
% --------------------- Macros ----------------------------
\documentstyle[12pt]{article}

\def\doit#1#2{\ifcase#1\or#2\fi}

% 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}

% 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}

% Calligraphic letters

\def\ca{{\cal A}}
\def\cb{{\cal B}}
\def\cc{{\cal C}}
\def\cd{{\cal D}}
\def\ce{{\cal E}}
\def\cf{{\cal F}}
\def\cg{{\cal G}}
\def\ch{{\cal H}}
\def\ci{{\cal I}}
\def\cj{{\cal J}}
\def\ck{{\cal K}}
\def\cl{{\cal L}}
\def\cm{{\cal M}}
\def\cn{{\cal N}}
\def\co{{\cal O}}
\def\cp{{\cal P}}
\def\cq{{\cal Q}}
\def\car{{\cal R}}
\def\cs{{\cal S}}
\def\ct{{\cal T}}
\def\cu{{\cal U}}
\def\cv{{\cal V}}
\def\cw{{\cal W}}
\def\cx{{\cal X}}
\def\cy{{\cal Y}}
\def\cz{{\cal Z}}


% 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\hp{\scriptsize{\dw}}
\def\hn{\scriptsize{\du}} 
\def\ph{\scriptsize{\wd}}
\def\nh{\scriptsize{\ud}} 

\def\bo{{\raise.15ex\hbox{\large$\Box$}}}		% D'Alembertian
\def\cbo{\Sc [}						% curly "
\def\pa{\partial}					% curly d
\def\de{\nabla}						% del
\def\dell{\bigtriangledown}				% hi ho the dairy-o
\def\su{\sum}						% summation
\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 $}}#1}	% 
\def\Tilde#1{{\widetilde{#1}}\hskip 0.03in}			% big tilde
\def\Hat#1{\widehat{#1}}			% big hat
\def\Bar#1{\overline{#1}}			% big bar
\def\bra#1{\left\langle #1\right|}		% < |
\def\ket#1{\left| #1\right\rangle}		% | >
\def\VEV#1{\left\langle #1\right\rangle}	% < >
\def\abs#1{\left| #1\right|}			% | |
\def\leftrightarrowfill{$\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)
\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}}
	\put(340,16){\line(1,0){4}}
	\put(335,0){\line(1,0){4}}
	\put(340,0){\line(1,0){4}}
	\put(345,0){\line(1,0){4}}
	\put(329,0){\line(0,1){16}}
	\put(330,0){\line(0,1){16}}
	\put(339,0){\line(0,1){16}}
	\put(340,0){\line(0,1){16}}
	\put(344,0){\line(0,1){16}}
	\put(345,0){\line(0,1){16}}
	\put(329,16){\oval(8,32)[bl]}
	\put(330,16){\oval(8,32)[br]}
	\put(339,0){\oval(8,32)[tl]}
	\put(345,0){\oval(8,32)[tr]}
	\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}
	\put(-20,-241.5){\ooo ababaighefdbfghgeahgdfgafagihdidihiidhiagfedhadbfdecdcdfagdcbhaddhbgfchbgfdacfediacbabab}
	\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}}
	\put(185,0){\line(1,0){4}}
	\put(169,0){\line(0,1){16}}
	\put(170,0){\line(0,1){16}}
	\put(179,0){\line(0,1){16}}
	\put(180,0){\line(0,1){16}}
	\put(184,0){\line(0,1){16}}
	\put(185,0){\line(0,1){16}}
	\put(169,16){\oval(8,32)[bl]}
	\put(170,16){\oval(8,32)[br]}
	\put(179,0){\oval(8,32)[tl]}
	\put(185,0){\oval(8,32)[tr]}
	\end{picture}
	\par\vskip-6.5mm
	\thicklines}
\def\title#1#2#3#4{\border\headpic {\hbox to\hsize{#4 \hfill UMDEPP #3}}\par
	\begin{center} \vglue .5in {\large\bf #1}\\[.6in] 
	{#2}\\[.1in] {\it Department of Physics and Astronomy}\\
	{\it University of Maryland, College Park, MD 20742}\\[1.5in] 
	{\bf Abstract}\\[.1in] \end{center} \begin{quotation}}	% title stuff
\def\Title#1#2#3#4#5#6#7{\border\headpic 
	{\hbox to\hsize{#7 \hfill UMDEPP #6}}\par
	\begin{center} \vglue .4in {\large\bf #1}\\[.4in] 
	{#2}\\[.1in] {\it Department of Physics and Astronomy}\\
	{\it University of Maryland, College Park, MD 20742}\\[.1in]
	{#3}\\[.1in] {\it {#4}}\\ {\it {#5}}\\[.5in] {\bf Abstract}\\[.1in]
	\end{center} \begin{quotation}}			% " for 2 authors
\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}

% --------------- csdef.tex -------------------
\def\doit#1#2{\ifcase#1\or#2\fi}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\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\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\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\E{{\cal E}}
\def\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\hati{{\hat{I}}}
\def\dt{$~D=10$~}
\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\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\szet{{${\scriptstyle \b}$}}
\def\ula{{\un a}} 
\def\ulb{{\un b}} 
\def\ulc{{\un c}}
\def\uld{{\un d}}
\def\ulA{{\un A}} 
\def\ulM{{\underline M}} 
\def\cdm{{\Sc D}_{--}}
\def\cdp{{\Sc D}_{++}}
\def\vTheta{\check\Theta}
\def\Pisl{{\Pi\!\!\!\! /}}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}

\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\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}

\def\Dot#1{\buildrel{_{_{\hskip 0.01in}\bullet}}\over{#1}}
\def\dt#1{\Dot{#1}}
\def\uln{{\underline n}}
\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}	
\def\Hat#1{\widehat{#1}}			
% ------------------- End of csdef.tex -----------------

% ------------------- 
\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\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\uln#1{\underline{#1}}
\def\ulp{{\underline p}} 
\def\ulq{{\underline q}} 
\def\ulr{{\underline r}}

\def\hatm{\hat m}
\def\hatn{\hat n}
\def\hatr{\hat r}
\def\hats{\hat s}
\def\hatt{\hat t}

\def\ul{\underline}
\def\un{\underline} 
\def\-{{\hskip 1.5pt}\hbox{-}}

\def\kd#1#2{\d\du{#1}{#2}}
\def\footnotew#1{\footnote{\hsize=6.5in {#1}}} 

\def\low#1{{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}

\def\ip{{=\!\!\! \mid}}
\def\unb{{\underline {\bar n}}}
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~~~
\vskip 0.07in

{\hbox to\hsize{
January 1996
\hfill UMDEPP 96--61}}
{\hbox to\hsize{~~~~~ ~~~~~~\hfill 
%(Revised Version)
}} 
\par 

\hsize=6.5in
\textwidth=6.5in

\begin{center}
\vglue 0.1in

{\large\bf Toward ~an ~Off - Shell ~11D ~Supergravity ~Limit ~of}
\vskip 0.02in
{\large\bf M - Theory}$\,$\footnote{This work is 
supported in part by NSF grant \# PHY-93-41926   
and by DOE grant \# DE-FG02-94ER40854.}  
\\[.1in]

\baselineskip 10pt 

\vskip 0.18in

\doit1{Hitoshi ~N{\small ISHINO}\footnote{E-mail: nishino@umdhep.umd.edu.  
Also at Department of Physics \& Astronomy, Howard University, 
Washington D.C.~20059, USA.}
~ and~ S.~James ~G{\small ATES}, $\,$Jr.\footnote{E-mail:  
gates@umdhep.umd.edu}   
\\[.25in]
{\it Department of Physics} \\ [.015in]
{\it University of Maryland at College Park}\\ [.015in]
{\it College Park, MD 20742-4111, USA} \\[.18in]
}


\vskip 1.5in

{\bf Abstract} \\[0.1in]  
\end{center}

\begin{quotation}

~~~We demonstrate that in addition to the usual fourth-rank superfield 
$(W_{a b c d})$ which describes the on-shell theory, a spinor superfield 
$(J_\a )$ can be introduced into the 11D geometrical tensors with 
engineering dimensions less or equal to one in such a way to satisfy 
the Bianchi identities in superspace.  The components arising from 
$~J_\a$~ are identified as some of the auxiliary fields required for a 
full off-shell formulation.  Our result indicates that eleven 
dimensional supergravity does not have to be completely on-shell.  
The $~\k\-$symmetry of the supermembrane action in the
presence of our partial off-shell supergravity background is also 
confirmed.  Our modifications to eleven-dimensional supergravity theory 
are thus likely relevant for M-theory.  We suggest our proposal as a 
significant systematic off-shell generalization of eleven-dimensional 
supergravity theory. 
   

\endtitle

\oddsidemargin=0.03in
\evensidemargin=0.01in
\hsize=6.5in
\textwidth=6.5in
\baselineskip 18.0pt 

\centerline{\bf 1.~~Introduction}

There has been a revival of interest in eleven-dimensional (11D) 
supergravity theory \ref\cremmer{E.~Cremmer, B.~Julia and J.~Scherk, 
\pl{76}{78}{409}; E.~Cremmer and Julia, \pl{80}{78}{48}; 
\np{159}{79}{141}.}. This revival
is occurring within the context of strong/weak duality 
\ref\wittencomments{E.~Witten, \np{443}{95}{85}.} between 10D type-II 
superstring theories \ref\gsw{{\it See e.g.}, M.~Green, J.H.~Schwarz and 
E.~Witten, {\it ``Superstring 
Theory''}, Vols.~I and II, Cambridge University Press (1987).}, and 11D
supermembrane theory \ref\bst{E.~Bergshoeff, E.~Sezgin, and  P.~Townsend,
\pl{189}{87}{75}; \ap{185}{88}{330}; E.~Bergshoeff and M.~de Roo,
\pl{133}{84}{67}; E.~Bergshoeff, M.J.~Duff, C.N.~Pope and E.~Sezgin,
\pl{224}{89}{71}; P.K.~Townsend, \pl{350}{95}{184}.}, and as a important
component of a newly proposed fundamental theory  called ``M-Theory'' 
\ref\schwarz{J.H.~Schwarz,  {\it ``The Power of M-Theory''}, Rutgers 
preprint, RU-95-68  (Oct.~1995), hep-th/9510086; {\it ``Superstring 
Dualities''},  Caltech preprint, CALT-68-2019 (Sep.~1995), hep-th/9509148.} 
suggested to provide a unifying paradigm from which perhaps all superstring 
and heterotic string theory and various known (as well as unknown 
dualities) can be derived.  If such an underlying theory  exists in 11D, 
we expect its background sector to have a much richer structures than the 
original 11D supergravity theory \cremmer.  This speculation looks natural, 
when we recall that 10D superstring theory \gsw\ generated chiral fermions 
with no cosmological constant {\it unlike} the original 11D supergravity 
\cremmer.    

As for any significant generalization or modification of 11D 
supergravity \cremmer, there had been tantalizing speculations on
the possibility of higher-derivative terms \ref\alvarez{L.~Alvarez-Gaum\'e, 
private communication (1982).} even prior to the re-birth
of string theory.  Within superstring theories it is known that higher
curvature terms, like the $~\a^{\,}{}'{}^3\z{\scst(3)}$~ correction
from  $~N=2A$~ superstring to 10D, $N=2$ supergravity, exist. In the
superspace approach \ref\cf{E.~Cremmer and S.~Ferrara,
\pl{91}{80}{61}; L.~Brink and P.~Howe, \pl{91}{80}{384}.} for example, the
search for higher-order terms {\it via} a method similar to that developed 
for superstring corrections to 10D, $N=1$ supergravity \ref\gnss{S.J.~Gates, 
Jr.~and H.~Nishino, \pl{173}{86}{46 and 52}; \np{291}{87}{205}.}\ref
\nishino{H.~Nishino, \pl{258}{91}{104}.}  
at first looks impracticably complicated, due to the $~32\times 32~$ matrix  
representation of the Clifford algebra in 11D, as well as the absence 
of a dilaton field that could simplify computations \gnss.  In a component 
formulation in ref.~\ref\peter{P. van Nieuwenhuizen, Stony Brook 
preprint, Print-84-0737 (Aug.~1984).}, some generalized Chern-Simons 
terms were tentatively added to the 11D supergravity Lagrangian
\cremmer, but unfortunately the supersymmetric invariance of the total
action was not confirmed as expected.  There have been some works dealing
with  auxiliary fields for 11D supergravity \ref\aux{A.~Van Proeyen,
Presented at 2nd Europhysics Study Conf.~on Unification of Fundamental
Interactions, Erice, Sicily  (Oct.~1981), in Erice EPS {\it ``Unification
1981''} 367;  A. Van Proeyen, \np{196}{82}{489}.}, but
they provide no systematic construction of the off-shell formulation.   At
the present time, almost twenty years after its initial construction 
\cremmer, no successful modifications of 11D supergravity with systematic 
(even perturbative) supersymmetric covariance exist to our knowledge.  

We mention, however, an intriguing ``glimmer of hope'' for
the off-shell formulation of 11D supergravity.  It was observed
that the on-shell superspace formulation of the 11D supergravity theory bore
a strange resemblance to the on-shell superspace formulation of the 4D, 
$N = 2$ supergravity theory \ref\gatesfourd{S.J.~Gates, Jr.,
\pl{95}{80}{305}.}.  It was also noted that the difference between the
on-shell and off-shell versions of 4D, $N = 2$ supergravity was the
presence or absence of an auxiliary spinor superfield.  On the basis of the
similarity between the on-shell theories, it was suggested that an off-shell
version of 11D supergravity would necessarily require the presence of a
similar spinorial superfield.  At that time it was proposed that a future
investigation would be undertaken in this direction.  

In this paper we take a significant first step toward the 
non-trivial off-shell generalization of 11D supergravity, motivated by the
above indication in 4D, $N=2$ supergravity.  We will prove that there exist
a solution of the 11D superspace Bianchi identities in terms of two
algebraically independent superfields denoted by ~$J_{\a}$~ and 
~$W_{a b c d}$.  The latter is the the on-shell field which in a certain
limit describes the purely physical and propagating degrees of freedom of
11D supergravity, and it is also an analog of the superfield $~W_{\a \b 
\g}$~ for 4D, $N=1$ supergravity \ref\ggrs{S.J.~Gates,
Jr., M.T.~Grisaru, M.~Ro\v cek and W.~Siegel, {\it Superspace}, Benjamin 
(1982).}.  From a geometrical point of view, this multiplet can be called 
the 11D supergravity ``Weyl multiplet''.  The second superfield ~$J_{\a}$~ 
is a superfield whose
presence implies that the 11D supergravity theory described by our 
superspace construction is {\it not} an on-shell construction.  It may be 
thought as the multiplet of auxiliary fields \aux\ for 11D supergravity. 
In the following we investigate some of the low dimensional auxiliary 
fields that it contains.  We will not, however, be able to give a complete 
description of this superfield by the end of this present work. 
      


\bigskip\bigskip\bigskip

%\newpage



\centerline{\bf 2.~~Partial Auxiliary Field Structure for 11D, $N=1$~ 
Supergravity}

Our guiding principle in superspace is as usual the satisfaction of the 
Bianchi identities (BIs):
$$ \eqalign{&\nabla_{\[ A} T\du{ B C)} D - T\du{\[A
B|} E T\du{E|C)} D -  \half R\du{\[A B| c} d ({\cal M} \du d c )\du{|C)} D
\equiv 0 ~~, \cr  & \fracm1{24} \nabla_{\[A_1} F_{A_2\cdots A_5 )} - \fracm
1{12} T \du{\[ A_1 A_2 |} B F_{B | A_3A_4A_5)} \equiv 0 ~~, \cr } 
\eqno(2.1) $$  
which we call $~(A B C,D)$~ and $~(A_1\cdots A_5)\-$type
BIs\footnotew{For our conventions and notations, see the next section.}   
Our purpose is to satisfy these BIs at engineering dimensions of $~d \le 1$,
as the usual fundamental step of solving them \gnss.  An important 
guiding principle is to follow the method for the non-minimal
4D, $N=1$ \gatesfourd\ theory, 
where a spinor superfield $~T_\a$~ was introduced to generalize the 
system, and contains some of the auxiliary fields in component approaches.
As an 11D, $N=1$ analog of the $~T_\a\-$superfield, we introduce a
spinorial superfield $~J_\a$, whose first derivative takes the form 
$$\nabla_\a J_\b = C_{\a\b} S + i (\g^a)_{\a\b} v_a + (\g^{a b})_{\a\b}  
t_{a b} + i(\g^{\[ 3 \]} )_{\a\b} U_{\[ 3 \]} + (\g^{\[ 4 \]} )_{\a\b} 
V_{\[ 4 \]} + i (\g^{\[ 5 \]})_{\a\b} Z_{\[ 5 \] } ~~.       
\eqno(2.2) $$
Here the subscript $~{\scst \[ n \]}$~ stands for the totally antisymmetric
indices such as $~{\scst a_1 \cdots a_n}$.  In addition to $~J_\a$~
we also introduce a fourth-rank tensor superfield $~W_{a b c d}$ that is 
independent of $~J_\a$ and contains the fields of the purely on-shell 
theory.  In order to go back to the usual on-shell theory, we can just 
identify $~W_{a b c d}~$ with the fourth-rank field strength $~F_{a b c d}$~, 
and set ~$J_\a$~ simply to zero.  

We are now ready to present our results for constraints of $~d \le 1$,
which constitute the foundation of our modified theory:      
$$\eqalign{T\du{\a\b} c = \, & + i (\g^c)_{\a\b} ~~, ~~~F_{\a\b c d} = +\half 
(\g_{c d})_{\a\b} ~~, ~~~~ F_{\a\b\g\d} = F_{\a\b\g d} = 0 ~~, \cr 
T_{\a\b}{}^\g =\, & - 8 (\g^a)_{\a\b} (\g_a)^{\g\d} J_\d \equiv - 8
  (\g^a)_{\a\b} (\g_a J)^\g ~~, \cr 
T\du{\a b} c  = \, &+ 8 (\g^c \g_b J)_\a ~~, ~~~~ 
  F_{a b c \d} = +12 i (\g_{a b c} J)_\d~~, \cr 
T\du{\a b} \g = \, & + \fracm i {144}
(\g\du b {c d e f} + 8 \d\du b e \g^{d e
  f} )\du\a\g W_{c d e f} \cr 
  & + 8i (\g_b)\du\a\g S  
   -8 (\g^c\g_b)\du\a\g v_c + 8i (\g^{c d} \g_b)\du\a\g t_{c d} \cr 
  & - 8 (\g^{c d e}\g_b)\du\a\g U_{c d e} + 12i (\g\du b{c d e
   f})\du\a\g V_{c d e f} - 8 (\g^{c d e f g}\g_b)\du\a\g Z_{c d e f g}\cr 
  & - 18 i (\g_b)\du\a\g Q - i(\g_b \g^{c d e}
  )\du\a\g Q_{c d e} - 12 i (\g^{c d})\du\a\g Q_{b c d} \cr 
  & - \fracm {17i}{12} (\g_b\g^{c d e f}) \du a g
  Q_{c d e f} - \fracm {20i} 3  (\g^{d e f})\du\a\g Q_{b d
  e f} ~~, \cr 
R_{\a\b c d} = \, & + \fracm 1 {72} (\g\du{c d}{e f g h} )_{\a\b} 
  (W_{e f g h} + 576 V_{e f g h})  + \fracm 1 3 (\g^{e f})_{\a\b} 
  (W_{c d e f} + 576 V_{a b c d}) \cr 
  & - 32 (\g^e)_{\a\b} Q_{c d e} - \fracm 8 3 ( \g\du{c d}{e f g h})_{\a\b} 
  Q_{e f g h} - 64 (\g^{e f})_{\a\b} Q_{c d e f} ~~, \cr 
  F_{a b c d} = \, & + W_{a b c d} + 576 \, V_{a b c d} ~~, \cr } {~~~~~}
\eqno(2.3) $$
The ~$Q$'s are {\it not} new superfields, but are just
products of two $~J^\a$'s defined by
$$\eqalign{& Q\equiv \left( {\Bar J} J\right)  \equiv 
J^\a  J_\a ~~, ~~~~  Q_{a b c} \equiv \left( {\Bar J}
\g_{a b c} J \right) \equiv  J^\a (\g_{a b c}) \du\a\b J_\b ~~,  \cr  
& Q_{a b c d} \equiv \left({\Bar J} \g_{a b c d} J \right) \equiv 
J^\a (\g_{a b c d}) \du\a\b J_\b ~~. \cr }  
\eqno(2.4) $$ 

Some remarks are now in order.  We mention that this set of constraints can
{\it not} be reduced to the original unmodified theory by Cremmer {\it et
al}.~\cremmer\ by any superfield redefinitions including super-Weyl 
rescaling \ref\gv{S.J.~Gates, Jr.~and S.~Vashakidze, \np{291}{87}{172}.}. 
This is critical for our system to really describe a new modification
that is not trivially related to the conventional on-shell system.  

Note that the superfields ~$S,~v_a,~t_{a b},~U_{\[3\]},~V_{\[4\]}$~
and $~Z_{\[ 5 \]}$~ are algebraically independent superfields, that play 
the roles of auxiliary fields in component formulations\footnote{In fact, 
the quantity ~$U_{\[3\]}$~ is the 11D analog of our 10D $A_{\[3\]}$-tensor
\gnss.}.  
In this sense, our modified system already gives an off-shell formulation of 
11D, $N=1$ supergravity.  Following 4D, $N=2$ analysis \gatesfourd, we
also introduce an independent superfield $~W_{a b c d}$, which is an 11D 
analog of the tensor superfield $~W_{a b}$ of the 4D, $N=2$ theory.  We have
determined the constants in (2.3) such that the BI's of $~d\le 1$ are 
satisfied.  To be more specific, the BIs at $~d\le 1/2$~ require the forms 
of constraints as  
$$ \eqalign{T\du{\a\b}\g = \,&  +
20(\a-1) \d\du{(\a}\g J_{\b)} + 2(5\a-9)  (\g^a)_{\a\b} (\g_a J)^\g - 3
(\a-1) (\g^{a b} )_{\a\b} (\g_{a b}J)^\g ~~, \cr  
T\du{\a b}c = \,& + 8 \d\du b c J_\a -
8 \a (\g\du b c J)_\a ~~, ~~~~  F_{a b c \d} = + 12i (\g_{a b c} J)_\d ~~, 
~~~~ F_{a b c d} = W_{a b c d} + x V_{a b c d}~~, \cr  
T\du{\a b} \g = \,&  + \fracm i
{144} \left( \g\du b{c d e f} + 8 \d\du b c \g^{d e f}\right)\du\a\g \, 
W_{c d e f}  \cr    
& + i a_0 (\g_b)\du\a\g S  
   + a_1 \d\du\a\g v_b + a_2 (\g_b{}^c)\du\a\g v_c + i a_3 (\g\du b{c d}
   ) \du\a\g t_{c d} + i a_4 (\g^c)\du\a\g t_{b c} \cr 
& + a_5 (\g\du b {c d e}) \du\a\g U_{c d e} 
   + a_6 (\g^{c d})\du\a\g U_{b c d} + i a_7 (\g\du b {c d e f}) \du\a\g 
   V_{c d e f} + i a_8 (\g^{c d e})\du\a\g V_{b c d e} \cr
&  + a_9 (\g\du b{c d e f g})\du\a\g Z_{c d e f g} + a_{10} (\g^{c d e
f})\du\a\g Z_{b c d e f} \cr   
&  + i \xi_1 (\g_b)\du\a\g Q +  \xi_2
(\g_b\g^{c d e}) \du\a\g Q_{c d e}  + \xi_3 (\g^{c d})\du\a\g Q_{b c d} ~~  
\cr 
& + i \xi_4 (\g_b\g^{c d e f})\du\a\g Q_{c d e f} + i \xi_5
(\g^{c d e})\du\a\g Q_{b c d e} ~~, \cr } 
\eqno(2.5) $$
where  $~\a,~x,~a_0,~\cdots,~a_{10},~ \xi_1,~\cdots,~\x_5$~ are unknown
constants.  At ~$d=1$, in the $~(\a\b c,d)\-$type BI the symmetric part
$~{\scst (c d)}$~ in $~R_{\a\b c d}$~ should be excluded, and this fixes 
some of the $~a$'s.  The $~(\a\b\g,\d)\-$type BI has terms linear in 
$~S,~v_a,~ t_{\[2\]},~ U_{\[3\]},~V_{\[4\]},~Z_{\[5\]}$, whose coefficients 
are required to vanish independently due to the algebraic independence of 
these superfields.  Also to be used are the identities in (3.6) in order to 
have only independent terms with the $~U_{\[3\]}$~ and $~V_{\[4\]}$~ 
superfields, as well as the similar identity 
$$ \half (\g^{\[a|c})_{(\a\b|} (\g\ud{|b\]}c)\du{|\g)}\d = - (\g_c)_{(\a\b|} 
(\g^c\g^{a b})\du{|\g)}\d + \half (\g^{\[a|})_{(\a\b|}(\g^{|b\]})\du{|\g)}\d 
+ (\g^{a b})_{(\a\b} \d\du{\g)}\d ~~, 
\eqno(2.6) $$
for $~t_{\[2\]}\-$terms.   

At this 
point all the $~a$'s are determined uniquely together with $~\a=1$:
$$ \eqalign{& a_0 = + 8 ~~, ~~~~ a_1 = -8~~, ~~~~a_2 = +8 ~~, ~~~~a_3 = 
+8~~, ~~~~ a_4 = -16 ~~, ~~~~a_5 = +8 ~~, \cr 
& a_6 = -24~~, ~~~~ a_7 = +12~~, ~~~~a_8 = 0 ~~, ~~~~a_9 = +8 ~~, ~~~~
a_{10} = -40~~, ~~~~\a= 1  ~~. \cr }
\eqno(2.7) $$  
The ~$(\a\b c d e)\-$type BI has also terms linear in these superfields,
vanishing consistently with (2.7), and in particular, the constant $~x$~ 
is now fixed to be $~x= 576 $.   Note the curious fact that if we rewrite 
$~T\du{\a b}\g$~ using $~F_{\[4\]}$~ instead of $~W_{\[4\]}$, then we will 
find that all the $~S,~\dots, ~ Z_{\[5\]}\-$terms in $~T\du{\a b}\g$~ can 
be re-combined exactly into a term $~-8i (\g_b)^{\g\e} \nabla_{\a} J_\e$.  
      
The $~J^2\-$terms can be fixed by the help of Tables 1 through 6 in the 
next section.  All the $~\xi$'s are uniquely determined as 
$$\xi_1 = - 18~~, ~~~~ \xi_2 = -1 ~~, ~~~~ \xi_3 =
-12  ~~, ~~~~ \xi_4 = -\fracm{17}{12} ~~, ~~~~ \xi_5 = - \fracm{20} 3 ~~,
\eqno(2.8) $$ 
yielding (2.3).  
  


\bigskip\bigskip\bigskip




\centerline{\bf 3.~~Useful Relationships and Identities}

Since the computations involved in our analysis are highly technical and 
lengthy, exposing some crucial identities will be of practical
importance.  First of all we give notational explanations about the 
products of our $~\g\-$matrices in 11D.  Our basic anticommutator is 
$~\{ \g^a, \g^b\} = + 2\eta^{a b} = \hbox{diag.}\,(+ - \cdots - )$. 
Accordingly we have $~\g^a\g^b = + \eta^{a b} + \g^{a b}$, where 
$~(\g^a\g^b)\du\a\b \equiv (\g^a)\du\a\g
(\g^b)\du\g\b, ~(\g^{a b})\du\a\b \equiv (1/2) (\g^a\g^b - \g^b
\g^a)\du\a\b$.  More generally we define 
$$ (\g^{a_1\cdots a_n})
\du\a\g \equiv \fracm 1{n!} \left[\, (\g^{a_1})\du\a{\b_2}
(\g^{a_2})\du{\b_2}{\b_3} \cdots (\g^{a_n})\du{\b_n}\g +
(n!-1~\hbox{perms.)}\, \right] ~~. 
\eqno(3.1) $$
Accordingly, we have $~(\g^a)_{\a\b} = (\g^a)\du\a\g C_{\g\b}= -
C_{\a\g} (\g^a)\ud\g\b$, {\it etc.}, where the last expression needs 
an extra sign as usual \ggrs.  Other important identities are the Fierz
identities 
$$ \li{ & \d\du{\[\a}\g \d\du{\b\]} \d = + \fracm1{16} \left[ \,
C_{\a\b} C^{\g\d} - \fracm 1 6 (\g_{a b c})_{\a\b} (\g^{a b c})^{\g\d} +
\fracm 1 {24} (\g_{a b c d})_{\a\b} (\g^{a b c d})^{\g\d} \, \right] ~~,        
& (3.2a) \cr  
& \d\du{(\a}\g \d\du{\b)}\d = - \fracm 1{16} \left[ \, (\g^a)_{\a\b}
(\g_a)^{\g\d} - \half (\g^{a b})_{\a\b} (\g_{a b})^{\g\d} + \fracm1{120}
(\g^{\[5\]})_{\a\b} (\g_{\[5\]})^{\g\d} \,  \right]  ~~.    
&(3.2b) \cr } $$
 
We summarize the most useful identities in Tables 1 through 6 below, 
which will be of great importance, once we have understood the way to use
them.\footnotew{To our knowledge, these results have never been 
published in literature for easy access.}    
              
\vskip 0.5in

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{123}$ && $X_{143}$ && $X_{233}$ && $X_{213}$ &&
$X_{523}$ &\cr\tablerule  
&& $Y_{23}$ &&   $-72$     && $0$    && $-40$ && $+16$ && $+1008$       
&\cr\tablerule && $Y_{43}$ && $+4$ && $-28$ && $+24$ && $-4$ && $-112$      
&\cr\tablerule \noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in
\hskip 1.0in Table 1: $~\g^g\-$Multiplication for $~Q_{\[3\]}$ 
\vskip 0.2in}

\vskip 0.3in

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{123}$ && $X_{143}$ && $X_{233}$ && $X_{213}$ &&
$X_{523}$ &\cr\tablerule  
&& $Y_{13}$ && $-32$ && $0$   && $+160$ && $+96$ && $-1152$       
&\cr\tablerule && $Y_{33}$ && $+8$ && $0$ && $+40$ && $+16$ && $-432$  
&\cr\tablerule     
&& $Y_{53}$ && $-4$ && $-28$ && $+16$ && $-4$ && $+32$ 
&\cr\tablerule 
\noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in 
\hskip 1.0in Table 2: $~\g^{g h}\-$Multiplication for $~Q_{\[3\]}$ 
\vskip 0.2in}

%\vskip 0.3in

\vfill\eject

~~~
\vskip 0.6in

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{123}$ && $X_{143}$ && $X_{233}$ && $X_{213}$ &&
$X_{523}$ &\cr\tablerule  
&& ${\Tilde Y}_{23}$ && $+4$ && $0$ && $+4$ && $0$ && $-72$       
&\cr\tablerule 
&& ${\Tilde Y}_{43}$ && $+8/3$ && $0$ && $-8/3$ && $-8/3$ && $0$      
&\cr\tablerule     
&& $Y_{63}$ && $-1/3$ &&  $0$ && $+1$ && $+2/3$ && $-6$       
&\cr\tablerule 
&& $Y_{83}$ && $+4$  && $+4$ && $-8$ && $-4$ && $+16$       
&\cr\tablerule \noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in
\hskip 1.0in Table 3: $~\g^{g h k l m}\-$Multiplication for $~Q_{\[3\]}$ 
\vskip 0.2in } 

%\vfill\eject

\vskip 0.3in

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{134}$ && $X_{154}$ && $X_{224}$ && $X_{244}$ &&
$X_{624}$ &\cr\tablerule  
&& $Y_{34}$ &&  $-72$ && $0$ && $+16$ && $-24$ && $-960$       
&\cr\tablerule 
&& $Y_{54}$ && $+4$ && $-28$ && $-4$ && $+20$ && $+72$      
&\cr\tablerule \noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in
\hskip 1.0in Table 4: $~\g^g\-$Multiplication for $~Q_{\[4\]}$ 
\vskip 0.2in}
\vskip 0.3in

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{134}$ && $X_{154}$ && $X_{224}$ && $X_{244}$ &&
$X_{624}$ &\cr\tablerule  
&& $Y_{24}$ && $-48$  && $0$ && $+64$ && $+144$ && $-1920$       
&\cr\tablerule 
&& $Y_{44}$ &&  $+8$ && $0$ && $+16$ && $+56$ && $-320$      
&\cr\tablerule     
&& $Y_{64}$ &&  $-4$ && $-28$ && $-4$ && $+12$ && $+8$ 
&\cr\tablerule 
\noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in
\hskip 1.0in Table 5: $~\g^{g h}\-$Multiplication for $~Q_{\[4\]}$ 
\vskip 0.2in } 

\vskip 0.3 in
%\vfill\eject 

\moveright0.5in\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 400pt{\strut#& \vrule#\tabskip=1em plus2em&
  \hfil# & \vrule# & \hfil#\hfil& \vrule#& 
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule#&
  \hfil#\hfil& \vrule# & \hfil#\hfil& \vrule# 
  \tabskip=0pt \cr\tablerule
&&  ~~~~    && $X_{134}$ && $X_{154}$ && $X_{224}$ && $X_{244}$ &&
$X_{624}$ &\cr\tablerule  
&& $Y_{14}$ && $-8$ && $0$ && $0$ && $-8$  && $-64$       
&\cr\tablerule 
&& ${\Tilde Y}_{34}$ &&  $+12$ && $0$ && $-8$ && $-12$ && $+288$      
&\cr\tablerule     
&& ${\Tilde Y}_{54}$ &&  $+4$ && $0$ && $-16/3$ && $-20$ && $+160$       
&\cr\tablerule 
&& $Y_{74}$ && $-1/3$  && $0$ && $+2/3$ && $-1/3$ && $-56/3$       
&\cr\tablerule     
&& $Y_{94}$ && $+4$  && $+4$ && $-4$ && $-12$ && $+136$       
&\cr\tablerule \noalign{\smallskip} &\multispan7 ~~~~~ \hfil\cr}
\vskip -0.035in
\hskip 1.0in Table 6: $~\g^{g h k l m}\-$Multiplication for $~Q_{\[4\]}$
\vskip 0.2in } 

The method to use these tables can be clarified as follows. 
First, the $~X$'s are defined by 
$$ \eqalign{& \Xs{123}{^a}{^{b c}}{_{a b c}} ~~,  ~~~~ 
\Xs{143}{^a}{_a\g^{b c d}}{_{b c d}} ~~, \cr 
& \Xs{233}{^{ab}}{_a{}^{c d}}{_{b c d}} ~~, ~~~~ \Xs{213}{^{a b}}{^c}
{_{a b c}} ~~, \cr 
& \Xs{523}{^{a b c d e}}{_{a b}}{_{c d e}} ~~, \cr 
&\Xs{134}{^a}{^{b c d}}{_{a b c d}} ~~, ~~~~ \Xs{154}{^a}{_a\g^{b c d e}}
{_{b c d e}} ~~, \cr  &\Xs{224}{^{a b}}{^{c d}}{_{a b c d}} ~~, ~~~~
\Xs{244}{^{a b}}{_a{}^{c d e}}{_{b c d e}} ~~, \cr  &\Xs{624}{^{a b c d e
f}}{_{a b}}{_{c d e f}} ~~.  \cr }
\eqno(3.3) $$
As is easily seen, the meaning of the indices on $~X's$~ denote the number 
of indices on the $~\g\-$matrices and $~Q$'s.  Note that we need two
$~\g\-$matrices for the second factors in $~X_{143}$~ and $~X_{154}$.  This
is necessary only when $~{\scst j > k} $~ for $~X_{i j k}$,
and $~Q_{\[k\]}~({\scst k < j})$~ has no free indices.      
Similarly $~Y$'s are defined by 
$$\eqalign{& \Ys{13}{^a}{_{g h a}} ~~,~~~~ \Ys{23}{^{b c}}{_{g b c}} ~~, 
~~~~ \Ys{33}{^{\[g| a b}} {^{|h\]}{}_{a b}}  ~~, \cr 
&\Ys{43}{^g\g^{a b c}}{_{a b c}} ~~, ~~~~
\Ys{53}{^{g h}\g^{a b c}}{_{a b c}} ~~, ~~~~
\Ys{63}{_{\[g h k l|}{}^{a b}}{_{|m\] a b}} ~~,  \cr     
&\TildeYs{23}{_{\[ g h|}}{_{ |k l m\]}} ~~, ~~~~
\TildeYs{43}{_{\[g h k|}{}^c} {_{|l m\] c}} ~~, ~~~~
\Ys{83}{_{g h k l m}\g^{a b c}} {_{a b c}} ~~, \cr  
&\Ys{14}{^{\[g|}} {^{|h k l m\]}} ~~, ~~~~
\Ys{24}{^{c d}}{_{g h c d}}~~, ~~~~
\Ys{34}{^{d e f}}{_{g d e f}} ~~, ~~~~ \cr 
& \Ys{44}{_{\[g|}{}^{b c d}}{_{|h\] b c d}} ~~, ~~~~
\Ys{54}{^g\g^{c d e f}}{_{c d e f}} ~~, ~~~~ 
\Ys{64}{_{g h}{}\g^{a b c d}}{_{a b c d}} ~~, \cr
& \Ys{74} {^{\[g h k l|}{}_{b c d}}{^{|m\] b c d}} ~~,  ~~~~
\Ys{94}{^{g h k l m}\g^{a b c d}}{_{a b c d}} ~~, \cr 
& \TildeYs{34}{_{\[ g h |} {}^a} {_{|k l m\] a }} ~~, ~~~~
\TildeYs{54}{_{\[ g h k|}{}^{c d}}{_{|l m \] c d}}~~, \cr }
\eqno(3.4) $$ 
The {\it tildes} on $~Y$'s in Table 3 or 6 are needed to distinguish them
from non-tilded ones in other tables.  Note also the special $~\g\-$matrix
structures for $~Y_{43},~Y_{53},~Y_{83},~Y_{54},~Y_{64}$~ and 
$~Y_{94}$.    

The meaning of Table 1 is now as follows.  The first column means that if we
multiply $~(\g^g)^{\a\b}$~ by $~X_{123}$~ with the  
$~{\scst \a}$~ and $~{\scst \b}\-$indices contracted, the result will be the
linear combination of $~Y_{23}$~ and $~Y_{43}$~ with the free indices
$~{\scst g,~\g}$~ and $~{\scst \d}$.  In particular, Table 1 tells that 
these terms have the coefficients $~-72$~ and $~+4$, respectively.  
More explicitly
$$ (\g^g)^{\a\b} (\g^a)_{(\a\b|} (\g^{b c})\du{|\g)} \d Q_{a b c}
= -72 (\g^{b c} )\du\g\d Q\ud g{b c} + 4 (\g^g\g^{a b c})\du\g\d Q_{a b c}
~~.  
\eqno(3.5) $$  
The advantage of these tables is their compactness to convey
so much information for long formulae as above.  This can be realized, once
we have fixed the convention for the indices on $~X$'s and $~Y$'s with their
contractions. 
Table 1 through Table 3 are for $~Q_{\[3\]}$, while Tables 4 through Table 6
are for $~Q_{\[4\]}$.  The factor $~Q_{\[n\]}$~ can be replaced by any 
arbitrary totally antisymmetric tensor.  

Notice the important fact that {\it not} all of these $~X$'s are really 
independent.  As a matter of fact, there exist two identities that relate 
these $~X$'s:
$$\li{& X_{523} \equiv +10 X_{143} -24 X_{123} - 30 X_{213} + 6 X_{233} ~~,
& (3.6a) \cr 
& X_{624} \equiv + 8 X_{134} + 2 X_{154} - 24 X_{224} ~~. 
&(3.6b) \cr } $$ 
This can be directly confirmed by the use of the identity such as
$$ (\g_{a b}) _{(\a\b} (\g^b)_{\g\d)} \equiv 0 ~~.     
\eqno(3.7) $$   
We can see that the identities (3.6) are the only relationships among
these $~X$'s, by solving simultaneous equations out of Tables 1 through 3
or Tables 4 trough 6, respectively for $~X_{i j 3}$~ and $~X_{i j 4}$.  

Another practical usage of these tables can be found even in the
$~J\-$independent sector of the BIs.  For example, in the 
$~(\a\b\g,\d)\-$type BI we encounter the crucial equation 
$$ \eqalign{ +\fracm1 3 (& \g^{\[f|})_{(\a\b|} (\g^{|g h k\]} )
\du{|\g)}\d + 2 (\g^a)_{(\a\b|} (\g_a \g^{f g h k} )\du{|\g)} \d
\cr        
& - (\g^{\[f g|} )_{(\a\b|} (\g^{|h k\]} )\du{|\g)} \d - 
(\g^{a b f g h k})_{(\a\b|} (\g_{a b})\du{|\g)} \d \equiv 0~~,   \cr } 
\eqno(3.8) $$ 
but this turns out to be equivalent to (3.6b).   

Finally we give here a useful applications of our symbols of $~X$'s to
important identities.  They arise in the $~(\a\b\g,\d)\-$type BI for
products of two $~J$'s: 
$$\li{& (\g^a)_{(\a\b} J_{\g)}  (\g_a J)^\d  
  = \fracm 1 {768} \left( + 24 X_{110} + 4 X_{143} - 24 X_{123} + X_{154} 
  - 8 X_{134} \right) ~~, \cr 
& (\g^{c d})_{(\a\b} J_{\g)} (\g_{c d} J)^\d    
  = \fracm 1{384} \left( - 120 X_{110} + 20 X_{143} + 24 X_{233} - 5
  X_{154} - 8 X_{244} \right)  ~~, \cr   
& (\g^a)_{(\a\b|} (\g_{a b} J)_{|\g)} (\g^b J)^\d  
  = \fracm 1{384} \left( + 120 X_{110}  + 8 X_{143} + 12 X_{123} +
  X_{154} + 4 X_{134} \right) ~~,  \cr   
& (\g^a)_{(\a\b|} (\g^a\g^{bc} J)_{|\g)} (\g_{b c} J)^\d 
  = \fracm 1{384} \left( - 1320 X_{110} + 28 X_{143} + X_{154} \right) 
  ~~, \cr
& (\g^a)_{(\a\b|} (\g^b J)_{|\g)} (\g_{a b} J)^\d 
  = \fracm 1{384} \Big( -120 X_{110} + 8 X_{143} - 60 X_{123} - X_{154}
+ 12 X_{134} \Big) ~~,  \cr    
& (\g^{a b})_{(\a\b|} (\g_a J)_{|\g)} (\g_b J)^\d  
  = \fracm 1{384} \Big( - 120 X_{110} - 20 X_{143} +24 X_{213}  
  - 12 X_{233} \cr  
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~
  - 5 X_{154} - 4 X_{244} + 12 X_{224}  \Big) ~~,  \cr  
& (\g^{a b})_{(\a\b|} (\g\du b c J)_{|\g)} (\g_{c a} J)^\d 
  =\fracm1 {384} \Big( +1080 X_{110} - 60 X_{123} + 24 X_{233} + 12
  X_{134} \cr
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~
  - 8 X_{244} - 7 X_{154}\Big) ~~. 
&(3.9) \cr} $$ 


%\bigskip\bigskip\bigskip

\vfill\eject



\centerline{\bf 4.~~Relationship with M-Theory}

We now mention a possible important link with M-theory \schwarz, {\it
via} the $~\k\-$symmetry \ref\pol{J.~Hughes, J.~Liu and J.~Polchinski, 
\pl{180}{86}{370}.} of supermembrane theory \bst.  As suggested
in the recent literature \ref\smtlink{C.~Schmidhuber, {\it {``D-brane 
Actions"}} Princeton Univ.~preprint PUPT-1585 (Jan.~1996), hep-th 
960103.}, M-theory seems to be related to the existence of the
supermembrane action in a supergravity background.  Thus, there is
an {\it {a}} {\it {priori}} possibility that our putative off-shell 
supergravity is in conflict with this requirement.  This is exactly
analogous to the non-trivial consistency confirmation between 
$\kappa$-symmetry of the 10D Green-Schwarz action and the low-energy 
{\it dual} formulation of 
the heterotic string first performed ten years ago \gnss. So a good test
of our off-shell supergravity proposal is to see if it is a consistent
background for the supermembrane action and likely M-theory.  

We recall that the supermembrane action \bst\ takes the form,\footnotew
{The notation here is self-explanatory.}      
$$ I = \int d^3 x \left[ \,
+ \half {\sqrt {- g}} g^{i j} \eta_{a b} \Pi\du i a \Pi\du j b - \half
{\sqrt{-g}} - \fracm 1 3 \e^{i j k} \Pi\du i C \Pi\du j B \Pi\du k A 
A_{A B C} \,  \right]  \equiv \int d^3 x \, L ~~.  
\eqno(4.1)  $$  
In our off-shell backgrounds (2.3), the $~\k\-$transformation takes the form
\pol
$$ \li{& \d_\k E^\a
\equiv (\d_\k Z^M) E\du M \a = ( I + \G) \ud \a\b \k^\b ~~,~~~~ \G \equiv +
\fracm i{6{\sqrt{-g}}} \e^{i j k} \Pi\du i a \Pi\du j b \Pi\du k c \g_{a
b c}~~,  
& (4.2a) \cr  
& \d_\k E^a \equiv (\d_\k Z^M) E\du M a = 0 ~~, 
~~~~\Pi\du i A \equiv (\partial_i Z^M) E\du M A~~,      
& (4.2b) \cr} $$
for the 11D superspace vielbein $~E\du M A$~ with the coordinates $~Z^M$.    
In the invariance check we adopt the 1.5-order
formulation, namely we can always use the algebraic $~g\low{i
j}\-$field equation as the embedding condition \bst 
$$ g\low{i j} = \eta_{a b}\Pi\du i a \Pi\du j b ~~. 
\eqno(4.3) $$                           
Now the $~\k\-$transformation of (4.1) yields new terms containing 
$~T\du{c\b} a$~ and $~F_{a b c \d}$:
$$\eqalign{\d_\k L = \, &  + {\sqrt{-g}} g^{i j} (\d_\k E^\b) \left[\, i
\Pi\du i\g  (\g^a)_{\g\b} \Pi_{j a} + \Pi\du i c T\du{c\b} a \Pi_{j a} \,
\right] \cr  & - \fracm 1 3 \e^{i j k} (\d_\k E^\d) \Pi\du j b \Pi\du k a
\left[\, \half \Pi\du i \g (\g_{a b})_{\g\d} + \Pi \du i c F_{a b c \d} \,
\right] ~~. \cr } 
\eqno(4.4) $$ 
Using (4.3) and (4.2a), as well as relations under (4.3) such as 
$$\fracm1{\sqrt{-g}} \e\du i {j k} \Pi\du j b \Pi\du k c \g_{b c} \, \G = -
2i \Pi\du i a \g_a ~~, ~~~~
g^{i j} \Pi\du i\g \Pi_{j a} \g^a \G = + \fracm i{2{\sqrt{-g}}} 
\e^{i j k} \Pi\du i\g \Pi\du j a \Pi\du k b \g_{a b} ~~,    
\eqno(4.5) $$ 
we see that all the $~J\-$independent terms cancel each other as the 
usual supermembrane case \bst, while all the $~J\-$dependent terms are
finally re-arranged into two kinds of terms proportional to $J^\a\G\du\a\b
\k_\b$~ and $J^\a\k_\a$, which turn out to cancel themselves.  

The above discussion constitutes a proof of the fact that our off-shell 
supergravity formulation (2.4) can also provide a consistent background 
for the supermembrane action. This is necessary for there to be a link
\smtlink\ of our off-shell system with M-theory \schwarz.       


\bigskip\bigskip\bigskip


%\vfill\eject


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

In this paper we have given partial but explicit results for the
beginning of a complete off-shell formulation of a 11D, $N=1$ supergravity 
theory that is consistent with the $~\k\-$symmetry of the supermembrane 
action, interpreted as the desirable zero-mass limit of M-theory
\schwarz.  At this point in our understanding, all the off-shell effects 
are controlled by the spinorial superfield $~J_\a$, which is an 11D analog 
of the superspace formulation of 4D, $N=2$ supergravity developed in 
1980 \gatesfourd.    

Our result is certainly ``a counter example'' to the conventional myth 
that there can exist {\it no} off-shell formulation for 11D
supergravity \cremmer\ with auxiliary fields. It has long been thought that
due to the strict supersymmetry in the maximal dimensions (where 
the complete on-shell theory contains only gauge fields with no matter 
fields), there would be no generalization to an off-shell formulation.  
It is remarkable that to the orders we have checked, we have succeeded in 
overcoming this taboo by introducing two algebraically independent 
superfields $~W_{a b c d}$~ and $~J_\a$.  The recent development in M-theory 
\schwarz\ has provided a strong motivation to put an end to such a myth, 
introducing some auxiliary fields that will be important to accommodate 
possible higher-order curvature corrections presumably generated from 
M-theory.      

Our results stress the importance of the superspace formulation \ggrs\
of supergravity, which still remains as the most powerful tool for 
supergravity and related subjects even more than twenty years after the 
first discovery of superspace formulations \ref\salamstrathdee
{A.~Salam and J.~Strathdee, \np{76}{74}{477}.} and supergravity itself 
\ref\ffn{D.Z.~Freedman, S.~Ferrara and P.~van Nieuwenhuizen, 
\pr{13}{76}{1324}; \ibid{\bf{D14}}{76}{912}.}. This is because it is 
always the case both in superstring and supermembrane theories 
that the critical zero mass sector of a total theory is controlled by 
supergravity, whose understanding plays a key role in the formulations 
as field theories with which we can not dispense.  Even though superspace 
formulations remain a tool, it contains still many mysteries 
that no one has completely mastered.

An important application of our result is its utilization {\it via} 
strong/weak duality \wittencomments, namely a kind of dimensional 
reduction into 10D, $N=2$ supergravity/superstring theory \gsw\ should 
exist.  In particular, we 
expect the structure of superstring corrections in 10D $N=2$ system differ
markedly from our previous proposals \gnss\nishino\ 
from those for the 10D, $N=1$ superstring. With
the reduction of our 11D results to 10D $N=2$ theory, we at last have a 
geometrical superspace structure to control the appearance of the $~\a\,'
{}^3 \z{\scst (3)}$~ terms \ref\gw{D.~Gross and E.~Witten, \np{277}{86}{1};
M.T. Grisaru, A.E.M. van de Ven and D. Zanon, \pl{173}{86}{423};  
\np{277}{86}{388}; \np{277}{86}{409}; \pl{177}{86}{347}.} as 
the lowest order corrections instead of curvature square terms at 
$~{\cal O}(\a^{\,}{}')$~ known to occur in the $N=1$ theory.    

We also mention another attempt by one of the authors 
\ref\nishinocs{H.~Nishino, Maryland preprint, UMDEPP, in preparation 
(Jan.~1996).} in order to accommodate generalized Chern-Simons terms into  
11D, $N=1$ supergravity.  In this formulation an eleventh-rank antisymmetric 
field expected in the M-theory arises naturally.  Even though we do not 
know the direct link of our present off-shell formulation to this 
Chern-Simons modification \nishinocs, we have an increased chance to find 
that all of these theories are related to each other in the context of 
M-theory \schwarz.               

In closing, we do so on a note of caution also.  
For although we believe our observation is important, we 
know of at least two arguments that suggest that there must exist at 
least one other tensor superfield that will be required to have a 
completely off-shell formalism. This is to be expected even from the
structure of the non-minimal 4D, N = 1 supergravity.  There it is known
that there are {\it three} algebraically independent tensors
$~W_{\alpha \beta \gamma}, ~G_{\alpha \Dot \beta}$~ and $~T_{\a}$. In
this work we have introduced the eleven dimensional analog of $~W_{\a\b\g}$~
and $~T_\a$, so apparently it remains to find the eleven dimensional
analog of the $G_{\alpha \Dot \beta}$.  In future works, these aspects of
the eleven dimensional theory will require further study.

We expect our explicit results to open completely new directions to be 
explored for superstring, super p-brane and supergravity theories. 
We have increased optimism for resolving long-standing problems 
in superstring theories such as the dualities and vacuum structures of 
superstring theory.  

\bigskip\bigskip\bigskip

The authors are grateful to J.H.~Schwarz for stimulating discussions that
motivated us to look again at our suggestion made in 1980 \gatesfourd.

\vfill\eject

\listrefs

\end{document} 

