hep-th0103224.tex
\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)
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	\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
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	\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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	\par\vskip-8mm}
\def\headpic{						% UM heading
	\indent
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
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	\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{\hskip2.5pt_{\scriptscriptstyle\bullet}}\over{#1}}
%\def\dt#1{\Dot{#1}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}\,} \def\Tr{\,\,{\rm Tr}\,}
\def\det{{\rm det}\,} 
\def\Norm{|\hskip -2.0pt|}
\def\Sgn{\hbox{\rm Sgn}\,}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} \def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
                              \scriptscriptfont0\fivebf
            \textfont1\tenmib \scriptfont1\sevenmib
                               \scriptscriptfont1\fivemib
            \textfont2\tenbsy \scriptfont2\sevenbsy
                               \scriptscriptfont2\fivebsy}
 %
\def\rlx{\relax\leavevmode}                  
 % Guess what this is for...
\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
                      {\hbox{$\scriptscriptstyle\BMfont#1$}}
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\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}
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\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
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\font\cmss=cmss10
\font\cmsss=cmss10 at 7pt
\def\ZZ{\rlx\leavevmode
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                    {\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}}}
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\def\IP{\rlx{\rm I\kern-.18em P}}
\def\IR{\rlx{\rm I\kern-.18em R}}
\def\IN{\rlx{\rm I\kern-.20em N}}
\def\Ione{\rlx{\rm 1\kern-2.7pt l}}
%
%%% apple lw
\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.59truein}
\def\speclscape{\special{ps: landscape}}

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

% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string\string##1}}}
\nolabels
%
% tagged sec numbers
\global\newcount\secno \global\secno=0
\global\newcount\meqno \global\meqno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi  %if desired
\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1\message{(\secsym\the\subsecno. #1)}
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\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
%
%%% DO NOT ERASE THE FOLLOWING definitions of \def\footnote
\doit0{
\def\footnote#1{\let\@sf=\empty 
 \ifhmode\edef\@sf{\spacefactor=\the\spacefactor}\/\fi
 #1\@sf\vfootnote{#1}}
\def\vfootnote#1{\insert\footins\bgroup
 \interlinepenalty=\interfootnotelinepenalty
 \splittopskip=\ht\strutbox
 \splitmaxdepth=\dp\strutbox \floatingpenalty=20000
 \leftskip=0pt \rightskip=0pt \spaceskip=0pt \xspaceskip=0pt
 \textindent{#1}\footstrut\futurelet\next\fo@t}
\def\fo@t{\ifcat\bgroup\noexpand\next \let\next\f@@t
 \else\let\next\f@t\fi \next}
\def\f@@t{\bgroup\aftergroup\@foot\let\next}
\def\f@t#1{#1\@foot}
\def\@foot{\strut\egroup}
\def\footstrut{\vbox to\splittopskip{}}
\skip\footins=\bigskipamount 
\count\footins=1000
\dimen\footins=8in 
This is a test for footnote.\footnote*{This is a footnote.}
\end{document} 
}

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

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

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

\count100=1  

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

\font\LargeLarge=cmr17 scaled \magstep5 

\def\alephnull{~$\large{\aleph_0}\,$~} 
\def\alephnulllarge{${\displaystyle\aleph_0}$~} 
\def\Bo{\bo{\hskip 0.03in}} 
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) }
\def\.{.$\,$} 
\def\alpr{\a{\hskip 1.2pt}'} 

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

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

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

\def\hepth#1{hep-th/{#1}}

\def\binomial#1#2{\left(\,{\buildrel 
{\raise4pt\hbox{$\displaystyle{#1}$}}\over 
{\raise-6pt\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\fracms#1#2{{{\small{#1}}\over{\small{#2}}}} 
\def\low#1{{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}} 

\def\ip{{=\!\!\! \mid}} 
\def\Lslash{${\rm L}{\!\!\!\! /}\, $} 
\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~} 
\def\eqstar{~{\buildrel * \over =}~} 

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

\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~}
\def\Check#1{\raise0pt\hbox{$\buildrel{\raise0pt\hbox{$_{_{_\smile}}$}}%
\over{\raise1pt\hbox{${#1}$}}$}}

% If we need the framing in the cover page, put 1 after the following 
% \framing-command, and put 0 otherwise: 
\framing{0} 

\doit1{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0103224$\,$}}\par 
{\vskip -0.06in}
{\hbox to\hsize{~~~~~ ~~~~~ \hfill UMDEPP 01--046}}\par
\vskip -0.15in 
\hfill 
{(Revised Version)}  \\ 
\begin{center} 
%\vglue .25in 

\vskip -0.23in

{\large\bf A $\,$Note $\,$on $\,$Embedding $\,$of $\,$M-Theory
$\,$Corrections} \\[0.026in] 
{\large\bf into $\,$Eleven-Dimensional $\,$Superspace}\footnote{This 
work is supported in part by NSF grant \# PHY-98-02551.} \\[.1in] 

\doit0{
{\bf\largetitle A ~Note ~on ~Embedding ~of ~M~-~Theory
~Corrections}
\\[0.026in] 
{\bf\largetitle into ~Eleven~-~Dimensional ~Superspace}\footnote{This 
work is supported in part by NSF grant \# PHY-98-02551.} \\[.1in] 
}

\baselineskip 9pt 

\vskip 0.20in 

Hitoshi ~N{\smallcmr ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
\\[.15in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland} \\ [.015in] 
{\it College Park, MD 20742-4111} \\[.20in]   

and \\[0.18in] 

Subhash ~R{\smallcmr AJPOOT}\footnote{E-Mail:
rajpoot@csulb.edu}
\\[.15in] 
{\it Department of Physics \& Astronomy} \\ [.015in] 
{\it California State University} \\ [.015in] 
{\it Long Beach, CA 90840} \\[0.02in]   


\vskip 1.2in

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

\baselineskip 14.0pt 



By analyzing eleven-dimensional superspace fourth-rank 
superfield strength $F$-Bianchi identities,
we show that M-theory corrections to eleven-dimensional supergravity can
not be embedded into the mass dimension zero constraints, such as the
$~(\g^{a b})_{\a\b} X_{a b}{}^c~$ or $~i (\g^{a_1\cdots a_5})_{\a\b}
X_{a_1\cdots a_5}{}^c\,\-$terms in the supertorsion constraint
$~T_{\a\b}{}^c$.  The only possible modification of superspace
constraint at dimension zero is found to be the scaling of $~F_{\a\b c
d}$~ like $~F_{\a\b c d} = (1/2) 
\big(\g_{c d}\big)_{\a\b} \, e^\Phi$~ for some real scalar superfield
$~\Phi$, which alone is further shown not enough to embed general
M-theory corrections.  This conclusion is based on the dimension zero
$F$-Bianchi identity under  the two assumptions:  (i) There are no
negative dimensional constraints on the $F$-superfield strength:
$F_{\a\b\g\d} = F_{\a\b\g d} =0$;  (ii) The supertorsion $T$-Bianchi
identities and $F$-Bianchi identities are not modified by Chern-Simons
terms. Our result can serve as a powerful tool for future exploration of
M-theory corrections embedded  into eleven-dimensional superspace
supergravity.  


\vskip 0.30in

\leftline{\small PACS: 04.50.+h, 04.65.+e, 11.30.Pb}
\vskip -0.05in
\leftline{\small Key Words: M-Theory, Supergravity, Superspace, 
Eleven-Dimensions}

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\centerline{\bf 1.~~Introduction} 

If M-theory 
%%% 
\ref\mtheoryrev{{\it For reviews of M(atrix)-theory, see, e.g.,}  
A.~Bilal, Fort.~f\"ur Phys.~{\bf 47} (1999)
5; T.~Banks, {\it `TASI Lecture Note on Matrix Theory'}, \hepth{9911068}; 
W.~Taylor IV, {\it The M(atrix) 
Model of M-Theory'}, Lectures for NATO school {\it `Quantum Geometry'}
(Iceland 1999), \hepth{0002016}; {\it and references therein}.}        
%%%
unifies superstring theories 
%%% 
\ref\gsw{M.~Green, J.H.~Schwarz and E.~Witten, {\it `Superstring
Theory'}, Vols.~I and II,  Cambridge University Press (1987).}, 
%%%  
such as type-I, heterotic and type-IIA superstring theories, and its low
energy limit is described by eleven-dimensional (11D) supergravity 
%%% 
\ref\cjs{E.~Cremmer, B.~Julia and J.~Scherk, \pl{76}{78}{409}.}, 
%%% 
it is natural to expect that there must be high energy corrections to
11D supergravity compatible with supergravity formulation itself, just as 
superstring corrections can be embedded into 10D supergravity \gsw
%%% 
\ref\gn{{\it See, e.g.,} S.J.~Gates, Jr.~and H.~Nishino, 
\pl{173}{86}{52}; \np{291}{87}{205}; 
S.J.~Gates, Jr.~and S.I.~Vashakidze, \np{291}{87}{173}.}.  
%%% 
Based on this principle, there have been attempts to embed such 
M-theory corrections into 11D supergravity, such as in component
formulation 
%%% 
\ref\peeters{K.~Peeters, P.~Vanhove and A.~Westerberg, 
{\it `Supersymmetric $~R^4$~ Actions and Quantum corrections to 
Superspace Torsion Constraints'}, Talk at NATO Advanced Research
Workshop on Noncommutative Structures in Math.~\& Phys., Kiev 
(Sept.~2000), hep-th/0010182; Class.~Quant.~Grav.~{\bf 18}
(2001) 843, \hepth{0010167}.}, 
%%% 
as well as in superspace formulation 
%%% 
\ref\ng{H.~Nishino and S.~J.~Gates, Jr., \pl{388}{96}{504},  
\hepth{9602011}.}% 
%%% 
\ref\howe{P.~Howe, \pl{415}{97}{149}, \hepth{9707184}.}% 
%%% 
\ref\cgnn{M.~Cederwall, U.~Gran, M.~Nielsen and B.~Nilsson,
{\it `Manifestly Supersymmetric M-Theory'}, JHEP {\bf 0010} (2000) 041,
\hepth{0007035}; {\it `Generalized 11-Dimensional Supergravity'}, 
hep-th/0010042.}%  
%%%  
\ref\gnrecent{S.J.~Gates, Jr., and H.~Nishino, {\it `Deliberations 
on 11D Superspace for the M-Theory Effective Action'},
Phys.~Lett.~B, in press, \hepth{0101037}.}.    
%%%  

In superspace formulation \howe\cgnn, it is so far commonly believed or 
expected that such M-theory corrections are most likely embedded into the
generalized symmetric matrix components of the mass dimension
zero ($d=0$) supertorsion component $~T_{\a\b}{}^c$, such as 
$~(\g^{a b})_{\a\b} X_{a b}{}^c~$ or 
$~i (\g^{a_1\cdots a_5})_{\a\b} X_{a_1\cdots a_5}{}^c~$ with some 
appropriate superfields $~X\du{a b} c$~ or $~X_{a_1\cdots a_5}{}^c$~ 
\howe\cgnn%
%%%
\ref\townsend{{\it See, e.g.,} 
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; \pl{412}{97}{265}, 
\hepth{9712004}.}.   
%%%
However, it is not clear whether we need only the
corrections of $~T\du{\a\b}c$~ or $~F_{\a\b c d}$~ at $~d=0$~ alone for
embedding M-theory corrections, or we also need any negative dimensional 
($d<0$) constraints, such as $~F_{\a\b\g d}$~ or 
$~F_{\a\b\g\d}$~ for such modifications for $F$-Bianchi identities
(BIs).    

In this Letter, we will present a `no-go theorem' for embedding M-theory 
corrections into constraints for superspace BIs with
modified constraints only at $~d=0$.  We will show that the $~d=0$~
corrections of constraints are not enough for embedding M-theory
corrections into 11D superspace supergravity.  Our conclusion is based on  
two assumptions:
(i) All the $F$-superfield strength constraints at $~d < 0$~ vanish;
(ii) The supertorsion $T$-BIs and $F$-BIs are not modified by
Chern-Simons terms.  In addition to these assumptions, our conclusion
also relies on the so-called   `conventional constraints' that relate
various superfields in the most general  expansions of the superspace
derivatives: $~E_\a$~ and $~E_a$~ \gnrecent.  
% These conventional constraints restrict, {\it e.g.,} 
% that one-gamma term in $~T\du{\a\b} c$~ is only the standard one: 
% $i~\big(\g^c\big)_{\a\b}$, while allowing the general two-gamma term
% $~\big(\g^{d e} \big)_{\a\b} X\du{d e} c$~ and the five-gamma term 
% $~i \big(\g^{d_1\cdots d_5} \big)_{\a\b} X\du{d_1\cdots d_5} c$~
% corrections.  Furthermore, these $~X$'s themselves are restricted to be 
% {\it e.g.,} $~X\du{a b}b=0$, {\it etc.}, as will be seen.  
 These conventional constraints are restrictive, {\it e.g.,} 
the one-gamma term in $~T\du{\a\b} c$~ is only the standard one: 
$i~\big(\g^c\big)_{\a\b}$, while  the  two-gamma term
$~\big(\g^{d e} \big)_{\a\b} X\du{d e} c$~ and the five-gamma term 
$~\big(\g^{d_1\cdots d_5} \big)_{\a\b} X\du{d_1\cdots d_5} c$~
corrections are of a general form.  Furthermore, the $~X$'s themselves are
restricted {\it e.g.,} $~X\du{a b}b=0$, {\it etc.}, as will be shown later.

A statement for the necessity of the $~F\-$constraints at $~d<0$~ has
been given in \cgnn, but without any proof.  
In the present paper, we provide explicit evidence for that claim.  
By studying the $F$-BI at $~d=0$,
we show that,  as long as the $F$-constraints at $~d<0$~ are absent,
there can be {\it no} such corrections as
$~\big(\g^{d e}\big)_{\a\b} X\du{d e} c$~ or $~\big(\g^{d_1\cdots d_5}
\big)_{\a\b} X\du{d_1\cdots d_5} c$~ possible in $~T_{\a\b}{}^c$~ other
than the standard one-gamma term that can embed M-theory corrections. 


\bigskip\bigskip\bigskip

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\centerline{\bf 2.~~Solving $F$-BI at $~d=0$}

We first give the most important part of our results here, namely 
we analyze whether the $~d=0$~ $F$-BI of the type
$~(\a\b\g\d e)$: 
$$ \li{ & \frac16 \nabla_{(\alpha} F_{\beta\gamma\delta) e}  
	     + \nabla_e F_{\alpha\beta\gamma\delta} \cr 
&~~~~~  -  \frac16 T_{ e ( \alpha | }{}^f F_{f | \beta \gamma\delta) } 
	     -  \frac16 T_{ e ( \alpha | }{}^\eta 
      F_{\eta | \beta \gamma\delta) }  
      - \frac14 T_{ ( \alpha\beta | }{}^f F_{f | \gamma\delta) e} 
 	      - \frac14 T_{ ( \alpha\beta | }{}^\eta 
 	      F_{\eta | \gamma\delta) e} \equiv 0 
&(2.1) \cr } $$  
allows any non-trivial solution possible for embedding 
M-theory corrections \cgnn.  
For simplicity (as the assumption of our `no-go theorem'), we put 
the $~d < 0$~ $F$-constraints to be zero:  
$$ \li{ & F_{\alpha\beta\gamma\delta} = 0~~, 
~~~~ F_{\alpha\beta\gamma d}=0 ~~. 
&(2.2) \cr } $$  
Accordingly, there remains only one term in (2.1) at $~d=0$:   
$$ \li{ & T_{ ( \alpha\beta | }{}^f 
	F_{f | \gamma\delta) e} \equiv 0 ~~. 
&(2.3) \cr } $$ 

It is now a purely algebraic question whether there can be any 
non-trivial solution to (2.3), when we postulate 
$$ \li{ & T_{\a\b}{}^c = i (\g^c)_{\a\b} 
     + \frac12 (\g^{a b})_{\a\b} X_{a b}{}^c 
	    + \frac i{120}  (\g^{a_1\cdots a_5})_{\a\b} 
	X_{a_1\cdots a_5}{}^c~~,   
&(2.4) \cr } $$ 
for the $~d=0$~ supertorsion constraint for $~T_{\a\b}{}^c$~ \cgnn.
Here we have no corrections for the first one-gamma term, due to the 
`conventional constraints' relating $~E_\a$~ and $~E_a$~ as in eqs.~(24) 
and (25) in \gnrecent.  On the other hand, $~X\du{a b} c$~ 
and $~X_{a_1\cdots a_5}{}^c$~ are some appropriate superfields that can 
possibly embed M-theory corrections \howe\cgnn.  Accordingly, we need to put
the most general corrections also into the $~d=0$~ 
$F$-constraint\footnotew{In this paper, the symbol $~{\scst \[ n\]}$~ 
in general  denotes the total antisymmetrization of $~n$~ indices, {\it e.g.,}
$~\big(\g^{[5]}\big)_{\a\b} \big(\g\low{\[5\]}\big)_{\g\d}
\equiv \big(\g^{a_1\cdots a_5}\big)_{\a\b} \big(\g\low{a_1\cdots
a_5}\big)_{\g\d}$.}  
$$ \li{& F_{\alpha\beta c d} = i (\gamma^e)_{\alpha\beta} U_{e c d} 
      + \frac12 (\gamma^{e f})_{\alpha\beta} U_{e f c d} 
	+ \frac i{120} (\gamma^{\[5\]})_{\alpha\beta} U_{ \[5\] c d}   ~~. 
&(2.5) \cr} $$ 
In particular, the lowest order on-shell physical superfield constraint in 
%%% 
\ref\cfbh{E.~Cremmer and S.~Ferrara, \pl{91}{80}{61}; 
L.~Brink and P.~Howe, \pl{91}{80}{384}.}  
%%% 
corresponds to the special case of 
$$ \li{ & F_{\a\b c d} = + \frac 12 \big(\g_{c d} \big)_{\a\b}~~, ~~~~
        U_{a b c d} = + \frac 14 \eta\low{a \[ c}\eta\low{ d \] b}~~,   
&(2.6) \cr } $$
with $~U_{a b c} = U_{a_1\cdots a_5 b c} = 0$.    
Since we are considering M-theory corrections, these $~X$~
and $U\-$superfields can be dealt as perturbation, namely we can first
consider the satisfaction of the BI (2.3) at the linear order, temporarily
ignoring the bilinear order (cross terms).  For this reason, we
concentrate on the analysis at the linear order in terms of $~X$'s 
and $~U$'s.   

Based on the fluctuation analysis \gnrecent, 
we can impose on the $~X$'s the following `conventional constraints'
$$ \li{ & X_{a b}{}^b = 0 ~~, ~~~~ X_{\[ a b c \]} = 0 ~~,~~~~ 
	X_{a_1 \dots a_4 b}{}^b = 0 ~~, ~~~~ 
	X_{\[ a_1 \dots a_5 b \]} = 0 ~~.
&(2.7) \cr } $$ 
Under these constraints, $~X\du{a b}c$~ has only 429 degrees of freedom,
while $~X\du{\[5\]} c $~ has 4,290 degrees of freedom.   
These constraints are analogous to the familiar torsion constraint $~T\du{a b}c
=0$~ commonly used in superspace, which does not delete any degrees of 
freedom.  To be more specific, we saw in the fluctuation analysis in \gnrecent\
that some components of superfield $~\Delta\du\a\b$~ \gnrecent\
entering in $~E_\a \equiv E\du\a M\partial_M$~ can be expressed in terms
of $~H\du\a b$~ under (2.7):  
$$\li{ & E_\a = \Psi^{1/2} \big\{ \exp\big( \frac12 \Delta\big) \big\}\du\a\b
	\big( D_\b + H\du\b m \partial_m \big) ~~, 
&(2.8\rma) \cr 
& \Delta\du\a\b \equiv \Big[\, i \Psi^a\g_a  + \frac12 \Psi^{a b} \g_{a b} 
	+ \frac i 6 \Psi^{\[3\]} \g\low{\[3\]}  
	+ \frac 1{24} \Psi^{\[4\]} \g\low{\[4\]}  
	+ \frac i{120} \Psi^{\[5\]} \g\low{\[5\]}\, \Big]\du\a\b ~~, 
&(2.8\rmb) \cr 
& X_{a b}{}^c  =  \fracm1{16} \Big[\, 
      (\g_{a b})^{\g\d} D_{\g}{\rm H}{}_{\d}{}^{c} +  16 
     \d_{\[a}{}^c \Psi_{b \]} -  16  \Psi_{a b}{}^c \, \Big] ~~, 
&(2.8\rmc)  \cr
& X_{a_1\cdots a_5}{}^c = \fracm1{16} \Big[\, 
     i (\g_{a_1\cdots a_5})^{\g\d} D_{\g}{\rm H}{}_{\d}{}^{c} +
      \fracm{2}{3} \d_{\[ a_1}{}^c \Psi_{a_2\cdots a_5 \]} 
      + \fracm 2{15}
\e{\,}^c{}_{ a_1\cdots a_5}{}^{\[5\]} \Psi_{\[5\]} \, \Big] ~~.   
&(2.8\rmd) \cr } $$ 
As seen from (2.8c) and (2.8d), the constraints (2.7) are just 
equivalent to determining $ ~\Psi^{\[1\]}, ~\Psi^{\[3\]}, ~\Psi^{\[4\]}$ 
and ~$\Psi^{\[5\]}$~ in terms only of $~H\du\a b$ ~\gnrecent.  
This situation is similar to the constraint $~T\du{a b} c =0$~ 
in order to express the Lorentz connection $~\phi\du{m a}b$~ in terms 
of the vielbein $~e\du a m$.  Therefore we stress that there is no loss of 
degrees of freedom under the covariant constraints (2.7).   

The original tensor superfields $~X$'s or $~U$'s are  reduced into more
fundamental irreducible components.  For example, the original 
$~U_{a b c}$~ has 605 components considering their symmetries, which can
be decomposed into $~U_{a b c} = {\cal A}^{\{ 429 \} }_{a b c} + {\cal
A}^{\{ 165 \}}_{a b c} + \eta\low{a\[ b} {\cal  A}^{\{ 11 \} }_{c\]}$,
where $~{\cal A}^{\{ 165 \} }_{a b c}$~ is totally antisymmetric,  while
the $~{\bf 429}\-$part is traceless: $~{\cal A}^{\{ 429 \} } {}\dud a a
c=0$, as the remainder degrees of freedom are out of the  original $~11 \times
55 = 605$~ components.  In a similar fashion, we can decompose the rest of
$~U$'s as 
$$\li{ & U_{a b c} = + {\cal A}^{\{429\}}_{a b c} + 
      {\cal A}^{\{165\}}_{a b c} 
      + \eta\low{a \[b} {\cal A}^{\{11\}}_{c\]} ~~, \cr 
& U_{a b}{}^{c d} = + {\cal A}^{\{2,574\}}_{~~~~~a b}{}^{c d} 
     + {\cal A}^{\{330\}}_{~~~~a b}{}^{c d} 
     + \d\du{\[a|}{\[c|} {\cal A}^{\{65\}}_{~~|b\]}{}^{|d\]}
     + \d\du{\[a|}{\[c|} {\cal A}^{\{55\}}_{~~|b\]}{}^{|d\]}
     + \d\du{\[a|}{\[c|} \d\du{|b\]}{|d\]} {\cal A}^{\{1\}} ~~, \cr  
& U_{a_1\cdots a_5 b c} = + {\cal A}^{\{17,160\}}_{a_1\cdots a_5 b c}
     + \frac1{120} \e\low{a_1\cdots a_5 \[b|}{}^{d_1\cdots d_5} 
     {\cal A}^{\{4,290\}}_{d_1\cdots d_5|c\]} 
     + \eta\low{\[b|\[a_1|} 
     {\cal A}^{\{3,003\}}_{|a_2\cdots a_5\] |c \]} \cr 
& ~~~~~ ~~~~~ ~~~~ + \eta\low{\[b| \[a_1|} 
     {\cal A}^{\{462\}}_{|a_2\cdots a_5\] | c\]} 
    + \frac1{24} \e\low{a_1\cdots a_5 b c}{}^{\[4\]} 
     {\cal B}^{\{330\}}_{\[4\]} 
    + \eta\low{b\[a_1|} \eta\low{|a_2| c} 
     {\cal B}^{\{165\}}_{|a_3 a_4 a_5\]} ~~. 
&(2.9) \cr } $$ 

Substituting each irreducible component back into (2.3), we get the set
of algebraic conditions to be satisfied for each irreducible components:
$$\li{ & i \big(\g^d \big)_{(\a\b|} \big(\g_{d e} \big)_{|\g\d)} 
          {\cal A}^{\{ 1 \}} = 0 ~~, 
&(2.10\rma)  \cr   
& \big(\g^d \big)_{(\a\b|} \big(\g_{\[d|} \big)_{|\g\d)} 
          {\cal A}^{\{ 11 \}}_{|e\]}  = 0 ~~, 
&(2.10\rmb)  \cr   
& i \big(\g^a \big)_{(\a\b|} \big(\g^{b c}\big)_{|\g\d)} 
          \Big[ \, \eta\low{a\[b|} {\cal A}^{\{ 55 \}}_{| c\] e} 
          - \eta\low{e \[b|} {\cal A}^{\{ 55 \}}_{|c\] a} \, \Big] 
          = 0 ~~,  
&(2.10\rmc)  \cr   
& i \big(\g^a \big)_{(\a\b|} \big(\g^{b c}\big)_{|\g\d)} 
          \Big[ \, \eta\low{a\[b|} {\cal A}^{\{ 65 \}}_{|c\] e} 
          - \eta\low{e \[b|} {\cal A}^{\{ 65 \}}_{|c\] a} \, \Big] 
        = 0 ~~, 
&(2.10\rmd)  \cr   
& \big(\g^d \big)_{(\a\b|} \big(\g_{d e}{}^{\[3\]}
      \big)_{|\g\d)}  {\cal B}^{\{ 165 \}}_{\[3\]} = 0 ~~, 
&(2.10\rme)  \cr   
& \frac i2 \big(\g^d \big)_{(\a\b|} \big(\g^{a b} \big)_{|\g\d)} 
          {\cal A}^{\{ 330 \}}_{a b d e} 
       - \frac1{24}\frac1{120} 
       \big(\g^d \big)_{(\a\b|} \big(\g^{\[5\]} \big)_{|\g\d)} 
       ~\e\low{\[5\] d e}{}^{\[4\]}{\cal B}^{\{ 330 \}}_{\[4\]} 
       = 0 ~~, 
&(2.10\rmf)  \cr 
& - \big(\g^a \big)_{(\a\b|} \big(\g^b \big)_{|\g\d)} 
          {\cal A}^{\{ 429 \}}_{a b e} 
      + \frac 14 \big(\g^{a b} \big)_{(\a\b|} \big(\g_{d e}\big)_{|\g\d)} 
      X\du{a b} d = 0 ~~, 
&(2.10\rmg)  \cr  
& \big( \g^d \big)_{(\a\b|} \big(\g^{b\[4\]} \big)_{|\g\d)} 
          \eta\low{\[d| b} {\cal A}^{ \{ 462 \}}_{\[4\] | e \] } = 0 ~~, 
&(2.10{\rm h})  \cr 
& i \big(\g^d \big)_{(\a\b|} \big(\g^{a b}
          \big)_{|\g\d)} {\cal A}^{\{ 2,574 \}}_{a b d e} = 0 ~~, 
&(2.10{\rm i})  \cr   
& \big(\g^d \big)_{(\a\b|} 
         \big(\g^{b\[4\]} \big)_{|\g\d)} 
         \eta\low{\[d| b} {\cal A}^{\{ 3,003\}}_{\[4\] e} = 0 ~~, 
&(2.10{\rm j})  \cr 
& \frac1{120} \big(\g^d \big)_{(\a\b|} \big(\g^{\[5\]}
        \big)_{|\g\d)} \e\low{\[5\] \[ d |}{}^{\[5\]'} 
       {\cal A}^{\{ 4,290 \}}_{\[5\]' |e\]}
      + \frac i 2 \big( \g^{\[5\] } \big)_{(\a\b|} 
     \big( \g_{d e} \big)_{|\g\d)} X\du{\[ 5 \]} d = 0 ~~,  
&(2.10{\rm k})  \cr 
& \big(\g^d \big)_{(\a\b|} \big(\g^{\[5\]}
         \big)_{|\g\d)} {\cal A}^{\{ 17,160 \} }_{\[5\] d e} = 0 ~~.   
&(2.10{\rm\ell})  \cr   } $$ 
The conceptually important ingredient here is that different irreducible
components in (2.10), such as $~{\bf 165}$~ vs. ${\bf 2,574}$~ will not 
interfere with each other.  Moreover, 
$~{\cal A}^{\{330\}}_{a b c d}$~ and $~{\cal B}^{\{330\}}_{a b c d}$~  in
(2.10f), $~{\cal A}^{\{429\}}_{a b c}$~ and $~X\du{a b} c$~ in (2.10g), or 
$~{\cal A}^{\{4,290\}}_{\[5\]a}$~ and $~X\du{\[5\]} c$~ in (2.10k) are
proportional to each other.  

From now on, we use heavily the Fierz-type identities (5.1) - (5.4) and
Lemmas (5.5) - (5.8) that will be given separately in section 6.  We
start with eq.~(2.10a).  This condition is identically satisfied for 
arbitrary 
$~{\cal A}^{\{1\}}$~ due to the well-known Fierz identity in 11D
$$ \li{ & \big( \g^{a b} \big)_{( \a\b|} \big(\g_b \big)_{|\g\d)} 
     \equiv  0 ~~. 
&(2.11) \cr } $$ 
Next, due to Lemma 2 in (5.5), eq.~(2.10b) implies that 
$$ \li{ & {\cal A}^{\{11\}}_a =0 ~~.   
&(2.12) \cr } $$
Eqs.~(2.10c) and (2.10d) satisfy the 
assumption of Lemma 2, when $~B_{a, b c}$~ is identified with 
$$ \li{ & B_{a, b c} ~~ \rightarrow ~~ 
    \eta\low{a c}{\cal A}^{\{n\}}_{b e} 
     - \eta\low{a b}{\cal A}^{\{n\}}_{c e} 
     - \eta\low{e c}{\cal A}^{\{n\}}_{b a} 
     + \eta\low{e b}{\cal A}^{\{n\}}_{c a}   
&(2.13) \cr } $$
both for $~n=55$~ and $~n=66$.  Here we use the `arrow' symbol instead of
`equality', due to the free index $~{\scst e}$, while $~B_{a, b c}$~
on the l.h.s.~can be arbitrary including any such `free' index.  
Therefore corresponding to (5.6a), we get the condition 
$$ \li{ & \eta\low{a\[b} {\cal A}^{\{n\}}_{c\] e} 
    - 8 \eta\low{e\[b} {\cal A}^{\{n\}}_{c\] a} 
     + \eta\low{e a} {\cal A}^{\{n\}}_{\[b c\]}  
	- \eta\low{e\[b|} \big({\cal A}^{\{n\}}_{a |c\]} 
	+ {\cal A}^{\{n\}}_{|c\] a} \big) = 0 ~~,  
&(2.14) \cr } $$
both for $~n=55$~ and $~n=66$.  Now the $~{\scst a e}\-$ 
and $~{\scst c e}\-$contractions respectively yield 
$$ \li{& {\cal A}^{\{55\}}_{b c} = 0~~, 
~~~~ {\cal A}^{\{66\}}_{b c} = 0~~.
&(2.15) \cr } $$   

Eqs.~(2.10e), (2.10h), (2.10j) and (2.10$\ell$) are solved based on 
Lemma 3 in (5.8), namely they imply nothing other than the vanishing of 
$$\li{ & {\cal B}^{\{165\}}_{\[3\]} = 0~~, ~~~~
       {\cal A}^{\{462\}}_{\[4\] a} =0 ~~, ~~~~ 
       {\cal A}^{\{3,003\}}_{\[4\] a} =0 ~~, ~~~~
       {\cal A}^{\{17,160\}}_{\[5\] a b} =0 ~~.  
&(2.16) \cr } $$ 
     
Eq.~(2.10f) is understood as a sum of (5.2) + (5.4), when $~B_{a, b c}$~
and $~B_{a, \[5\]}$~ are identified with
$$ \li{& B_{a, b c} ~~\rightarrow~~ 24 A^{\{330\}}_{b c a e} ~~, ~~~~
     B_{a,\[5\]} ~~ \rightarrow ~~ 
    - \e\low{\[5\] a e}{}^{\[4\]}{\cal B}^{\{330\}}_{\[4\]}  ~~. 
&(2.17) \cr } $$ 
The like terms of the types $~\big(\g^a \big)_{\a\b}
\big(\g^{\[5\]} \big)_{\g\d}$~ and $~\big(\g^{\[2\]} \big)_{\a\b}
\big(\g^{\[2\]'} \big)_{\g\d}$~ in the sum (5.2) + (5.4) 
yield respectively the conditions 
$$ \li{ & 9 B_{a, b c_1\cdots c_4} 
     - \frac1{24} \eta\low{a \[ b| } B_{d,}{}^d{}_{| c_1\cdots c_4 \] } 
     + \frac1{24} B_{\[b|, a |c_1\cdots c_4\]} = 0 ~~, 
&(2.18\rma) \cr 
& B_{a,}{}^a{}_{\[4\]} = 0 ~~,       
&(2.18\rmb) \cr } $$ 
both of which have contributions only from $~B_{a, \[5\]}$.  Eq.~(2.18b) 
deletes the middle term in (2.18a).  
Other non-trivial like terms are of the type $~\big( \g^{\[2\]}
\big)_{\a\b} \big( \g^{\[6\]} \big)_{\g\d}$, which have contributions
both from the $~B_{a, b c}$~ and $~B_{a, \[4\]}$~ terms, yielding the
condition 
$$ \li{& \frac5 2 \big(\d\du a c\d\du b d - \d\du b c\d\du a d \big)
    B_{\[f, g h\]} - \frac1{30} \, \e\dud b {c d e\[4\]}{f g h} 
    B_{a,e\[4\]} =0 ~~.  
&(2.19) \cr } $$ 
Now if we look into only the $~{\scst\[a b c d\]}\-$component of this 
equation, and multiply it by $~\e\du{c d}{a b g h k_1\cdots k_5}$,
we  get the condition 
$$ \li{ & B_{f,k_1\cdots k_5} = 0 ~~,   
&(2.20) \cr } $$  
up to terms that vanish upon using eq.~(2.18b).  This implies 
consistently with (2.18) that  
$$ \li{ & B_{a, b_1\cdots b_5} = 0 ~~\Longrightarrow ~~
    {\cal B}^{\{330\}}_{\[4\]} = 0 ~~. 
&(2.21) \cr } $$ 
Now once $~{\cal B}^{\{330\}}$'s does not contribute, then only the first
term in  (2.10f) remains, which in turn implies {\it via} (5.7) in Lemma
2 that 
$$ \li{& {\cal A}^{\{ 330 \}}_{a b c d}  = 0 ~~, 
&(2.22) \cr } $$  
because $~{\cal A}^{\{ 330 \}}{}\dud{a c} a d = 0$~ manifestly, 
upon the identification $~B_{a, b c}\rightarrow {\cal A}{}
^{\{330\}}_{b c a d}$.  

Eq.~(2.10i) has the $~\g\-$matrix structure of (5.2) with $~B_{a, b c}$~
identified with 
$$ \li{ & B_{a, b c} ~~ \rightarrow ~~ {\cal A}^{\{2,574\}}_{b c e a}~~,
     ~~~~ B\dud{a,}a c ~~ \rightarrow ~~ 
    {\cal A}^{\{2,574\}}\du{~~~~~a c e}a = 0 ~~, 
&(2.23) \cr} $$ 
so that the assumption of (5.7) in Lemma 2 is satisfied, and therefore 
$~B_{a, b c} = 0$, {\it i.e.,} 
$$ \li{ & {\cal A}^{\{2,574\}}_{b c e a} = 0 ~~.  
&(2.24) \cr } $$

At this stage, eqs.~(2.10g) and (2.10k) are the only remaining 
conditions to be solved.  Eq.~(2.10g) is regarded as the sum 
(5.1) + (5.3), when $~A_{a, b}$~ for the former, and $~A_{a b, c d}$~
for the latter are respectively identified with 
$$ \li{ & A_{a, b} ~~\rightarrow ~~ -24 {\cal A}^{\{429\}}_{a b e} ~~, 
    ~~~~ A \du{a b,}{c d} ~~\rightarrow ~~ 
     - 6 \eta\low{e\[a| } X\ud{c d} {| b\]}
     - 6 \d\du e{\[c} X\du{a b} {d\]} ~~. 
&(2.25) \cr } $$ 
Accordingly, it is convenient to rewrite this (5.1) + (5.3) in terms
of $~A_{a, b}$~ and $~A_{a b, c d}$~ instead of $~{\cal A}$'s: 
$$ \li{ \frac1{24} & \big( \g^a \big)_{(\a\b|}  
      \big( \g^b \big)_{|\g\d)} A_{a, b} 
     + \frac1{96} \big( \g^{a b} \big)_{(\a\b|}  
      \big( \g^{c d} \big)_{|\g\d)} A_{a b , c d}  \cr 
= & + \frac1{48} \big( \g^a \big)_{\a\b} \big( \g^b \big)_{\g\d} 
     \big( 18 A_{a, b} 
     - \eta\low{a b} A\du{c,} c + 2 A_{a c, b}{}^c \big) \cr
& + \frac1{96} \big( \g^{a b} \big)_{\a\b} \big( \g\low{c d} \big)_{\g\d} 
     \Big[ + \d\du{\[a|}{\[c|} A\du{|b\]}{|d\]} 
     -\frac12 \d\du{\[a} c \d\du{b\]} d A\du{f,}f  \cr 
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ + 9 A_{a b,}{}^{c d}  + \frac12
     A\dud{\[a|}{\[c|}{,}{}_{|b\]}{}^{|d\]}
     - \d\du{\[a|}{\[c|} A\du{|b\]f,}{|d\]f} \, \Big] \cr
& + \frac1{5,760} \big(\g^{\[3\]a b} \big)_{\a\b} 
     \big(\g\du{\[3\]}{c d} \big)_{\g\d} 
     \Big[ + \frac52 \d\du{\[a}{\[c} A\du{b\],}{d\]} 
     -\frac12 \d\du{\[a}c \d\du{b\]}d A\du{f,}f \cr 
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~  -10 A_{a b,}{}^{c d} 
     - 5 A\dud{\[a|}{\[c|}{,}{}_{|b\]}{}^{|d\]} 
     + \frac5 2 \d\du{\[a|}{\[c|} A\du{|b\] f,}{|d\] f} \, \Big]  = 0
     {~~. ~~~~~ ~~~~~ } 
&(2.26) \cr } $$ 
Since each of the different $~\g\-$matrix structure is independent, 
we have the following three conditions
$$ \li{ & 18 A_{a, b} -\eta\low{a b} A\du{c,}c + 2 A\du{a c, b} c
     = 0 ~~, 
&(2.27\rma) \cr 
& \d\du{\[a|}{\[c|} A\du{|b\]}{|d\]} 
     -\frac12 \d\du{\[a} c \d\du{b\]} d A\du{f,}f 
     + 9 A_{a b,c d} + \frac12 A\dud{\[a|}{\[c|}{,}{}_{|b\]}{}^{|d\]}
     - \d\du{\[a|}{\[c|} A\du{|b\]f,}{|d\]f} = 0 {~~,~~~~~ ~~~~~}
&(2.27\rmb) \cr 
& \frac52 \d\du{\[a}{\[c} A\du{b\],}{d\]} 
     -\frac12 \d\du{\[a}c \d\du{b\]}d A\du{f,}f 
    -10 A_{a b, c d} - 5 A\dud{\[a|}{\[c|}{,}{}_{|b\]}{}^{|d\]}
     + \frac5 2 \d\du{\[a|}{\[c|} A\du{|b\] f,}{|d\] f}  = 0 
     {~~. ~~~~~ ~~~~~} 
&(2.27\rmc) \cr } $$
Obviously, (2.27a) implies that $~A\du{a,} a =0$, already satisfied by the
tracelessness of $~{\cal A}^{\{429\}}_{a b c}$.  
Using this back in (2.27a) implies that 
$$ \li{ &  9 A_{a, b} + A\du{a c, b} c = 0 ~~. 
&(2.28) \cr } $$ 
On the other hand, by contracting the $~{\scst b d}\-$indices 
in (2.27b), we get 
$$\li{ & 9 A\du{a,}c  - 7 A\du{a b ,c} b = 0 ~~. 
&(2.29) \cr } $$
Obviously, (2.29) and (2.28) lead to $~A_{a b} = 0, 
~A_{a b}{}^{c b}=0$, which {\it via} 
(2.27b) and (2.27c) implies also that $~A_{a b c d}=0$.  Therefore we get 
$$ \li{ & A_{a, b} = 0 ~~\Longrightarrow~~
     {\cal A}^{\{429\}}_{a b c} = 0 ~~,  
&(2.30\rma)  \cr
& A_{a b , c d} = 0 ~~\Longrightarrow ~~ X_{a b}{}^c = 0 ~~. 
&(2.30\rmb)  \cr } $$

We are now left with the condition (2.10k).  As mentioned before, 
we can regard $~{\cal A}^{\{4,290\}}_{\[5\]a} $~ and $~X_{\[5\]}{}^c$~ as 
proportional to each other: $~{\cal A}^{\{4,290\}}_{\[5\] a} =
\hbox{const.}~X_{\[5\] a}$.  Then finding a non-trivial solution for
(2.10k) is equivalent to deducing a non-trivial solution for the unknown 
parameters $~a~$ and 
$~b$~ other than $~a = b = 0$~ in the equality 
$$ \li{ & i a \big( \g^d \big)_{(\a\b| } 
     \big( \g\du{\[d|}{\[5\]} \big)_{|\g\d)} X_{\[5\] | e\]}  
     + i b \big( \g^{\[5\]} \big)_{( \a\b | } 
     \big( \g_{d e}\big)_{| \g\d)} X_{\[5\] }{}^d = 0 ~~,   
&(2.31) \cr } $$
for an arbitrary $~X\du{\[5\] } c \neq 0$.  The simplest way to get 
more explicit conditions from (2.31) is to multiply it by 
$~\big(\g^{a b} \big)^{\a\b}$, and contract the indices $~{\scst\a\b}$:
$$ \li{ \big( \g^{a b} \big)^{\a\b} 
    \, \big[&\hbox{LHS of}~(2.31)_{\a\b\g\d} \, \big] \cr 
     = & + 20 i (3a + b) \big( \g^{\[a | \[4\] } \big) _{\g\d} 
     X\dud{\[4\]}{|b\]} e 
     - 8 i (a + 5b) \d\du e{\[a|}\big( \g^{\[5\] } \big)_{\g\d} 
     X\du{\[5\]} {|b\]} \cr 
& + 40i (a-b) \big( \g\du e{\[4\]} \big)_{\g\d} X\du{\[4\]}{\[a b\]} 
   + 40i b \big( \g^{\[4\]\[a|} \big)_{\g\d}
     X\du{\[4\] e}{|b\]} = 0 ~~.   
&(2.32) \cr } $$
We next multiply (2.32) by $~\big(\g^b\big)\du\e\g$, to get  
the only solutions $~a=b=0$~ as  
$$ ( 60a + 28b ) \big( \g^{a\[5\]} \big)_{\e\d} X_{\[5\]e}
      + ( 40a + 8b ) \big( \g^{e\[5\]} \big)_{\e\d} X_{\[5\]a}
      + (320a + 160b) \big(\g^{\[4\]} \big)_{\e\d} X_{\[4\]}{}^{a e} 
      = 0  $$  
$$ \Longrightarrow ~~ a = b = 0 ~~.
\eqno(2.33) $$
This is because the $~\g^{[4\]}\-$term yields $~2a + b = 0$, while
the multiplication of the $~\g^{\[6\]}\-$terms by $~i \g^a$~ yields 
$~25a + 11 b =0$.  Therefore the only solutions to the condition (2.10k) are 
$$ \li{ & {\cal A}^{\{4,290\}}_{\[5\] a}  = 0 ~~, ~~~~
     X\du{\[5\]} c = 0 ~~. 
&(2.34) \cr } $$

Collecting all the results above, {\it i.e.,} (2.12), (2.15), (2.16), 
(2.21), (2.22), (2.24), (2.30), and (2.34), we reach the conclusion that 
among all the components of $~U$'s entering (2.5), {\it except for} 
the singlet component $~{\cal A}^{\{1\}}$~ in (2.9), as well as 
all the $~X$'s in (2.4), should be zero, in order to satisfy the $F$-BI 
(2.1) at $~d=0$~ under the conditions (2.2).  Therefore, the only
possible form for $~F_{\a\b c d}$~ is 
$$ \li{ & F_{\a\b c d} = + 2 \big( \g_{c d}\big)_{\a\b} 
     {\cal A}^{\{1\}} 
     + i \big( \g^b \big)_{\a\b} {\cal A}^{\{165\}}_{b c d} ~~. 
&(2.35) \cr } $$
The remaining component $~{\cal A}^{\{165\}}_{a b c}$~ here 
is due to the fact that this component 
does {\it not} enter any of the conditions in (2.10).  
In other words, we can still have
a term  proportional to $~{\cal A}^{\{165\}}_{a b c}$~ in (2.35).  

However, we point out some degrees of freedom of superfield redefinition 
of the potential superfield $~A_{A B C}$.  This is associated with 
the definition of the superfield strength 
$~F_{A B C D}$.  In fact, consider the shift\footnotew{Notice 
the crucial difference of the symbol $~A$'s from $~{\cal A}$'s which
should not be confused with the former.}  
$$\li{ & A_{a b c} ~~ \rightarrow ~~ A_{a b c} 
     + {\cal A}^{\{165\}}_{a b c} ~~, 
&(2.36) \cr } $$
keeping other components among $~A_{A B C}$~ intact.  This can absorb the 
$~{\cal A}^{\{165\}}\-$term in (2.35), while any of the
constraints at $~d< 0$~ in (2.2) are maintained.  For
example for $~F_{\a\b\g\d}$, we have 
$$ \li{ & F_{\a\b\g\d} \equiv \frac 1 6 \nabla_{(\a} A_{\b\g\d)} 
     - \frac 1 4 T\du{(\a\b|}\e A_{\e|\g\d)} 
    - \frac 1 4 T\du{(\a\b|} e A_{e |\g\d)} ~~, 
&(2.37) \cr } $$
which is intact under the shift (2.36).  The same is also true for 
$~F_{\a\b\g d}$, which we skip here.  

Based on these results and considerations, we conclude that the 
only degree of freedom possibly embedding M-theory corrections for the
superspace constraints at $~d=0$~ is the singlet component $~{\cal
A}^{\{1\}}$~ in (2.35), which implies that 
$~F_{\a\b c d}$~ can be only of the form 
$$ \li{ & F_{\a\b c d} = \frac 12 \big(\g_{c d} \big)_{\a\b} \, e^\Phi ~~
&(2.38) \cr } $$
proportional to the on-shell physical superfield constraint \cfbh, scaled
by some real scalar superfield $~\Phi$.  

We mention here the importance of the conventional constraints (2.7).  
If these constraints {\it were not} imposed, we would have such 
corrections as 
$$ \li{& T_{\a\b}{}^c = 
     i \big( \g^c \big)_{\a\b} + i \big( M \g^c \big)_{( \a\b )} ~~, ~~~~
     F_{\a\b c d} = \frac 12 \big( \g_{c d}\big)_{\a\b} 
     + \frac 12 \big( M \g_{c d} \big)_{( \a\b )} ~~, 
&(2.39) \cr } $$ 
for an {\it arbitrary} $~32\times 32$~ matrix $~M\du\a\b$~  satisfying
(2.3) at the linear order in $~M$.  However, these corrections can not
embed M-theory corrections, because they contribute as redundant degrees
of freedom.  
We can also show that the gravitational superfield equation will not be 
modified at the linear order in $~M$, because 
the $~(\a\b c ,\d)\-$type BI is not modified by $~M$.  Therefore, such a
matrix $~M$~ is not enough for embedding M-theory corrections.  We skip 
the details here, leaving them for a future publication.  

Our analysis so far is concerned only with the linear order terms in
(2.3) in the fluctuations in $~F_{\a\b c d}$~ and $~T_{\a\b}{}^c$~ for
M-theory corrections.  However, even if we include the bilinear-order
terms in (2.3), our conclusion above remains intact.  In other words, the
linear-order satisfaction of the $F$-BI at $~d=0$~ 
only by the limited form (2.38) will not be affected by the inclusion
of next bilinear order terms.  To put it differently, once the most
general correction of the $~F_{\a\b c d}$~ has determined as in (2.3) at
the linear order, in particular with no corrections for $~T_{\a\b}{}^c$,
then it is straightforward to confirm that (2.38) satisfies also the
$F$-BI at $~d=0$~ to `all orders' in the expansion in terms of $~\Phi$, 
because the only correction is 
just a scaling of $~F_{\a\b c d}$, while there is no derivative involved 
in the $~d=0$~ BI (2.3).  

This `no-go theorem' established here is not so surprising
from the viewpoint that (2.11) is the only available Fierz identity of the
type $~\big(\g^{\[m\]} \big)_{(\a\b|} \big(\g^{\[n\]} \big)_{|\g\d)} =
0$~ allowing arbitrary integers $~m$~ and $~n$.  In other words, such
conditions as (2.3) has too many free indices to allow more degrees of 
freedom other than the singlet component $~{\cal A}^{\{1\}}$~ as in
(2.10a).   


 


\bigskip\bigskip\bigskip 

% \vfill\eject



\vbox{
\centerline{\bf 3.~~Effect of Scalar Superfield $~\Phi$}  

We have seen that the only possible correction of superspace
constraints at $~d=0$~ is the scaling of $~F_{\a\b c d}$~ in  
(2.38) by a real scalar superfield $~\Phi$.  The next question is what 
sort of M-theory corrections can be embedded into 
this scalar superfield $~\Phi$.  
}


The answer can be easily deduced from dimensional considerations.  First,
since the scalar superfield $~\Phi$~ is at $~d=0$, its spinorial derivative
enters  into the $~d=1/2$~ constraints $~T\du{\a\b}\g,~T\du{\a b}c$~ and
$~F_{\a b c d}$.  Therefore we have again the spinorial superfield
$~J_\a$~ 
\ng\ related to $~\Phi$~ by\footnotew{We do not take the standpoint in 
\howe\ that there is no auxiliary spinorial superfield in $~d=1/2$~ 
constraints at least temporarily, for the sake of argument here.} 
$$ \li{ & \nabla_\a \Phi \equiv \xi J_\a ~~, 
&(3.1) \cr } $$ 
with an appropriate constant $~\xi$~ like 
in refs.~\gnrecent\ng.  The most general forms for  these
constraints are now 
$$ \li{ &T\du{\a b} c = \a_1 \d\du b  c J_\a 
     + \a_2 \big(\g\du b c\big)\du\a\b J_\b ~~, 
&(3.2\rma) \cr 
&F_{\a b c d} = i \eta e^\Phi \big( \g_{b c d}\big)\du\a\b J_\b ~~, 
&(3.2\rmb) \cr 
& T\du{\a\b}\g = \b_1 \d_{(\a}{}^\g J_{\b)} 
     + \b_2 \big( \g^a \big)_{\a\b} \big( \g_a\big)^{\g\d} J_\d 
     + \b_3 \big( \g^{\[2\]} \big)_{\a\b} 
     \big( \g\low{\[2\]}\big)^{\g\d} J_\d ~~, 
&(3.2\rmc) \cr  } $$ 
with the unknown coefficients $~\a_1, ~\a_2, ~\eta, ~\b_1, ~\b_2, ~\b_3$.  
The satisfaction of $~d=1/2$~ BIs yields the following 
relationships among the coefficients
$$ \li{ & \b_1 = - \frac 5 2 \big( \a_1 + \a_ 2 \big) + \xi~~,  ~~~~
      \b_2 = - \frac 14 \big( 9 \a_1 + 5\a_2\big) + \frac 3 4 \xi ~~, \cr 
&\b_3 = + \frac 3 8 \big( \a_1 + \a_2  \big) - \frac 1 8 \xi ~~, ~~~~  
     \eta = + \frac 3 2 \a_1 - \frac 12 \xi ~~. 
&(3.3) \cr } $$ 

Note the important fact that the exponential function $~e^\Phi$~ is needed
in $~F_{\a\b\g d}$, while such a factor is absent in $~T\du{\a b}c$~ and 
$~T\du{\a\b}\g$~ in order to satisfy the $F$-BI at $~d=1/2$.  For
example, the $~(\a\b c d e)\-$type $F$-BI tells us that all the $~F_{a b c
d}\-$linear terms in $~T\du{\a b}\g$~ stay the same with {\it no}
exponential function $~e^\Phi$:
$$ \li{& T\du{\a b}\g \, \big|_F 
     = \frac i{144} \big( \g\du b{\[4\]} F_{\[4\]} 
     + 8 \g^{\[3\]} F_{b\[3\]} \big)\du\a\g   ~~,  
&(3.4) \cr } $$
where the symbol $~\big|_F$~ denotes the $~F_{a b c d}\-$linear part of 
$~T\du{\a b} \g$.  As is well-known, the $~(\a b c d e)\-$type $F$-BI at
$~d=3/2$~ gives the expression for $~\nabla_\a F_{b c d e}$~ in
terms of $~T\du{a b}\g$~ and $~F_{\a b c d}$, where the latter contains
the linear $~J$'s as   
$$ \li{&\nabla_\a F_{b c d e}\big|_{\nabla J} 
	= \frac i 6 \eta e^\Phi \big( \g_{\[b c d | }\big)\du\a\b 
	\nabla_{| e\]} J_\b ~~, 
&(3.5) \cr } $$
Here the factor $~e^\Phi$~ is involved {\it via} 
(3.2b).  On the other hand, the $~(\a b c d)\-$type $T$-BI gives
the relationship 
$$\li{ & R_{\a b c d}\big|_{\nabla J} 
	= \a_1 \eta\low{b \[c} \nabla_{d\]} J_\a
	+ \a_2 \big(\g_{c d} \big)\du\a\b \nabla_b J_\b ~~, 
&(3.6) \cr } $$
with {\it no} exponential factor $~e^\Phi$~ with the $~J$'s.
When (3.5) and (3.6) are used in the $~(a\b\g,\d)\-$type BI, the former 
produces an exponential factor $~e^\Phi$, while the latter does {\it not}.   
In order to avoid this mismatch, we are forced to put $~\eta = 0 $, 
which in turn {\it via} (3.3) implies that 
$$ \li{ & \eta = 0 ~~, ~~~~\a_1 = \frac 1 3 \xi ~~, ~~~~
	\b_1 = - \frac 5 2\a_2  + \frac 1 6 \xi~~,  
     ~~~~ \b_2 = - \frac 5 4 \a_2 ~~,  ~~~~
	\b_3 = + \frac 3 8 \a_2 ~~.  	 
&(3.7) \cr } $$

The gravitino superfield equation can be obtained from the 
$~(a \b \g , \d)\-$type BI, by contracting spinorial indices in 
several different ways, which should 
be consistent with each other.  One way is to multiply this BI by  
$~i\big(\g^a \big)\du\a\b \d\du\d\g$~ to get the trace part 
of the gravitino superfield equation 
$$ \li{ & - \frac {185} 8 \big( \g^{a b} \big)_{\a\b} 
	T\du{a b}\b + \frac{23 i} 6 \xi \big( \g^a \big)_{\a\b}\nabla_a 
	J^\b = 0 ~~, 
&(3.8) \cr } $$ 
while another way is to multiply the same BI by $~i\big(\g^a\big)\du\d\g$~ 
to get 
$$ \li{ & \frac {15} 8 \big( \g^{a b} \big)_{\a\b} 
	  T\du{a b}\b + \frac{11 i} 6 \xi \big( \g^a \big)_{\a\b}\nabla_a 
	  J^\b = 0 ~~, 
&(3.9) \cr } $$ 
up to terms, such as $~J^2$~ or the $~J$'s with fundamental physical 
superfields ignored as higher orders.  Note that all the 
$~\a_2\-$dependent terms cancel  each other in these equations. 
obviously, (3.8) and (3.9) lead to   the conclusion that $~\xi = 0$~ as
the only possible solution.  Unfortunately, this is a trivial solution,
because this implies that $~J_\a =0$~ in (3.1), so that $~\Phi =
\hbox{const}$.  Even though we did not mention, there are also other
additional conditions on the independent parameters in (3.7), that will
not change the conclusion here.  This is because they provide {\it more}
stronger conditions on the unknown parameters, 
but they never avoid the conclusion
$~\xi =0$~ above.  Note also that our  result here is in agreement
with the argument about the absence of the off-shell $~J\-$superfield in
\howe.   

From these considerations, we conclude that the scalar superfield
$~\Phi$~  embedded into $~F_{\a\b c d}$~ as the exponent above is {\it
not} enough to embed M-theory corrections, as long as the $F$-BIs are
not modified by Chern-Simons terms.  


\bigskip\bigskip\bigskip

% \vfill\eject


\centerline{\bf 4.~~Fermionic $~\k\-$Symmetry and Chern-Simons 
Modification}  

We next consider the fermionic $~\k\-$symmetry of supermembrane action
%%% 
\ref\bst{E.~Bergshoeff, E.~Sezgin and P.K.~Townsend,
\pl{189}{87}{75}; \ap{185}{88}{330}.},   
%%% 
which justifies our assumption (i) about the vanishing $F$-constraints
at $~d<0$.  The standard supermembrane action is \bst\ 
$$ \li{ I \equiv  \int d^3 \s \Big[ & \! + \frac12 {\sqrt {-g}} g^{i j} 
	\eta\low{a b} \Pi\du i a \Pi\du j b - \frac12 {\sqrt {-g}} 
    - \frac1 3 \e^{i j k} \Pi\du i A \Pi\du j B \Pi\du k C A_{C B A}\,\Big]~~.
&(4.1) \cr } $$ 
with the pull-backs $~\Pi\du i A \equiv \big(\partial_i Z^M\big) E\du M A$, 
for the superspace coordinates $~Z^M$~ and the 
inverse vielbein $~E\du M A$~ in the 11D superspace we are dealing with.
The fermionic $~\k\-$symmetry is dictated by \bst 
$$ \li{ & \d_\k E^\a = \big( I + \G \big)\ud\a\b \k^\b ~~, ~~~~  
	\d_\k E^a = 0 ~~, ~~~~
        \G^{\a\b} \equiv \frac i 6 \e^{i j k} 
        \Pi\du i a \Pi\du j b \Pi\du k c 
	\big(\g_{a b c} \big)^{\a\b} ~~, 
&(4.2) \cr } $$
where $~\d_\k E^A \equiv \big(\d_\k Z^M\big) E\du M A$.  
The general variation formula under $~\d_\k E^a = 0$~ is 
$$ \li{ & \d_\k I = {\sqrt{-g}} 
     g^{i j}\big(\d_\k E^\a \big) \Pi\du i B T\du{B\a} d \Pi_{j d} 
	 + \frac 1 3 \e^{i j k} \big(\d_\k E^\a \big) 
	 \Pi\du i B \Pi\du j C \Pi\du k D F_{D C B \a} ~~. 
&(4.3) \cr } $$ 
The first term is from the variation of the 
kinetic term, while the second one is 
from the Wess-Zumino-Novikov-Witten (WZNW) term in (4.1).
 
We now study whether the non-zero $F$-constraints  
$~F_{\a\b\g\d} \neq 0, ~F_{\a\b\g d} \neq 0$~ at $~d < 0$~ are
compatible with this fermionic symmetry (4.2).  Using (4.3), we easily
see that if there are such non-trivial $F$-constraints at $~d<0$, they
will contribute only to the variation of the WZNW-term in (4.3),
that are {\it not} simply cancelled by the variation of the
kinetic term: 
$$ \li{ \d_\k I = & + \frac1 3 \e^{i j k} \big( \d_\k E^\a \big) 
     \Pi\du i\b \Pi\du j\g \Pi\du k\d F_{\d\g\b\a} 
     + \e^{i j k} \big( \d_\k E^\a \big) 
     \Pi\du i\b \Pi\du j\g \Pi\du k d F_{d\g\b\a} ~~.   
&(4.4) \cr } $$
	
It is  unlikely that new corrections due to 
other radical and non-conventional corrections such as 
$~\d_\k E^a =0$~ itself, or due to the addition of some other terms to the
action $~I~$ itself 
can lead to the fermionic $~\k\-$invariance of the conventional 
supermembrane action (4.1), because such corrections occur only at 
$~d<0$, which do not seem to communicate with other $~d\ge 0$~ 
constraints.\footnotew{One can, of course, give up such fermionic
$~\k\,$-symmetry of the supermembrane action (4.1) entirely, but we do not
argue about the `legitimacy' of fermionic symmetry itself in this
paper.}



\bigskip\bigskip






\bigskip\bigskip\bigskip

\centerline{\bf 5.~~Useful Algebraic Lemmas}

In what follows, we list up some useful algebraic lemmas and relationships 
that play decisive roles for our conclusions in this paper.  
We start with the Fierz identity 
$$ \li{\frac1{24} \big( \g^a \big)_{(\a\b | } \big( \g^b \big)_{| \g\d)} 
	A_{a, b} 
= & + \big( \g^a \big)_{\a\b} \big( \g^b \big)_{\g\d} 
	\Big( \frac 3 8 A_{a ,b} 
	- \frac 1{48} \eta\low{a b} A\du{c, } c \Big) \cr 
& + \big( \g_{a b} \big)_{\a\b} \big( \g^{c d} \big)_{\g\d} 
	\Big[ \, \frac1{96} \d\du{\[c| } {\[ a|} A\du{|d\], } {|b\]} 
	- \frac1{192} \d\du{\[c} a \d\du{d\]} b A\du{c, } c \, \Big] \cr 
& + \big( \g^{\[4\]a} \big)_{\a\b} \big( \g\low{\[4\]}{}^b \big)_{\g\d} 
	\Big( \frac1{576} A_{a, b} - \frac1{5,760} \eta\low{a b} 
	A\du{c,} c  \Big)  ~~,   
&(5.1) \cr } $$
where $~A_{a, b}$~ are any arbitrary symmetric tensor superfield.  
In a similar fashion 
for an arbitrary tensor superfield $~B_{a, b c}$~ with the property
$~B_{a, b c} = - B_{a, c b}$, we have 
\vbox{
$$ \li{ \frac i {48} \big( \g^a \big)_{(\a\b| } \big(\g^{b c}
     & \! \big)_{|\g\d)} B_{a, b c} \cr 
= \, & + \frac i {96} 
   \Big[\, \big( \g^a \big)_{\a\b} \big(\g^{b c} \big)_{\g\d}
     + \big( \g^{b c} \big)_{\a\b} \big(\g^a \big)_{\g\d} \, \Big] 
	\Big( 9 B_{a, b c} - B_{b ,c a} - B_{c, a b} 
     - \eta\low{a\[b|} B\dud{d,}d {|c\]} \Big) \cr 
& + \frac i {2,304} \big( \g^{\[4\]a} \big)_{\a\b} 
	\big(\g\low{\[4\] }{}^{b c} \big)_{\g\d}
	\Big( B_{a, b c} - B_{b ,c a} - B_{c, a b} 
      - \frac15 \eta\low{a\[b|} B\dud{d,}d {|c\]} \Big) \cr 
& + \frac i {1,152} \Big[\, \big( \g^{\[2\]} \big)_{\a\b} 
	\big(\g\low{\[2\]}{}^{a b c} \big)_{\g\d}
	+ \big( \g^{\[2\]a b c} \big)_{\a\b} 
	\big(\g\low{\[2\]}\big)_{\g\d} \, \Big] 
	B_{\[a , b c\]} ~~.  
&(5.2) \cr } $$
}
Similarly, for any arbitrary tensor superfields with the properties 
$~A_{a b, c d} = + A_{c d , a b} = (1/4) A_{\[ a b\], \[c d\] }, ~
A_{\[a b, c d \]} =0, ~A\du{a b,}{a b} = 0$, we have 
$$ \li{ 
\frac1{96} \big( \g^{a b} \big)_{(\a\b|} & \big(\g^{c d} \big)_{|\g\d)} 
	A_{a b, c d}  \cr 
= & + \frac1{24} \big( \g^a \big)_{\a\b} \big(\g^b \big)_{\g\d } 
	A_{a c , b}{}^c \cr  
& + \frac1{96} \big( \g^{a b} \big)_{\a\b} \big(\g^{c d} \big)_{\g\d } 
	\Big[ \, 9 A_{a b, }{}^{c d} + A_{\[ a|}{}^c{}_{, | b \]}{}^d  
	- \eta\low{\[a|}{}^{\[ c | } A\du{| b\] e, }{| d \] e} \, \Big] \cr  
& + \frac1{2,304} \big( \g^{\[3\] a b} \big)_{\a\b} 
	\big(\g\low{\[3\] c d} \big)_{\g\d } 
	\Big[\, - 4 A_{a b, }{}^{c d} - 4 A_{\[ a |}{}^c{}_{, |b\]}{}^d 
	+\d\du{\[a | }{\[ c | } A\du{|b\] e,}{| d\] e}  
	\, \Big]  ~~.    
&(5.3) \cr } $$
For an arbitrary tensor superfield $~B_{a , b_1\cdots b_5}$~ 
with the totally antisymmetric indices $~{\scst b_1\cdots b_5}$, 
we have
$$ \li{ \frac 1{2,880} \big( \g^a \big)_{(\a\b|} 
	& \big( \g^{\[5\]} \big)_{|\g\d)} B_{a, \[5\] }  \cr  
= & + \frac 1{5,760} \big(\g^a \big)_{\a\b}  
	\big(\g^{b c_1\cdots c_4} \big)_{\g\d} 
	\Big[ \, 9 B_{a, b c_1\cdots c_4} 
	- \frac1{24} \eta\low{a \[ b|} B_{d,}{}^d{}_{| c_1\cdots c_4 \] } 
	+ \frac1{24} B_{ \[b | , a |c_1\cdots c_4\] } \, \Big] \cr 
& - \frac 1{5,760} \big( \g^{a b} \big)_{\a\b} 
	\big(\g\du b{c d_1\cdots d_4} \big)_{\g\d} 
	\Big[ \, B_{a, c d_1\cdots d_4} 
	-\frac1{300} \eta\low{a \[ c| } B_{e,}{}^e{}_{| d_1\cdots d_4 \]} 
	+ \frac 1{24} B_{\[ c| , a | d_1\cdots d_4\] } \, \Big] \cr 
& + \frac 1{192} \big( \g^{b c} \big)_{\a\b} 
	\big(\g^{\[2\]}\big)_{\g\d} B_{a,}{}^a{}_{b c \[2\]} 
   + \big[ \, \big( \g^{\[5\]}\big)_{\a\b} \hbox{-terms} ~\big]  
   	{~~, ~~~~~ ~~~~~} 
&(5.4) \cr } $$
where we have omitted the terms with the structure of 
$~\big(\g^{\[5\]} \big)_{\a\b}$, because they are independent from the terms 
explicitly given here, and moreover, they are too messy whose structures 
are not decisive for our lemma below.  

Using (5.1) - (5.4), we can get the following important lemmas:

\noindent Lemma 1:  {\it If the l.h.s.~of (5.1) vanishes, then it 
follows that $~A_{a , b} = 0$.} \hfill (5.5) 

\noindent Lemma 2:  {\it The vanishing of 
the l.h.s.~of (5.2) implies that the following 
two conditions hold:} 
$$ \li{ & 9 B_{a, b c} - B_{b, c a} - B_{c, a b} 
	- \eta\low{a \[b|} B_{d,}{}^d{}_{|c\]}  = 0 ~~,  
&(5.6\rma) \cr 
& B_{a, b c} + B_{b, c a} + B_{c, a b} = 0 ~~. 
&(5.6\rmb) \cr } $$ 
{\it In particular, when $~B_{a,}{}^a{}_b =0$, it follows that $~B_{a, b
c}=0$.} \hfill (5.7) 
 
\noindent Lemma 3:  {\it The vanishing of the l.h.s.~of (5.4) implies
that $~B_{a, b_1\cdots b_5} =0$.} \hfill(5.8)  

Some remarks are in order for these lemmas:  First, Lemma 1 is based
on the fact that each sector of different structure of
$~\gamma\-$matrices in (5.1) for the two pair of indices
$~{\scst \a \b}$~ and $~{\scst \g \d}$~ is to be independently zero.  
This leads to the condition $~(3/8) A_{a, b} 
- (1/48)\eta\low{a b} A\du{c, } c = 0$, 
whose trace gives $~A_{c ,}{}^c =0$, which in turn yields $~A_{a, b} =0$,
when  re-substituted into this original equation. 
Second, Lemma 2 is also easy under (5.2), because we
can require each of the two sectors with the $~\g\-$matrix structures 
$~\big(\g^a \big)_{\a\b} \big(\g^{b c} \big)_{\g\d}$~ and   
$~\big(\g^{\[2\]})_{\a\b} \big(\g\low{\[2\]}{}^{a b c}\big)_{\g\d}$~ 
should vanish independently.  Note here that a simple contraction of two 
indices in (5.6a) does {\it not} lead to $~B\dud{a, }a b=0$, due to 
the vanishing trace component.  We did not write the condition of
vanishing of the second line on the r.h.s.~of (5.2), because it is just
a necessary condition of the conditions (5.6a) and (5.6b).  
We also mention that the well-known 
identity (2.11) is nothing else than a special case of $~B_{a , b c} =
\eta\low{a \[b} v\low{c\]}$~ satisfying both (5.6a) and (5.6b).  Third,
Lemma 3 is straightforward, because each sector in the r.h.s.~of
(5.4) is to vanish independently.  The last sector with 
$~\big(\g^{\[2\]} \big)_{\a\b} \big(\g^{\[2\]'}\big)_{\g\d}$~ yields 
$$ \li{ & B\dud {a,} a {b_1\cdots b_4} = 0 ~~,  
&(5.9) \cr } $$ 
which combined with the vanishing of the first and second lines of the
r.h.s.~of (5.4) implies immediately $~B_{a, b_1\cdots b_5}=0$.  This is 
due to the difference in the coefficient in the first terms in these two 
sectors.


\bigskip\bigskip\bigskip


%\vfill\eject

\vbox{
\centerline{\bf 6.~~Concluding Remarks}

In this Letter, we have shown that the $~d=0$~ $F$-BI can not be 
satisfied by any correction of the type 
$~(\g^{a b})_{\a\b} X_{a b}{}^c~$ or 
$~i (\g^{a_1\cdots a_5})_{\a\b} X_{a_1\cdots a_5}{}^c~$ in dimension zero
supertorsion constraint $~T_{\a\b}{}^c$, based on the 
two assumptions:  (i) M-theory corrections to $F$-superfield strength 
at $~d<0$~ are absent;  (ii) The $T$ and $F$-BIs are not modified
by Chern-Simons terms.  Additionally, we relied upon the 
`conventional constraints' that relate and delete unnecessary freedom in the 
expansion of $~E_\a$~ and $~E_a$.  These conventional constraints 
restrict the structure of corrections in $~T_{\a\b}{}^c$, such that the 
one-gamma term $~i\big( \g^c\big)_{\a\b}$~ in
$~T\du{\a\b}c$~ receives no corrections, while the $~\g^{\[2\]}$ or
$~\g^{\[5\]}$ terms can, satisfying the conditions in (2.7).  
We have seen that this result is valid for 
$F$-BI without Chern-Simons modifications.   
}

Subsequently, we have also analyzed the BIs at $~d\ge 1$, and obtained 
some conditions on the constraints of $~T\du{\a\b}\g, ~T\du{\a b}c$~ 
and $~F_{\a b c d}$, at least for the case that the $F$-BIs are not modified 
by Chern-Simons terms.  In particular, we have found that 
$~F_{\a b c d}=0$~ in order to satisfy the matching exponential functions
$~e^\Phi$~ in the $~(a \b\g,\d)\-$type BI at $~d=3/2$.  
On the other hand, the consistency of gravitino superfield equation 
out of the same $~(a \b\g, \d)\-$type BI leads to the condition of 
vanishing of $~\nabla_\a \Phi = 0$, leading to 
the trivial solution $~\Phi = \hbox{const}$.  
This validates our conclusion, because this scalar superfield 
$~\Phi$~ was the only possible modification at $~d=0$.  In other words, 
the modification of the constraint $~F_{\a\b c d}$~ at $~d=0$~ is 
not enough for embedding M-theory corrections.  

As has been mentioned, our result is not so 
surprising, but reasonable enough from the following viewpoints. 
Namely, the only Fierz identity of the form 
$~\big(\g^{\[m\]}\big)_{(\a\b}\big(\g^{\[n\]}\big)_{\g\d)}$~ is nothing
other than (2.11).  This is also understandable from the fact that the
$F$-BI (2.3) at $~d=0$~ has four spinorial indices and one vectorial index
as free indices, and therefore the vanishing of (2.3) gives such a strong
condition as all the  components in $~U$'s and $~X$'s are zero, 
except for the singlet $~{\cal A}^{\{1\}}$~ in $~U_{a b c d}$.  

Our main conclusion in this paper can be bypassed by avoiding at least one
of the two assumptions (i) and (ii) above.  The assumption (i) seems
very difficult to avoid, due to the fermionic symmetry of supermembrane
action that seems to prevent the introduction of any $F$-constraints at
$~d<0$.  On the other hand, the assumption (ii) is also difficult,
because there has been no other known example of such supertorsion
$T$-BIs modified by Chern-Simons terms.   
We mention also that our result here does {\it not} contradict 
the works in \cgnn, because the $F$-BIs we dealt with in our paper 
has not been analyzed explicitly in \cgnn.  In fact, a  
statement about the necessity of the $F$-constraints at $~d<0$~ was given in  
\cgnn\ without explicit proof.  Our result in this paper provides 
a supporting proof with evidence for this statement, in terms
of explicit computation of the $F$-BI at $~d=0$.

We believe that not only the conclusion presented in this 
paper, but also the technical ingredient of Fierz identities will be of
great importance in the future, for exploring any possible M-theory 
corrections into 11D superspace supergravity.   



\bigskip\bigskip


\centerline{\bf Acknowledgements}  

We are grateful to M.~Cederwall, S.J.~Gates, Jr., U.~Gran and
B.E.W.~Nilsson for considerable help and discussions for crucial points
in this paper.  


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