 \big) \equiv 0$~ if $~\psi_1$~ and
$~\psi_2$~ share the same chirality.  The Fierzing (2.11) is possible
without a total symmetrization to be used in the next section,
thanks to the constancy of the parameter $~\e$.  After applying the  
Fierzing (2.11) to (2.10), the question $~\d_Q I_{\rm NCSYM} \eqques 0$~
is now equivalent to 
$$\li{ 0 & \eqques + \int d^{10} x\, 
     f^{I J K} \big( \Bar\l{}^I \star \g^{\r\s\t} \l^J \big) \star 
      \big( \Bar\e\g_{\r\s\t} \l^K \big) \cr 
& ~~ = + 2 \big( \g^{\r\s\t} \big)_{\a \[ \b | }
	 \big( \g_{\r\s\t} \big)_{ | \g \] \d} \, \e^\d 
    \int d^{10} x\, \tr \big( \l^a \star\l^\b \star \l^\g \big) ~~.
&(2.12) \cr } $$ 
This term with three-gamma matrix sandwiched by the $~\l$'s was absent
in the {\it commutative} case, because of the $~{\scst (\a\b)}$~
symmetry of $~f^{I J K} \l^{I \, \a} \l^{J \, \b}$.  But now we have 
this because the latter is to be replaced by $~f^{I J K} \l^{I \, \a}
\star \l^{J \, \b}$, which has no such symmetry in $~{\scst (\a\b)}$~ \
due to the $~\star$~ product.  This problem is solved by the
$~\g\-$matrix identity
$$ \li{ & \big(\g^{\r\s\t} \big)_{\a \[\b | }  
      \big( \g_{\r\s\t} \big)_{ | \g\d\]} \equiv 0 ~~,	  
&(2.13) \cr } $$ 
so that (2.12) equals 
$$\li{ 0 & \eqques + \int d^{10} x\, 
     f^{I J K} \big( \Bar\l{}^I \star \g^{\r\s\t} \l^J \big) \star 
      \big( \Bar\e\g_{\r\s\t} \l^K \big) \cr 
& ~~ = + 2 \big( \g^{\r\s\t} \big)_{\a \[ \b | }
	 \big( \g_{\r\s\t} \big)_{ | \g \] \d} \, \e^\d 
    \int d^{10} x\, \tr \big( \l^a \star\l^\b \star \l^\g \big) \cr 
& ~~ = + \big( \g^{\r\s\t} \big)_{\a \b}
	 \big( \g_{\r\s\t} \big)_{ \g \d} \, \e^\d 
    \int d^{10} x\, \tr \big( \l^a \star\l^\b \star \l^\g \big) \cr 
& ~~ = + \frac 12 \int d^{10} x\, 
       f^{I J K} \big( \Bar\l{}^I \star \g^{\r\s\t} \l^J \big) \star 
      \big( \Bar\e\g_{\r\s\t} \l^K \big) ~~,	 
&(2.14) \cr } $$ 
proportional to the first line with the constant 1/2.  This implies that
the first line itself is to vanish.  The important ingredient here is 
that even though there is a three-gamma matrix sandwiched by the
$~\l$'s, such a term vanishes by itself due to the $~\g\-$algebra
(2.13).  This concludes the confirmation of superinvariance of 
the action $~I_{\rm NCSYM}$.   



\bigskip\bigskip\bigskip

% \newpage 


\leftline{\bf 3.~~Noncommutative Supersymmetric Theory	
with Higher-Derivatives in 10D}     

We next consider the inclusion of higher-order derivatives into 
the supersymmetric Abelian gauge theory in 10D.  Since the resulting
lagrangian will contain the bosonic noncommutative DBI lagrangian, we can
regard this  also as the supersymmetrization of noncommutative DBI
theory.  In this sense, we call our action noncommutative supersymmetric
DBI action  represented by $~I_{\rm NCSDBI}$.  

In this paper we fix our lagrangian for $~I_{\rm NCSDBI}$~ with all the  
quartic terms at $~{\cal O}(\a^2)$~ including also $~\l^4\-$terms. 
However, the terms quintic in fields are ignored at $~{\cal
O}(\a^2)$, just for simplicity of computation.	These quintic terms had
been also ignored in \brs, but has been analyzed recently in superspace
formulation \cederwall.  These quintic terms arise only in the
commutators of gauge-covariant derivatives that always contains the
structure constant of the gauge group \cederwall.  In this paper, these
quintic terms are ignored just for simplicity, such that
partial integrations can be done rather easily.  As for  
the transformation rule for the Yang-Mills field $~A_\m$~ and the 
gaugino $~\l$, we fix terms only up to $~\l^3$~ and $~\l^2 F\-$terms,
respectively.	

We first summarize our result here.  Our total action
$~I_{\rm NCSDBI}$~  is 
$$ \li{ I_{\rm NCSDBI} \equiv & \int d^{10} x\, \Lag_{\rm NCSDBI} ~~, 
&(3.1) \cr 
\Lag_{\rm NCSDBI} \equiv & -\frac 14 F_{\m\n}{}^I \star F^{\m\n}{}^I 
    - 2 \big( \Bar \l{}^I \star \g^\m D_\m \l{}^I \big) \cr 
& + \a^2 \tr {\cal S}^\star \Big[ 
     - \frac 14 \big( F\star F\star F\star F \big)\du\m\m
       + \frac 1{16} \big( F\star F \big)\du\m\m \star 
	      \big( F\star F \big)\du\n\n \cr 
& ~~~~~ ~~~~~ ~~~~~\,	+ 2 \big( F\star F \big)^{\m\n} 
     \star\big(\Bar\l \star\g_\m D_\n \l\big) 
      + \frac 12 F\du\m\l \star \big( D_\l F_{\n\r}\big) 
     \star \big( \Bar\l \star \g^{\m\n\r} \l \big) \cr	
& ~~~~~ ~~~~~ ~~~~~\,  - \frac 4 3 \big( \Bar\l \star \g_\m D_\n \l \big) 
     \star \big( \Bar\l \star \g^\m D^\n \l\big) \, \Big] 
    + {\cal O}(\a^2\varphi^5) + {\cal O}(\a^3) ~~.  
&(3.2) \cr } $$ 
Here $~\a$~ is a constant with the dimension of $~(\hbox{length})^{2}$~
in order to keep track of the higher-derivative terms.	For example, we
sometimes call the first line in (3.2) $~\Lag_{\a^0}$, while the rest
$~\Lag_{\a^2}$.  The symbol $~{\cal O}(\a^2\varphi^5)$~ represents
any terms quintic in fields at $~{\cal O}(\a^2)$, while $~{\cal
O}(\a^3) $~ is for terms at order higher than $~{\cal
O}(\a^2)$.  

The operator
$~{\cal S}^\star$~ is for the total symmetrization operator, defined by 

\vbox{
$$ \li{ & {\cal S}^\star \big( A \star B\star C \star D\big) \cr 
& ~~~~~ \equiv \frac 14 \Big[ \, A\star {\cal S}^\star 
      \big( B \star C \star D\big) 
       + (-1)^{A(B+C+D)} B\star {\cal S}^\star \big(C\star D\star A\big)
\cr  & ~~~~~ ~~~~~ ~~ + (-1)^{(C+D)(A+B)} C\star 
     {\cal S}^\star \big(D\star A\star B\big)
     + (-1)^{D(A+B+C)} D\star 
     {\cal S}^\star\big(A\star B\star C\big) \Big] {~~, ~~~~~
~~~~~}	 
&(3.3\rma) \cr 
& {\cal S}^\star \big( A\star B\star C\big) \cr 
& ~~~~~ \equiv \frac 16 \Big[ \, A\star B\star C 
     + (-1)^{A(B+C)} B\star C\star A 
    + (-1)^{C(A+B)} C\star A\star B \cr 
& ~~~~~ ~~~~~ ~~ + (-1)^{C B} A\star C\star B 
     + (-1)^{A B} B\star A\star C 
     + (-1)^{C(A+B) + A B} C\star B\star A \, \Big] ~~,  
&(3.3\rmb) \cr } $$ }
\vskip -0.2in
\noindent 
where the superscripts $~{\scst A,~B,~C,~D}$~ are for the Grassmann 
parities of each field in $~A,~B,~C,~D$.  Effectively, under the  
$~{\cal S}^\star\-$operations, the ordering problem with the $~\star$~
product
disappears \gps, because of the total symmetrization by definition.  
Note also that the $~\tr\-$operation acts on the anti-hermitian generators
$~T^I$, after the total symmetrization by the $~{\cal S}^\star$'s.    

The symbols such as $~\big(F\star F\star F\star F\big)\du\m\n$~ are
defined by 
$$ \li{ & \big(F\star F\big) \du\m\n \equiv F\du\m\r \star F\du\r\n ~~, 
~~~~ \big(F\star F\star F \star F\big)\du\m\n
    \equiv F\du\m\r \star F\du\r\s \star F\du\s\t \star F\du\t\n ~~. 
&(3.4) \cr } $$  
Compared with the commutative case \brs\cederwall, all the coefficients in
(3.2) are in agreement with the commutative case, {\it except for} the 
special usage of the total symmetrization 
$~{\cal S}^\star$~ for $~\star$~ products. 
If we look only at purely bosonic terms, they are of the form
$$ \li{ & + \frac 14 \tr \big(F\star F\big)\du\m\m 
     - \frac 1 4 \a^2 \tr {\cal S}^\star 
      \Big[ \, \big( F\star F\star F\star F\big)\du\m\m \, \Big] ~~, 
&(3.5) \cr } $$
which are the terms at $~{\cal O}(\a^0)$~ and $~{\cal O}(\a^2)$~ in the 
noncommutative DBI lagrangian \gps\footnotew{The noncommutative square 
root $~{\root\star\of{1+x}}$~ is defined by the expansion
$~{\root\star\of{1+x}} \equiv 1 + \sum_{n=1}^\infty (1/n!) (1/2) (1/2-1)
\cdots	(3/2-n) \, \overbrace{x\star x\star \cdots \star x}^n$~ \gps,
while the $~\hbox{det}^\star$~ is the noncommutative determinant.}  
$$ \li{ & \Lag_{DBI} \equiv b^{-2} \a^{-2}\tr {\cal S}^\star \left[\,
{\root\star\of{\det^\star \big(\d\du\m\n 
     + b \a  F\du\m\n \big) }} \, \right] ~~.  
&(3.6) \cr } $$  
Therefore, our action $~I_{\rm NCSDBI}$~ can also be regarded as the
supersymmetrization of that of the noncommutative DBI theory.  

Our $~I_{\rm NCSDBI}$~ is invariant under supersymmetry 
$$ \li{  \d_Q A_\m = & - \big( \Bar\e\g_\m \l\big) \cr 
& + \a^2 {\cal S}^\star \Big[ +\frac 38 \big(F\star F\big)\du\n\n
     \star \big(\Bar\e\g_\m\l\big) 
     - \big(F\star F\big)\du\m\n \star \big(\Bar\e \g_\n \l\big) \cr 
& ~~~~~ ~~~~~ ~\, - \frac 14 F_{\m\n}\star F_{\r\s} \star 
     \big( \Bar\e \g^{\n\r\s} \l\big) 
     + \frac1{16} F^{\r\s} \star F^{\t\l} \star
     \big(\Bar\e\g_{\m\r\s\t\l} \l\big) \, \Big] 
     + {\cal O}(\a^2\l^3) {~~, ~~~~~ ~~~~~}  
&(3.7\rma) \cr	
\d_Q \l = & + \frac 18 \big(\g^{\m\n} \e\big) F_{\m\n} \cr 
& + \a^2 {\cal S}^\star 
     \Big[ +\frac 1{64} \big( F\star F\big)\du\n\n \star
     F_{\s\t} \big( \g^{\s\t}\e\big) 
     - \frac 1{16} \big( F\star F\star F \big)_{\m\n} 
     \big(\g^{\m\n}\e\big) \cr 
& ~~~~~ ~~~~~ ~\,   - \frac 1{384} F_{\m\n}\star F_{\r\s} 
     \star F_{\t\l} 
     \star \big(\g^{\m\n\r\s\t\l}\l \big) \, \Big] 
     + {\cal O} (\a^2 \l^2 F\big) ~~,	  
&(3.7\rma) \cr } $$
where the adjoint indices are suppressed, but are taken for granted.  

The confirmation of the supersymmetric invariance of our action $~I_{\rm
NCSDBI}$~ up to $~{\cal O}(\a^2\varphi^5)$~ and $~{\cal
O}(\a^3)\-$terms is performed as follows.  First, note that there arise
two sorts of terms, when we vary
$~I_{\rm NCSDBI}$: (I) $~F^3\l\-$terms, and ~ (II)  $~F\l^3\-$terms. 
Next, we take care of these two categories in turn:

As for the (I) $~\a^2 F^3\l\-$terms, there are three sources of these
terms: (i) $~\d_Q\Lag_{\a^2 F^4}\big|_{\a^2 F^3\l}$, (ii) ~$\d_Q
\Lag_{\a^2 F^2\l^2} \big|_{\a^2 F^3\l}$, and ~(iii)
$~\d_Q\Lag_{\a^0}\big|_{\a^2 F^3\l}$.  

We start with the sectors (i) and (ii).  To this end, we first establish
the  convenient lemma for the variation of $~\d_Q A_\m$:
$$ \li{ & \a^2 {\cal S}^\star 
     \Big[\, \big\{ \d_Q \big( F\star F\big)_{\m\n} \big\}
     \star X^{\m\n} \, \Big] \cr 
& ~~~~~ = {\cal S}^\star \Big[ \, 2\a^2 \big(\Bar\e \g^\r \l\big) 
     \star D_\m \big( F_{\r\n} \star X^{\m\n} \big) 
     - 2 \a^2 \big(\Bar\e \g_\m\l\big) 
     \star D_\r \big( F\ud\r\n \star X^{\m\n} \big) \,\Big] 
    + {\cal O}(\a^2\varphi^4) {~~, ~~~~~} 
&(3.8) \cr } $$ 
where $~X^{\m\n}$~ is an arbitrary field or $~\star$~ products of
fields.  Eq.~(3.8) is up to a total divergence in 10D, as well as
quartic terms in fields at $~{\cal O}(\a^2)$.  By the aid of this lemma,
we can arrange all  the contributions in (i) and (ii) as
$$\li{ \d_Q \Lag_{\a^2} \big|_{\a^2\l F^3} & \cr 
= \a^2 \tr {\cal S}^\star \Big[ & + \frac 12 \big( \Bar\e\g^\r \l\big) 
     \star \big(F\star F\big)^{\l\n} \star D_\l F_{\n\r} 
     + \frac 3 2 \big( \Bar\e\g^\r \l\big) 
     \star \big( F\star F\big)_{\r\t} \star D_\n F^{\n\t}  \cr 
& - \frac 12 \big( \Bar\e\g_\r \l\big) \star 
    \big(F\star F\big)\du\s\s \star D_\m F^{\m\r} 
     + \frac 14 \big( \Bar\e\g\du{\m}{\r\s} \l\big) 
     \star \big(F\star F\big)^{\m\n} D_\n F_{\r\s} \cr 
& - \frac 18 \big( \Bar\e\g^{\m\r\s} \l\big) \star
     F_{\r\s} \star F_{\t\n} \star D_\m F^{\t\n} 
       - \frac 14 \big( \Bar\e\g^{\m\r\s} \l \big) \star
     F_{\m\t} \star F_{\r\s}\star D_\n F^{\n\t} \cr 
& + \frac 12 \big( \Bar\e \g^{\m\s\n} \l\big) \star 
    F\du\m\l \star F_{\s\r} \star D_\l F\du\n\r 
     - \frac 18 \big( \Bar\e\g^{\s\t\m\n\r} \l\big) \star
    F\du\m\l \star F_{\s\t} \star D_\l F_{\n\r}\, \Big] 
    {~~.~~~~~ ~~~~~}  
&(3.9) \cr } $$
The guiding principle for arranging these terms is that any derivative 
hitting on the $~\l\-$field should be partially integrated, such that 
only one of the ~$F$'s in the three $~F$'s should be hit by a
derivative.  We can always perform such a partial integration, yielding 
only those terms in (3.9).  There was also an exact cancellation between
the like terms of the type $~\big(\Bar\e\g^\r \l\big) \star F_{\n\r} 
\star F_{\m\s}\star D^\n F^{\m\s}$, which is thus absent
in (3.9).    

We next look into the sector (iii).  For this sector, we need a
special lemma related to the $~{\cal S}^\star\-$operation.  This is
because when we  vary the kinetic terms at $~{\cal O}(\a^0)$, we need to
substitute the $~{\cal O}(\a^2)\-$terms in (3.7).  However, the $~{\cal
S}^\star\-$operation in such terms in (3.7) symmetrizes only three
fields, while these terms from this sector (iii) are supposed to cancel
those terms in (3.9), where all the $~{\cal S}^\star\-$operations
symmetrize all the four fields.  A convenient lemma to solve this problem
is 
$$ \li{ & \int d^{10} x\, \tr \Big[\, 
     {\cal S}^\star(A\star B\star C\big) \star D\, \Big] 
    \equiv \int d^{10} x\, \tr {\cal S}^\star 
     \big( A\star B \star C\star D\big) ~~. 
&(3.10) \cr } $$
This lemma means that any $~\star$~ product of a $~{\cal
S}^\star\-$symmetrized
three  fields with a field equals the total symmetrization of the
$~\star$~ product of all the four fields.  This 
lemma is easily confirmed by the use of the fundamental identity,
such as (2.6), considering also the Grassmann parities of 
all the fields.  In fact, the l.h.s.~of (3.10) is 
$$ \li{ & {\hskip -0.3in} \hbox{(r.h.s.)} \cr  
= & \int d^{10} x \, \frac 1{24} \Big[ 
	 + \starproduct A B C D 
	 + \grass{A(B+C)} \starproduct B C A D 
	 + \grass{C(A+B)} \starproduct C A B D \cr  
& + \grass{A B} \starproduct B A C D 
	 + \grass{A(B+C)+ B C} \starproduct C B A D   
	 + \grass{B C} \starproduct A C B D \cr 
& + \grass{A (B+ C + D)} \starproduct B C D A 
       + \grass{A(B + C + D) + B(C + D)} \starproduct C D B A \cr 
& ~~~~~ ~~~~~ + \grass{A(B + C + D) + D (B+C)} \starproduct D B C A \cr 
& + \grass{A(B + C + D) + B C} \starproduct C B D A 
       + \grass{A(B + C + D) + B(C + D) + C D} \starproduct D C B A \cr  
& ~~~~~ ~~~~~ + \grass{A(B + C + D) + C D} \starproduct B D C A \cr 
& + \grass{(A+B)(C+D)} \starproduct C D A B 
	+ \grass{(A+B)(C+D)+ C(D+A)} \starproduct D A C B \cr 
& ~~~~~ ~~~~~ + \grass{B(C+D)} \starproduct A C D B \cr 
& + \grass{(A+B)(C+D)+ CD} \starproduct D C A B 
	+ \grass{B(C+D)+CD} \starproduct A D C B \cr 
& ~~~~~ ~~~~~ + \grass{B(C+D)+AC} \starproduct C A D B \cr 
& + \grass{D(A+B+C)} \starproduct D A B C 
	+ \grass{C D} \starproduct A B D C  \cr 
& ~~~~~ ~~~~~ + \grass{C D+A(B+D)} \starproduct B D A C \cr 
& + \grass{D(B+C)} \starproduct A D B C
	+ \grass{A B + C D}\starproduct B A D C \cr 
& ~~~~~ ~~~~~  + \grass{(B+C) D + A(B+D)} \starproduct D B A C 
     \, \Big] ~~,   
&(3.11) \cr } $$ 
where $~(-1)^A$~ represent the usual Grassmann parity of the
$~A\-$field, namely, $~(-1)^0 = +1$~ when the field
$~A$~ is bosonic, while it is $~(-1)^{+1} = - 1$~ when fermionic.  Next
we use the  property (2.6) for the r.h.s.~of (3.11), such that the field
$~D$~ is always at the end:  
$$\li{ & {\hskip -0.1in} \hbox{(r.h.s.)} \cr 
&  = \int d^{10} x\, \frac1{24} \Big[ 
+ 6 \calS^\star \big(A \star B\star C \big) \star D \cr
& + \starproduct A B C D 
    + \grass{B A} \starproduct B A C D 
    + \grass{A(B+C)} \starproduct B C A D \cr 
& + \grass{B C} \starproduct A C B D 
    + \grass{A (B+C) + BC} \starproduct C B A D 
    + \grass{C(A+B)} \starproduct C A B D {~~~~~ ~~~~~} \cr 
& + \starproduct A B C D + \grass{B C} \starproduct A C B D 
     + \grass{B A} \starproduct B A C D  \cr 
& + \grass{C(A+B)} \starproduct C A B D 
     + \grass{A(B + C) + B C} \starproduct C B A D 
     + \grass{A(B+C)} \starproduct B C A D {~~~~~} \cr 
& + \starproduct A B C D + \grass{C(A+ B)} \starproduct C A B D 
     + \grass{B C} \starproduct A C B D \cr 
& + \grass{A(B+C)} \starproduct B C A D 
     + \grass{A(C + B) + B C} \starproduct C B A D \cr 
& {\hskip 3in}	
     + \grass{A B} \starproduct B A C D \, \Big] ~~,  
&(3.12) \cr } $$ 
Now the second and third lines in (3.12) are combined to yield 
$~\calS^\star \big(A \star B\star C\big) \star D$.  The same is also 
true for the fourth and fifth lines of (3.12), and the remaining 
lines.	After all, we get from (3.12) that the r.h.s.~of (3.10) is 
$$ \li{ \hbox{(r.h.s.)} = & \int d^{10} x \, 
     \frac 1{24} \Big[ + 6 \calS^\star \big( A\star B\star C\big) 
     \star D
     + 6\calS^\star \big( A\star B \star C\big) \star D \cr
& ~~~~~ ~~~~~ ~\quad + 6\calS^\star \big(A \star B\star D\big) 
    + 6\calS^\star\big( A\star B\star C\big) \star D \, \Big] \cr 
= & \int d^{10} x\, \calS^\star \big( A\star B\star C\big) \star D
    = \hbox{(l.h.s.)} ~~. 
&(3.13) \cr } $$
agreeing with the l.h.s.~of (3.10).  

After using the lemma (3.10), we can arrange all the contributions from
(iii) $~\d_Q\Lag_{\a^0}\big|_{\a^2 F^3\l}$~ in such a way that all of
them cancel exactly all the terms in  (3.9).  In other words, we get
$~\d_Q \Lag_{\rm NCSDBI} \big|_{\a^2 \l F^3}  = \d_Q \big( \Lag_{\a^2}
+ \Lag_{\a^0} \big) \big|_{\a^2 \l F^3}= 0$~ up to a total divergence.	

As for the (II) $~\a^2 F\l^3\-$terms, we need to proceed with special
care.  As has been mentioned, we do not fix the $~{\cal O}(\a^2
\l^3)\-$terms in
$~\d_Q A_\m$~ or $~{\cal O}(\a^2 \l^2 F)\-$terms in $~\d_Q \l$~ in this
paper.	Consequently, any variation of the $~{\cal O}(\a^0)$~ kinetic
terms that potentially contribute to the $~\a^2 F \l^3\-$terms {\it via}
these terms in $~\d_Q A_\m$~ or $~\d_Q\l$~ will not concern us here. 
Accordingly, we can also ignore  terms with the factor $~D_\n F^{\m\n}$~
and/or
$~\Dsl\l$~ which can be absorbed into the modification of $~\d_Q A_\m$~
and $~\d_Q\l$, {\it via} the variation of the $~{\cal O}(\a^0)$~ kinetic
terms.	Considering these points, there are only two sources for this
sector:  (i) $~\d_Q \Lag_{\a^2 \l^4}\big|_{\a^2 F\l^3}$~ and (ii) 
$~\d_Q \Lag_{\a^2 F^2 \l^2}\big|_{\a^2 F\l^3}$.  Let us now consider the
sectors (i) and (ii) in turn.  

A simple consideration reveals, the sector (i) can produce only the 
terms of the type $~{\cal S}^\star \big[\, \big(D F\big) \star \big(
\Bar\e \g  D \l\big) \star \big( \Bar\l \g\low{\[3\]}\l\big) \,
\big]$, after appropriate partial integrations, where $~\g$~ in
$~\big(\Bar\e \g D\l\big)$~ can be any odd number of $~\g$'s. 
This is due to the chiralities of these fermionic fields, and the 
fact that under the $~{\cal S}^\star\-$operation, only three-gamma can be
sandwiched by the two $~\l$'s  at the end.  A further consideration shows
that there are only  five categories of such terms defined by  
$$\li{ & (1A) \equiv \a^2 \tr {\cal S}^\star \Big[ \, 
      \big( D_\n F^{\r\s} \big) \star 
      \big( \Bar\e\g^\t D^\n \l \big) \star 
      \big( \Bar\l \star\g_{\r\s\t} \l \big) \, \Big]  ~~, 
&(3.14\rma)  \cr 
& (3A) \equiv \a^2 \tr {\cal S}^\star \Big[ \, 
      \big( D_\n F_{\r\s} \big) \star 
      \big( \Bar\e\g^{\r\t\l} D^\n \l \big) \star 
      \big( \Bar\l \star\g\ud\s{\t\l}  \l \big)\, \Big] ~~, 
&(3.14\rmb)  \cr 
& (3B) \equiv \a^2 \tr {\cal S}^\star \Big[ \, 
      \big( D_\n F\du\m\r \big) \star 
      \big( \Bar\e\g^{\m\t\l} D_\r \l \big) \star 
      \big( \Bar\l \star\g\du{\t\l}\n \l \big)\, \Big]	 ~~, 
&(3.14\rmc)  \cr 
& (3C) \equiv \a^2 \tr {\cal S}^\star \Big[ \, 
      \big( D_\m F_{\n\r} \big) \star 
      \big( \Bar\e\g^{\m\t\l} D^\r \l \big) \star 
      \big( \Bar\l \star\g\du{\t\l}\m \l \big)\, \Big]	~~, 
&(3.14\rmd)  \cr 
& (5A) \equiv \a^2 \tr {\cal S}^\star \Big[ \, 
      \big( D_\m F_{\r\s} \big) \star 
      \big( \Bar\e\g^{\r\s\t\l\o} D^\m \l \big) \star 
      \big( \Bar\l \star\g_{\t\l\o} \l \big) \, \Big]  ~~.  
&(3.14\rme) \cr } $$
We can further see that these five terms are not really independent of 
each other.  There is a relationship up to a term proportional to 
$~D_\n F^{\m\n}$~ and $~\Dsl\l$~ for the reason already
mentioned: 
$$ \li{ & (5A) - 12 (3A) - 42 (1A) \eqdot{\cal O}(\a^2)~~. 
&(3.15) \cr } $$ 
where $~\eqdot$~ stands for an equality up to the factor of 
$~D_\n F^{\m\n} \eqdot{\cal O}(\a^2)$~ or $~\Dsl\l \eqdot {\cal
O}(\a^2)$.  Eq.~(3.15) can be confirmed by the aid of Fierz identity
for arbitrary Majorana-Weyl spinors $~\psi_1,~\cdots,~\psi_4$: 
$$ \li{ {\cal S}^\star \Big[\,	\big( \Bar\psi_1 \! & \star \psi_2 \big)
     \star \big( \Bar\psi_3\star \psi_4\big) \, \Big]  \cr 
     = {\cal S}^\star \Big[ \!\! & - \frac1{16} \rppppo
	 - \frac1{16} \rpppp{\m}   \cr 
& + \frac1{32} \rpppp{\m\n}    
	 + \frac1{96} \rpppp{\m\n\r}  \cr 
& - \frac1{384} \rpppp{\m\n\r\s}   
	 - \frac1{1920} \rpppp{\m\n\r\s\t} \, \Big] {~, ~~~~~ ~~~~~}  
&(3.16) \cr } $$ 
which is a totally symmetrized version of (2.11), now with the 
space-time dependent $~\psi$'s which necessitates the $~{\cal
S}^\star\-$operation.  In order to get the relationship (3.15), we
consider  and simplify the following term 
$$ \li{ & \tr {\cal S}^\star \Big[ \, \big(D_\l F_{\n\r} \big) \star
     \big( \Bar\e\g_\m \l \big) \star 
     \big( \Bar\l \star \g^{\m\n\r} D^\l \l \big)\, \Big]  \cr 
& ~~ = \tr {\cal S}^\star 
    \Big[ + \frac 12 \big( D_\l F_{\n\r} \big) \star 
     \big( \Bar\e\g_\m \l \big) \star 
     D^\l \big( \Bar\l \star \g^{\m\n\r} \l\big) \, \Big] \cr
& ~~ \eqdot \tr {\cal S}^\star \Big[- \frac 12 \big( D_\l F_{\n\r} \big)
     \star \big( \Bar\e\g_\m D^\l \l \big) \star
     \big( \Bar\l \star \g^{\m\n\r} \l\big) \, \Big]  
     = - \frac 12 (1A) ~~.  
&(3.17) \cr } $$
On the other hand, by the use of the Fierzing (3.16) applied to 
the case of $~\Bar\psi_1 \equiv\Bar\e\g_\m,~ \psi_2 \equiv \l, ~
\Bar\psi_3\equiv \Bar\l, ~\psi_4 \equiv \g_{\m\n\r} D_\l\l$,  
we get
$$\li{ & \tr {\cal S}^\star \Big[ \, \big(D_\l F_{\n\r} \big) \star
     \big( \Bar\e\g_\m \l \big) \star 
     \big( \Bar\l \star \g^{\m\n\r} D^\l \l \big)\, \Big]  \cr 
& ~~ = \tr {\cal S}^\star \Big[ 
     + \frac 1{96} \big( D^\l F\ud\n\r \big) \star
      \big( \Bar\e \g^\m \g^{\s\t\o} \g_{\m\n\r} D^\l \l \big)
    \star \big( \Bar\l \star \g_{\s\t\o} \l\big) \, \Big] \cr 
& ~~ = - \frac 1{48} (5A) + \frac 14 (3A) + \frac 3 8 (1A) ~~.
&(3.18) \cr } $$
Equating (3.17) with (3.18), we get (3.15). 

By the frequent use of these relations, we get the contribution
from the sector (i) as 
$$ \li{ & \d_Q \Lag_{\a^2 \l^2}\big|_{\a^2 F\l^3} 
   = \a^2 \Big[ + \frac 1{36} (5A) 
      + \frac 1 6 (3A) - \frac 1 6 (1A) \, \Big] ~~,  
&(3.19) \cr } $$ 
while for the contribution from the sector (ii) as 
$$ \li{& \d_Q \Lag_{\a^2 F^2 \l^2} \big|_{\a^2 F\l^3}  
 = - \frac 1{24}\a^2 (5A) + \frac 3 4 \a^2 (1A) ~~. 
&(3.20) \cr } $$  
To reach this result, we also used the fact that  
$$ \li{ & \a^2 \tr {\cal S}^\star \Big[ F^{\r\n} \star 
     \big(\Bar\e \g_\m\l\big) \star \big( D_\r \Bar\l\big) 
     \star \g^\m D_\n \l \, \Big] \eqdot 0 ~~,	
&(3.21) \cr } $$ 
as is easily confirmed by a Fierzing, up to the terms vanishing upon 
$~D_\n F^{\m\n} \eqdot {\cal O}(\a^2)$~ or $~\Dsl \l \eqdot {\cal
O}(\a^2)$, as well as up to  a total divergence.  Now, by the use of
(3.15), we see that the two contributions from (3.19) and (3.20) exactly
cancel each other.  This concludes the summary of the invariance 
check of our total action $~I_{\rm NCSDBI}$~ up to $~{\cal
O}(\a^2\varphi^5)$~ and $~{\cal O}(\a^3)\-$terms.  













\bigskip\bigskip\bigskip


\leftline{\bf 4.~~Ambiguities Associated with Field Redefinitions}    

It is well-known in commutative supersymmetric gauge theory in 10D that
there is some ambiguity about the coefficients of certain lagrangian
terms caused by the freedom of field redefinitions \brs\cederwall. 
This is also true with our noncommutative action $~I_{\rm NCSDBI}$.  To
see this, we consider the field redefinitions of the type 
$$ \li{ & A_\m \equiv A_\m' - \frac 14 c_2 \a^2 
    {\cal S}^\star \Big[\, F_{\n\r} \star \big( \Bar\l \star
     \g\du\m{\n\r} \l\big)\, \Big]  ~~, 
&(4.1\rma) \cr 
& \l \equiv \l' + \frac 1{16} \a^2 {\cal S}^\star 
    \Big[\, c_1 \big( F \star F\big)\du\m\m \star \l 
     + c_3 F_{\m\n} \star F_{\r\s} \star \g^{\m\n\r\s} \l \, \Big] ~~,	
&(4.1\rmb) \cr } $$
with arbitrary real constants $~c_1,~c_2$~ and $~c_3$.	If we
substitute  (4.1) into our lagrangian (2.2),  there arise some new
terms with $~c_1,~c_2,~c_3$~ up to $~{\cal O}(\a^4)\-$terms, which we
call 
$~\D\Lag_{\rm NCSDBI}$.  The explicit form of $~\Lag_{\rm NCSDBI}$~ is 
easily computed to be 
$$\li{ \D\Lag_{\rm NCSDBI} = \a^2 \tr {\cal S}^\star 
     \Big[ \! & - \frac 14 c_2 F_{\m\n} \star\big( D_\l 
   F\ud \l \r \big) \star \big( \Bar\l \star \g^{\m\n\r} \l \big) \cr 
& - \frac 14 c_1 \big( F\star F\big) \du\m\m \star 
     \big( \Bar\l \Dsl \l\big)	
    - \frac 14 c_3  F_{\m\n} \star F_{\r\s} \star
    \big( \Bar\e \g^{\m\n\r\s} \Dsl\l\big) \, \Big] ~~. 
&(4.2) \cr } $$
For the reason already mentioned, the $~{\cal S}^\star\-$operation on 
three fields in (4.1) is converted into the total symmetrization on 
all the four fields at the lagrangian level in (4.2).	

Meanwhile the supersymmetry transformation rule (3.7) is also modified, 
as 
$$\li{ \d_Q \!\!\!{}' ~\l ' = & + \frac 18 \g^{\m\n} \e
F_{\m\n}\!\!\!{}' \cr 
& + \a^2 {\cal S}^\star \Big[ - \frac 1{128} (c_1 + 4c_3 - 2) 
     \big( F\, ' \star F\, ' \big)\du\n\n \star F_{\r\s}
     \!\!\!{}'~  \g^{\r\s} \e 
      + \frac 1{16} (c_3-1) \big( F \, ' \star F\, ' 
     \star F\, ' \big)_{\m\n} 
    \g^{\m\n} \e \cr  
& ~~~~~ ~~~~~ ~~ - \frac1{384} (3c_3+ 1) F_{\m\n}\!\!\!{}'\,  
	 \star F_{\r\s} \!\!\!{}' \, \star  
	 F_{\t\l}\!\!\!{}' {} ~ \g^{\m\n\r\s\t\l} \e \, \Big] 
    +{\cal O}(\a^2 \l^3) ~~, 
&(4.3\rma) \cr 
\d_Q \!\!\!{}' ~ A_\m ' = & - \big( \Bar\e\g_\m \l ' \big) \cr 
& + \a^2 {\cal S}^\star \Big[\, - \frac1{16} (c_1+2c_2 - 6) 
     \big( F \,' \star F\, ' \big)\du\n\n \star
      \big(\Bar\e\g_\m \l ' \big) 
     + \frac 14 (c_2-4) \big(F\, ' \star F\, ' {}\big)\du\m\n 
     \big( \Bar\e \g_\n \l ' \big) \cr 
& ~~~~~ ~~~~~ ~~  - \frac 18 (-c_2 +2c_3 + 2) 
     F_{\m\n}\!\!\!{}' \, \star F_{\r\s} \!\!\!{}'\, \star \big( \Bar\e
     \g^{\n\r\s} \l ' \big) \cr  
& ~~~~~ ~~~~~ ~~  - \frac 1{16} (c_2+c_3-1) F_{\r\s}\!\!\!{}' \, \star
      F_{\l\t}\!\!\!{}' \, \star
     \big( \Bar\e \g\du\m{\r\s\l\t}\l ' \big) \, \Big] 
     + {\cal O}(\a^2 F\l^2)  ~~,    
&(4.3\rmb) \cr } $$
up to $~{\cal O}(\a^2 \l^3)$~ or $~{\cal O}(\a^2 F\l^2)$~ as in 
(3.7).	Since not only the fields $~A_\m$~ and $~\l$~ but also the 
supersymmetry transformation rule itself is changed, we need to put 
the prime also on $~\d_Q$~ itself.  To be more specific, the
modification in $~\d_Q A_\m$~ is understood as 
$$ \li{ & \D \big( \d_Q A_\m\big) \equiv 
     \d_Q \!\!\!{}' ~ A_\m ' - \d_Q A_\m \equiv 
     \big( \d_Q \!\!\!{}' ~ - \d_Q \big) A_\m ' 
     + \d_Q \big( A_\m ' - A_\m\big)  ~~,  
&(4.4) \cr } $$
where the first term is the modification of the transformation
rule, interpreted as  
$$\li{ & \big( \d_Q \!\!\!{}' ~ - \d_Q \big) A_\m ' 
     = \big[ - \big( \Bar\e \g_\m \l ' \big) \, \big] -   
       \big[ - \big( \Bar\e \g_\m \l \big) \, \big]  ~~.
&(4.5) \cr } $$ 
These results (4.2) through (4.5) are in agreement with 
\brs\cederwall, despite the noncommutativity inherent in our system.   




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\leftline{\bf 5.~~Concluding Remarks}

In this paper, we have presented the noncommutative version
of supersymmetric non-Abelian gauge theory in 10D given by the action
$~I_{\rm NCSYM}$~ (2.1).  We have seen that  despite the noncommutativity
yielding a potentially non-vanishing term (2.12) in the invariance
check under (2.4), it actually vanishes, thanks to the $~\g\-$matrix
algebra (2.13), as well as the basic property (2.6) of noncommutativity.  

We have next presented an action $~I_{\rm NCSDBI}$~ with higher-derivative
terms (3.2) up to quintic terms, as a
noncommutative generalization of a supersymmetric DBI action in 10D,
which is also regarded as  a supersymmetrization of noncommutative a DBI
action.  With the frequent aid of basic equations of noncommutative
geometry, in addition to the usage of the $~{\cal S}^\star\-$operator for
the total symmetrization of $~\star$~ products, we have found that all the
$~{\cal O}(\a^2)$~ terms with $~\star$~ products of four fields cancel
each other, in the supersymmetric variation of our action $~I_{\rm
NCSDBI}$~ under supersymmetry (3.7).  

We have also clarified possible ambiguities of coefficients in 
certain terms in the action, in terms of field redefinitions of 
the $~A_\m$~ and $~\l\-$fields.  This situation is completely the same 
as the commutative case \brs\cederwall, even with the exact matching of
coefficients.	 

Our result leads to the next natural trial of all the 
possible dimensional reductions into dimensions lower than 10D, 
acquiring all the known and possibly unknown noncommutative gauge
theories in these lower dimensions, including those in 4D.  To put 
it differently, we can take the advantage of high dimensions 
as in 10D, {\it via} dimensional reductions that 
generate more possibilities compared with the direct formulations in
4D.  

Once the most fundamental case of $~U(N)$~ Yang-Mills group has been 
established for noncommutative supersymmetric DBI action, it is much 
easier to apply the recent technique of anti-automorphism of
$~\star\-$matrix algebra \bsjv, in order to get other gauge algebras,
{\it e.g.,} the $~o_\star(N)$~ or $~usp_\star(N)$~ algebra, with more  
phenomenological applications.	

Our result also indicates that there is no fundamental obstruction
for  constructing noncommutative supersymmetric DBI action presumably 
in any space-time dimensions.  As we have seen, the system somehow
arranges itself, and automatically avoids any new problem caused by the
noncommutativity, as long as we use the total symmetrization operator
$~{\cal S}^\star$, even in such  subtle computations with
higher-derivatives at
$~{\cal O}(\a^2)$.  This result also strongly suggests that the
possibility of noncommutative supergravity that has never been
established in the past, even though there seems some
potential problem with defining spinors in space-time with complex
metric.  Our recent formulation of noncommutative gravity 
based on teleparallelism 
%%%
\ref\nrtel{H.~Nishino and S.~Rajpoot, {\it `Teleparallel Complex Gravity 
as Foundation for Noncommutative Gravity'}, CSULB-PA-01, hep-th/0107216.}
%%% 
might well be a good starting point for such a purpose.  

Once our noncommutative supersymmetric gauge theory containing the
next-leading terms in noncommutative DBI action is established in 10D, 
all other lower-dimensional descendant theories are generated by
dimensional reductions.  In this sense, our theory plays a role of the
master theory for noncommutative supersymmetric DBI theory.  
 



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\end{document} 





