hep-th0107155.tex
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\doit0{
% Horizontal alignment of accents: bars, hats, tildes, etc.
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\doit{#1}{\font\twfvmi  = ammi10   \@magscale5 % math italic
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\font\twfvsy  = amsy10   \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018			% Gothic
\font\sc=font005			% script
\def\Go#1{{\hbox{\go #1}}}	% Gothic for single characters 
%  in equations
\def\Sc#1{{\hbox{\sc #1}}}	% script for single characters 
%  in equations
\def\Sf#1{{\hbox{\sf #1}}}	% sans serif for single characters 
%  in eqs
\font\oo=circlew10	  % thick circles (hollow: ` and a-n , 
%  solid: p-z)
\font\ooo=circle10			% thin circles   
\font\ro=manfnt				% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for 
%  left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

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

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


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

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

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

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

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

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} 
\raisebox{-.45ex}{$\hskip .48ex { |}$}}	
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} 
\raisebox{.45ex}{$\hskip .48ex { 
|}$}}	
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~} % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}}		% D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}}		% D'Alembertian
\def\cbo{\Sc [}						% curly "
\def\pa{\partial}					% curly d
\def\dell{\bigtriangledown}				% hi ho the dairy-o
\def\pr{\prod}						% product
\def\iff{\leftrightarrow}				% <-->
\def\conj{{\hbox{\large *}}}				% complex conjugate
\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}} %< or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}	%> or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu %
\llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu %
\llap {$\ddot
	\smile$}}}					% happy face
\def\dg{\sp\dagger}					% hermitian 
% conjugate
\def\ddg{\sp\ddagger}					% double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}				% superscript (unaligned)
\def\sb#1{{}_{#1}}				% sub"
\def\oldsl#1{\rlap/#1}				% poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1}	% good slash for lower case
\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}	% " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}	% " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}	% 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}	% 

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

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

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
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	\else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous				% eqalignno
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	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}
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% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
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%   (LaTeX)

% Title page
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%   (LaTeX)
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        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
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Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
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\def\endtitle{\end{quotation}\newpage}			% end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
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\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
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\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}}
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\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}}
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\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}
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\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}
\def\calX{{\cal X}}
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\def\rmd{{\rm d}} 
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\def\Mpl{M_{\rm Pl}}
\def\mw{m\low{\rm W}}
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\def\Re{{\rm Re}\,\,}  \def\Im{{\rm Im}\,\,} 
\def\tildef{{\tilde f}}
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\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}} 
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\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}} 
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%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
	\hbox{$\scriptstyle{#2}$}}}}}			% fraction
%\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A 
{\buildrel\leftrightarrow\over D}_{#1} B\right)} 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}

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

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

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

\begin{document}

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

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

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

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

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

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

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

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

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

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

% \input defrr.tex
\font\smallcmr=cmr6 scaled \magstep2 

\def\alephnull{~$\large{\aleph_0}\,$~} 
\def\spinorip#1{\left({#1}\right)} 
\def\Check#1{\raise02pt\hbox{\Large\v{}}{\hskip -8.5pt}{#1}} 
\def\kd#1#2{\d\du{#1}{#2}} 
\def\jgtp#1#2#3{Jour.~of Group 
Theory for Physicists {\bf{#1}} (19{#2}) {#3}}

\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\Dsl{D\!\!\!\! /{\hskip 2pt}} 
\def\calDsl{{\cal D}\!\!\!\!{\hskip 1.5pt}/{\hskip 2.0pt}} 
\def\lhand{\low{\rm L}} \def\rhand{\low{\rm R}} 
\def\calM{{\cal M}} \def\calK{{\cal K}} \def\calU{{\cal U}} 
\def\calF{{\cal F}} \def\calG{{\cal G}} \def\calC{{\cal C}}
\def\unC{\underline C} 
\def\lanu{\langle\n|\,} \def\ranu{\,|\n\rangle} 
\def\laminu{\langle-\n|\,} \def\raminu{\,|\!-\n\rangle} 
\def\dashs{{}\hskip 1pt '}

\def\Pup{P_\uparrow} \def\Pdown{P_\downarrow} 
\def\Pupdown{P_{\uparrow\downarrow}} \def\Pdownup{P_{\downarrow\uparrow}}  
\def\na{n_a} \def\nb{n_b}
\def\nc{n_c} \def\nd{n_d} \def\ne{n_e} \def\nf{n_f}
\def\ng{n_g} \def\ma{m_a} \def\mb{m_b} \def\mc{m_c}
\def\md{m_d} \def\me{m_e} \def\mf{m_f} \def\mg{m_g}  
\def\nsl{{n\!\!\!\!\hskip1.2pt/}\,} \def\msl{{m\!\!\!\!/}\hskip1.8pt}
\def\Du#1#2{\low{#1}{}^{#2}} \def\uD#1#2{^{#1}{}\low{#2}} 
\def\phia{\nabla_a\varphi} \def\phib{\nabla_b\varphi} 
\def\phic{\nabla_c\varphi} \def\phid{\nabla_d\varphi}
\def\phie{\nabla_e\varphi} \def\phif{\nabla_f\varphi} 
\def\phig{\nabla_g\varphi}
\def\phim{D_m\varphi} \def\phin{D_n\varphi} 
\def\phir{D_r\varphi} \def\phis{D_s\varphi} 
\def\phimu{\partial_\m\varphi} \def\phinu{\partial_\n\varphi}
\def\phiroh{\partial_\r\varphi} \def\phisigma{\partial_\s\varphi}
\def\phitau{\partial_\t\varphi} \def\philambda{\partial_\l\varphi}
\def\phiomega{\partial_\o\varphi}  \def\phipsi{\partial_\psi\varphi} 
\def\tilphim{D_m\Tilde\varphi} \def\tilphin{D_n\Tilde\varphi} 
\def\tilphir{D_r\Tilde\varphi} \def\tilphis{D_s\Tilde\varphi} 
\def\tilphia{\nabla_a\Tilde\varphi} \def\tilphib{\nabla_b\Tilde\varphi} 
\def\tilphic{\nabla_c\Tilde\varphi} \def\tilphid{\nabla_d\Tilde\varphi}
\def\tilphie{\nabla_e\Tilde\varphi} \def\tilphif{\nabla_f\Tilde\varphi} 
\def\tilphig{\nabla_g\Tilde\varphi}
\def\phimu{\partial_\m\varphi} \def\phinu{\partial_\n\varphi}
\def\tilphiroh{\partial_\r\Tilde\varphi} 
\def\tilphisigma{\partial_\s\Tilde\varphi}
\def\tilphitau{\partial_\t\Tilde\varphi} 
\def\Pisl{\Pi\!\!\!\!/\hskip2.0pt}   
\def\nablasl{\nabla\!\!\!\!/} 
\def\calM{{\cal M}}
\def\TildeM{\Tilde{\cal M}}


%\def\det{{\rm det}\,} 
\def\Det{{\rm Det}\,} \def\calDet{{\cal D}\rm et\,} 
%\def\Sgn{\hbox{\rm Sgn}\,}
%\def\Norm{|\hskip -2.0pt|}
\def\hskp{\hskip -5pt}

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\rightline{CSULB--PA--01--2}
\rightline{hep-th/0107155} 
\rightline{(Revised Version)}

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\baselineskip 18pt 

\centerline{\large\bf Teleparallel Superspace in Eleven Dimensions} 

\centerline{\large\bf Coupled to Supermembranes$\,$\footnote{This
work is supported in part by NSF grant \# PHY-93-41926.}}


\vskip 0.5in


\baselineskip 10pt 


\centerline{S.~James ~G{\smallcmr ATES}, Jr.,$^\dagger$~
Hitoshi ~N{\smallcmr ISHINO}$^\star$~ and ~ 
Subhash ~R{\smallcmr AJPOOT}$^\#$}

\vskip 0.3in

\centerline{$\dagger$~ {\it Dept.~of Phys., Univ.~of Maryland,  
College Park, MD 20742-4111}} 
\vskip 0.05in 
\centerline{gatess@wam.umd.edu}
\vskip 0.15in 
\centerline{$\star$~{\it Dept.~of Phys.~\& Astr., 
California State Univ., Long Beach, CA 90840}}
\vskip 0.05in 
\centerline{hnishino@csulb.edu} 
\vskip 0.15in 
\centerline{$\,\#$~ {\it Dept.~of Phys.~\& Astr., 
California State Univ., Long Beach, CA 90840}}
\vskip 0.05in 
\centerline{rajpoot@csulb.edu}  


\vskip 1.6in


\hsize=5.5in

\centerline {\bf Abstract} 
\vskip 0.1in  

\baselineskip 16pt 

~~~We present a superspace formulation of $~N=1$~ eleven-dimensional
supergravity with no manifest local Lorentz covariance, which we call
teleparallel superspace.  This formulation will be of great importance,
when we deal with other supergravity theories in dimensions higher than 
eleven dimensions, or a possible formulation of 
noncommutative supergravity.  As an illustrative example, we 
apply our teleparallel superspace formulation to the case of 
$~N=1$~ supergravity in twelve-dimensions.  
We also show the advantage of teleparallel 
superspace as backgrounds for supermembrane action.   


\vskip 0.6in

\noindent{\small PACS: ~04.65, ~02.40.H, ~02.40.K,~02.40.M}
\vskip -0.1in

\noindent{\small Key Words: Superspace, Supergravity, Teleparallelism, 
Lorentz Invariance}


\vfill\eject


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\vskip 0.1in                                                       
\centerline{\bf 1.~~Introduction}          
         
Local Lorentz covariance is usually taken for granted in the 
conventional supergravity theories 
%%%
\ref\pvn{P.~van Nieuwenhuizen, 
\prep{68}{81}{189}.}%
%%%
\ref\ggrs{S.J.~Gates, Jr., M.T.~Grisaru, M.~Ro\v cek and
W.~Siegel, {\it `Superspace'}, Benjamin/Cummings, Reading, 
MA (1983).},
%%%
starting with the $~N=1$~ supergravity in four dimensions
(4D)\footnotew{We also use the notation $~D=4$~ such as in $~D=4,\,N=1$~
supergravity.} 
%%%
\ref\sgoriginal{D.Z.~Freedman, P.~van Nieuwenhuizen and
S.~Ferrara, \pr{13}{76}{3214}; S.~Deser and B.~Zumino, \pl{62}{76}{335}.}, 
up to the $~N=1$~ supergravity in 11D 
%%%
\ref\cjs{E.~Cremmer, B.~Julia 
and N.~Scherk, \pl{76}{78}{409}; E.~Cremmer and B.~Julia, \pl{80}{78}{48};
\np{159}{790}{141}.}%
%%%
\ref\cf{E.~Cremmer and S.~Ferrara, \pl{91}{80}{61}; L.~Brink and P.~Howe,
\pl{91}{80}{384}.}.    
%%%
However, the recent developments in higher dimensional supergravity in
$~D\ge 12$~ 
%%%
\ref\nishone{H.~Nishino, \pl{428}{98}{85}.}, 
%%%
or of an $~N=2$~ supergravity theory 
%%%
\ref\nishtwo{H.~Nishino, hep-th/9706148, \pl{437}{98}{303}.}%
%%%
\ref\sgall{H.~Nishino, hep-th/9807199, \np{542}{99}{217}.}
%%%
motivate formulations in which the local Lorentz symmetry is not
manifest.  As a matter of fact, in the consideration of 
generalized algebra called M-algebra 
%%%
\ref\malgebra{E.~Sezgin, \hepth{9609086}, \pl{392}{97}{323}.} 
%%%
for M-theory 
%%%
\ref\mtheory{{\it For reviews, see e.g.,}
P.K.~Townsend, {\it `M-Theory from its Superalgebra'}, \hepth{9712004}  
{\it and references therein};  
{\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}.}, 
%%%
the manifest Lorentz covariance is not crucial, but is realized only as
hidden symmetry in terms of composite connections.   We call such
superspace, in which no local Lorentz symmetry is manifest, 
`teleparallel superspace'.\footnotew{We adopt this terminology for 
superspace, in which local Lorentz covariance is not manifest, even if it
is  {\it not} explicitly broken.  In this sense, our terminology covers 
a wider set of formulations in superspace.   This formulation is also
similar to the superspace  for $~D=4,\,N=1$~ supergravity in 
%%%
\ref\taylor{J.G.~Taylor, \pl{78}{78}{577}.}.}      

In conventional supergravity theories, there has been no strong
motivation of  teleparallelism formulation.  This is because in component
formulation of  conventional supergravity, the local Lorentz covariance
is manifest from  the outset anyway, and moreover, there is no strong
reason to consider its explicit breakings\footnotew{See,  {\it e.g.,}
subsection 1.5 of ref.~\pvn.}, because of the need of local Lorentz
covariance for removing unwanted ghosts in the system.   This situation
in component turns out to be similar in superspace formulations
\taylor.\footnotew{Some superspace formulation in 4D in this direction 
was presented in \taylor, using only the  torsion superfield  with no
curvature superfields, in order to reduce possible  counter-terms for
renormalizability of supergravity.}~~As such, there has been so far no
spectacular development for teleparallel supergravity, until  the recent
construction of higher-dimensional supergravity in $~D\ge 12$~ with no
manifest Lorentz covariance \nishone\nishtwo\sgall.   Even though these
higher-dimensional supergravity theories have  no built-in local Lorentz
covariance, they have become important nowadays, motivated also by
M-theory \mtheory\ and F-theory 
%%% 
\ref\ftheory{C.~Vafa, \np{469}{96}{403}.},
%%%
or S-theory 
%%%
\ref\stheory{I.~Bars, \pr{55}{97}{2373}.},
%%%
and so forth.   

Another important motivation of considering the lack of manifest Lorentz 
covariance is associated with the recent development of noncommutative 
geometry, in which the introduction of `constant tensor' 
$~\theta^{\m\n}$~ explicitly breaks Lorentz covariance 
%%%
\ref\noncomm{A.~Connes, M.R.~Douglas and A.~Schwarz, JHEP {\bf 9802} 
(1998) 003; 
Y.K.E.~Cheung and M.~Krogh, Nucl.~Phys.~{\bf B528} (1998) 185;
C.-S.Chu and P.-M.~Ho, Nucl.~Phys.~{\bf B528} (1999) 151; 
V.~Schomerus, JHEP {\bf 9906} (1999) 030; 
F.~Ardalan, H.~Arfaei and M.M.~Sheikh-Jabbari, JHEP {\bf 9902} (1999) 016;
J.~Hoppe, Phys.~Lett.~{\bf B250} (1990) 44;
D.B.~Fairlie, P.~Fletcher and C.K.~Zachos, Phys.~Lett.~{\bf B218} 
(1989) 203; 
N.~Seiberg and E.~Witten, JHEP {\bf 9909} (1999) 032,
hep-th/9908142.}.    
%%%
For example, in a recent paper 
%%% 
\ref\nrnoncomm{H.~Nishino and S.~Rajpoot,  
{\it `Complex Teleparallel Gravity as 
Foundation for Noncommutative Gravity'}, 
CSULB--PA--01--1, \hepth{0107216}.} 
%%% 
we have shown that teleparallel gravity can be the foundation of 
noncommutative gravity, in order to delete the undesirable 
negative energy ghosts in the antisymmetric component $~B_{\m\n}$~ 
in the complex metric $~g_{\m\n}$~ in the Lorentz covariant 
formulation 
%%% 
\ref\chamseddine{A.~Chamseddine, \hepth{0005222}, 
Commun.~Math.~Phys.~{\bf 218} (2001) 283; 
\hepth{0010268}, Int.~Jour.~of Mod.~Phys.~{\bf A16} (2001) 759.}.      
%%% 
From the viewpoint that the existence of the constant tensor
$~\theta^{\m\n}$~  already breaks the Lorentz symmetry, it is even 
imperative to consider the supersymmetric extension, {\it i.e.,}
teleparallel supergravity. 


In this paper we present a teleparallel superspace formulation with a
consistent set of superspace constraints for Bianchi identities, together
with the physical superfield equations.  We also couple this background to
supermembrane action, and confirm its fermionic ~$\k\-$symmetry.  
Interestingly, we find that the fermionic $~\k\-$symmetry becomes even
simpler  and more natural in the teleparallel superspace formulation. 
Such a formulation will be of great importance when we deal with  higher
dimensional supergravity \nishone\nishtwo\sgall\ in which the local
Lorentz symmetry is not manifest.  As an illustrative example, we 
describe how to reformulate $~N=1$~ superspace supergravity in 12D 
\nishone\sgall\ in teleparallel superspace.    
 
 

\bigskip\bigskip\bigskip



\vbox{
\centerline{\bf 2.~Notational Preliminaries} 

Before giving our constraints, we first set up our notations, which are 
conceptually different from the conventional supergravity.  
Our superspace covariant derivative $~E_A $~ is simply defined by 
}
$$ \li{ &E_A \equiv E\du A M \partial_M ~~, 
&(2.1) \cr } $$
with no term for Lorentz connection $~\phi\du{A \, b} c$~ \ggrs.  
Relevantly, the anholonomy coefficients $~C\du{A B} C$~ are defined by 
$$ \li{& C\du{A B} C\equiv \big( E_{\[ A} E\du{B)} M\big) E\du M C ~~, 
&(2.2) \cr } $$
with no explicit Lorentz connection here, either.  Similarly, 
the superfield strength $~F_{A B C D}$~ is defined by
$$ \li{ & F_{A B C D} \equiv \frac1 6 E_{\[A} A_{B C D)} - 
     \frac 14 C\du{\[A B|} E A_{E |C D)} ~~.
&(2.3) \cr } $$
with no Lorentz connection.  
Accordingly, the superspace Bianchi identities  
are only $~C\-$ and $~F\-$types:
$$ \li{& E_{\[A} C\du {B C)} D - C\du{\[A B|} E C\du{E| C)}D \equiv 0~~, 
&(2.4) \cr 
& \frac1{24} E_{\[A} F_{B C D E)} -\frac1{12} C\du{\[A B|} F F_{F |C D E)} 
     \equiv 0~~.   
&(2.5) \cr } $$
We sometimes refer them respectively as $~(A B C,D)\-$ and 
$~(A B C D E)\-$type Bianchi identities.  Even though we do not give a
detailed computation, the l.h.s.~of both (2.4) and (2.5) are covariant
under the following local Lorentz transformation  with the parameter
$~\L^{a b}$~ \ggrs, despite the absence of Lorentz connection
$~\phi\du{A\, b}c$:
$$ \li{&\d_{\rm L} E\du A M = \frac 12 \L^{b c} \big( \calM_{b c} \big)\du A B 
     E\du B M ~~, 
&(2.6\rma) \cr 
&\d_{\rm L} C\du{A B} C = + \frac 12 \L^{b c} 
     \big( \calM_{b c} \big)\du{\[A|} D C\du{D|B)} C 
     + \frac 12 (-1)^{(A+B)(C+D)} 
     \L^{b c} \big( \calM_{b c} \big)^{C D} C_{A B D} \cr 
& ~~~~~ ~~~~~ ~~~~\, + \frac 12 \big( E_{\[A|} \L^{b c} \big) 
     \big(\calM_{b c} \big)\du{|B)} C ~~, 
&(2.6\rmb) \cr 
& \d_{\rm L} F_{A B C D} = \frac1{12} \L^{a b} 
    \big(\calM_{a b}\big)\du{\[A|}E F_{E|B C D)} ~~, 
&(2.6\rmc) } $$  
where (2.6b) follows from (2.6a).  The local Lorentz covariance of the 
l.h.s.~of (2.4) and (2.5) is confirmed by the mutual cancellations of 
terms generated by the local Lorentz transformation (2.6).  To put it 
differently, the local Lorentz symmetry is a `hidden' symmetry in this
formulation.  
  
Accordingly, the Ricci tensor component $~R_{a b}(\phi)$~ in the
conventional superspace can be re-expressed in terms of the anholonomy
coefficients by  
$$ \li{R_{a b}(\phi) = & - E_a C_b - \frac12 E_c C\du{a b}c 
     - \frac12 E_c C\ud c{(a b)}  - C\du{a c}\g C\du{\g b} c\cr
& + \frac12 C\du a{c d} C_{b d c} + \frac12 C\du a{c d} C_{b c d} 
     + \frac12 C\du{a b} c C_c   
     + \frac12 C\ud c{( a  b )} C_c - \frac14 C_{c d a} C\ud{c d} b ~~, 
&(2.7) \cr } $$
Here $~C_a \equiv C\du{a b} b$~ is the vectorial component of $~C\du{a
b}c$.  Note that (2.7) is an equality only to understand the total local
Lorentz covariance, because the l.h.s.~makes sense only in the
conventional superspace with the explicit Lorentz connection, while the
r.h.s.~in terms of the anholonomy coefficients is useful only in our
teleparallel superspace.  In other words, we should not use such
expressions as the l.h.s.~of (2.7) in our teleparallelism. 
Interestingly, the antisymmetric component $~R_{\[a b\]}$~  is shown to
vanish identically due to the $~(a b c,c)\-$type Bianchi identity:
$$ \li{ & R_{\[a b\]} = - E_{\[a} C_{b\]} - E_c C\du{a b} c 
     + C\du{a b} c C_c - C\du{\[a| c}\g C\du{\g|b\]} c \equiv 0~~, 
&(2.8) \cr } $$ 
therefore (2.6) has effectively only the symmetric component in $~R_{(a b)}$.  



\bigskip\bigskip\bigskip





\centerline{\bf 3.~Constraints and Superfield Equations} 

There are basically two methods to fix superspace constraints for our
teleparallel superspace.  The first method is to use the relationships 
between the torsion superfields in the conventional superspace and the 
teleparallel superspace, and another way is the direct way, writing down 
all the possible term with unknown coefficients to be fixed by the 
satisfactions of all the Bianchi identities (2.1) and (2.2).  Both of these
methods give the following consistent set of constraints:   
$$\li{& C\du{\a\b} c = + i \big( \g^c \big)_{\a\b}~~, 
&(3.1\rma) \cr 
& F_{\a\b c d} = + \frac 12 \big( \g_{c d} \big)_{\a\b} ~~, 
&(3.1\rmb) \cr 
& C\du{\a\b}\g = + \frac14 \big( \g_{d e} \big)\du{(\a}\g C\du{\b)}{d e} ~~, 
   ~~~~ C\du\a{b c} = - C\du\a{c b} ~~, 
&(3.1\rmc) \cr 
& C\du{\a b} \g = + \frac i{144} \big(\g\du b{\[4\]} F_{\[4\]} 
     + 8 \g^{\[3\]} F_{b\[3\]} \big)\du\a\g 
     - \frac 18 \big( \g^{c d} \big)\du\a\g 
     \big(2C_{b c d} - C_{c d b} \big) ~~, 
&(3.1\rmd) \cr 
& E_\a C_{\b c d} = + \frac1{144} \big( \g\du{c d}{\[4\]} \big)_{\a\b} 
     F_{\[4\]} 
     + \frac16 \big(\g^{\[2\]} \big)_{\a\b} F_{c d \[2\]} 
     - \frac i4 \big( \g^e \big)_{\a\b} C_{c d e} 
     + \frac i 4 \big( \g^e \big)_{\a\b} C_{e \[c d \]} \cr 
& ~~~~~ ~~~~~ ~~\,\, + \frac14 \big(\g^{a b} \big)\du\a\g C_{\b a b} C_{\g c d} 
     + C\du{\a c} e C_{\b e d} ~~, 
&(3.1\rme) \cr
& E_\a C_{b c d} = + E_{\[b|} C_{\a|c\] d} + C\du{\a \[ b|}\e C_{\e|c\] d} 
     + C\du{\a\[b|}e C_{e |c\] d}      
     - i \big( \g_d C_{b c}\big)_\a + C\du{b c}e C_{e \a d} {~~,~~~~~ ~~~}  
&(3.1\rmf) \cr 
& E_\a F_{b c d e} = - \frac18 \big(\g_{\[b c} C_{d e\]} \big)_\a 
     + \frac16 C\du{\a\[b|} f F_{f|c d e\]} ~~, 
&(3.1{\rm g}) \cr 
& E_\g C\du{a b} \d = + E_{\[a|} C\du{\g|b\]}\d + C\du{a b}e C\du{e \g}\d 
     + C\du{a b}\e C\du{\e\g}\d       
     + C\du{\g\[a|}e C\du{e|b\]}\d 
     + C\du{\g\[a|}\e C\du{\e|b\]}\d {~~.~~~~~ ~~~~~} 
&(3.1{\rm h}) \cr  } $$ 
We sometimes use the symbol 
$~F_{\[4\]}$~ for quantities with totally antisymmetric bosonic indices to save
space, namely $~F_{\[4\]}$~ is equivalent to $~F_{a b c d}$.  
Eq.~(3.1a) through (3.1e) satisfy the Bianchi identities of
the engineering dimensions $~d=0,~1/2$~ and $~d=2$, while (3.1f) through
(3.1h) are from $~d=3/2$~ and $~d=2$.  

Let now us give some remarks about these constraints.   Note first that
even though the particular component $~C\du{\a b}c$~ remains in many of
these constraints, it is recombined to form a locally Lorentz covariant
expression in the physical superfield equations to be seen later.   As
was mentioned, even though there is the second direct method to obtain
these constraints, it is easier to use the first method using the
relationships between the conventional superspace and our teleparallel
superspace.  For example, a set of superspace constraints for torsion
superfields in the former \cf%
%%% 
\ref\ng{H.~Nishino and S.J.~Gates, Jr., 
\pl{388}{96}{504}.} 
%%%
are expressed in terms of anholonomy coefficients as   
$$ \li{&T\du{\a\b} c = C\du{\a\b} c~~, 
&(3.2\rma) \cr 
&T_{a b c} = C_{a b c} - \phi_{\[a b\]c} = 0  ~~~~ \Longleftrightarrow~~~~
     \phi_{a b c} = \frac12 \big( C_{a b c} - C_{a c b} - C_{b c a} \big) ~~, 
&(3.2\rmb) \cr 
&T\du{\a b} c = C\du{\a b} c - \phi\du{\a b} c =0  ~~~~ \Longrightarrow
     ~~~~C\du{\a b}c = \phi\du{\a b}c ~~, ~~~~
      \phi_{\a (b c)} = C_{\a (b c)} = 0 ~~,  
&(3.2\rmc) \cr 
&T\du{\a\b}\g 
      = C\du{\a\b}\g 
       + \frac14 \big( \g\low{d e} \big)\du{(\a|} \g \phi\du{|\b)} {d e} 
      = 0 ~~~~ \Longrightarrow ~~~~
      C\du{\a\b} \g = - \frac 14 \big( \g\low{d e} \big)\du{(\a|}\g 
     \phi\du{|\b)}{d e}  ~~, 
&(3.2\rmd) \cr 
&T\du{\a b} \g = C\du{\a b} \g - \phi\du{b \a}\g 
     = C\du{\a b}\g + \frac14\big( \g^{c d} \big)\du\a\g \phi_{b c d} ~~, 
&(3.2\rme) \cr 
& T\du{a b}\g = C\du{a b}\g ~~. 
&(3.2\rmf) \cr } $$ 
For example, from (3.11) we see that we can impose the condition that the 
symmetric component in $~C_\a{}^{c d}$~ is zero as in (3.1c).  
Therefore if we adopt essentially the 
same constraints for the conventional superspace as in
\cf\ng, we see that $~C\du{\a\b} c$~ is exactly 
the same as $~T\du{\a\b} c$~ as in (3.1) from (3.2), $~\phi_{a b c}$~ 
is completely expressed in
terms of $~C_{a b c}$~ as in (3.10) as usual, 
$~C\du{\a b}\g$~ is  expressed in
terms of $~F_{\[4\]}$~ and $~C_{a b c}$~ as in (3.1e) {\it via} (3.2f), 
$~C\du{\a\b}\g$~ is 
expressed in terms of $~C\du{\a b}c$, while $~C\du{a b}\g$~ can be regarded as 
the superfield strength in our superspace, equivalent to $~T\du{a b}\g$~
in the conventional superspace.  

Once we comprehend these correspondences, the satisfaction of all the Bianchi
identities (2.4) and (2.5) is clear.  However, we can also confirm all the
Bianchi identities by the usual direct computation.    

Prepared with these constraints, we can also get the superfield equations 
from the $~d=3/2$~ and $~d=2$~ Bianchi identities, as 
$$ \li{& i(\g^b)_{\a\b} C\du{a b}\b = 0 ~~, 
&(3.3\rma) \cr   
\noalign{\vskip 0.1in\vfill\eject}   
& E_a C_b + \frac12 E_c C\du{a b} c + E_d C\ud d {(a b)} 
     - \frac12 C\du a{c d} C_{b d c} - \frac12 C_{a c d} C\du b{c d} 
     - \frac12 C\du{a b}c C_c \cr 
& ~~~~~ ~~- \frac12 C\ud c{( a b) } C_c + \frac14 C_{c d a} C\ud{c d}b 
      + C\du{a c}\g C\du{\g b} c \cr 
& ~~~~~ ~~ + \frac13F_{a \[3\]} F\du b{\[3\]} 
      - \frac1{36} \eta_{a b} F_{\[4\]}{}^2  = 0 ~~, 
&(3.3\rmb) \cr   
\noalign{\vskip 0.1in}   
& E_d F\du{a b c} d - \frac14 C_{d e\[a} F\du{b c\]}{d e} 
     - \frac12 C^d F_{a b c d} + \frac1{576} \e\du{a b c}{\[4\]\[4\]'} 
       F_{\[4\]} F_{\[4\]'} = 0 ~~.   
&(3.3\rmc) \cr } $$   
Eqs.~(3.3a), (3.3b) and (3.3c) are respectively the gravitino,
gravitational and $~F\-$superfield equations.   The $~E 
C,~C^2$, and $~C\du{a c}\g 
C\du{\g b} c\-$terms\footnotew{The $~C$'s in 
$~EC$~ and $~C^2$~ are for $~C\du{a b}c $~ with purely bosonic indices.}
in the first two lines in (3.3b) are all arranged themselves to be
equivalent to the Ricci tensor (2.7) in the conventional superspace, and
therefore the total expression (3.3b) is local Lorentz covariant.  This
explains also why the 
$~C\du{a c}\g C\du{\g b} c\-$term is needed in there.    As usual in any
11D superspace formulation, the gravitino superfield equation (3.15) is
obtained by the $~(a \b\g,\d)\-$Bianchi identity at $~d=3/2$, while  the
gravitational superfield equation (3.3b) is obtained by the spinorial
derivative  $~\big(\g^c
\big)^{\g\a} E_\g \big[\,i \big(\g^a\big)_{\a\b} C\du{a b}\b \,
\big]=0$, and similarly the $~F\-$field equation (3.3c) is from 
$~\big(\g_{\[a b|} 
\big)\du\a\b E_\b \big[\, i \big(\g^d\big)\ud\a\g C_{|c\] d}{}^{\g}
\,\big]=0$.  The last two are also based on the  constraints (3.1g) and
(3.1h).  Recall that all the terms antisymmetric in $~{\scst \[a b\]}$~
in (3.3b)  vanish identically due to (2.8).


\bigskip\bigskip




\centerline{\bf 4.~Couplings to Supermembranes} 

Once our superspace has been established, our next natural step is to 
consider its couplings to supermembrane, in particular with fermionic
symmetries.  

Our total action for supermembrane theory is the same as that in the 
conventional one: 
$$ \li{ & I \equiv I_\s + I_A~~, 
&(4.1) \cr 
& I_\s \equiv \int d^3 \s\, \left( + \frac12 {\sqrt{-g}} g^{i j} 
     \eta_{a b} \Pi\du i a \Pi\du j b - \frac12{\sqrt{-g}} \right) ~~, 
&(4.2) \cr 
& I_A \equiv \int d^3 \s \, \left(- \frac1 3 \,\e^{i j k} \,
     \Pi\du i A \Pi\du j B \Pi\du k C A_{C B A} \right) ~~. 
&(4.3) \cr} $$
As usual, we have $~\Pi\du i A \equiv \big(\partial_i Z^M \big) E\du M A$, and 
the indices $~{\scst i,~j,~\cdots~=~0,~1,~2}$~ are for the 3D world-volume 
for supermembrane.  

In the conventional formulation, we need to vary $~\Pi\du i A$~ under 
the general transformations $~\d E^A \equiv (\d Z^M) E\du M A$.  In our 
teleparallel superspace, this variation is given by 
$$ \li{ & \d\Pi\du i A =\partial_i \big( \d E^A \big) 
     - \Pi\du i B \big( \d E^C \big)  C\du{C B} A~~. 
&(4.4) \cr } $$
Note that there is no Lorentz connection $~\phi\du{A b}c$~ explicitly
involved in here, as is usually the case with our teleparallel
superspace.  On the other hand, in the conventional superspace
%%%
\ref\gsw{M.~Green, J.H.~Schwarz and E.~Witten, {\it `Superstring
Theory'}, Vols.~I and II,  Cambridge University Press (1987).}%
%%%
\ng\ we have  
$$ \li{ & \d\Pi\du i A = \nabla_i \big( \d E^A \big) 
     - \Pi\du i B \, \big( \d E^C \big) 
     \big( T\du{C B} A + \phi\du{C B}A\, \big)  ~~,  
&(4.5) \cr } $$ 
where $~\nabla_i \big( \d E^A\big) \equiv \partial_i \big( \d E^A\big) +
(1/2)\Pi\du i B \phi\du{B c} d \big( \calM\du d c\big)^{A D} 
\big( \d E_D\big)$, with the explicit Lorentz connection $~\phi\du{A b}
c$.  The explicit Lorentz connection term in (4.5), which looks in a
sense unnatural, automatically decouples from the variation of the total
action, due to $~\d_\k E^a = 0$~ for the fermionic $~\k\-$transformation
\gsw.  In our teleparallel superspace, on the other hand, the Lorentz
connection  term is absent from the outset, and there is no need for such
a rearrangement. 

Once this subtlety is clarified, the confirmation of fermionic 
$~\k\-$invariance \gsw\ of our action: $~\d_\k I = 0$~ under the
transformation 
$$ \li{& \d_\k E^\a = \big(I + \G\big)\ud \a\b \k^\b ~~, 
&(4.6) \cr 
&\d_\k E^a = 0 ~~, 
&(4.7) \cr 
& \G \equiv \frac i {6{\sqrt{-g}}} \, \e^{i j k} \Pi\du i a 
     \Pi\du j b \Pi\du k c \,  
     \big( \g_{a b c} \big) ~~,   
&(4.8) \cr } $$
becomes straightforward.  Here $~\G$~ satisfies the following usual
relationships under the embedding condition  $~g_{i j} = \Pi\du i a
\Pi_{j a}$~ equivalent to the $~g_{i j}\-$field equation:
$$ \li{&\G^2 = I ~~, 
    ~~~~\e\du i{j k} \Pi\du j a \Pi\du k b \g_{a b} \G 
    = - 2 i {\sqrt{-g}} \,  \Pi\du i a \g_a~~, 
&(4.9) \cr } $$

To summarize our supermembrane couplings, the fermionic
$~\k\-$transformation becomes simpler and more natural in our
teleparallel superspace, where there is no need of explicit  Lorentz
connection terms.  We regard this feature as one of the advantages of
considering teleparallelism in superspace formulation, as well as the 
first signal of the naturalness of teleparallel superspace for  
the couplings to supermembrane.        

\bigskip\bigskip\bigskip



\centerline{\bf 5.~Supergravity in 12D as an Example}

Once we have established the teleparallel formulation of superspace,  the
next natural step is to apply it to supergravity theories, in particular,
to higher-dimensional supergravity in which Lorentz covariance is not 
built-in \nishone\nishtwo\sgall.  In this paper, we give the case of
$~N=1$~ supergravity  in 12D \nishone\sgall\ as an illustrative example.  

We review first the {\it non}-teleparallel formulation of
$~N=1$~ supergravity in 12D \nishone\sgall\ 
with important relationships.  The most crucial ones is the definitions of the 
null-vectors $~m^a$~ and $~n^a$:    
$$ \big( n^a \big) 
  = \bordermatrix{ & {\scst (0)} & {\scst (1)} & \cdots & {\scst (9)} 
    & {\scst (11)} & {\scst (12)} \cr  
  & 0, & 0, & \cdots, & 0 , & + \frac1{\sqrt2}, & - \frac1{\sqrt2} \cr} 
~, ~~~ 
\big( \na \big) 
    = \bordermatrix{& {\scst (0)} & {\scst (1)} & \cdots & {\scst (9)} 
    & {\scst (11)} & {\scst (12)}  \cr  
    & 0, & 0, & \cdots, & 0 , & + \frac1{\sqrt2}, &  + \frac1{\sqrt2}\cr } 
  ~, ~~~ $$   
$$ \big( m^a \big) 
  = \bordermatrix{ & {\scst (0)} & {\scst (1)} & \cdots & {\scst (9)} 
    & {\scst (11)} & {\scst (12)} \cr  
  & 0, & 0, & \cdots, & 0 , & + \frac1{\sqrt2}, & + \frac1{\sqrt2} \cr} 
~, ~~~ 
\big( \ma \big) 
    = \bordermatrix{& {\scst (0)} & {\scst (1)} & \cdots & {\scst (9)} 
    & {\scst (11)} & {\scst (12)}  \cr  
    & 0, & 0, & \cdots, & 0 , & + \frac1{\sqrt2}, & - \frac1{\sqrt2}\cr } 
  ~.  ~~~  $$ 
$$ \eqno (5.1) $$  
As in \nishone\sgall, we use the indices $~{\scst a,~b,~\cdots ~=~
(0),.~(1), ~\cdots ,~(9)., ~(10), ~(11)}$~ with parentheses for 
local Lorentz indices.  Only in this section, we adopt the signature 
$~(-,+,+,\cdots,+,-)$~ in order to 
comply with the notation in \nishone\sgall.  We can also use the 
light-cone coordinates $~V_{\pm} \equiv \frac1{\sqrt2} 
\big( V_{(11)} \pm V_{(12)} \big)$.   Relevantly, we have 
$~n^a\na = m^a\ma = 0 ~~,~~~~ m^a\na = m^+ n_+ = m_- n^- 
= + 1$.\footnotew{For other notational details, see refs.~\nishone\sgall.}
Other important quantities are the projection operators $~\Pup,~\Pdown, ~
\Pupdown$~ defined by  
$$ \li{& \Pup \equiv + \frac12
\nsl\msl = + \frac12\g^+ \g^- ~~, ~~~~ \Pdown \equiv + \frac12 \msl\nsl 
     = + \frac12 \g^- \g^+ ~~,  
&(5.2\rma) \cr 
& \Pup\Pup = +\Pup~~, ~~~~ \Pdown\Pdown = + \Pdown~~, ~~~~ 
     \Pup + \Pdown = + I~~, ~~~~\Pup\Pdown = \Pdown\Pup = 0 
     {~~, ~~~~~ ~~~~~}
&(5.2\rmb) \cr  
&\Pupdown \equiv \Pup - \Pdown = \g^{+ -}~~.   
&(5.2\rmc) \cr } $$ 
Relevantly, we have ~$(\nsl)_{\a\Dot\b} = - (\nsl)_{\Dot\b\a}, 
~(\msl)_{\a\Dot\b} = - (\msl)_{\Dot\b\a},  ~ 
(\Pup)_{\a\b} = - (\Pdown)_{\b\a}, ~ (\Pupdown)_{\a\b} \newline = +
(\Pupdown)_{\b\a}$.   
In supergravity in 12D, we can define the Lorentz generators formally as
\nishone\sgall 
$$\li{&\big({\Tilde{\cal M}}_{a b}\big)^{c d} 
\equiv + {\Tilde\d}\du{\[a} c \, {\Tilde\d}\du{b\]} d ~~, 
&(5.3\rm a)\cr ~~~~ 
& \big({\Tilde{\cal M}}_{a b}\big) \du\a\b \equiv +\half \big(
    \g_{a b}  \Pup \big) \du\a\b~~,  ~~~~ \big( \Tilde{\cal M}_{a b}\big)
    \du{\Dot\a}{\Dot\b} \equiv 
     +\half\big(\Pdown\g_{a b}\big)\du{\Dot\a}{\Dot\b}~~. 
&(5.3\rm b) \cr} $$
where 
$$ \Tilde\d\du a b \equiv \d\du a b -m_a n^b 
     =\cases{ \d\du i j ~~& \hbox{(for ~${\scst a~=~i,~~b~=~j}$)}~~,  \cr 
     \d\du++ = 1 ~~&\hbox{(for ~${\scst a~=~+,~~b~=~+}$)} ~~, \cr
     0~~&\hbox{(otherwise)} ~~. \cr } 
\eqno(5.4) $$ 

The Bianchi identities in the {\it non}-teleparallel superspace 
in 12D are \nishone\sgall
$$ \li{&\frac12\nabla_{\[A} T\du{B C)}D 
    - \frac12T\du{\[A B|} E T\du{E| C)}D
    - \frac14 R\du{\[A B| e}f 
      \big(\Tilde\calM \du f e \big)\du{|C)}D\equiv0~~,   
&(5.5) \cr
& \frac16 \nabla_{\[A} G_{B C D)} - \frac1 4 T\du{\[A B|} E G_{E| C D)}
\equiv 0   ~~,  
&(5.6) \cr 
& \frac12\nabla_{\[A} R\du{B C) d} e 
     - \frac12 T\du{\[A B|} E R\du{E| C) d} e\equiv 0~~. 
&(5.7) \cr} $$ 
One set of solutions for constraints at mass dimensions $~0\le d \le 1$~ 
satisfying these Bianchi identities is \nishone\sgall 
$$ \li{&T\du{\a\b} c = + \big( \g^{c d} \big)_{\a\b} 
      \phid + \left( \Pupdown \right)_{\a\b} \nabla^c\varphi 
     {~~,~~~~~ ~~~~~} 
&(5.8\rma) \cr 
&G_{\a\b c} = + T_{\a\b c} ~~, 
&(5.8\rmb)  \cr 
& T\du{\a\b} \g = + (\Pup)\du{(\a|}\g \left( \g^c\Bar\chi
\right)_{|\b)}\phic - \big( \g^{a b} \big)_{\a\b} \left( \Pdown\g_a\Bar\chi
\right)^\g \phib ~~, 
&(5.8\rmc) \cr 
& \nabla_\a \Phi = + \left( \g^c\Bar\chi \right)_\a \phic ~~, 
&(5.8\rmd) \cr 
& \nabla_\a\Bar\chi_{\Dot\b} = -\fracm1{24} \big( \g^{c d e}\Pup
\big)_{\a\Dot\b} G_{c d e} +\half \big( \g^c \Pup \big)_{\a\Dot\b} 
\nabla_c \Phi - \big( \g^c\Bar\chi \big)_\a \Bar\chi_{\Dot\b} \phic ~~, 
&(5.8\rme)\cr 
& G_{\a\b\g} = 0 ~~, ~~~~
     T\du{\a b} c = 0 ~~, ~~~~ T\du{\a b}\g = 0 ~~,  
     ~~~~ G_{\a b c} = 0 ~~, 
&(5.8\rmf) \cr 
& T\du{a b} c = - G\du{a b} c ~~, 
&(5.8\rmg)  \cr  } $$  
where $~\nabla_a\varphi$~ and $~\nabla_a\Tilde\varphi$~ respectively 
satisfy the properties of $~\na$~ and $~\ma$~ in (5.1), 
under the extra constraints 
$$\li{ &\nabla_{\ul\a}\varphi = \nabla_{\ul\a}\Tilde\varphi = 0 ~~, 
      ~~~~~ \big(\nabla_a\varphi\big)^2 = 
      \big(\nabla_a\Tilde\varphi\big)^2 = 0 ~~, ~~~~
      \big(\nabla_a \varphi \big) \big(\nabla^a\Tilde\varphi\big) = 1 ~~, 
&(5.9\rma) \cr 
& \nabla_a\nabla_b\varphi = \nabla_a\nabla_b\Tilde\varphi = 0 ~~.  
&(5.9\rmb)  \cr }  $$ 
Here the underlined spinorial indices $~{\scst\un\a,~\un\b,~
\cdots}$~ represent both dotted and undotted spinors: 
$~{\scst \un\a ~\equiv~(\a,\Dot\a), ~\un\b ~\equiv~(\b,\Dot\b), 
\cdots}$~ \nishone\sgall.  

We have also extra constraints to delete {\it some} extra 
components of relevant superfields \nishone\sgall:
$$ \li{ & T\du{A B} c \nabla_c\varphi = 0 ~~, ~~~~ 
     G_{A B c} \nabla^c\varphi =
     0~~, ~~~~   T\du{a B}C \nabla^a\varphi = 0 ~~, 
&(5.10\rma) \cr 
& R\du{A B c} d \nabla_d\varphi =  0 ~~, ~~~~ 
     R\du{a B c} d \nabla^a\varphi = 0 ~~, 
&(5.10\rmb) \cr 
& (\nabla^a\varphi) \nabla_a\Phi = 0 ~~,  
     ~~~~ (\nabla^a\varphi) \nabla_a \Bar\chi_{\Dot\b}= 0 ~~,
&(5.10\rmc) \cr 
& (\g^c)\du\a{\Dot\b} \Bar\chi_{\Dot\b} \nabla_c\Tilde\varphi = 0 ~~, ~~~~ 
    T\du{a b} \g (\g^d)\du \g {\Dot\a}\nabla_d\Tilde\varphi = 0 ~~,  
&(5.10\rmd) \cr
& \phi\du{A b} c \phic = 0 ~~, ~~~~
      \phi\du{a b} c \nabla^a\varphi = 0 ~~. 
&(5.10\rme) \cr } $$ 
As has been stated in \nishone, these constraints will {\it not} 
delete {\it all} the extra components in 12D, and this is the 
non-trivial part of higher-dimensional supergravity \sgall. 

We now consider the reformulation of teleparallel superspace of this 
$~N=1$~ supergravity in 12D \nishone\sgall.  
In the {\it non}-teleparallel case, we had the supercurvature 
Bianchi identity (5.8) due to the existence of Lorentz connection 
$~\phi\du {A b} c$.  Moreover, we had to confirm the consistency of
this $~R\-$Bianchi identity (5.7) with other $~T~$ and $~G\-$Bianchi
identities (5.5) and (5.6) \nishone.  However, in the present 
teleparallel superspace, we do not have the $~R\-$Bianchi 
identity.  Instead we have only two Bianchi identities corresponding 
to (5.5) and (5.6):
$$ \li{& E_{\[A} C\du {B C)} D - C\du{\[A B|} E C\du{E| C)}D \equiv 0~~, 
&(5.11) \cr 
& \frac 1 6 E_{\[A} G_{B C D)} 
    -\frac1 4 C\du{\[A B|} F G_{F |C D)}  \equiv 0~~.    
&(5.12) \cr } $$ 
The disappearance of the $~R\-$Bianchi identity is the first advantage
for our teleparallel formulation for 12D supergravity, in which Lorentz
covariance is not built-in  off-shell from the outset.  For example, even
though  the null-vectors $~\ma$~ and $~\na$~ are replaced by more Lorentz
`covariant' gradients of scalar superfields in (5.9), these scalar
superfields satisfy  all the properties (5.1) only `on-shell' as extra
constraints, and therefore  this formulation still has the drawback of
lacking Lorentz covariance  at the off-shell level.  This gives a 
natural justification of considering teleparallel superspace formulation
with no manifest local Lorentz covariance from the outset.  

A set of constraints at mass dimensions $~0\le d \le 1$, 
satisfying the Bianchi identities (5.11) and (5.12) can be 
obtained in a way similar to (3.2).  Namely, we first 
write down the relationships between the supertorsion $~T\du{A B}C$,  
anholonomy coefficients $~C\du{A B} C$, and Lorentz connection
$~\phi\du{A b} c$, {\it i.e.}, $~T\du{A B} C \equiv C\du{A B} C
+ (1/2)\phi\du{\[ A | d}e \big( {\Tilde{\calM}}\du e d \big)\du{| B)} C$, 
and next we solve this for $~\phi\du{A b} c$.  Afterwards, we treat 
the anholonomy coefficients $~C\du{a b} c$~ and $~C\du{\a b} c$~ as 
independent superfields.  The most important `bridge' relationships 
between the {\it non}-teleparallel superspace and teleparallel
superspace are 
$$ \li{ & \phi_{\a b c} = C_{\a b c} 
     + \phi\du{\a \[b|} d \big( \nabla_{| c\]} \varphi\big)
      \big( \nabla_d{\Tilde\varphi} \big) = - C_{\a c b}~~, 
&(5.13\rma) \cr 
& \phi_{a b c} = + \frac12 \big( C_{a \[ b c\]} 
     - C_{b c a} + G_{a b c} \big) 
    + \phi\du{a \[ b|} d\big( \nabla_{| c\]} \varphi\big)
    \big( \nabla_d\Tilde\varphi \big) ~~,
&(5.13\rmb) \cr} $$ 
obtained from $~T\du{\a b}c$~ and $~T\du{a b}c$~ in (5.8f,g).    

Compared with conventional superspace formulations \ggrs, the last  extra
terms in these equations reflect the Lorentz non-covariance in our 12D. 
At first glance, these extra terms seem to cause a problem, we can not
get closed forms for $~\phi\du{\a b} c$~ or $~\phi\du{a b} c$.  However,
it does not matter in practice, because the extra terms in (5.13a,b) do
{\it not} enter the combination $~\phi\du{A d} e \big( \TildeM\du e
d\big)\du B C$~ effectively,  as can be easily confirmed under the
definition of $~\TildeM$~  in (5.3).  

After all of these, we get the set of constraints from (5.8) as 
$$ \li{&C \du{\a\b} c = + \big( \g^{c d} \big)_{\a\b} 
      \, E_d\varphi 
     + \left( \Pupdown \right)_{\a\b} E^c\varphi {~~,~~~~~ ~~~~~} 
&(5.14\rma) \cr 
&G_{\a\b c} = + C_{\a\b c} ~~, 
&(5.14\rmb)  \cr 
& C\du{\a\b} \g = + (\Pup)\du{(\a|}\g \left( \g^c\Bar\chi
    \right)_{|\b)}\, E_c\varphi - \big( \g^{a b} \big)_{\a\b}
    \left( \Pdown\g_a\Bar\chi \right)^\g \, E_b\varphi
     + \frac 1 4 \big(\g^{d e} \Pup\big) \du{(\a|} \g 
     C_{| \b) d e} ~~,  
&(5.14\rmc) \cr 
& E_\a \Phi = + \left( \g^c\Bar\chi \right)_\a \, E_c\varphi ~~, 
&(5.14\rmd) \cr 
& E_\a\Bar\chi_{\Dot\b} 
    = -\fracm1{24} \big( \g^{c d e}\Pup \big)_{\a\Dot\b} 
     G_{c d e} +\half \big( \g^c \Pup \big)_{\a\Dot\b} \, E_c \Phi 
    - \big( \g^c\Bar\chi \big)_\a \Bar\chi_{\Dot\b} \, E_c\varphi 
    + \frac14 C\du\a{c d} \big(\g_{c d} \Bar\chi\big)_{\Dot\b} 
    {~~, ~~~~~ ~~~~~} 
&(5.14\rme) \cr 
& G_{\a\b\g} = 0 ~~, ~~~~ G_{\a b c} = 0 ~~, 
&(5.14\rmf) \cr 
& C\du{\a b}\g = - \frac 18 \big( \g^{c d} \Pup \big)\du\a\g 
    \big( 2 C_{b c d} - C_{c d b} + G_{b c d} \big) ~~, 
&(5.14\rmg) \cr
& E_\a C\du{\b c} d = + \frac12 C\du{\a\b}\e C\du{\e c} d 
    + \frac 12 C\du{c (\a|} e C\du{e| \b) } d 
    + \frac 14 \big(\g^{a b} \big)_{\a\b} 
    \big( G\du{c a} d - C\du{c a} d + C\ud d{c a} 
    + C\ud d{a c} \big) \, E_b \varphi {~. ~~~~~ ~~~~~ ~~} 
&(5.14{\rm h})  \cr } $$  
To be consistent within our teleparallel superspace, we use 
$~E_a\varphi,~E_a\Tilde\varphi$~ instead of 
$~\nabla_a\varphi,~\nabla_a\Tilde\varphi$.  However, 
they are actually the same, so that (5.9) is replaced by 
$$\li{ & E_{\ul\a}\varphi = E_{\ul\a}\Tilde\varphi = 0 ~~, 
      ~~~~~ \big(E_a\varphi\big)^2 = 
      \big(E_a\Tilde\varphi\big)^2 = 0 ~~, ~~~~ 
      \big(E_a \varphi \big) \big(E^a\Tilde\varphi\big) = 1 ~~, 
&(5.15\rma) \cr 
& E_a \big(E_b\varphi\big) 
    = E_a\big(E_b\Tilde\varphi\big) = 0 ~~.  
&(5.15\rmb)  \cr } $$   
Compared with the {\it non}-teleparallel case (5.8), the last terms in
(5.14c) and (5.14e) are new ones, coming from the substitutions of 
(5.13a) for the Lorentz connection in the {\it non}-teleparallel 
equations.  Eq.~(5.14g) is also new here, because $~T\du{\a b}\g =0$~ in
(5.8f), but generated by the substitution of (5.13b).  Eq.~(5.14h) is
another new one needed to satisfy the $~(\un\a\un\b c, d)$~ and
$~(\a\b\g,\d)\-$type Bianchi identities at 
$~d=1$.  This is also because the superfield $~C\du{\a b} c$~ is treated
as an independent superfield in our teleparallel superspace, similarly
to the 11D case in (3.1e).  

The extra constraints in our teleparallel superspace, 
corresponding to (5.10) are 
$$ \li{ & C\du{A B} c E_c\varphi = 0 ~~, ~~~~ 
     G_{A B c} E^c\varphi =
     0~~, ~~~~   C\du{a B}C E^a\varphi = 0 ~~, 
&(5.16\rma) \cr 
& (E^a\varphi) E_a\Phi = 0 ~~,  
     ~~~~ (E^a\varphi) E_a \Bar\chi_{\Dot\b}= 0 ~~,
&(5.16\rmb) \cr 
& (\g^c)\du\a{\Dot\b} \Bar\chi_{\Dot\b} 
    E_c\Tilde\varphi = 0 ~~, ~~~~ 
    C\du{a b} \g (\g^d)\du \g {\Dot\a}\, E_d\Tilde\varphi = 0 ~~.   
&(5.16\rmc) \cr } $$ 
Needless to say, there is no constraints involving the curvature 
tensor or the Lorentz connection, such as (5.10b) or (5.10e).  

Our constraints above can be derived from the {\it non}-teleparallel 
system \nishone\sgall, but we can independently confirm the satisfaction
of Bianchi identities (5.11) and (5.12) at $~0\le d \le 1$.  
In this connection, useful relationships are 
$$\li{ & C\du{\a b} c \, E_c \Tilde \varphi = 0 ~~, 
&(5.17\rma) \cr 
& \big( C_{a b c} - C_{a c b} - C_{b c a} 
     + G_{a b c} \big) \, E^c\Tilde\varphi = 0 ~~, 
&(5.17\rmb) \cr} $$ 
which are needed for confirming the $~(a b \un\a \un\b)$~
and $~(\un\a\un\b c, d)\-$type Bianchi identities at $~d=1$.  
Note that $~E_a\Tilde\varphi$~ is used here instead of $~E_a\varphi$.  
These constraints are not additional ones, but are just 
necessary conditions of (5.13).   
      
We can mimic this procedure in other supergravity formulations in 
higher-dimensions in $~D\ge 12$~ \nishone\nishtwo\sgall, 
but we skip their details in this paper.  
 



\bigskip\bigskip\bigskip



\centerline{\bf 6.~Concluding Remarks} 

In this paper we have presented a new formulation of superspace in which 
local Lorentz covariance is not manifest.  We have set up all the
necessary superspace constraints, and derived all the superfield equations 
as a result of satisfaction of all the Bianchi identities.  There seems to be
no fundamental obstruction for such a formulation in superspace.      

The possibility of supergravity formulations in component that lack
manifest local Lorentz covariance has been already known for some time. 
For example, in subsection 1.5 in \pvn, a component  formulation using 
only torsions with no use of curvature tensors is mentioned as `flat 
supergravity with torsion'.  In this Letter, we have confirmed similar 
formulation is also possible in superspace for $~D=11, \, N=1$~
supergravity.  

The important ingredients we have found in our work are summarized as
follows: First, we have found that all the superspace constraints and
superfield equations are re-formulated completely in terms of anholonomy 
coefficients $~C\du{A B}C$~ with no explicit usage of the Lorentz
connection,  as a parallel result to component  formulation \pvn. 
Second, we have found that the particular anholonomy  coefficient
component $~C\du{\a b} c$~ is to be treated as an independent 
superfield, because all the Bianchi identities are satisfied for any
arbitrary form of $~C\du{\a b}c$.  Third, this particular component  is
eventually combined with $~C_{a b c}$~ to form locally Lorentz covariant 
terms, such as the Ricci tensor combination (2.7), in our superfield
equations.   Fourth, we have found that this teleparallel superspace has
no problem with  the fermionic $~\k\-$symmetry of supermembranes,  or it
has even simpler and more natural results.  Fifth, the teleparallel 
formulation is possible not only in 11D, but also in other dimensions.  
As an explicit example, we have applied this formulation to 
$~D=12, \, N=1$~ supergravity \nishone\sgall.  

Our teleparallel superspace formulation will be of great importance, 
when considering supergravities in higher-dimensions given in 
\nishone\nishtwo\sgall, in which Lorentz covariance is not built-in from
the outset.  As an illustrative example, we have re-formulated  
$~D=12,\,N=1$~ superspace supergravity \nishone\sgall\ in terms of  
teleparallel superspace.  We have found there are `bridge'  relationships
(5.13), that give important links between the  original {\it 
non}-teleparallel superspace \nishone\sgall\ and our   present 
teleparallel superspace.  It is also important that the  second terms in
(5.12) will not enter the combination $~\phi\du{A d} e \big( \TildeM\du e
d\big)\du B C$.  Therefore the lack of closed forms for $~\phi\du {a b}
c$~ or $~\phi\du{\a b} c$~ will {\it not} matter in practice to derive 
the set of constraints for teleparallel superspace.  We 
expect that this basic structure is common to other higher-dimensional
supergravities  in $~D\ge 12$~
\nishtwo\sgall.  

The fact that the fermionic $~\k\-$symmetry of supermembrane in the 
Green-Schwarz formulation is simplified in teleparallel superspace
indicates something deeper in general supergravity backgrounds for
extended objects.  It seems that the local Lorentz symmetry is not
crucial even in conventional supergravity, not to mention
higher-dimensional ones in 12D \nishone\nishtwo\ related to F-theory
\ftheory\ or 13D \sgall\ related to S-theory \stheory.  In other words,
when we need to study non-perturbative nature of M-theory \mtheory, 
F-theory \ftheory, S-theory \stheory, or higher-dimensional theories,
local Lorentz symmetry is not the dominating symmetry, like the case of
the original superstring theory formulated in the light-cone gauge
\gsw.  As a matter of fact, there are other examples of this kind for
superstring physics, {\it e.g.,} in ref.~%
%%% 
\ref\gnz{M.T.~Grisaru, H.~Nishino and D.~Zanon, \pl{306}{88}{625};
\np{314}{89}{363}.}, 
%%%
null-vectors for constraints in superspace for Green-Schwarz 
$~\s\-$model $~\b\-$functions were introduced.  It seems that the loss of
Lorentz symmetry in the Green-Schwarz formulation is inherent in
superstring theories, which were originally formulated in the light-cone
gauge.  It is not surprising that this feature of superstring theories
pops up in M-theory \mtheory, F-theory \ftheory, S-theory \stheory, or
other higher-dimensional and more fundamental theories.  As some readers 
have already noticed, another important supporting fact is the recent
development of noncommutative geometry \noncomm\ associated with
D-branes, in which the constant tensor $~\theta^{\m\n}$~ breaks the  
Lorentz covariance.  These recent developments give more than enough 
motivation to consider teleparallel supergravity formulation in 
superspace.  

Our result of teleparallel superspace in this paper is directly
applicable to other conventional supergravity theories in other
dimensions.  Now that we have at hand  the explicit example of teleparallel
superspace applied to {\it non}-teleparallel $~D=12, \, N=1$~ superspace
\nishone\sgall\ an application, it is much easier to repeat similar
analyses in other supergravity theories in  higher-dimensions in $~D\ge
12$~ \nishone\nishtwo\sgall\ as well as lower-dimensions $~D\le 10$.    





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We are grateful to W.~Siegel for helpful discussions.     


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