hep-th0002029.tex
%%%%%%%%%%%%% Macros (defrr.tex, revised 06/22/97) %%%%%%%%%%%%%%%%%%%
\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 equations
\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
\topmargin=0in				% top margin (less 1") (LaTeX)
\headheight=0in				% height of heading (LaTeX)
\headsep=0in			% separation of heading from body (LaTeX)
\textheight=9in				% height of body (LaTeX)
%\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)
\belowdisplayskip=1em plus.3em minus.5em	% " below
\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)
\def\oldheadpic{				% old UM heading
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(349,16)
	\put(325,16){\line(1,0){4}}\put(330,16){\line(1,0){4}}
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	\put(329,16){\oval(8,32)[bl]}\put(330,16){\oval(8,32)[br]}
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	\end{picture}
	\par
	\thicklines
	\vskip.2in}
\def\border{						% border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
        \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}
\def\bordero{						% alternate border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
	\put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhdhebaaahjhhdahbahgdedgehgfdiehhgdigicba}
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	\end{picture}
	\par\vskip-8mm}
\def\headpic{						% UM heading
	\indent
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
	\put(170,16){\line(1,0){4}}
	\put(180,16){\line(1,0){4}}
	\put(175,0){\line(1,0){4}}
	\put(180,0){\line(1,0){4}}
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	\put(169,0){\line(0,1){16}}
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	\put(179,0){\oval(8,32)[tl]}
	\put(185,0){\oval(8,32)[tr]}
	\end{picture}
	\par\vskip-6.5mm
	\thicklines}

\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{_{_{\hskip 0.01in}\bullet}}\over{#1}}
%\def\Dot#1{\buildrel{_{_{\hskip 0.01in}{_\bullet}}}\over{#1}}
\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}

%%%%%%%%%%%%%%%%%%%%%%%% Additional Fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
%%%%%%%%%%%%%%%%%%%%%%%% End of Additional Fonts %%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%% harvmactest.tex %%%%%%%%%%%%%%%%%%%%%%%%
%
%%% 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
%
%%%%%%%%%%%%%%%%%%%%%%%%% End of harvmac.tex %%%%%%%%%%%%%%%%%%%%%%%%
\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} 

%        10        20        30        40        50        60        70   
\count100=1
%\input defrr.tex 

\font\smallcmr=cmr6 scaled \magstep2 
\font\largecmrseventeen=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\Christoffel#1#2#3{\left\{ {\buildrel #1 
\over{\scst #2 #3}} \right\} } 

\def\binomial#1#2{\left(\,{\buildrel 
{\raise4pt\hbox{$\displaystyle{#1}$}}\over
{\raise-6pt\hbox{$\displaystyle{#2}$}}}\,\right)}

%\def\binomial#1#2{{{#1}\choose{#2}}}
%\def\binomial#1#2{\left({\scst\matrix{#1\cr#2\cr}}\right)}
%\def\binomial#1#2{\left(\,{\buildrel 
%{\displaystyle{#1}}\over
%\def\binomial#1#2{\left({\buildrel
%{\scriptstyle{#1}}\over{_{\scriptstyle{#2}}}}\right)}
%{\raise-4pt\hbox{$\displaystyle{#2}$}}}\,\right)}

\font\smallcmr=cmr6 scaled \magstep2 

\def\plpl{{+\!\!\!\!\!{\hskip 0.009in}{\raise -1.0pt\hbox{$_+$}} 
{\hskip 0.0008in}}} 
\def\mimi{{-\!\!\!\!\!{\hskip 0.009in}{\raise -1.0pt\hbox{$_-$}} 
{\hskip 0.0008in}}} 

\def\ul{\underline} 
\def\un{\underline} 
\def\-{{\hskip 1.5pt}\hbox{-}} 
\def\fracmm#1#2{{{#1}\over{#2}}} 
\def\low#1{{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}} 

\def\ip{{=\!\!\! \mid}} 
\def\Lslash{${\rm L}{\!\!\!\! /}\, $}

\def\iiastar{IIA$^*$}  \def\iibstar{IIB$^*$} 
\def\typeiiastar{type \iiastar}  \def\typeiibstar{type \iibstar}

\def\typeiia{type IIA}  \def\typeiib{type IIB}  
\def\geleven{\g\low{11}} 

\def\framing#1{\doit{#1} 
{\framingfonts{#1} 
\border\headpic 
}} 

\framing{0} 

{\hbox to\hsize{\hfill hep-th/0002029}} \par 
{\hbox to\hsize{~~~~~ ~~~~~ \hfill UMDEPP 00--037}}\par\vskip -0.1in 
\hfill 
\\

\begin{center} 

\vskip -0.5in

{\large\bf Non$\,$-$\,$Linear~ Realization~ of~ 
$\aleph_0\,$-$\,$Extended~ Supersymmetry}\footnote{This 
work is supported in part by NSF grant \# PHY-93-41926.} \\[.1in] 

% {\bf\largecmrseventeen in Arbitrary Dimensions}

\baselineskip 9pt 

\vskip 0.2in 

Hitoshi ~N{\smallcmr ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
% ~and ~S. James ~G{\smallcmr ATES}, Jr.,\footnote{E-Mail: 
% gatess@wam.umd.edu} 
\\[.2in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland at College Park} \\ [.015in] 
{\it College Park, MD 20742-4111, USA} \\[.020in]   


\vskip 0.65in 

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

\baselineskip 15.5pt 

~~~As generalizations of the original Volkov-Akulov action in 
four-dimensions, actions are found for all space-time dimensions $\,D$
invariant under $\,N$ non-linear realized global supersymmetries.   We
also give other such actions invariant under the global non-linear
supersymmetry.  As an interesting consequence, we find a non-linear
supersymmetric Born-Infeld action for a non-Abelian gauge group for
arbitrary $\,D\,$ and $\,N$, which coincides with the linearly 
supersymmetric Born-Infeld action in $~D=10$~ at the lowest order.  For
the gauge group $~U({\cal N})$~ for M(atrix)-theory, this model has 
$~{\cal N}^2\-$extended non-linear supersymmetries, so that its large
$~{\cal N}$~ limit corresponds to the infinitely many
($\large{\aleph_0}$) supersymmetries.  We also perform a duality
transformation from $~F_{\m\n}$~ into its Hodge dual
$~N_{\m_1\cdots\m_{D-2}}$.  We next point out that any Chern-Simons
action for any (super)groups has the non-linear supersymmetry as a
hidden symmetry.  Subsequently, we present a superspace formulation for
the component results.  We further find that as long as superspace
supergravity is consistent, this generalized Volkov-Akulov action can
further accommodate such curved superspace backgrounds with local
supersymmetry, as a super $~p\-$brane action with fermionic
kappa-symmetry.  We further elaborate these results to what we call
`simplified' $~(\hbox{Supersymmetry})^2\-$models, with both linear and
non-linear representations of supersymmetries in superspace at the same
time.  Our result gives a proof that there is no restriction on $~D$~ or
$~N$~ for  global non-linear supersymmetry.  We also see that the
non-linear realization of supersymmetry in `curved' space-time can be
interpreted as `non-perturbative' effect starting with the `flat'
space-time.     



\begin{quotation} 
~~~
\endtitle 

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\centerline{\bf 1.~~Introduction}

The study of non-linear supersymmetry realizations has a
very long history dating to the first effort by Volkov and Akulov 
%%%
\ref\va{D.V.~Volkov and V.P.~Akulov, JETP Lett.~{\bf 16} (1972) 438; 
\pl{46}{73}{109}.}, 
%%%
who gave the initial example of supersymmetry in four-dimensions
($D=4$).  This initial work was further elaborated by its 
linearization 
%%%  
\ref\rocek{M.~Ro\v cek, \prl{41}{78}{451}.}, 
%%%
and generalized to extended local supersymmetries in $~D=4$~  
%%%
\ref\ferrara{S.~Ferrara, L.~Maiani, P.C.~West \zp{19}{83}{267}; 
S.~Ferrara, {\it `Problems in Unification and Supergravity'}, in 
Proceedings of `La Jolla Unification Workshop' (Jan.~1983).}.        
%%% 
The importance of non-linear realizations of supersymmetry has 
been revealed in many different contexts.  For example, during the first
revolution of string theory 
%%%
\ref\gsw{M.~Green, J.H.~Schwarz and E.~Witten, {\it `Superstring
Theory'}, Vols.~I and II,  Cambridge University Press (1987).}, 
%%%
there was strong motivation for supersymmetrization  
%%%
\ref\cf{S.~Cecotti and S.~Ferrara, \pl{187}{87}{335}.}
%%%
of the Born-Infeld action
%%%
\ref\bi{M.~Born and L.~Infeld, Proc.~Roy.~Soc.~(London) 
{\bf A144} (1934) 425.}%
%%%
\ref\birev{{\it For recent reviews, see, e.g.,} 
A.A.~Tseytlin, {\it `Born-Infeld Action, Supersymmetry
and String Theory'}, \hepth{9908105}; {\it and references therein}.}, 
%%%
and initial results were summarized in   
%%%
\ref\brs{E.~Bergshoeff, M.~Rakowski and E.~Sezgin, \pl{185}{87}{371}.}
%%%
where the lower-order terms were explicitly and systematically 
presented.  

The recent discovery of gauge-fixed supersymmetric Born-Infeld action  
in $~D\le 10$~ to all orders 
%%%
\ref\aps{M.~Aganagic, C.~Popescu, and J.H.~Schwarz,
\pl{393}{97}{311}; Nucl.~Phys.~\hfill
\newline{\bf B495} (1997) 99.} 
%%% 
has drawn much attention in the context of D-brane dynamics 
%%%
\ref\pol{J.~Polchinsky, {\it `TASI Lectures on D-Branes'}, 
\hepth{9611050}.}% 
%%%
\ref\dpbranes{I.A.~Bandos, D.P.~Sorokin and M.~Tonin, 
\np{497}{97}{275}; R.~Kallosh, {\it `Volkov-Akulov Theory and D-Branes'}, 
\hepth{9705118}; V.A.~Akulov, I.~Bandos, W.~Kummer and V.~Zima,
\np{527}{98}{61}.}% 
%%% 
\ref\nilsson{M.~Cederwall, A.~von Gussich, B.E.W.~Nilsson 
and A.~Westerberg, \np{490}{97}{163};
M.~Cederwall, A.~von Gussich, B.E.W.~Nilsson, P.~Sundell and 
A.~Westerberg, \np{490}{97}{179};
T.~Adawi, M.~Cederwall, U.~Gran, M.~Holm and B.E.W.~Nilsson,
\ijmp{13}{98}{4691}.}% 
%%%
\ref\bt{E.~Bergshoeff and P.~Townsend, \np{490}{97}{145}.}%   
%%%
\ref\sorokin{{\it For recent reviews, see, e.g.,} D.~Sorokin, 
{\it `Superbranes and Superstrings'}, \hepth{9906142};
{\it and references therein}.}    
%%% 
related to superstrings \gsw\ and M-theory
%%%
\ref\hulltownsendwitten{C.~Hull and P.K.~Townsend, \np{348}{95}{109};
E.~Witten, \np{443}{95}{85}; P.K.~Townsend, {\it `Four
Lectures on M-Theory'}, in {\it `Proceedings of ICTP Summer School on
High Energy Physics and Cosmology'}, Trieste (June 1996), 
hep-th/9612121; {\it `M-theory from its Superalgebra'},
hep-th/9712004.}. 
%%%
The non-Abelian generalization of these formulations 
was also considered in   
%%%  
\ref\tseytlin{A.A.~Tseytlin, \np{501}{97}{41}.}.
%%%
In such a formulation, there is already non-linear realization of
supersymmetry, interpreted as the signal of `broken' supersymmetry. 
In $~D=4$~ formulations, non-linear supersymmetry has been
interpreted in terms of partially broken
$~N=2$~ supersymmetry 
%%% 
\ref\bw{J.~Bagger and J.~Wess, \pl{138}{84}{105}; J.~Hughes and
J.~Polchinski, \np{278}{86}{147}; J.~Bagger and A.~Galperin,
\pr{55}{97}{1091}.}, 
%%% 
with Nambu-Goldstone (NG) multiplets.  The relationship between 
the non-linear and linear supersymmetry in superspace has been 
also studied in 
%%% 
\ref\ik{E.A.~Ivanov, A.A.~Kapustnikov, \np{333}{90}{439}; 
\pl{143}{84}{379}; J.~Phys.~{\bf G8} (1982) 167;
J.~Phys.~{\bf A11} (1978) 2375; 
T.~Uematsu and C.~Zachos, \np{201}{82}{250}.}.    
%%% 
These global supersymmetric systems have been further applied to 
the question of spontaneous breaking of local supersymmetry
(supergravity)  
%%%
\ref\sg{U.~Lindstrom and M.~Ro\v cek, \pr{19}{79}{2300};
\hbox{A.A.~Kapustnikov, Theor.~Math.} 
Fiz.~{\bf 47} (1981) 198;
S.~Samuel and J.~Wess, \np{221}{83}{153}; \ibid{233}{84}{488}.}.
%%% 

Despite of these developments over the decades since 1970's, however, 
including the recent D-brane models, there still has been no universal  
guiding principle that can systematically yield new invariant actions 
for non-linear realization of supersymmetry in arbitrary space-time
dimensions.  This seems true also for the interpretation of non-linear
realization as `broken' supersymmetry \bw.  On top of this, even though
the  original spirit of Volkov-Akulov formulation of non-linear
supersymmetry \va\ was `off-shell' without auxiliary fields present and
no physical superpartner fields in $~D=4$, this aspect has not been 
maintained in many formulations recently presented.  Perhaps one of
the reasons is that D-brane theory \pol\bt\aps\ necessitates the
superpartner vector field for the gaugino.   

In the case of linear realizations of supersymmetry, it is well-known 
that there are restrictions on the space-time dimensions $~D$~ and the 
number $\,N$~ of extended supersymmetries, such as $~N\le 8$~ in $~D=4$.
However, very little has been stated about such restrictions on $~D$~ or
$\,N$~ for extended supersymmetries in the case of non-linear
realizations.    

In this paper, we will take a step toward a universal understanding 
of non-linear realization of supersymmetry in arbitrary dimensions.  
We will present simple actions that are invariant under non-linearly
realized supersymmetry.   We also give some matter lagrangians that are
invariant under such global but non-linear supersymmetry.  Such matter
lagrangians coupled to NG$\,$-fermion within $~D=4$~ have been studied
in many contexts of phenomenological models
%%% 
\ref\cl{T.E.~Clark and S.T.~Love, \pr{39}{89}{2391};
\ibid{D54}{96}{5723}; T.E.~Clark, T.~Lee, S.T.~Love and G.-H.~Wu,
\pr{57}{98}{5912}.},  
%%%
or non-linear realizations for extended supersymmetries \ferrara. 
We stress that our result is valid in any space-time dimensions 
$~D$~ as well as any number $\,N$~ of extended supersymmetries.   We 
find results  applicable to arbitrary non-linearly realized
$~N\-$extended supersymmetries, as long as the space-time allows 
the definition of spinor fields.  As a by-product, we also give a
supersymmetric Born-Infeld action for an arbitrary non-Abelian gauge
group $~^\forall D$~ and $~^\forall N$.  We also establish a superspace
formulation for these component results, whose geometrical
significance is clearer than in a component formulation.    
We further give the super $~p\-$brane 
%%% 
\ref\pbrane{A.~Achucarro, J.M.~Evans, P.K.~Townsend and D.L.~Wiltshire,
\pl{198}{87}{441}.}  
%%% 
interpretation of some of these actions, on supergravity backgrounds.  
This will be achieved by introducing the maximal-rank superpotential 
$~C_{A_1\cdots A_D}$~ in an arbitrary even dimensions $~D=2k$, which was 
first formulated in the context of super eight-brane 
%%% 
\ref\eightbrane{H.~Nishino, \pl{457}{99}{51}.}.    
%%% 
Motivated by this, we also 
present a possible (Supersymmetry)$^2\-$formulation 
%%% 
\ref\supersquare{S.J.~Gates, Jr.~and H.~Nishino, 
\cqg{3}{86}{391}.},    
%%% 
in which the non-linear and linear realizations of supersymmetries coexist
simultaneously within the same superspace.



     
\bigskip\bigskip\bigskip

%\vfill\eject

\vbox{\centerline{\bf 2.~~Non-Linear Realization of Supersymmetry}

\bigskip

\leftline{\bf 2.1~~Generalized Volkov-Akulov Action}

We first setup our supersymmetry transformation which is the starting 
point in this paper.  As mentioned above, there is no restriction on 
the space-time dimensions $~D$~ in our formulation, except that we have 
the usual signature $~(D-1,1)$~ with the flat metric $~(\eta_{m n})
=\hbox{diag.}\,(-,-,\cdots,-,+)$, with only one time coordinate.  Our
system initially has only one  NG$\,$-fermion, which may be Majorana,
Weyl, Majorana-Weyl,  or Dirac spinor.  As will be seen, there is no
restriction on such spinorial property for our formulation.   
}

The rule for a supersymmetry transformation of the NG$\,$-fermion is
essentially  the same as those in \va\ or \aps, but we have to keep in
mind that  the space-time dimension $~D$~ is arbitrary:\footnotew{We do
not introduce any dimensionful constants in this paper.  We just put
dimensionful constants like the Planck constant to unity.}   
$$ \li{ & \d_Q \l =  \e + i \big(\Bar \e
          \g^\m \l \big) \partial_\m \l 
          \equiv  \e + \xi^\m \partial_\m \l ~~, 
&(2.1) \cr } $$ 
where the parameter $~\e^{\un\a}$~ of the supersymmetry generator 
$~Q_{\un\a}$~ does not depend on space-time coordinates $~x^\m$, 
since we are dealing only with global supersymmetry.  The quantity
$~\xi^\m$~ defined by\footnotew{Here the gamma-matrix $~\g^\m$ is 
the same as $~\g^m\d\du m\m$.  The usage of the index $~{\scst \m}$~
instead of $~{\scst m}$~ becomes clearer shortly.}\footnotew{Here 
the underlines for the indices $~{\scst\un\a,~\un\b,~\cdots}$~ 
are for any additional indices  for spinors, such as `dotted-ness' 
for chiralities, or $~\hbox{\it USp}(N)$-indices needed in some 
space-time dimensions $~D$~ 
%%%
\ref\kt{T.~Kugo and P.K.~Townsend, \np{211}{83}{157}.}.   
%%%
Accordingly, the $~\g$-matrix may also
contain any relevant metric for such indices.}    
$$ \li{&\xi^\m \equiv  i \big(\Bar\e \g^\m\l \big) 
          = - i \e^{\un\a} \big( \g^\m \big)_{\un\a\un\b} \l^{\un\b} ~~,
&(2.2)  \cr} $$
behaves like the parameter for general coordinate transformation.  
It is easy to see the closure of supersymmetry on $~\l$, as a routine 
check for consistency of the system: 
$$ \li{ & \[ \, \d_Q(\e_1) , \d_Q(\e_2)  \, \] \l = 
          2i \big(\Bar\e_2\g^\m \e_1 \big) \partial_\m\l  
	          = \d_P(2i\Bar\e_2\g^m\e_1) \l 
          \equiv \d_P \big(\zeta_{12}^m \big)\l~~, 
&(2.3) \cr } $$
where $~\d_P$~ is the usual translation, and $~\z_{12}^m \equiv 
2i\big(\Bar\e_2 \g^m \e_1 \big) $, reflecting the underlying 
supersymmetry algebra\footnotew{The index for the $\,N$-extended
supersymmetry can be implicit in the underlined indices.  We will 
return to this point later.}
$$ \li{ & \{ Q_{\un\a} , Q_{\un\b} \} 
          = 2 \big( \g^m \big)_{\un\a\un\b} P_m ~~, 
&(2.4) \cr } $$ 

A fundamental observation is that the quantity defined by
$$ \li{& E\du\m m \equiv \d\du\m m 
          + i \big( \Bar\l\g^m \partial_\m \l \big) 
          \equiv \d\du\m m + \L \du\m m~~, 
&(2.5) \cr } $$
transforms under ~$\d_Q$~ as  
$$\li{\d_Q E\du\m m & = i \big(\Bar\e\g^\n\l \big)  \partial_\n
          E\du\m m + i \big(\Bar\e\g^\n \partial_\m\l \big) E\du\n m \cr 
& = \xi^\n \partial_\n E\du\m m 
          + \big(\partial_\m\xi^\n \big) E\du\n m 
          = \d_{\rm G} (\xi^\r) E\du\m m ~~. 
&(2.6) \cr} $$
This is a significant result, because it means that the field  $~E\du\m
m$~ transforms under supersymmetry, as if it were a vielbein 
transforming under the general coordinate transformation  $~\d_{\rm G}$~
with the parameter $~\xi^\m$~ not only in $~D=4$~ \va\ but also in any
space-time dimensions!  Now the advantage of  using the two distinct sets
of indices $	~{\scst m,~n,~\cdots}$~ and ~${\scst \m, ~\n,~\cdots}$~ on 
$~E\du\m m$~ is also clear:  They respectively behave like the curved
frame and local Lorentz frame indices.  

Once this is understood, it is straightforward to confirm the 
invariance of the action 
$$ \li{ & I_E \equiv \int d^D  x \, \det \big( E\du\m m \big)
          \equiv \int d^D  x \, E 
          \equiv \int d^D x \, \Lag_E ~~, 
          ~~~~ E \equiv \det \big( E\du\m m \big)~~, 
&(2.7) \cr } $$
under the variation in (2.1), as a total 
divergence:\footnotew{The existence of the inverse vielbein $~E\du m\m$~ 
is taken for granted otherwise the action is ill-defined.}  
$$ \li{ \d_Q \Lag_E & \, = E E\du m \m \d_Q E\du\m m 
          = E E\du m \m \left[\, \xi^\n \partial_\n E\du\m m 
          + \big( \partial_\m\xi^\n \big) E\du \n m \, \right]  
          = \partial_\m\big( E \xi^\m \big) ~~. 
&(2.8) \cr } $$
We can reconfirm this `formal' proof by the direct   
variation with respect to the NG-fermion
whose details are to be skipped here. 

As the $\l\-$field equation or the original lagrangian
(2.7) reveals, the action $~I_E$~ contains the standard kinetic term for
the NG$\,$-fermion, upon the expansion $~\det\,(E\du\m m) \approx 1 + i
\big(\Bar\l\g^\m\partial_\m\l \big) + {\cal O}(\l^4)$. 	Therefore, 
the NG$\,$-fermion is regarded as a `physical' field, also
with self-interaction terms that make our system highly non-trivial.   


\bigskip\bigskip

\vbox{
\leftline{\bf 2.2~~More Generalizations}

As the reader may have already noticed, once the transformation 
rule of the vielbein is as appears in (2.5), we can write other 
lagrangian densities constructed from the vielbein.  For example, 
we can write lagrangians in terms of the scalar curvature made
 up of the `metric'\footnotew{We use $~G_{\m\n}$~ instead of
$~g\low{\m\n}$~ as a reminder that this is a  `composite' metric 
but not a fundamental one.  Relevantly, $~G_{\m\n}$~ should {\it 
not} be confused  with the Einstein tensor $~R_{\m\n} - (1/2)
G_{\m\n} R$.} 
$~G_{\m\n} \equiv\eta_{m n} E\du\m m E\du\n n$
with its inverse metric $~G^{\m\n}: ~G_{\m\n} G^{\n\r} =
\d\du\m\r$, such as the `Hilbert action':
}
$$ \li{& I_R \equiv \int d^D x \, {\sqrt{-G}} R (G) ~~, \cr 
& R (G) \equiv G^{\m\n} \, R_{\m \n} (G) ~~,
	~~~~ R_{\m\n} (G) \equiv R\du{\m\r\n}\r (G) ~~,  \cr 
& R\du{\m\n\r}{\s} (G) \equiv 
          \partial_\m \left\{ {\buildrel \s \over{\scst\n\r}} \right\}
          - \partial_\n \left\{ {\buildrel \s \over{\scst\m\r}} \right\}
          - \left\{ {\buildrel \t \over{\scst\m\r}}\right\} 
          \left\{ {\buildrel \s \over{\scst\t\n}} \right\}  
          + \left\{ {\buildrel \t \over{\scst\n\r}}\right\} 
          \left\{ {\buildrel \s \over{\scst\t\m}} \right\} ~~,  \cr 
& \left\{ {\buildrel \r\over{\scst\m\n}} \right\} 
          \equiv \frac12 G^{\r\s} \big( \partial_\m G_{\n\s}  
          + \partial_\n G_{\m\s} - \partial_\s G_{\m\n} \big)~~,    
&(2.9) \cr } $$ 
because $~G_{\m\n}$~ transforms as
$$ \li{ & \d_Q G_{\m\n} = \xi^\r \partial_\r G_{\m\n} 
          + \big(\partial_\m\xi^\r \big) G_{\r\n} 
          + \big( \partial_\n\xi^\r \big) G_{\m\r} 
          = \d_{\rm G} (\xi^\r) G_{\m\n} ~~, 
&(2.10) \cr } $$
as if it were a `metric' tensor.  
Accordingly, the curvature `tensors' above transform as
$$ \li{& \d_Q R\du{\m\n\r}\s (G) = \d_{\rm G} (\xi^\t) 
	R\du{\m\n\r}\s (G) ~~, \cr 
& \d_Q R_{\m\n} (G) = \d_{\rm G} (\xi^\t) R_{\m\n} (G) ~~, ~~~~
	\d_Q R (G) = \d_{\rm G} (\xi^\t) R (G) ~~.  
&(2.11) \cr } $$

In the case we need more `canonical' kinetic term for the NG$\,$-fermion
$~\l$, we can simply add the action $~I_E$~ (the analog of a
`cosmological term') to $~I_R$, so that a kinetic term for
the $~\l\-$field is present, while higher-order terms are regarded as
interaction terms.  In fact, if the total action is $~I_E +
I_R$, the lowest-order term in $~I_R$~ is already at the fourth order 
in $~\l$~ as an interaction term.  

Some readers may wonder if the Riemann or Ricci tensor built out of the
metric $~G_{\m\n}\equiv \eta_{mn} E\du\m m E\du\n n$~ vanishes by the
use of the ~$\l\-$field equation.  We show this is not the case, {\it
e.g.,} for the Ricci tensor
$$ \li{& R_{\m\n} (G) = - 2i \big[\, \Bar\l,{}_\r \g^\r \l,{}_{\m\n}
	- \Bar\l,{}_{(\m|}\g^\r \l,{}_{ | \n) \r} 
	- \Bar\l,{}_\r \g_{(\m|}\l,{}_{|\n)}{}^\r 
	+ \Bar\l,{}_{(\m}\g_{\n)}\l,{}\du\r\r 
	\,\big] + {\cal O}(\l^4) ~~.  {~~~~~ ~~~~~} 
&(2.12) \cr } $$ 
Here $~\l,{}_\m \equiv \partial_\m \l,~\l,{}_{\m\n} \equiv
\partial_\m\partial_\n \l$, {\it etc.}\footnotew{Since only lowest-order
terms are under question here, the raising/lowering indices  by the
metric $~G_{\m\n}$~ or $~\eta_{\m\n}$~ does not matter here.} 
Obviously, even though $~ i\g^\m\partial_\m \l \approx {\cal O}(\l^3)$~ 
by the NG$\,$-fermion equation at the lowest order, there
still remains non-vanishing component even for the scalar curvature
$~R(G)$.  We can further elaborate our result, introducing more 
higher-derivative lagrangians, such as $~\big(R_{\m\n\r}{}^\s (G)
\big)^2,~ (R_{\m\n} (G) \big)^2$, or $~R(G)^2$, etc.   

We stress here that the invariance of $~I_E $~ or $~I_R$~ relied only on
the particular transformation rule (2.1)  of the vielbein, but did not
use any property related to the space-time dimensions, such as Fierz
arrangements.  This implies that the non-linear
realizations of supersymmetry should be valid in $~^\forall D$, where
we can define a spinor field with the usual kinetic term.  In this
context, the specific nature of the spinors (Weyl, Majorana, or Dirac)
does not matte.  

The generalization to extended supersymmetry is also straightforward,   
assigning an explicit $~N\-$supersymmetry index $~{\scst i}$~ 
on $~\l^i$.  Accordingly, (2.1) and (2.2) are now 
$$ \li{ & \d_Q \l^i = \e^i + \xi^\m \partial_\m \l^i ~~, 
          ~~~~ \xi^\m \equiv i \big( \Bar\e{}^i \g^\m \l^i \big) ~~. 
&(2.13) \cr } $$
If we suppress the explicit contraction index $~{\scst i}$~ as
understood, then formally the same equations as (2.4) through (2.9)
follow, and there is no restriction on the value of $\,N$.  We have thus
a non-linear realization of arbitrarily extended global supersymmetries.

Note that we are dealing with two different space-times, the a `flat'
space-time with no gravity and with no curvature, while 
the other space-time has non-vanishing curvature (2.12), in which we can
define general coordinate transformations.  Interestingly, the action
$~I_E$~ is invariant in both of these space-time simultaneously.   

Our result applies even to $~D\ge 12$~ for the invariant actions under
non-linear supersymmetry, including of course includes $~D=12$~ of
F-theory  
%%% 
\ref\ftheory{C.~Vafa, \np{469}{96}{403}.} 
%%%
or $~D=13$~ of S-theory
%%%
\ref\stheory{I.~Bars, \pr{55}{97}{2373}.}, 
%%%
and so forth, as well as the M(atrix)-theory in $~D=11$~ 
%%%
\ref\bfss{T.~Banks, W.~Fischler, S.H.~Shenker and L.~Susskind, 
\pr{55}{97}{5112};  {\it For reviews, see, e.g.,}
A.~Bilal, {\it `M(atrix)-Theory: A Pedagogical 
Introduction'}, \hepth{9710136}; 
T.~Banks, {\it `TASI Lecture Note on Matrix Theory'}, \hepth{9911068}; 
W.~Taylor IV, {\it The M(atrix) 
Model of M-Theory'}, \hepth{0002016}; {\it and references therein}.}.     
%%% 
As a matter of fact, we will see a more explicit link with the 
M(atrix)-theory \bfss\ in section 3.3.    



\bigskip\bigskip\bigskip



\vbox{\centerline {\bf 3.~~Matter Lagrangians}  

\bigskip

\leftline{\bf 3.1~~Bosonic Lagrangians}

We have so far offered a possible meaning of our `vielbein', only with 
the NG$\,$-fermion.  The next natural question is whether or not we can
introduce other matter fields.  In the case of $~D=4$, we already know
that such lagrangians are easily constructed \ik\ferrara\cf.   However,
because of the latter applications to super $~p\-$branes,  {\it etc.}, 
we generalize these into $~^\forall D$~ and $~^\forall N$, even though
such a generalization looks straightforward.  We show that the answer to
this question is in  the affirmative, {\it i.e.,} we can introduce other
matter fields, as long as our general covariance $~\d_{\rm G}$~ is
maintained.     
}

First, we can show that the algebra of supersymmetry closes on the
following scalars $~\varphi^a$, a non-Abelian gauge field $~A_\m{}^I$,
an $~n\-$th rank covariant tensor field $~T_{\m_1\cdots\m_n}$~ with no
restriction on symmetry or antisymmetry: 
$$ \li{ & \d_Q \varphi^a = i \big( \Bar\e\g^\m
          \l \big) \partial_\m \varphi^a
          = \xi^\m \partial_\m \varphi^a = \d_{\rm G} (\xi^\r)
\varphi^a~~, 
&(3.1\rm a)  \cr 
& \d_Q A_\m{}^I = \xi^\n \partial_\n A_\m{}^I  
	          + \big(\partial_\m \xi^\n \big) A_\n{}^I 
	          = \d_{\rm G} (\xi^\r) A\du\m I ~~, 
&(3.1\rm b) \cr 
& \d_Q T_{\m_1\cdots \m_n} = \xi^\r \partial_\r T_{\m_1\cdots \m_n}
	          + \big(\partial_{\m_1} \xi^\r \big) T_{\r\m_2\cdots\m_n}
	          + \cdots + \big(\partial_{\m_n} \xi^\r \big)
	          T_{\m_1\cdots\m_{n-1}\r} 
           = \d_{\rm G} (\xi^\r) T_{\m_1\cdots \m_n} 
           ~~.{~~~~~ ~~~~~}   
&(3.1\rm c) \cr } $$ 
Here and throughout this paper we use exactly the same $~\xi^\m$~ as
(2.1).  We use the indices $~{\scst a,~b,~\cdots}$~ for an appropriate
representation of a certain compact non-Abelian gauge group $~G$~ for 
the scalars, while $~{\scst I,~J,~\cdots}$~ for the adjoint
representation of $~G$.  Needless to say, if some of the indices on 
the tensor are contravariant ones, we can change the signs of terms
corresponding to those contravariant indices in (3.1c), following 
the usual textbook result.  Interestingly, the 
closure on the vector or tensors generates {\it no} gauge
transformations as in (2.3), in addition to the standard translation:  
$$ \li{ & \[ \,\d_Q(\e_1), \d_Q(\e_2) \, \] \phi 
          = \d_P \big(\zeta_{12}^m \big) \phi ~~, 
&(3.2) \cr } $$
where $~\phi$~ is an arbitrary field in (3.1).  This is another aspect
of the non-linear supersymmetry different from the linear supersymmetry,
in which such gauge transformations are induced in the commutator
algebras  
%%%
\ref\wz{J.~Wess and B.~Zumino, \np{70}{74}{39}.}.    
%%%

The construction of some invariant `matter' actions is now an easy 
task, as long as our general covariance is maintained {\it via}
$~G_{\m\n}$~ or $~E\du\m m$.  A typical bosonic action is like 
$$ \li{& I_{\rm B} = \int d^D x \, {\sqrt{-G}} \, \left[\,  
          + \frac 12 G^{\m\n} \big(D_\m\varphi^a \big)
          \big(D_\n\varphi^a \big)
          - \frac14 G^{\m\r} G^{\n\s} F_{\m\n}{}^I F_{\r\s}{}^I \,
          \right]~~,  
&(3.3) \cr } $$  
where $~D_\m\varphi^a$~ is the usual gauge covariant derivative 
$~D_\m\varphi^a \equiv \partial_\m \varphi^a + A_\m{}^I  \big( T^I
\big)^{a b} \varphi^b$~ for an appropriate representation of the
generators $~T^I$  of the gauge group $~G$, and $~F_{\m\n}{}^I\equiv
\partial_\m A_\n{}^I-\partial_\n A_\m{}^I + f^{I J K} A_\m{}^J A_\n{
}^K$~ is the usual field strength.  These gauge covariant combinations
transform under $~\d_Q$~ as    
$$ \li{ & \d_Q \big( D_\m \varphi^a \big) 
          = \xi^\n \partial_\n \big( D_\m \varphi^a \big) 
          + \big(\partial_\m\xi^\n \big) D_\n \varphi^a
          =  \d_{\rm G} (\xi^\r) \big( D_\m \varphi^a \big) ~~, 
&(3.4\rm a) \cr 
& \d_Q F_{\m\n}{}^I  =
          \xi^\r \partial_\r F_{\m\n}{}^I 
	          + \big(\partial_\m\xi^\r \big) F_{\r\n}{}^I 
	          + \big(\partial_\n\xi^\r \big) F_{\m\r}{}^I  
          = \d_{\rm G} (\xi^\r) F_{\m\n}{}^I ~~.
&(3.4\rm b) \cr} $$ 
It is not a problem at all to generalize the action (3.3) to more
sophisticated interactions.  If the above result is only for 
$~D=4$, it is nothing new \ik\cf\ferrara\cl, but 
our point here is that these actions are valid for 
$~^\forall N\-$extended supersymmetry in $~^\forall D$.  



\bigskip\bigskip

\leftline{\bf 3.2~~Non-Abelian Supersymmetric Born-Infeld Lagrangians}

With these relations at hand, it is now straightforward even to 
supersymmetrize a Born-Infeld lagrangian.  This can be easily done, 
because the combination $~H_{\m\n} \equiv G_{\m\n} + 
F_{\m\n}$~ for an  Abelian field strength $~F_{\m\n}$~ transforms
`covariantly': $~\d_Q H_{\m\n} = \d_Q \big(G_{\m\n} + F_{\m\n} \big) 
	= \d_{\rm G} (\xi^\r) H_{\m\n}$, therefore 
the action\footnotew{We need the negative signs in the square roots, 
because $~\det \big( \eta_{m n}\big) <0 $.}
$$ \li{ & I_{\rm S B I} \equiv \int d^D x \, 
	    \left[ \, - \det\, \big( G_{\m\n} + F_{\m\n} \big)\,
	\right]^{1/2} = \int d^D x\, 
	          \left[ \, - \det\, \big( H_{\m\n} \big)\, \right]^{1/2} 
&(3.5) \cr } $$ 
is invariant under supersymmetry: $~\d_Q I_{\rm S B I}=0$.  
This sort of supersymmetric Born-Infeld lagrangian in particular
dimensions, such as in $~D=10$~ is nothing new \brs\aps, but our
lagrangian is valid for $~^\forall D$~ and $~^\forall N$.  

It is not too difficult to generalize this result to the  
non-Abelian gauge group.  Following the non-supersymmetric non-Abelian
Born-Infeld action in \tseytlin, we can easily postulate a non-Abelian
analog of the supersymmetric Born-Infeld action (3.5), as  
$$ \li{ & I_{\rm N A S B I} = \int d^D x~ \hbox{STr}\, 
          \left[ \,- \det\, \big( G_{\m\n} I + F_{\m\n}{}^I T^I \big)\,
          \right]^{1/2} \equiv \int d^D x ~ 
          \hbox{STr} \, \left[ \, - \det\, \big( H_{\m\n} \big) 
          \, \right]^{1/2} ~~,{~~~~~ ~~~~~} 
&(3.6) \cr } $$ 
where the generators $~T^I$~ are in an appropriate matrix
representation, and $~I$~ in $~G_{\m\n} I $~ is the unit matrix for such
a representation.  The determinant operation here is only for the indices
$~{\scst \m\n}$, but not for the group indices.  The $~\hbox{STr}$~ is
the symmetrized trace defined by $~\hbox{STr}\,\big( A_1\cdots A_n \big)
\equiv (1/n!) \Tr \,\big( A_1 \cdots A_n + \hbox{`all permutations'}
\big)$.  Due to the $~\d_Q\-$transformation property  
$$ \li{ & ~\d_Q H_{\m\n} = \d_Q \big(G_{\m\n} I
          + F_{\m\n}{}^I T^I \big) 
           = \d_{\rm G} (\xi^\r) H_{\m\n}~~, 
&(3.7) \cr } $$  
of the combination $~H_{\m\n} \equiv G_{\m\n} I + F_{\m\n}{}^I T^I$,
together with the property of $~\hbox{STr}$, it is clear that the action
$~I_{\rm N A S B I}$~ is invariant under $~\d_Q$.  As before, since
$~I_{\rm NASBI}$~ is valid for $~^\forall D$~ and $~^\forall N$, our
result is a generalization of \brs\aps, up to discrepancy in 
higher-order terms.  In fact, it is easy to see that the lowest-order
terms in our (3.6) agree with those in \brs\aps.    

Needless to say, we can also add the antisymmetric potential 
$~b_{\m\n}$~ to (3.5):
$$ \li{ & H_{\m\n} = G_{\m\n} + F_{\m\n} + b_{\m\n} ~~, 
&(3.8) \cr } $$
while the standard kinetic term 
for its field strength $~G_{\m\n\r} \equiv 3\partial_{\[\m} b_{\n\r\]}$~
is 
$$ \li{ & I_b = \int d^D x \,\big[\,\frac1{12} {\sqrt {-G}} 
	G^{\m\s} G^{\n\t} \, 
	G^{\r\upsilon} G_{\m\n\r} G_{\s\t\upsilon} \,\big] ~~,     
&(3.9) \cr } $$ 
as in super D-branes \bt.  In other words, there can be a general 
antisymmetric component $~b_{\m\n}$~ 
for the `metric' $~G_{\m\n}$, maintaining the
$~\d_Q\-$invariance as well as the $~b\-$field gauge invariance: 
$~\d_\a b_{\m\n} = 2\partial_{\[\m} \a_{\n\]}$.  Adding further 
an appropriate minimal couplings of $~A_\m$~ to scalars like $~I_{\rm B}$~
(3.3), there is no worry that the field strength $~F_{\m\n}$~ 
is completely gauged away in (3.8) by the field redefinition of
$~b_{\m\n}$.   

For the $~N\-$extended supersymmetry for $~N\ge 2$, we can introduce an
$~SO(N)$~ gauge field $~A\du \m{i j} ~{\scst
(i,~j,~\cdots~=~1,~2~,\cdots,~N)}$~ minimally coupled to $~\l^i$, when
all the components of $~\l$~ are Majorana spinors or (Majorana-)Weyl
spinors in the same chirality.  Such a vector field is also subject to the
`covariant' transformation rule (3.1b).  Relevantly, the derivative 
in (2.5) is replaced by the covariant derivative $~D_\m\l^i \equiv
\partial_\m \l^i + A\du\m{i j} \l^j$: 
$$ \li{& E\du\m m = \d\du \m m + i \big(\Bar\l{}^i \g^m D_\m \l^i \big)~~.
&(3.10) \cr } $$

One important case is when $~{\scst i,~j,~\cdots}$~ 
indices are adjoint ones 
$~{\scst I,~J,~\cdots~=~1,~\cdots,~\dim G}$~ 
\hbox{for} $~D_\m\l^I \equiv$ {~~}
$\partial_\m\l^I + f^{I J K} A\du\m J \l^K$: 
$$ \li{& E\du\m m = \d\du\m m + i \big(\Bar\l{}^I \g^mD_\m\l^I \big) ~~. 
&(3.11) \cr } $$
Then the lowest-order terms of our action (3.6) in $~D=10$~ 
coincide with those in linearly supersymmetric Born-Infeld action \aps. 
This seems to imply that our action (3.6) can be of equal importance for
the M(atrix)-theory \bfss\ as the D-brane actions 
\pol\dpbranes\nilsson\bt.\footnotew{See, however, the recent development
about subtlety at higher orders 
%%%
\ref\mtheoryrecent{I.L.~Buchbinder, S.M.~Kuzenko, 
A.A.~Tseytlin, {\it `On Low-Energy Effective Actions in N = 2, 4 
Superconformal Theories in Four Dimensions'}, \hepth{9911221}; 
Z.~Lalak, S.~Lavignac, H.P.~Nilles, 
{\it `Target-Space Duality in Heterotic and Type I Effective Lagrangians'},
\hepth{9912206}.}.   
%%%
We do not get into these details in this paper.}  Interestingly, the
number $~N$~ of  extended supersymmetries in our formulation will be $~N
= \dim G$, {\it  e.g.}, $~N = {\cal N}^2$~ for $~G = U({\cal N})$~ used
for M(atrix)-theory  \bfss.  Therefore the usual large $~{\cal N}$~
limit for $~U({\cal N})$~ \bfss\ corresponds to $~N ={\cal
N}^2\rightarrow\infty$, namely infinitely many ($\aleph_0$)
extended supersymmetries in our formulation!     



\bigskip\bigskip

\leftline{\bf 3.3~~Fermionic Lagrangians}

We can incorporate other fermionic fields rather easily, because we need
only global Lorentz invariance for our action, much like the case of
general covariance, which is not really present.  In fact, we can
show that the action\footnotew{In order to avoid any misinterpretation,
we always use $~E\du\m m$~ or $~E\du m\m$~  explicitly in this paper,
{\it instead of} $~\g^\m \equiv  \g^m E\du m\m$, {\it etc}.}  
$$ \li{& I_{\rm F} = \int d^D x \, \frac i2 E 
          E\du m \m \big(\Bar\psi{}^a\g^m D_\m\psi^a \big)  ~~, 
&(3.12) \cr } $$ 
for a fermion $~\psi^a$~ is actually invariant under supersymmetry:  
$$ \li{ & \d_Q \psi^a = i \big( \Bar\e\g^\m \l\big) 
          \partial_\m \psi^a = \xi^\m \partial_\m \psi^a 
          = \d_{\rm G} (\xi^\m) \psi^a ~~. 
&(3.13) \cr } $$ 
We use the indices $~{\scst a,~b,~\cdots}$~ for some appropriate
representation of the group $~G$~ that $~\psi$~ belongs to,
and $~D_\m$~ is the  gauge covariant derivative $~D_\m \psi^a \equiv
\partial_\m \psi^a + A_\m{}^I (T^I)^{a b} \psi^b$.  
The invariance $~\d_Q I_{\rm F}=0$~ is due to the transformations 
$$ \li{& \d_Q \big(D_\m\psi^a \big) 
          = \xi^\n \partial_\n \big(D_\m\psi^a \big) 
          + \big( \partial_\m \xi^\n \big) D_\n\psi^a 
          = \d_{\rm G} (\xi^\n)  \big(D_\m\psi^a \big) ~~, 
&(3.14) \cr } $$  
like a `general covariant vector', with respect to the index
$~{\scst\m}$.  As in the case of $~\l$, there is no
essential restriction on the property of the spinor $~\psi$, such as
Weyl, Majorana, Majorana-Weyl, or Dirac spinor, with any
additional group indices in arbitrary $~^\forall D$.  

Once we have a fermionic field $~\psi$, then we can modify (2.5) as 
$$ \li{ & E\du \m m = \d\du \m m 
     + i \big( \Bar\l\g^m \partial_\m \l \big) 
     + i \big( \Bar\psi{}^a\g^m D_\m \psi^a \big)~~, 
&(3.15) \cr } $$ 
This is because $~\big(\Bar\psi{}^a\g^m D_\m \psi^a\big)$~ transforms as
a `covariant vector' with respect to the index $~{\scst \m}$.  Accordingly
we have the alternative fermionic action 
$$ \li{ & I_{\rm E}' = \int d^D x ~ 
     \det \Big[\, \d\du\m m 
     + i \big( \Bar\l\g^m \partial_\m \l \big)
     + i \big( \Bar\psi{}^a\g^m D_\m \psi^a \big) \, \Big]~~.  
&(3.16) \cr } $$
We have thus two actions in good contrast, $~I_E'$~ for fermions, and 
$~I_{\rm N A S B I}$~ for bosons.  


\bigskip\bigskip


\vbox{
\leftline{\bf 3.4~~Chern-Simons Lagrangians}

Our results above have a wide range of applications.  Since $~I_{\rm
N A S B I}$~ needs no explicit `metric', but the GN$\,$-field forms 
a composite metric, the first interesting application is  
the Chern-Simons action for an arbitrary (super)group in $~D=2n+1$~ 
%%%
\ref\cham{A.H.~Chamseddine, \pl{233}{89}{291}; \np{346}{90}{213}.}%  
%%% 
\ref\zanelli{M.~Ba\~nados, R.~Troncoso and J.~Zanelli, 
\pr{54}{96}{2605}; R.~Troncoso and J.~Zanelli, \pr{58}{98}{101703}; 
{\it `Higher Dimensional Gravity and Local Anti-de Sitter Symmetry'},
hep-th/9907109.}%  
%%% 
\ref\phvnw{C.R.~Preitschopf, T.~Hurth, P.~van Nieuwenhuizen 
and A.~Waldron, Nucl.~Phys.~Proc.~Suppl.~
\newline{\bf 56B} (1997) 310.}:  
%%% 
}
$$ \li{ I_{\rm C S} & = \int_{M^D} \,  \hbox{STr} \, 
          \big( \overbrace{F\wedge F\wedge 
          \cdots \wedge F}^n \wedge \, A +
          \hbox{`Chern-Simons~  completion'} \big) {~~~~~ ~~~} 
&(3.17\rm a)  \cr 
& = \int_{M^{D+1}} \hbox{STr} \, 
     \big( \underbrace{\Hat F \wedge \Hat F \cdots 
     \wedge \Hat F}_{n+1} \big)  ~~, 
&(3.17\rm b) \cr } $$ 
to be added to the above actions: $~I \equiv I_{\rm C S} + I_{\rm N A S
B I} + I_{\rm E}' + I_{\rm B}$.  Here $~\hbox{STr}$~ 
acts on fermionic generators as an 
antisymmetrized trace, as a supergroup generalization of the symmetrized
trace for ordinary groups.  In (3.17b), the {\it hatted} field strength
$~\Hat F_{\hat\m\hat\n}$~ is defined in the extended space-time
$~D+1=2n+2$~ with one additional Vainberg coordinate $~0\le y \le 1$~ 
%%% 
\ref\vainberg{M.M.~Vainberg, {\it `Variational Methods for the 
Study of Non-Linear Operators'}, (Holden Day, San Fransisco, 1964).}%  
%%% 
\ref\gnssp{S.J. Gates, Jr. and H. Nishino, \pl{173}{86}{46}.}
%%%
with the indices $~\big(\Hat x^{\hat\m} \big) \equiv
\big(\Hat x^0,~\Hat x^1, \cdots ,~\Hat x^{2n}, ~ \Hat x^{2n+1} 
\equiv y\big) \equiv (x^\m, y) $~ by 
$$\li{ & \Hat F_{\hat\m \hat\n} \equiv 
     \Hat \partial_{\hat\m} \Hat A_{\hat\n}
     - \Hat \partial_{\hat\n} \Hat A_{\hat\m} 
     + \[ \Hat A_{\hat\m}, \Hat A_{\hat\n} \} \equiv  
     \cases{\Hat F_{\m\n} \equiv \partial_\m \Hat A_\n -
     \partial_\n \Hat A_\m 
     + \[ \Hat A_\m, \Hat A_\n \} ~~, \cr 
     \Hat F_{y\n} \equiv \partial_y \Hat A_\n = - \Hat F_{\n y} ~~, \cr } \cr 
& \Hat A_\m (x^\n,1) \equiv A_\m (x^\n) ~~, ~~~~ 
	\Hat A_\m(x^\n,0) = 0 ~~,  
      ~~~~ \Hat A_y = 0 ~~, 
&(3.18) \cr } $$ 
in such a way that the leading term (3.17a) is generated by the
$~y\-$integration $~\int_0^1 dy$~ \vainberg.  The gauge group here can 
be even a supergroup, such as $~\hbox{\it OSp}(N|m)$~ for Chern-Simons
supergravity in $~^\forall D = 2n+1$~ \cham\zanelli\phvnw.  Accordingly,
the generators $~T^I$~ in (3.6) should be those of the gauge (super)group
under consideration.  Due to our transformation rule (3.1b), the 
invariance $~\d_Q I_{\rm C S}=0$~ is clear.    

The important point is that in the conventional Chern-Simons
formulations including those with supergroups \cham\zanelli\phvnw, the
introduction of additional `matter' fields has been problematic, because
the kinetic terms of such matter fields need some `metrics' or
`vielbeins'.  However, since $~I_{\rm N A S B I}$~  has no fundamental
metric, it is compatible with such formulations without a metric by
construction.  In this sense, the non-linear realization of
supersymmetry is more natural for Chern-Simons theories.  Due to the
minimal couplings in $~I_{\rm F}$~ or $~I_{\rm B}$, there are still
non-trivial interactions of the gauge fields to other `matter' bosonic
or fermionic fields.  

To put this differently, the important conclusion is that {\it any}
Chern-Simons action (3.17) formulated in $~^\forall D = 2n+1$~ for a
(super)group is invariant under {\it hidden} global  non-linear
supersymmetry dictated by (3.1b).  Moreover, once the action
$~I_E$~ (2.7) is added to $~I_{\rm CS}$, then the GN$\,$-field becomes
physical, and the total action is no longer invariant under arbitrary 
general coordinate transformations, so that the GN$\,$-fermion is 
no longer gauged away.  Therefore the global supersymmetry is not a
superficial symmetry any longer by the presence of $~I_E$.  This has a
very far-reaching conclusion, because any Chern-Simons theory for a
supergroup \cham\zanelli\phvnw\ {\it except in} ~$D=3$~ 
%%% 
\ref\ngchern{H.~Nishino and S.J.~Gates, Jr., \ijmp{8}{93}{3371}.}   
%%% 
has been supposed to have no direct link with the Poincar\'e 
supersymmetry with diffeomorphism, in particular in the context of
M-theory \zanelli% 
%%%
\ref\horava{P.~Ho\v rava, \pr{59}{99}{046004}; M.~Ba\~nados, {\it
`Gravitons and Gauge Fields in 5D Chern-Simons  Supergravity'},
\hepth{9911150}.}.\footnotew{See, however, ref.~%
%%%    
\ref\mn{P.~Mora and H.~Nishino, {\it `Fundamental Extended Objects for 
Chern-Simons Supergravity'}, UMDEPP 00-052.} 
%%% 
in which extended objects 
are introduced to Chern-Simons supergravity to study such a direct 
link with Poincar\' e supergravity.}
%%%
This missing link has been a major drawback with the
Chern-Simons formulation \cham\zanelli\phvnw\ in the context of
space-time supersymmetry.  In other words, the non-linear representation
of supersymmetry may well provide a `missing link' between Chern-Simons
theory for supergroups \cham\zanelli\phvnw\ and space-time supergravity
theories  
%%%
\ref\pvn{P.~van Nieuwenhuizen, \prep{68}{81}{189}.}
%%%
with diffeomorphism.      



\bigskip\bigskip

\leftline{\bf 3.5~~Generalized $~\s\-$Model Lagrangians}

We can try even a more sophisticated example.  Suppose we have a coset
$~G/H$, whose coordinates are parametrized by the scalars
$~\varphi^a$.  Now instead of $~H_{\m\n}$~ in (3.5), we consider 
$$ \li{ & H_{\m\n} \equiv G_{\m\n} + F_{\m\n}  
	+ g\low{a b}(\varphi)\big(\partial_\m\varphi^a \big) 
\big(\partial_\n\varphi^b \big)~~, 
&(3.19) \cr } $$
where $~g\low{a b}(\varphi)$~ is the metric on this coset.  
Due to the transformation property of $~\partial_\m\varphi^a$~
under $~\d_Q$, it is clear that the action 
$$ \li{ & I_\s = \int d^D x \, \Big[ -  \det \Big\{ \, 
     G_{\m\n} + F_{\m\n}  
	    + g\low{a b}(\varphi) \big(\partial_\m\varphi^a \big)
     \big(\partial_\n\varphi^b \big) \Big\} \, \Big]^{1/2}~~, 
&(3.20) \cr } $$ 
is invariant under $~\d_Q$.  As we easily see, the lowest order-term
in the determinant contains the same kinetic term for the scalars
$~\varphi^a$~ as the conventional $~\s\-$model.  We can further add
some Wess-Zumino-Novikov-Witten (WZNW) term to $~I_\s$, as an analog of
WZNW-terms used in D-brane theories \bt\aps\dpbranes.     


\bigskip\bigskip

\vbox{
\leftline{\bf 3.6~~Duality Transformation of $~F_{\m\n}$~ into
$~N_{\m_1\cdots\m_{D-2}}$}

As another interesting by-product, we mention the duality 
transformation 
%%% 
\ref\nt{H.~Nicolai and P.K.~Townsend, \pl{98}{81}{257}.} 
%%%
for the field strength $~F_{\m\n}$~ into $~N_{\m_1\cdots \m_{D-2}}$~ in
$~^\forall D$.  We can show this is not too difficult, even with the 
general `metric tensor' like that in (3.20).  Such a duality 
transformation has been done in $~D=4$
%%% 
\ref\dualityinv{G.W.~Gibbons and D.A.~Rasheed, \np{454}{95}{185};
\pl{365}{96}{46}; M.K.~Gaillard and B.~Zumino, {\it `Non-Linear 
Electromagnetic Self-Duality and Legendre Transformations'}, in 
{\it `Duality and Supersymmetric Theories'}, P.C.~West and D.~Olive,
eds., Cambridge University Press, \hepth{9712103};
D.~Brace, B.~Morariu and B.~Zumino, {\it `Duality
Invariant Born-Infeld Theory'}, \hepth{9905218}.},     
%%% 
but our formulation here is complimentary, applicable to
$~^\forall D\neq 4$~ and $~^\forall N$.
}

We start with the generalized supersymmetric Born-Infeld action 
$$\li{ & I_{\rm S B I}' 
     \equiv \int d^D x \, \left[ - \det\big(g_{\m\n} + F_{\m\n} \big) \,
\right]^{1/2} ~~, 
&(3.21) \cr } $$
for an Abelian field strength $~F_{\m\n}$.  Here $~g_{\m\n}$~ can be any
generalized metric containing any  fields other than $~A_\m$~ or
$~F_{\m\n}$, {\it e.g.,}   $$ \li{ & g\low{\m\n} \equiv G_{\m\n}  
	    + g\low{a b} (\varphi) \big(\partial_\m\varphi^a \big)
     \big(\partial_\n \varphi^b \big) ~~, 
&(3.22) \cr } $$ 
in the case of (3.20).  The crucial point here is that $~g_{\m\n}$~ 
should have {\it no} coupling to $~A_\m$~ for the duality 
transformation \nt\ to work.  

The lagrangian (3.21) is not very convenient for duality transformations,
because the lowest-order term in $~F$ is not manifestly quadratic in $~F$,
is not manifest.  In order to make the $~F^2\-$term more manifest, we
first introduce the generalized vielbein  
$$ \li{ & g\low{\m\n} = e\du \m m \eta_{m n} e\du \n n ~~.   
&(3.23) \cr } $$
Note that this $~e\du\m m $~ is more generalized than (2.5), and it 
contains at least the latter as a part of it.  
Using such a vielbein, we see that 
$$ \li{ & \det \big(g_{\m\n} + F_{\m\n} \big) 
      = \det\left[ \, e\du\m m \big(\d\du m n+ F \du m n \big)
        e_{\n n} \right] \cr 
& ~~~ = \big(\det e\du \m m \big) \, \big(\det e_{\n n} \big)  
       \det \big(\d\du m n + F\du m n \big) 
      = \det\big(g_{\m\n} \big) \,  
       \det\big(\d\du m n + F\du m n \big) ~~,    
&(3.24) \cr } $$ 
where as usual $~F_{m n} \equiv e\du m \m e\du n \n F_{\m\n},~ 
F\du m n \equiv \eta^{n r} F_{m r}$, {\it etc}.  Note also that 
the antisymmetry $~F^t = - F$~ implies that 
$$ \li{ & \det\big( \d\du m n + F\du m n)  = \det \big( I + F \big) 
       = \det\big\{ \big( I + F \big)^t \big\}  
     = \det \big(I - F\big) \cr 
& = \left[\,\det\big( I + F \big)\, \det\big( I - F \big)\, \right]^{1/2}
       = \left[ \, \det\big\{ \big( I + F\big) 
      \big( I - F\big) \big \} \,\right]^{1/2}   \cr
& = \big[ \, \det \big( I - F^2 \big) \, \big]^{1/2} 
     =  \big[ \, \det \big\{ \d\du m n  
        - \big( F^2 \big) \du m n \big\} \, \big]^{1/2}  ~~,   
&(3.25) \cr } $$ 
with $~\big( F^2 \big)\du m n\equiv F\du m r F\du r n$.  Hence (3.21) is
equivalent to  
$$\li{ & I_{\rm SBI}' = \int d^D x \, {\sqrt {-g}}\,  
     \big[ \, \det\big\{ \d\du m n - \big( F^2 \big)\du m n\big\}
      \,\big]^{1/4} ~~,   
&(3.26) \cr } $$
where $~g \equiv \det\big( g_{\m\n} \big)$~ and the $~F^2\-$term
is now manifest. 

We next `linearize' the determinant operation, in such a way that 
the $~F^2\-$term is involved only as a quadratic term, by introducing a
symmetric matrix auxiliary field $~P\du m n$: 
$$ \li{ & I_{\rm S B I}' = \int d^D x  \, 
        a {\sqrt{-g}}\left[ \, c P^{1/4} 
       + P^{1/4} \big( P^{-1} \big) \du m n   
       \big\{ \d\du n m - \big( F^2 \big)\du n m \big\} \, \right] ~~, 
&(3.27) \cr } $$
where $~P\equiv \det\big(P\du m n\big)$, and the constants $~a$~ and $~c$~
are
$$ \li{ & a = \frac14~~, ~~~~ c = 4 - D~~, 
&(3.28) \cr } $$ 
in order to get the normalized algebraic field equation for $~P\du m n$:
$$\li{ & P\du m n = \d\du m n - \big( F^2 \big)\du m n ~~.  
&(3.29) \cr } $$ 
Here the matrix multiplication for $~F^2$~ is done with the local
Lorentz indices $~{\scst m,~n,~\cdots}$.  

By these preliminaries, it is now easy to perform the duality 
transformation \nt\ from $~F_{\m\n}$~ into $~N_{\m_1\cdots m_{D-2} }$.  
As usual, we introduce the lagrange multiplier potential $~M_{\m_1\cdots
\m_{D-3}}$~ into a constraint action, so that the total action is now
$$ \li{ I_{\rm S B I}'' = \int d^D x\, 
	\Big[ \, & a c {\sqrt{-g}}
	P^{1/4} + a {\sqrt{-g}} P ^{1/4} \big( P^{-1} \big)\du m n 
      \big\{ \d\du n m - \big({\Hat F}{}^2 \big)\du n m\big\}  \cr 
& + \frac 2{(D-1)!} 
      \e^{\n\r\s\m_1\cdots \m_{D-3}} M_{\m_1\cdots \m_{D-3}} 
     \partial_\n \Hat F_{\r\s}  \, \Big] 
&(3.30\rm a)  \cr 
= \int d^D x\, 
     \Big[ \, & a c {\sqrt{-g}} P^{1/4} 
	+ a {\sqrt{-g}} P^{1/4} \big( P^{-1} \big)\du m n 
      \big\{ \d\du n m - \big({\Hat F}{}^2 \big)\du n m\big\}  \cr 
& - \frac 2{(D-2)!}\e^{\m_1\cdots \m_{D-2}\r\s} 
     N_{\m_1\cdots \m_{D-2}} \Hat F_{\r\s}  \, \Big] ~~.  
&(3.30\rm b)  \cr } $$ 
Here $~\Hat F_{\m\n}$~ is no longer a field strength of $~A_\m$, but is
regarded as an independent field variable \nt, while
$~N_{\m_1\cdots\m_{D-2}} \equiv (D-2) \partial_{\[\m_1}
M_{\m_2\cdots\m_{D-2}\]}$.  The constraint term at the end of (3.30a)
forces the Bianchi identity  for $~\Hat F_{\m\n}$~ for consistency.  
Our next step is to obtain the algebraic field equation
of 
$~\Hat F_{\m\n}$, as 
$$ \li{ & \fracm \d {\d \Hat F_{\m\n}} I_{\rm S B I} '' = 
     Q^{\m\r} \Hat F\du\r\n + \Hat F^{\m\r} Q\du\r\n  
     - 2 \Tilde N^{\m\n} = 0 ~~, 
&(3.31) \cr } $$ 
where 
$$ \li{ & Q\du\m\n \equiv P^{1/4} \big( P^{-1} \big)\du\m\n 
      \equiv P^{1/4} e\du\m m e\du n \n \big( P^{-1} \big) \du m n~~, 
&(3.32) \cr } $$ 
and 
$$ \li{ & \Tilde N^{\m\n} \equiv \frac1{\sqrt{-g}} 
         \frac1{(D-2)!} 
     \e^{\m\n\r_1\cdots\r_{D-2}} N_{\r_1\cdots \r_{D-2}} ~~.  
&(3.33) \cr } $$
The field equation (3.31) is conveniently expressed as a 
$~D\times D$~ matrix equation 
$$ \li{& \{ Q , \Hat F\}\du m n = 2 \Tilde N\du m n ~~.     
&(3.34) \cr } $$ 

In order to solve (3.34) for $~\Hat F$~ in terms of $~P$~ and $~\Tilde
N$, we split $~Q$~ into $~Q = I - 2 q$, and get the 
solution as an infinite series in terms of $~q$:
$$ \li{ \Hat F & = \sum_{r=0}^\infty \sum_{s= 0}^r
	   \binomial r s q^s {\Tilde N} q^{r-s}  
	= \sum_{r=0}^\infty \sum_{s= 0}^r \fracm 1{2^r} 
    \binomial r s \big( I - Q \big)^s \Tilde N 
     \big( I - Q \big)^{r-s} ~~.    
&(3.35) \cr } $$
Even though this is an infinite series,
it is a closed form, starting with the lowest-order term  
$~\Hat F^{\m\n} = \Tilde N^{\m\n} + {\cal O}(q)$, as desired.  
Our final step is to substitute (3.35) into
(3.30b), in order to eliminate $~\Hat F$.  After this, all the $~\Hat
F\-$terms are  replaced by its dual $~N_{\m_1\cdots\m_{D-2}}$, and thus
we have  achieved the duality transformation on a generalized metric 
(3.23).  

Eq.~(3.28) indicates that the case of $~D=4$~ is exceptional or
singular.  This is because the original field strength and its Hodge-dual
have the same rank.  This seems also related to the possible 
$~SL(2,\IR)$~ duality invariance between these two field strengths 
\dualityinv.  Other than this exceptional case of $~D=4$,   our action is
still superinvariant, because of the `general coordinate invariance'
valid in $~^\forall D\neq 4$~ and $~^\forall N$.




\bigskip\bigskip\bigskip



% From gatess@wam.umd.edu Thu Dec 30 20:54:59 1999

\centerline {\bf 4.~~Superspace Formulations}

\bigskip\bigskip

\leftline{\bf 4.1~~Second and Third-Rank Superfield Strengths}

Once we have understood the component formulation of the non-linear 
realization of supersymmetry, our next question is how to re-formulate 
the same results in superspace %%% 
\ref\superspace{S.J.~Gates, Jr., M.T. Grisaru, M. Ro\v cek and W. Siegel, 
{\it `Superspace'}, Benjamin/Cummings, Reading, MA, 1983.}.   
%%%   
This is a non-trivial question, because even though we know that such 
a formulation is definitely possible in $~D=4$~ with auxiliary fields \bw,
or without auxiliary fields in $~D=10$~ \aps, $~D=6$~ or $~D=7$~ \nilsson,
it is no longer trivial that we can repeat it in {\it arbitrary}
space-time dimensions $~^\forall D$.  In particular, it is no longer
automatic that superspace Bianchi identities \superspace are satisfied,
when supersymmetry is realized non-linearly.  This is due to possible  
`extra' symmetries present in the system, that prevent us from
formulating the component results in superspace 
\ngchern.  

We answer this question, by the actual investigation of conventional
superspace Bianchi identities \superspace.   As the first 
non-trivial example, we study the Bianchi identity\footnotew{In this
section of superspace, we use the indices $~{\scst A~=~(a,\un\a),~
B~=~(b,\un\b),~\cdots}$~ for superspace coordinates.   The underlined
spinorial indices $~{\scst \un\a,~\un\b,~\cdots}$~ include any possible
implicit indices such as `dottedness' for chiralities, or some
$~\hbox{\it USp}(N)$~ indices needed for  extended supersymmetry,
depending on $~D$~ \kt.  Our symmetrizations in this section are not
normalized,  {\it i.e.,} $~M_{\[A} N_{B)} \equiv M_A \mp
N_B$.}             
$$ \li{ &\nabla_{\[A} F_{B C)}{}^I - T\du{\[A B | }D F_{D| C)}{}^I  
	\equiv 0 ~~, 
&(4.1) \cr } $$
for the superfield strength $~F_{A B}{}^I$~ of a non-Abelian gauge 
vector superpotential $~A\du A I$, corresponding to (3.1b).  

Our superspace constraints are summarized as   
$$\li{ & T\du{\un\a\un\b} c = - 2i\big( \g^c \big)_{\un\a\un\b} ~~,
&(4.2\rm a) \cr 
& \nabla_{\un\a} \l^{\un\b} 
          = - \d\du{\un\a}{\un\b} 
          + i \big(\g^c\big)_{\un\a\un\g}\l^\g \nabla_c \l^{\un\b}
          \equiv - \d\du{\un\a}{\un\b} 
          - i \big(\g^c \l\big)_{\un\a} \nabla_c \l^{\un\b}~~, 
&(4.2\rm b) \cr 
& F\du{\un\a b} I  
          = + i \big( \g^c )_{\un\b\un\g} \, \l^{\un\g} \, F\du{c b} I
          \equiv - i \big( \g^c \l)_{\un\a} F\du{c b} I 
          = - P\du{\un\a} a F_{a b}{}^I ~~, 
&(4.2\rm c) \cr 
& F\du{\un\a\un\b} I = - \frac12 \big( \g^a\l)_{ ( \un\a}\, 
          \big( \g^a\l)_{\un\b )} F\du{a b} I 
          = + P\du{\un\a} a P\du{\un\b} b F\du{a b} I ~~, 
          ~~~~~ P\du{\un\a} b 
	 \equiv i \big(\g^b\l \big)_{\un\a} ~~.
&(4.2\rm d) \cr } $$
There is no {\it a priori} restriction on $~D$~
here.\footnotew{Depending on $~D$, we have to
switch between the antisymmetry and symmetry of the metric tensor for 
spinorial indices \kt.  For such cases,  we need to switch some signs for
contracted indices in the constraints here, but the basic structure is
still universal.}  The most important constraint is
(4.2d), which is similar to the postulates given in 
%%% 
\ref\gv{S.J.~Gates, Jr.~and S.~Vashakidze, \np{291}{87}{172}.}% 
%%%
\brs\ in $~D=10$~ as a special case of $~D$.  This is because
$~F\du{\un\a\un\b}
I$~ can be rewritten as 
$$ \li{ & F\du{\a\b} I = 
          - \frac1{16} \big(\g_c\big)_{\a\b} \, 
          \big(\Bar\l\g^{a b c}\l \big) \, F\du{a b} I  
          + \frac1{1920} \big(\g_{c_1\cdots c_5} \big)_{\a\b} \,
      \big(\Bar\l\g^a\g^{c_1\cdots c_5} \g^b \l\big)\, F\du{a b} I~~,  
&(4.3) \cr } $$
in terms of undotted chiral indices  in $~D=10$~ after Fierzing, where
the second term has been suggested in \gv\brs.  For an arbitrary
space-time dimension $~D$, we keep the original form (4.2d), because
forms like (4.3) needs Fierzing depending on $~D$.  

The confirmation of Bianchi identities (4.1) goes in a way similar to
the conventional case, once we know the structure of $~F\du{\un\a\un\b}
I$.   However, there are also important differences.  For example, it
turns out to be easier to proceed the conformation of Bianchi identities
from the highest mass dimension two ($d=2$) to the lowest $~d=1/2$~
backward, as opposed to the conventional case, where the
lowest-dimensional one is the simplest.  The reason for this is that the
lowest dimensional constraint (4.2c) is more involved than the higher
ones.   Another important aspect of this system is that the Bianchi
identities with these constraints do not yield any field equation for the
NG$\,$-fermion $~\l$.  This is natural, because as we saw in
component results, the transformation rule for $~\l$~ is independent of
the choice of the total action, {\it e.g.,} $~I_E$~ versus $~I_E +
I_R$, {\it etc.}, namely the system of non-linear  supersymmetry is
essentially `off-shell'.    

Another instructive example is the third-rank superfield strength 
$~G_{A B C}$, satisfying the Bianchi identity
$$ \li{& \frac 1 6 \nabla_{\[A} G_{B C D)} 
          - \frac 1 4 T\du{\[A B|} E G_{E| C D)} \equiv 0 ~~. 
&(4.4) \cr } $$
Our superspace constraints are  
$$ \li{ & G_{\un\a b c} = - P\du{\un\a} a G_{a b c} ~~, 
          ~~~~ G_{\un\a\un\b c} = - i \big( \g_c \big)_{\un\a\un\b} 
          + P\du{\un\a} a P\du{\un\b} b G_{a b c} ~~ , 
&(4.5\rm a)  \cr  
& G_{\un\a\un\b\un\g} 
    = - P\du{\un\a} a P\du{\un\b} b P\du{\un\g} c G_{a b c} ~~, 
&(4.5\rm b) \cr } $$
in addition to (4.2a) and (4.2b).

Note the interesting patterns of involvement of the factor 
$~P\du{\un\a} b$~ in (4.5) as well as (4.2), namely all the spinorial
index $~{\scst\un\a,~\un\b,~\cdots}$~ in the superfields are converted
into the bosonic ones $~{\scst a,~b,~\cdots}$~ by the factor 
$~P\du{\un\a} a$.  By reading these patterns, it is easy to
generalize these results to more higher-rank superfield strengths.  The
role of such a factor has been noted, {\it e.g.,} in $~D=6$~
\nilsson, but our point is that our constraints are valid for 
$~^\forall D$~ and $~^\forall N$.  Even though we skip the details for
these higher-rank superfield strengths in this paper, the geometrical
structure of this superspace formulation is much clearer than component
formulation.  In the next subsection, we address ourselves to the 
question of coupling to curved superspace backgrounds with 
supergravity, based on this particular role for 
the factor $~P\du{\un\a} b$.  



\bigskip\bigskip

\vbox{
\leftline{\bf 4.2~~Super $~(2k-1)\-$Brane Interpretations\footnotew{We
avoid such terminology as $~(D-1)$-brane, because it is confusing 
with the D-branes.}}

As the pattern of the factor $~P\du{\un\a} a$~
suggests, all of the effects of the GN$\,$-fermions can be
re-interpreted as part of the vielbein superfields.  For example, the
first form (2.5) of $~E\du\m m$~ suggests that the GN$\,$-fermion can 
be interpreted as nothing else than the $~\Theta\-$coordinates in {\it
flat} superspace, as has been also pointed out by several authors
\bt\nilsson\aps\ik\ for special cases of $~D=10$~ \aps, ~$D=6$, 
$~D=7$~ \nilsson, or $~D=4$~ \ik.  In our paper, we generalize this
feature to more general dimensions, because this process seems
imperative, if we also want to introduce `curved' backgrounds with
supergravity, and we need to consider more geometrical superspace
formulation.  
}

If this scenario really works in more general dimensions, we  must have
the geometric structure of a base manifold and a target superspace at the
same time, as special cases \aps\nilsson\ indicate. Accordingly, we must
have an analog of Green-Schwarz type super $~p\-$brane action \pbrane\ 
in which the base manifold (world-supervolume) 
and the bosonic coordinates in the target superspace have the 
same dimensions, with appropriate so-called fermionic $~\k\-$symmetry 
%%%
\ref\brink{L.~Brink and J.H.~Schwarz, \pl{100}{81}{310}; W.~Siegel,
\cqg{2}{85}{L95}.}%  
\pbrane.  
%%%
Such a formulation can accommodate `curved' superspace supergravity
backgrounds, whose classical superfield equations can be satisfied, 
only when the action has the $~\k\-$invariance.  The subtlety here 
is that there are two bosonic coordinates involved both for the
world-supervolume and the target superspace with the same dimensionality,
so that  we need to avoid the confusion between them.  This spirit
of having the same dimensionality both for the world-supervolume and the
target space-time was also presented in the context of doubly-supersymmetric 
$~p\-$branes or L-branes 
%%%
\ref\howesezgin{P.S.~Howe and E.~Sezgin, 
\pl{390}{97}{133}; P.S.~Howe, O.~Raetzel, I.~Rudychev,
E.~Sezgin, \cqg{16}{99}{705}.},  
%%%
or superembeddings
%%%
\ref\superembeddings{P.S.~Howe, E.~Sezgin, and P.C.~West,  
Contributed to {\it `Kharkov 1997, Supersymmetry and
quantum field theory'} p.~64, \hepth{9705093}.}% 
%%%
\sorokin.  

There has already been a general formulation for such a Green-Schwarz
$~\s\-$model action in general even dimensions $~D=2k$~ in 
\eightbrane\ in which the $~D=2k\,$-th rank antisymmetric potential
superfield $~C_{A_1\cdots A_D}$~ is introduced, as one more generalization
of super $~p\-$branes in \pbrane, and its $~\k\-$invariance has been
confirmed.  As a matter of fact, the generalized Volkov-Akulov action
$~I_E$~ (2.7) with the determinant form of $~E\du\m m$~ already suggests
the involvement of such maximally-ranked $~C_{A_1\cdots
A_D}\-$superfield.  In ref.~\eightbrane, we have seen that a similar
putative Green-Schwarz action in odd dimensions  $~D=2k-1$~ instead of
$~D=2k$~ leads to a trivial $~\k\-$symmetry, due to the absence  of
chirality projection $~\g\-$matrix like $~\g_5$.   

Following the results in \eightbrane, our $~(2k-1)\-$brane  
action in  general $~D=2k\-$dimensions should be 
$$\li{ S_{\rm GS} = \int d^{2k} \s \Big[ \!  
	& + \frac 12 {\sqrt {-g}} g^{i j} \eta\low{a b} \Pi\du i a \Pi\du
j b 
     - (k-1) {\sqrt{-g}}  \cr 
& + \frac 1{(2k)!} \e^{i_1\cdots i_{2k}} \Pi\du{i_1}{A_1} 
     \cdots \Pi\du{i_{2k}}{A_{2k}} C_{A_{2k}\cdots A_1} \, \Big] ~~, 
&(4.6) \cr } $$
where $~{\scst i,~j,~\cdots~=~0,~1,~2,~\cdots,~D-1~=~2k-1}$~ are the curved
indices for the world-supervolume with the coordinates $~\s^i$~ and the
metric $~g\low{i j}$, while $~\Pi\du i A\equiv \big(\partial_i Z^M \big)
E\du M A$~ are the ordinary pull-backs from the target superspace with
the coordinates $~(Z^M) \equiv (X^m, \Theta^{\un\m})$~ into the
world-supervolume.   The superpotential $~C_{A_1\cdots A_{2k}}$~ has its
superfield strength $~H_{A_1\cdots A_{2k+1}}$, satisfying the  Bianchi
identities
$$ \li{ & \frac1{(2k+1)!} \nabla_{\[A_1} H_{A_2\cdots A_{2k+2})} 
       - \frac1{2[(2k)!]} T\du{\[A_1A_2|}B H_{B|A_3\cdots A_{2k+2})}
       \equiv 0 ~~,  
&(4.7) \cr } $$
while the supertorsion satisfies the $~T\-$Bianchi identities \superspace
$$ \li{& \nabla_{\[ A} T\du{B C)} D 
     - T\du{\[ A B|} E T\du{E | C)} D 
     - \frac 12 R\du{\[A B| e} f \big({\cal M}\du f e \big)\du{|C)} D
     \equiv 0 ~~.  
&(4.8) \cr } $$
The $~H\-$Bianchi identities (4.7) at mass dimensions $~d=1/2$~ and $~1$~ 
\eightbrane\ require the conditions:   
$$ \li{&
T\du{(\un\a\un\b|}{\un\e} H_{\un\e|\un\g) d_1\cdots d_{D-1}} = 0 ~~, 
~~~~ T\du{\[c_1|(\un\a|}{\un\d} H_{\un\d|\un\b)|c_2\cdots c_D\]} = 0
~~.     
&(4.9) \cr  } $$ 
There are of course other Bianchi identities satisfied by other 
superfield strength, depending on the supergravity multiplet for a given
$~D$~ and $\,N$, {\it e.g.,} the fourth-rank superfield strength $~F_{A
B C}$~ for $~D=10, ~N=2A$, that we do not present here explicitly.  

The superspace constraints at mass dimension $~d=0$~ relevant
to our $~\k\-$transformation below are    
$$ \li{ & T\du{\un\a\un\b} c = - 2i
\big(\g^c \big)_{\un\a\un\b}~~,   
&(4.10\rm a) \cr 
& H_{\un\a\un\b c_1 \cdots c_{2k-1}} = 
      + i \big(\g_{c_1\cdots c_{2k-1}} 
      \g\low{2k+1}\big)_{\un\a\un\b} ~~,  
&(4.10\rm b) \cr } $$
where $~\g\low{2k+1}$~ is an analog of $~\g_5$~ in $~D=4$, 
and all other independent components in $~H_{A_1\cdots A_{2k+1}}$~
are zero \eightbrane.  

Even though our superspace constraints (4.10) look so simple, we
can accommodate all the curved supergravity effects in the 
system, such as the gravitino $~E\du a {\un\b} \approx \psi\du
a{\un\b}$, as long as the $~H\-$ and $~T\-$Bianchi identities above,  
as well as those for other superfield strengths are satisfied, with
curved supertorsions, supercurvatures and superfield strengths.  

As an important note to avoid confusion, we no longer use the flat case
constraints (4.2b) - (4.2d), (4.5),  once we have introduced the
Green-Schwarz action.  They will be recovered, though, when we consider 
the special flat superspace limit we mention shortly.   

It is easy to see that $~S_{\rm GS}$~ has the $~\k\-$invariance under
$$\li{& \d_\k E^{\un\a} = \frac12 \big(I + \G\big)\ud{\un\a}{\un\b}
     \, \k^{\un\b}~~, ~~~~ \d_\k E^a = 0 ~~, 
&(4.11\rm a) \cr 
& \G\ud{\un\a}{\un\b} \equiv
\frac{(-1)^{(k-1)(2k+1)/2}}{(2k)!{\sqrt{-g}}} 
    \e^{i_1\cdots i_{2k}} \Pi\du{i_1}{a_1} \cdots
    \Pi\du{i_{2k}}{a_{2k}} \big(\g_{a_1\cdots a_{2k}} \big)
    \ud{\un\a}{\un\b}  
    = \big(\g\low{2k+1} \big)\ud{\un\a}{\un\b} ~~,{~~~~~ ~~~~~}       
&(4.11\rm b) \cr } $$ 
following ref.~\eightbrane, whose details we do not repeat here.  
As in \eightbrane, the field equation of $~g\low{i j}$~ 
$$ \li{ & g\low{i j} = \eta\low{a b} \Pi\du i a \Pi\du j b~~,  
&(4.12) \cr } $$ 
is algebraic, so we do not have to worry about its variation under 
$~\d_\k$.  

In order to go back to `flat' superspace, we perform the
identifications $~\Theta^{\un\a}\equiv \l^{\un\a},~X^m 
\equiv \s^i$,\footnotew{This identification makes sense, because the
dimension of the $~\s^i$'s and that of $~X^m$'s coincide.} so that
$~S_{\rm GS}$~ coincides with the generalized Volkov-Akulov action
(2.7), after eliminating $~g\low{i j}$~ by its algebraic field equation
(4.12).  In fact, the $~\Pi$'s coincides with the $~E$'s in (2.5):  
$$\li{& \Pi\du i a      
         \rightarrow \d\du i a 
      + i \big(\Bar\Theta\g^a\partial_i \Theta \big) 
      \equiv \d\du i a + i \big(\Bar\l\g^a\partial_i \l\big) ~~,  
     ~~~~ \Pi\du i {\un\a} \rightarrow \partial_i \Theta^{\un\a} 
	     \equiv \partial_i \l^{\un\a} ~~. 
&(4.13) \cr} $$ 
In other words, in the flat target superspace, the fermionic  coordinates
are nothing else than the GN$\,$-fermions \aps\nilsson.  On the other
hand, the $~\k\-$transformation stays formally the same as (4.11), except
that we no longer have {\it e.g.,} the gravitino in the target
superspace: $~E\du a{\un\b} | \approx \psi\du a{\un\b} \rightarrow 0$.
Accordingly, our previous flat space constraints (4.2b) - (4.2d) or 
(4.5) will be recovered, due to the identification $~\Theta\approx \l$~ 
in the superspace vielbeins $~E\du m a$ \superspace.  Finally, the 
WZNW-term in (4.6) is shown to be equivalent to $~{\sqrt{- g}}$, as
follows: First, we can identify $~C_{a_1\cdots a_{2k}}\approx \e_{a_1
\cdots a_{2k}} C(Z)$~ with a scalar superfield $~C(Z)$.  Second, due to
$~H_{\un\a b_1\cdots b_{2k}} =0$, we can set  \hbox{$~C(Z) = \hbox{
const}$}.  Hence the WZNW-term is proportional to $~[(2k)!]^{-1}
\e^{i_1\cdots i_{2k}} \e_{a_1\cdots a_{2k}} \Pi\du{i_1}{a_1} \cdots
\Pi\du{i_{2k}}{a_{2k}}  = \det\big( \Pi\du i a\big) = {\sqrt{-g}}$.  

The introduction of the $~U(1)$~ field strength $~F_{i j}\equiv
\partial_i A_j - \partial_j A_i$~ with the fundamental vector $~A_i$~
on the  world-supervolume is not difficult.   For this purpose, we
modify $~S_{\rm GS}$~ as 
$$ \li{ S_{\rm G S}' = \int d^{2k} \s \, \Big[ \, &  \frac 12 
	{\sqrt{-\Tilde g}} \Tilde g^{i j} 
	\big( \eta_{a b} \Pi\du i a \Pi\du j b + F_{i j} \big) 
	- (k-1) {\sqrt{-\Tilde g}}  \cr 
& + \frac 1{(2k)!} \e^{i_1\cdots i_{2k}} \Pi\du{i_1}{A_1} 
     \cdots \Pi\du{i_{2k}}{A_{2k}} C_{A_{2k}\cdots A_1} \, \Big] ~~, 
&(4.14) \cr } $$
where $~\Tilde g\low{i j}$~ is an auxiliary field both with symmetric 
and antisymmetric components, playing a role of a `generalized' 
metric, and $~\Tilde g^{i j}$~ is its inverse with $~\Tilde g 
\equiv \det \big(\Tilde g\low{i j} \big)$.  Since the field equation 
of $~\Tilde g\low{i j}$~ is algebraic:
$$\li{ & \Tilde g\low{i j} = \eta_{a b} \Pi\du i a \Pi\du j b 
						+ F_{i j}~~, 
&(4.15) \cr } $$
we can eliminate $~\Tilde g_{i j}$~ to get the D-p-brane type 
action \bt\aps\nilsson\sorokin 
$$\li{ S_{\rm G S}' \rightarrow \int d^{2k} \s \Big[ \, \big\{ \! - 
	   \det \big( \eta_{a b} \Pi\du i a \Pi\du j b 
	   +F_{i j} \big) \big\}^{1/2}   
	   + \frac 1{(2k)!} \e^{i_1\cdots i_{2k}} \Pi\du{i_1}{A_1} 
     \cdots \Pi\du{i_{2k}}{A_{2k}} C_{A_{2k}\cdots A_1} \, \Big] 
     	~, {~} 
&(4.16) \cr } $$ 
also with the last WZNW-term.  The $~S_{\rm G S}'$~ has the
$~\k\-$invariance under (4.11) and $~\d_\k A_i = 0$, as is easily
confirmed.  We repeat here that these  actions are valid not only in
$~D=10$~ like type IIA or IIB superstrings \dpbranes\nilsson\aps\sorokin,
but also in any arbitrary even dimensions $~D=2k$~ where we can build
supergravity backgrounds  consistently in superspace
\eightbrane.\footnotew{This allows us to  even beyond $~D=2k\ge 12$~  
%%% 
\ref\twelve{H.~Nishino and E.~Sezgin, \pl{388}{96}{569};
H.~Nishino, \pl{428}{98}{85}; \ibid{437B}{98}{303};
\ibid{452B}{99}{265};  \np{523}{98}{450}; \ibid{B542}{99}{217}.},  
%%% 
if we can give up Lorentz covariance.}   

In this section, we have shown how to accommodate curved supergravity
backgrounds into our system, and in particular, we have seen how the
generalized Volkov-Akulov action (2.7) can be interpreted as a special
case of Green-Schwarz super $~(2k-1)\-$brane action in general $~D=2k$~
dimensions.  At the point of introducing supergravity backgrounds, we 
lose the universality on $~D$, due to the satisfaction of $~T\-$Bianchi
identities, which needs consistent supergravity constraints.  To put
this differently, we have seen in this section that, when the background
superspace is  flat, supersymmetry is global and non-linear, with no
restriction on $~D$~ or $\,N$, and we can always define Green-Schwarz
super $~p\-$brane actions.  However, when superspace is curved with 
supergravity with Lorentz covariance, there is a restriction, such as
$~D\le 11$~ by the ordinary Bianchi identities for the curved
backgrounds.  



\bigskip\bigskip 





\leftline{\bf 4.3~~Simplified (Supersymmetry)$^2\-$Models} 

The degeneracy of the dimensionalities between the target 
space-time and the world-supervolume suggests some formulation similar 
to what is called $~(\hbox{Supersymmetry})^2$$\-$models \supersquare.
Namely we expect two kinds of supersymmetries realized simultaneously  in
the target space-time and the world-supervolume.\footnotew{Similar
formulation was also presented for L-brane models \howesezgin, in which
various off-shell field strengths dual to scalars or vectors on the
world-supervolume are introduced.}  In this sub-section, we show that 
such a formulation indeed makes sense in general superspace, with the
simplification that these two supersymmetries are realized within the 
same space-time dimensions.  Such a sharing of the common space-time  is
possible, because of the degeneracy of the dimensionalities between  the
base and target space-time.  Accordingly, we have only one set of
superspace coordinates $~\big(Z^M \big) \equiv (x^m, \theta^{\un\m})$,
which we  temporarily call  `simplified'
(Supersymmetry)$^2\-$model.\footnotew{We can in principle elaborate this
formulation by introducing the second set of coordinates
$~(x'{}^m,\theta'{}^{\un\m})$, in order to distinguish the base and
target space-times, but we skip such a formulation in this paper.}   

Our starting point is to postulate the superspace analog of our 
component vielbein (2.5) by 
$$\li{& E\du M a = E^{(0)} \du M a 
     + (-1)^M i \big( \Bar\L \g^a \partial_M \L \big) 
     \equiv E^{(0)} \du M a 
     + (-1)^M i \L^{\un\b} \big(\g^a \big)\du{\un\b}{\un\g} 
     \partial_M \L_{\un\g} ~~, {~~~~~ ~~~} 
&(4.17\rm a) \cr  
& E\du M{\un\a} = E^{(0)} \du M{\un\a} = \d\du M{\un\a} ~~.   
&(4.17\rm b) \cr } $$ 
Here $~E^{(0)} \du M A$~ is the ordinary flat superspace vielbein 
\superspace: 
$$\li{ & \big(E^{(0)} \du M A \big)
    \equiv \left( \matrix{ \d\du m a  & ~~O \cr 
                    - i\big( \g^a \theta \big)_{\un\m} &
                            ~~ \d\du{\un\m} {\un\a} \cr } \right) ~~, 
    ~~~~ \big(E^{(0)} \du A M \big)
         \equiv \left( \matrix{ \d\du a m  & ~~O \cr 
                    + i\big( \g^m \theta \big)_{\un\a} &
                            ~~ \d\du{\un\a} {\un\m} \cr } \right) ~~.
   {~~~~~ ~~~}   
&(4.18) \cr } $$ 
The form for $~E\du M a $~ in (4.17a) is a simple generalization of (2.5)
into superspace, while there might be an additional term like 
$~\partial_M \L^{\un\a}$~ in (4.17b), which is excluded here just for
simplicity.  The Grassmann parity $~(-1)^M$~ is needed in (4.17a),
because of the contracted spinorial indices on $~\L$.  The $~\L^{\un\a}$~
in (4.17) is a spinor superfield, whose non-linear supersymmetry
transformation rule is   
$$ \li{ & \d_Q \L^{\un\a} 
       = \e^{\un\a} + \xi^M \partial_M \L^{\un\a} ~~,  
&(4.19) \cr } $$ 
where 
$$ \li{& \xi^m = i \big(\Bar\e \g^m \L \big) ~~, ~~~~ \xi^{\un\m} = 0 ~~.
&(4.20) \cr } $$ 
Here as in component case, $~\g^m \equiv \d\du a m \g^a$, while we 
avoid the usage of $~\g^m \equiv E\du a m \g^a $.  

We can also obtain the inverse matrix components perturbatively 
for $~E\du A M$, as  
$$ \li{ & E\du A m = E^{(0)} \du A m 
     - (-1)^A i \big( \Bar\L \g^m D_A \L \big) + {\cal O}(\L^4)~~, 
&(4.21\rm a) \cr 
& E\du A{\un\m} = \d\du A {\un\m}~~,  
&(4.21\rm b) \cr }  $$ 
up to $~{\cal O}(\L^4) \-$terms ignored compared with the $~{\cal
O}(\L^0)\-$terms.  Note the absence of $~\L\-$dependent terms in (4.21b),
which is confirmed to all orders  by the general method to get an inverse
of a supermatrix \pvn.   

We can easily show that (4.19) induces the supersymmetry 
transformation of the vielbeins   
$$ \li{ &\d_Q E\du M A = \xi^N \partial_N E\du M A + 
          \big( \partial_M \xi^N \big) E\du N A~~, ~~~~
         \d_Q E\du A M = \xi^N \partial_N E\du A M  
         - E\du A N \big( \partial_N \xi^M \big) ~~,{~~~~~}  
&(4.22) \cr } $$ 
due to the restriction (4.20).  Here we can confirm (4.22b) only up
to $~{\cal O}(\L^3)\-$terms because of the ignored terms in (4.21).  
Eq.~(4.22) implies nothing else than the general coordinate transformation
of the vielbein in superspace \superspace.    

When we consider some other superfields, such as a real scalar 
superfield $~\Phi$, its supersymmetry transformation rule is 
$$ \li{ & \d_Q \Phi = \xi^N \partial_N \Phi 
       = \d_{\rm G} (\xi^M) \Phi ~~, 
&(4.23) \cr } $$ 
to comply with (4.21).  It is not difficult to show that the commutator 
of two such non-linear supersymmetries on $~\L^{\un\a}$~ or $~\Phi$~ 
induces the desirable translations: 
$$ \li{ & \[ \, \d_Q(\e_1) , \d_Q(\e_2) \, \] = 
      + \d_P \big(\zeta_{12}^a \big) ~~,  
&(4.24) \cr } $$ 
with $~\zeta_{12}^a \equiv + 2i \big(\Bar\e_2\g^a \e_1)$~ like 
the component case (2.3).  

Note that since we are dealing with superspace, we have an additional
manifest linear supersymmetry $~\d_{Q'}$~ dictated by \superspace 
$$ \li{ & \d_{Q'} \L^{\un\a} = -
     \e{\,}'{}^{\un\b} D^{(0)}_{\un\b} \L^{\un\a}~~, ~~~~
\d_{Q'} \Phi = - \e{\,}'{}^{\un\b} D^{(0)}_{\un\b} \Phi ~~,     
&(4.25) \cr } $$  
with $~D^{(0)}_A \equiv E^{(0)} \du A M \partial_M$.  
The commutator of two such linear supersymmetries
induces the familiar translation operator:
$$ \li{ & \[ \, \d_{Q'}(\e{\,}'_1) , \d_{Q'}(\e{\,}'_2) \, \] = 
      + \d_P \big(\zeta\,'{}_{12}^a \big) ~~,  
&(4.26) \cr } $$
where $~\zeta\,'{}_{12}^a \equiv + 2i \big(\Bar\e{\,}'_2\g^a\e{\,}'_1)$~
similarly to  our non-linear supersymmetry (2.3).  Interestingly, by
simple commutator computation on $~\L^{\un\a}$~ and $~\Phi$, we see that
these two sorts of supersymmetries are commuting each other: 
$$ \li{ & \[\,  \d_Q(\e) , \d_{Q'}(\e{\,}') \] = 0 ~~, 
&(4.27) \cr } $$ 
as desired, with no interference.   

Having seen the existence of two sorts of supersymmetries, a 
natural question is whether or not these two supersymmetries are 
equivalent to each other after all, and if not, how the usual
theorem about maximal number of supersymmetries 
%%%
\ref\nahm{W.~Nahm, \np{135}{78}{149}.}%
%%%
\ref\hls{R.~Haag, J.~{\Lslash}opuszanski and M.F.~Sohnius,
\np{88}{75}{61}.} 
%%%
in each space-time 
dimension can be avoided?  Our answer to the first question is that 
these two supersymmetries are not equivalent to each other connected  
by super Weyl rescaling (field redefinitions), because the vielbeins
(4.17) for the supersymmetry $~\d_Q$~ gives geometry in
superspace different from the `flat' one $~\d_{Q'}$.  As for the second
question, we understand that the usual argument on maximal number of 
supersymmetries is based on the `linear' representation that mixes
physical fields with helicity differences of 1/2.  Since the 
non-linear representation does not shuffles such 
helicities, the usual restriction on the maximal number of extended
supersymmetries seems to be avoided here.  This is the reason why we
can set  up two different supersymmetries at the  same time, as a
$~(\hbox{Supersymmetry})^2\-$model.  

We can also postulate a possible action as a superspace
generalization of $~I_E$~ in (2.7):\footnotew{To comply with the 
superspace notation in \superspace, we use $~E \equiv \hbox{sdet}\,
\big( E\du A M\big)$~ with the inverse power compared with our
component convention.} 
$$ \li{ & I_E \equiv \int d^D x d^{\Tilde N} \theta \, 
     \hbox{sdet}\, \big( E\du M A\big) 
     \equiv \int d^D x d^{\Tilde N} \theta \, E^{-1} ~~,   
&(4.28) \cr } $$   
where $~\Tilde N \equiv 2^{\left[ D/2 \right]}\, N$~ is the number
of the $~\theta\-$coordinates for $~^\forall N$~ supersymmetry in
$~^\forall D$, counting each component of a Majorana spinor as unity.  
However, we immediately notice that such an action will produce
higher-derivative terms, as the mass dimension of the 
$~\theta\-$integral indicates, as usual in superspace
\superspace.  In fact, the superdeterminant is expanded as 
$$\li{& E^{-1}= 1+i\big(\Bar\L\g^a D_a \L\big)+{\cal O}(\L^4) ~~, 
&(4.29) \cr } $$ 
with $~D_a\equiv E\du a M \partial_M$.  
As a simple dimensional analysis reveals, some of the component fields in
$~\L^{\un\a}$~ have higher-order derivatives in their kinetic terms.  

In order to overcome this problem, we propose two models.  The first
one is to introduce some `spurion' scalar superfields
%%%
\ref\gg{L.~Girardello and M.T.~Grisaru, \np{194}{82}{65}.}
%%%
$~\Phi$~ that is
to be inserted into the above lagrangian:  
$$ \li{ & I_{E\Phi} = \int d^D x d^{\Tilde N} \theta \,  E^{-1} \Phi 
     = \int d^D x \, \big[ \, i d_0
    \big(\Bar\l\g^a D_a\l\big)+{\cal O}(\L^4) \, \big]   ~~,    
&(4.30) \cr } $$ 
with 
$$ \li{ & \Phi = \theta^{\Tilde N} d_0~~,  
&(4.31) \cr } $$ 
where $~d_0\neq 0$~ is an analog of the $~D\-$component in $~D=4$~ for the
highest sector component field of $~\Phi$, and we identify $~\L | \equiv
\l$.  The kinetic term in (4.30)  is physical with no negative energy
ghosts or higher-derivatives involved.   Even though this first model 
has the drawback due to the explicit breaking of both supersymmetries, 
it may still be of some mathematical interest, because of the co-existence
of two sorts of  supersymmetries.  We expect more improved models to be
developed in  order to avoid such explicit breakings.  

Our second model is much more advanced.  We introduce a constraint
action with a Lagrange multiplier superfield $~L^{\un\a M\un\b N}$~ for a
bilinear combination of  $~\L^{\un\a}$~ 
%%%
\ref\siegel{W.~Siegel, \np{238}{84}{307}.}:
%%%
$$ \li{& I_{L\L^2} \equiv \int d^D x d^{\Tilde N}\theta\, 
     \frac12 E^{-1} L^{\un\a a \un\b b} 
     \big(D_b \L_{\un\b} \big) \big( D_a\L_{\un\a} \big)
     ~~,  
&(4.32) \cr } $$  
where $~L$~ has no (anti)symmetry other than $~L^{\un\a a \un\b b }
= - L^{\un\b b\un\a a}$.  
Accordingly, the $~L$~ transforms in the standard fashion: 
$$ \li{& \d_Q L^{\un\a a\un\b b} = 
       \xi^M \partial _M L^{\un\a a \un\b b}
    = \d_{\rm G}(\xi^M) L^{\un\a a\un\b b} ~~, {~~~}\cr 
&\d_{Q'} L^{\un\a a\un\b b} = - \e{\,}'{}^{\un\g} D^{(0)}_{\un\g}
       L^{\un\a a\un\b b} ~~, 
&(4.33) \cr } $$ 
so that the total action $~I \equiv I_E + I_{L \L^2}$~ is invariant 
under both $~\d_Q$~ and $~\d_{Q'}$.  Notice here that the derivative 
in $~I_{L\L^2}$~ deletes the effect of the inhomogeneous term in the
transformation of $~\L$~ (4.19).   Clearly, the superfield equation
for $~L$~ from the total  action $~I$~ implies the vanishing of 
$~D_a\L_{\un\a}$:
$$\li{ & \fracm\d{\d L^{\un\a a\un\b b}} I 
     = \big( D_b \L_{\un\b} \big) \big(D_a \L_{\un\a} \big) = 0 ~~
	\Longrightarrow ~~ D_a \L_{\un\a} = 0 ~~,  
&(4.34) \cr } $$ 
because all the indices $~{\scst a,~\un\a, ~b, ~\un\b}$~ are free.  
This superfield equation preserves both the non-linear $~\d_Q$ and
linear $~\d_{Q'}$~ supersymmetries.  In particular, the effect of the
inhomogeneous term in (4.19) disappears under the derivative.  
On the other hand, the $~\L\-$superfield equation is 
$$ \li{ \fracm\d{\d \L_{\un\a}} I 
     = \, & E^{-1} (-1)^M 
     \Big( \fracm{\d E\du M{\un\b}}{\d\L_{\un\a}} \Big) 
     \left[ \, E\du{\un\b} M + \frac12 E\du{\un\b} M  
      L^{\un\g c\un\d d} \big( D_d \L_{\un\d}\big)   
     \big(D_c \L_{\un\d}\big) \, \right] \cr 
& - D_a \big( E^{-1} L^{\un\a a \un\b b} D_b 
     \L_{\un\b} \big) + {\cal O}(\L^5) ~~.   
&(4.35) \cr } $$ 
Since $~D_a \L_{\un\a}=0$~ on-shell, each term in (4.35) vanishes, 
and there arises no problem of higher-derivative kinetic term for 
$~\L$.  As usual in this sort of formulation, $~L^{\un\a a\un\b b}$~ 
does not enter any superfield equations effectively,  once (4.34) is
satisfied. Even if we add other `matter' actions to the above total
action $~I$, the NG$\-$fermion is `frozen' with no dynamics.  

The effective disappearance of the multiplier $~L^{\un\a a\un\b b}$~ from
the  set of physical field equations suggests the existence of
appropriate  gauge transformation that can eliminate $~L^{\un\a a\un\b
b}$~ \siegel.   In fact, it is easy to show that the total action $~I =
I_E + I_{L\L^2}$~ is invariant up to $~{\cal O}(\L^4)\-$terms under the
local $~\eta\-$symmetry:  
$$\li{ &\d_\eta \L_{\un\a} = \eta^b D_b
\L_{\un\a} ~~, \cr   & \d_\eta L^{\un\a a\un\b b} 
     = \big[ \, 2i\eta^a \big(\g^b \big)^{\un\a\un\b} 
     - \big( D_c \eta^a\big) L^{\un\b b \un\a c} 
     + \frac12 D_c \big( \eta^c L^{\un\a a\un\b b} \big) \, \big] 
     - {\scst ( \un\a a \leftrightarrow \un\b b) } + {\cal O}(\L^2) ~~,  
&(4.36) \cr } $$
where $~\eta^a$~ is a local vectorial parameter.   

This second model (4.32) with the constraint action $~I_{L\L^2}$~ is
more complicated than the first one (4.30), due to the supersymmetric
invariance maintained at the action as well as the superfield equation
level.  Similarly to the component case in section 2 for ordinary
space-time, the co-existence of two sorts of superspaces is common to
these two models, one with the usual flat superspace metric, and the
other with non-vanishing curvature with non-trivial vielbeins.  To our
knowledge, these models here are the first examples of this kind with the
realization of two different supersymmetries at the same time.  In
particular, in the second model, both of them are exact at the action
level.  

We draw readers' attention to the point that our formulation with the
unbroken non-linear supersymmetry is distinguished from other 
formulations with the non-linear realization as a manifestation  of
spontaneously `broken' linear supersymmetry, such as in refs.~\bw\ik. 
The interesting aspect of our formulation is that  both non-linear and
linear supersymmetries coexist in superspace with no mutual
inconsistency, that was not emphasized before.   



\bigskip\bigskip\bigskip


\vbox{\centerline {\bf 5.~~Concluding Remarks}

In this paper we have first studied non-linear realization of $~^\forall
N\-$extended global supersymmetry in arbitrary dimensions $~D$, and we
have given actions invariant under non-linear supersymmetry
transformations.   These actions and transformations are valid in {\it
arbitrary} dimensions $~D$~ and for {\it arbitrary} $\,N$, including of
course $~D\ge 12$, or $~N\ge 9$~ in $~D=4$.  This is because the 
vielbein field (2.5) transforms like the general coordinate
transformation with the parameter $~\xi^\m\equiv i
\big(\Bar\e\g^\m\l\big)$~ under the non-linear supersymmetry, even
though supersymmetry is global.  The action $~I_E$~ has non-trivial
features, because of the kinetic term of the NG$\,$-fermion, with
infinitely many interaction terms, despite of its superficially simple
form.  Even though it seems almost straightforward to generalize the
original VA action \va\ to $~^\forall D$~ and $~^\forall N$, these
generalizations will be of importance for later sections.  
}

These formulation have no need for the bosonic superpartner fields that
are indispensable in other formulations, such as in \bw, independent of 
$~D$~ or $\,N$.  This feature is very similar to the original model by
Volkov-Akulov \va\ with the single NG$\,$-fermion.  Reviewing also the 
transformation rule given in \aps, we notice that the vector field
does not enter the transformation rule of the NG$\,$-fermion, and this
signals the decoupling of the vector field from the NG$\,$-fermion
system.  From this viewpoint, it is natural that such a vector field can
be completely decoupled from the NG$\,$-fermion system, in some choice of
frame without spoiling global supersymmetry.  

For linear representation of supersymmetry, interacting models
are possible only $~D\le 11$, unless Lorentz symmetry is broken, such as
in those formulations in \twelve.  Therefore it has been believed that
any putative interacting supersymmetric model in $~D\ge 12$~ with Lorentz
covariance is `doomed' to fail.  The result in this paper have broken
this `taboo', namely we have seen that it is certainly possible to
formulate global non-linear supersymmetry in $~D\ge 12$, as long as
spinor fields can be defined.  In other words, global supersymmetry and
Lorentz symmetry can co-exist for any given $~D$~ and $\,N$, only at the
expense of the linear realization of local supersymmetry.  We now know
that interacting models with non-linear realization of global
supersymmetries are possible even for $~N\ge 9$~ in $~D=4$, or $~N\ge
2$~ in $~D=11$, or in $~D\ge 12$.  Since the non-linear realization of
supersymmetry is possible even in $~D\ge 12$~ with global Lorentz
symmetry, we do not take the standpoint that non-linear 
supersymmetry is a reflection of `broken' supersymmetries \bw.   

Although it emerged just as an interesting by-product, the
supersymmetric Born-Infeld action for a non-Abelian gauge group for
$~^\forall D$~ and $~^\forall N$~ we gave is a new result in this paper.  
Like other actions, this action also has non-trivial but consistent
interactions with the NG$\,$-fermion, and  it agrees in the special case
of $~D=10$~ with the result in \aps\ for the lowest-order terms. 
When the NG-fermion carries the adjoint index of the non-Abelian 
group, the lowest-order terms of our Born-Infeld action (3.6) coincide
with those in the linearly supersymmetric action \aps\ in $~D=10$.  
This seems to imply that our Born-Infeld action (3.6) provides an 
equally important foundation for the M(atrix)-theory \bfss\ as the 
linearly supersymmetric Born-Infeld action \aps\ for D-branes 
\pol\dpbranes\nilsson\sorokin, up to some recently-discovered subtlety 
at higher orders \mtheoryrecent.  Interestingly enough, since we have 
$~\dim G\-$extended supersymmetry, the usual large $~{\cal N}$~ limit 
for $~G=U({\cal N})$~ \bfss\ corresponds to 
$~N = {\cal N}^2\rightarrow\infty$, {\it i.e.,} infinitely many
($\large{\aleph_0}$)  extended supersymmetry limit for our non-linear
realization.  

Additionally, we have shown how to perform duality transformation from an
Abelian field strength $~F_{\m\n}$~ in a Born-Infeld (type) action into
its Hodge-dual field strength $~N_{\m_1\cdots \m_{D-2}}$~ in $~^\forall
D\neq 4$~ and $~^\forall N$, excluding the case of $~D=4$, in which
certain subtlety arises related to the $~SL(2,\IR)$~ duality
\dualityinv.        

As another important application, we have pointed out that our 
supersymmetric Born-Infeld action $~I_{\rm N A S B I}$~ is a suitable
`matter' action compatible with Chern-Simons actions, in which there is
no fundamental metric from the outset.  The gauge groups can be even a 
supergroup such as $~\hbox{\it OSp}(N|m)$, {\it etc}.  Another 
important aspect is the fact that any Chern-Simons action
(3.16) for any supergroup formulated in $~^\forall D = 2n+1$~
\cham\zanelli\phvnw\ has hidden global supersymmetry under (3.1b)
realized non-linearly.  This may well provide the long-standing `missing
link' between the Chern-Simons theories of supergroup and Poincar\' e
supergravity with diffeomorphism \pvn, especially in the context of the
superalgebra in M-theory \zanelli\phvnw\mn.      

In our formulation, we found that the minimal field content for 
supersymmetric Born-Infeld action is the NG$\,$-fermion and 
a vector field, both for Abelian and non-Abelian cases.  If we perform 
simple dimensional reduction from the starting dimensions $~D$~
into one dimension lower: $~D-1$, then one component of the vector 
field becomes a scalar field, so that more and more scalars are
generated, if we go further down \aps.  However, this is not the only
scenario we can think of, because in our formulation, at each dimension
$~D$~ we can formulate a supersymmetric Born-Infeld action with the
minimal set of fields, {\it i.e.,} one NG$\,$-fermion and a vector
field, but no scalars or any other fields.  This is also one of the
distinctive features of our formulation compared with other D-brane
formulations
\aps\dpbranes.  

Our result of extended supersymmetries on 
$~^\forall N$~ has other analogs.  
For example, similar situations have been already presented 
for infinitely many local supersymmetries in $~D=3$~ 
%%%
\ref\ngaleph{H.~Nishino and S.J.~Gates, Jr., \np{480}{96}{573}.}
%%% 
or arbitrarily many global supersymmetries in $~D=1$~  
%%%
\ref\gr{S.J.~Gates, Jr.~and L.~Rana, \pl{352}{95}{50}; 
\ibid{369B}{96}{269}.}.  
%%% 
These so-called \alephnull aspects of supersymmetries 
have already shown up in many different contexts in the past, and now  
we have explicit models at hand that elucidate such 
`universality' more explicitly in $~^\forall D$.  

We have also reformulated the component results in superspace, and in
particular, confirmed the Bianchi identities for a second-rank
superfield strength for a non-Abelian group and a third-rank superfield
strength, as explicit examples.  We have seen that the components
$~F\du{\un\a\un\b}I$~ or $~G_{\un\a\un\b\un\g}$~ are needed, and the
special factor $~P\du{\un\a}b \equiv i\big(\g^b \l \big)_{\un\a}$~ is
involved in a peculiar way, as a generalization of those given in
\nilsson.  Our constraints have essentially off-shell structure, namely
the  NG$\,$-fermion equations are not implied by the Bianchi identities,
as opposed to the usual on-shell superspace formulations in
higher-dimensions, or constrained superfield formulations in $D=4$~ 
\rocek\bw.  Remarkably, these constraints are valid for ~$^\forall D$~
and $~^\forall N$, which, to our knowledge, have not been emphasized in
the literature.  

As another important result, we have presented an universal 
Green-Schwarz $~\s\-$model super $~(2k-1)\-$brane action 
\pbrane\eightbrane\ in $~D=2k\-$dimensions that can accommodate even
curved superspace backgrounds, when we need to include supergravity,  by
introducing the maximally antisymmetric potential superfield 
$~C_{a_1\cdots a_{2k}}$~ \eightbrane.  The requirement of satisfaction
of $~T\-$Bianchi identities for supergravity with Lorentz covariance
restricts the dimensionality $~D$~ to be $~D=2k \, ~(1 \le k\le 5)$.  

This $~(2k-1)\-$brane formulation suggests the significance of the
target space-time whose dimensionality coincides with that of the
world-supervolume.  Motivated by this, we have given a `simplified'
$~(\hbox{Supersymmetry})^2\-$models as another by-product of our
formulations, in which linear and non-linear representations of
supersymmetries coexist at the same time.  
This is a simplified version of more
general $~(\hbox{Supersymmetry})^2\-$models, initiated in 
2D \supersquare.  We have seen that the original ansatz of forming a
vielbein in terms of NG$\,$-fermion is directly generalized into
superfields, and we have also given some explicit models with the
simultaneous realization of both linear and non-linear supersymmetries. 
In particular, the second model has both of these supersymmetries
explicit at the action level.  In this sense, our formulation is
distinguished from others, such as refs.~\bw\ik, in which non-linear
supersymmetry is regarded as spontaneous `breaking' of linear
supersymmetry.  

We saw that neither the metric nor the vielbein field in our formulation
is an elementary field, but they are effectively composite in terms of
the NG$\,$-fermion.  From this viewpoint, our theory is similar to the
Bars and MacDowell theory   
%%%  
\ref\bm{I.~Bars and W.S.~MacDowell, \pl{71}{77}{111}.}
%%%
in which the metric tensor is not an elementary field, but is a
composite bilinear form of gravitino which is the fundamental field. 
The main difference, though, is that the formulation in \bm\ contains a
gravitino as a gauge field for the local supersymmetry, while in our
theory supersymmetry is global but is realized non-linearly.   Most of
our actions in general space-time dimensions are only globally
supersymmetric in flat $~D$~ dimensions, except the super
$~(2k-1)\-$brane actions which can accommodate curved superspace
backgrounds.  

One might say that a possible drawback in our formulation is the loss of
`uniqueness' of actions for particle physics.  This is because
there are essentially no restrictions on possible invariant lagrangian
by supersymmetry, as long as the general covariance $~\d_{\rm G}$~ is
maintained in terms of our `metric' or `vielbein' for $~^\forall D$~ and
$~^\forall N$.  However, we instead stress the importance of
`universality' of supersymmetry more than `uniqueness' of a particle
theory.  Our result here opened a completely new avenue, {\it i.e.},
supersymmetry does not necessarily restrict the space-time dimensions for
possible interacting models for physics.  To our knowledge, the
significance of such universality has have been overlooked, or  
at least not emphasized during the decades since the first work in \va.  

One final point to be stressed is the co-existence of two sorts of 
space-times or even superspaces simultaneously, which are not equivalent
to each other:  One with flat metric, while the other with highly
non-trivial vielbeins with general coordinate transformations.  We have
seen this for component as well as superspace formulations.  However,
this is in a sense puzzling, because it is well-known in mathematics
to be impossible to embed a `curved' space-time into a `flat' space-time
of the same dimensionality.  Our solution to this puzzle is that the
curved space-time is `non-perturbatively' realized from the `flat'
space-time,  by adding up infinite series in terms of NG$\-$fermions.

Moreover, the co-existence of two different superspaces is also related
to the co-existence of two sorts of supersymmetries, one in linear and
the other in non-linear representations.  These aspects of non-linear
supersymmetry may well be different manifestations of the intricate
feature of M-theory \hulltownsendwitten\ for curved supergravity in
$~D=11$~ realized first as a global supersymmetry in flat space-time,
{\it via} a simple matrix theory \bfss.   

We believe that the results given in this paper can provide a first 
step into the studies of more general features of non-linear
supersymmetries in $~^\forall D$~ and $~^\forall N$, motivated also by
the recent development of D-branes \pol.  We have even seen that the
non-linear supersymmetry can co-exist with linear supersymmetry as simple
$~\hbox{(Supersymmetry)}^2\-$model in superspace.  We have also seen
that the non-linear realization as compared with the linear realization
of supersymmetry has more freedom for accommodating a much wider class of
models.



\bigskip\bigskip

We are grateful to J.~Bagger, S.J.~Gates, Jr., F.~Gonzalez-Rey, M.~Luty, 
M.~Ro\v cek, J.H.~Schwarz and W.~Siegel for helpful discussions.  
Special acknowledgement is for S.J.~Gates, Jr.~for 
many suggestions to improve the paper.  


\bigskip\bigskip

{\it Note Added:}  ~After completing this paper, we have encountered 
a recent work by P.C.~West 
%%%
\ref\west{P.~West, {\it `Automorphisms, Nonlinear Representations, and 
Branes'}, \hepth{0001216}.}, 
%%%
in which superembedding \sorokin\ for non-linear realizations 
is performed by enlarging the automorphism group of supersymmetry.      


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