hep-th0108123.tex
%%%%%%%%%%%%%% Latex File for CQG/132159/PAP %%%%%%%%%%%%
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\def\doit#1#2{\ifcase#1\or#2\fi}

% Underline for text or math
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\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
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\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}}

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\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
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% Math stuff with one argument
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\def\sb#1{{}_{#1}}                              % sub"
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\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}      % " upper
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\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}   % 
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\def\ket#1{\left| #1\right\rangle}              % | >
\def\VEV#1{\left\langle #1\right\rangle}        % < >
\def\abs#1{\left| #1\right|}                    % | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow 
 \mkern-6mu
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\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
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       %alternate fraction
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\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

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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}} %
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\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
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\def\ud#1#2{^{#1}{}_{#2}}
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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}\,\,} 
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\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}}
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\def\ulq{{\underline q}} \def\ulr{{\underline r}}
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\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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\begin{document}

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%%%%%%%%%%%%%%%%% Worksheet %%%%%%%%%%%%%%%%%%%%%%%
% \end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

% The following should be eliminated in the final version.  
% {\bf Preliminary Version (FOR YOUR EYES ONLY!)\hfill\today} \\[-0.3in] 

\doit0{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0108123}}\par 
\vskip -0.03in
{\hbox to\hsize{\hfill CSULB--PA--01--4}}\par 
\vskip -0.03in 
{\hbox to\hsize{\hfill (Revised Version)}}\par
% \\ 

\begin{center} 
%\vglue .25in 

%\vskip -0.6in 
\vskip 0.3in 

{\large \bf Generalized ~BF ~Theory ~in ~Superspace~ } 
\\[0.026in] 
{\large\bf as ~Underlying~ Theory~ of ~11D~ Supergravity} \\[.1in]  

\baselineskip 9pt 

\vskip 0.26in 

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


\vskip 2.3in 

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

\baselineskip 14.5pt 

~~~We construct a generalized $~B F$~ theory in superspace that
can embed eleven-dimensional supergravity theory.  Our topological  
$B F$ theory can accommodate all the necessary Bianchi identities for
teleparallel superspace supergravity in eleven-dimensions, as the
simplest but nontrivial solutions to superfield equations for our
superspace action.  This indicates that our theory may have 
solutions other than eleven-dimensional supergravity, accommodating
generalized theories of eleven-dimensional supergravity.  Therefore our
topological theory can be a good candidate for the low energy limit of
M-theory, as an underlying fundamental theory providing a `missing link'
between eleven-dimensional supergravity and M-theory.  


\vskip 0.65in

\leftline{PACS: ~02.40.-k, ~04.65.+e, ~11.25.-w, ~11.15.-q}
\vskip -0.03in
\leftline{Key Words: Topology, $BF$ Theory, Supergravity, 
M-Theory, Teleparallelism} 

\vfill\eject 

\baselineskip 17pt 

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\pageno=1 



\leftline{\bf 1.~~Introduction} 

It has been well-known that Chern-Simons theories in three-dimensions
(3D) or topological $~B F$~ field theories 
%%%
\ref\topological{{\it For reviews, see, e.g.,} 
D.~Birmingham, M.~Blau, M.~Rakowski and
G.~Thompson, \prep{209}{91}{129}; {\it and references therein}.}
%%% 
have important physical significance in physics.  For example, 3D gravity
theory is nothing other than a  Chern-Simons theory in 3D, which is
exactly soluble 
%%% 
\ref\wittencs{E.~Witten, \np{311}{88}{46}.}.  
%%% 
On the other hand, there have been considerable developments
associated  with M-theory 
%%%  
\ref\mtheory{C.~Hull and P.K.~Townsend, \np{348}{95}{109}; 
E.~Witten, \np{443}{95}{85}; 
P.K.~Townsend, {\it `Four Lectures on
M-Theory'}, in {\it `Proceedings of ICTP Summer School on High Energy
Physics and Cosmology'}, Trieste (June 1996), hep-th/9612121; 
{\it `M-theory from its Superalgebra'}, hep-th/9712004; 
{\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}.}  
%%% 
as the most fundamental theory unifying most of the known superstring
theories 
%%%
\ref\gsw{M.B.~Green, J.H.~Schwarz and E.~Witten, {\it `Superstring
Theory'}, Vols.~{\bf I} and {\bf II}, Cambridge University Press
(1987).}       
%%% 
in 10D, as well as lower-dimensional strings, whose low
energy limit is supposed to be described by 11D supergravity theory
%%% 
\ref\cjs{E.~Cremmer, B.~Julia and N.~Scherk, \pl{76}{78}{409};
E.~Cremmer and B.~Julia, \np{159}{79}{141}.}.   
%%% 

Considering such developments in M-theory \mtheory, a common 
expectation nowadays seems that there must exist a more explicit
formulation in 11D as a generalization of 11D supergravity \cjs, based on 
topological formulation, or on more enlarged gauge groups, 
dualities or higher-derivative $~R^2\-$theories.  For example,
Chern-Simons supergravity formulation 
%%% 
\ref\achu{A.~Achucarro and P.K.~Townsend, \pl{180}{86}{89};
E.~Witten, \np{311}{88}{46}; \ibid{323}{89}{113}.}% 
%%%
\ref\chamseddine{A.H.~Chamseddine, 
\pl{233}{89}{291}; \np{346}{90}{213}; 
M.~Ba\~nados, R.~Troncoso and J.~Zanelli, 
\pr{54}{96}{2605}; 
R.~Troncoso and J.~Zanelli, \pr{58}{98}{101703}; Cl.~\&
Quant.~Grav.~{\bf 17} (2000) 4451, hep-th/9907109.}% 
%%%
\ref\green{M.B.~Green, \pl{223}{89}{157}; 
G.~Moore and N.~Seiberg, \pl{220}{89}{422}; 
P.~Ho\v rava, \pr{59}{99}{046004}.}
%%% 
is one of such trials for exploring an underlying theory of 
M-theory \mtheory.  For example, in our recent paper 
%%% 
\ref\mn{P.~Mora and H.~Nishino, Phys.~Lett.~{\bf 482B} (2000) 222,
\hepth{0002077}.},  
%%% 
it has been shown that an action for fundamental extended objects for
Chern-Simons supergravity \achu\chamseddine\green\ in 11D for the group
$~OSp(32|1)$~ coincide with Type II Green-Schwarz superstring action
\gsw\ in a certain limit.  Another example is the reformulation of 11D
supergravity \cjs\ as a first-order topological field theory  
with certain constraints 
%%%
\ref\linsmolin{Y.~Ling and L.~Smolin, 
Nucl.~Phys.~{\bf B601} (2001) 191, hep-th/0003285.}.  
%%%   
This formulation yields the action proposed by D'Auria and Fr\' e
some time ago  
%%%
\ref\dauriafre{P.~Fr\' e, Class.~Quant.~Grav.~{\bf 1}
(1984) L81;  R.~D'Auria and P.~Fr\' e, 
Nucl.~Phys.~{\bf B201} (1982) 101,
Erratum-{\it ibid}.~{\bf B206} (1982) 496.}.  
%%%  
As for searching for the fundamental gauge group of 11D
supergravity \cjs, it is argued in ref.~% 
%%% 
\ref\westosp{P.~West, JHEP {\bf 0008} (2000) 007, \hepth{0005270}.} 
%%% 
that the symmetry of M-theory \mtheory\ is to be the group 
$~OSp(64|1)$, and is further developed in 
%%% 
\ref\westeleven{P.~West, Class.~Quant.~Grav.~{\bf 18} 
(2001) 4443, hep-th/0104081.}   
%%% 
that 11D supergravity \cjs\ is described by a non-linear realization based
on the group $~E_{11}$, and the gravitational degrees of freedom can be 
described by two fields related by duality.  In ref.~%
%%% 
\ref\rsquare{C.R.~Preitshopf, T.~Hurth, P.~van Nieuwenhuizen and 
A.~Waldron, Nucl.~Phys.~Proc.~Suppl.  
\newline{\bf 56B} (1997) 310, hep-th/9702123.}, 
%%% 
a MacDowell-Mansouri $~R^2\-$type action for the superalgebra 
$~OSp(8|1)$~ was proposed as possible low-energy limit of M-theory
\mtheory.  Even though a particular link with M-theory \mtheory\ or 10D
superstring
\gsw\ was shown in a certain limit in Chern-Simons supergravity \mn, or
some enlarged gauge group formulations \westosp\westeleven\ provide some
scenarios, there still seems to be some gap between these formulation and
M-theory \mtheory, in particular 11D supergravity in superspace 
%%% 
\ref\cf{E.~Cremmer and S.~Ferrara, \pl{91}{80}{61}; L.~Brink and
P.~Howe, \pl{91}{80}{384}.}.  
%%% 
Moreover, the
drawback of the topological formulation in \linsmolin\ is the usage of 
component language which is not always convenient to control certain 
generalization of 11D supergravity \cjs\cf, such as M-theory corrections, 
compared with superspace formulation
%%% 
\ref\mthcorrections{M.~Cederwall, U.~Gran, M.~Nielsen, B.E.W.~
Nilsson, JHEP {\bf 0010} (2000) 041S; 
J.~Gates, Jr.~and H.~Nishino, Phys.~Lett.~{\bf B508} (2001) 155,
hep-th/0101037; 
H.~Nishino and S.~Rajpoot, hep-th/0103224, Phys.~Rev.~{\bf D64} (2001)
124016.}.  
%%%
For such a superspace formulation of M-theory, there is a recent trial 
%%%
\ref\moffat{J.W.~Moffat, {\it `M-Theory'}, \hepth{0106031}.}, 
%%%
of construction of eleven-dimensional superspace with superspace
coordinates and a finite M-theory using non-anticommutative geometry. 
However, the most crucial supersymmetric invariance of the action in
\moffat\ remains to be confirmed.  The need of superspace formulation may
be attributed to the most common expectation that a desirable fundamental
underlying theory must have a very simple appearance possibly formulated
in superspace, with topological/geometrical features, but at the same
time, it should be rich and intricate enough to accommodate such
complicated theory as 11D supergravity \cjs.  

In our present paper, we present a simple topological `generalized' 
$~B F$~ theory formulated in superspace 
%%% 
\ref\superspace{{\it See, e.g.,} S.J.~Gates, Jr., 
M.T.~Grisaru, M.~Ro\v cek and  W.~Siegel, {\it `Superspace or One Thousand 
and One Lessons in Supersymmetry'}, Benjamin Cummings (Addison-Wesley), 
Reading, MA (1983).}    
%%% 
that can possibly unify the two important recent theories, {\it i.e.,}
topological $~B F$~ or Chern-Simons theory \topological\ and M-theory
\mtheory\ in terms of superspace language \superspace.  We present a
simple superspace lagrangian, in which all the necessary superspace
Bianchi identities in 11D \cf\ are `automatically' accommodated, as
solutions of the superfield equations obtained from our superspace
action.  To this end, we also utilize the recent result in teleparallel
superspace formulation  
%%% 
\ref\gnr{S.J.~Gates, Jr., H.~Nishino and S.~Rajpoot, \hepth{0107155}, 
Phys.~Rev.~{\bf D65} (2002) 024013.},      
%%% 
in which the local Lorentz symmetry in 
the conventional 11D superspace 
\cf\ is no longer manifest.  This is because in teleparallel superspace,
we do not need to introduce the supercurvature $~R\du{A B c} d$~ which
had  been the main obstruction of embedding certain Bianchi identity 
components in topological superfield equations \gnr.  Since our
formulation is  based on superspace, all the usual features of space-time
supersymmetry are built-in from the outset, in sharp contrast with other
formulations  such as Chern-Simons supergravity
\achu\chamseddine\green\mn\ in which space-time supersymmetry is not
manifest.  


 
  

\bigskip\bigskip\bigskip

% \newpage


\leftline{\bf 2.~~Review of Teleparallel Superspace}  

Before presenting our lagrangian, we briefly review the important
ingredients in our teleparallel superspace \gnr\ for 11D supergravity 
\cf\ which plays a central role in our formulation.  In the teleparallel 
superspace \gnr, we have no manifest local Lorentz covariance, and 
therefore these is no need of supercurvature $~R\du{A B c}d$, 
in contrast to the conventional superspace formulation \cf\superspace.
As will be seen, the absence of supercurvature in such
superspace \gnr\ enables us to consider the embedding of all the necessary
Bianchi  identities into the superfield equations in our topological
superfield theory.  As such, we have only two basic geometrical 
superfield strengths, {\it i.e.,} the anholonomy coefficients which is 
equivalent to a torsion superfield denoted by $~C\du{A B} C$~ and 
a 4-th rank antisymmetric superfield strength $~F_{A B C D}$~ which 
is similar to the conventional 11D superspace \cf\superspace.  

The Bianchi identities in our teleparallel superspace supergravity 
are \gnr
$$ \li{ & \frac 12 E_{\[ A } C \du{B C)} D 
     - \frac 12 C \du{\[ A B | } E  C \du{ E| C)} D \equiv 0 ~~, 
&(2.1\rma) \cr 
& \frac 1{24} E_{\[A} F_{ B C D E ) }
     - \frac 1{12} C \du{\[ A B | } F F_{ F | C D E)} \equiv 0 ~~,    
&(2.1\rmb) \cr } $$ 
where the derivative $~E_A$~ has no Lorentz
connection.  It replaces the Lorentz covariant derivative $~\nabla_A$~ in
the conventional superspace formulation \superspace, and all the
torsion superfields $~T\du{A B} C$~ in the conventional superspace
\superspace\ are replaced by the anholonomy coefficients $~C\du{A B} C$. 
To be more specific,  
$$\li{ & E_A \equiv E\du A M \partial_M~~, ~~~~
     C\du{A B} C \equiv \big( E_{\[ A} E\du{B)} M \big) \,  
     E \du M C  ~~, ~~~~  
    \[ E_A , E_B \} 
    \equiv C\du{A B} C E_C { ~~. ~~~~~ ~~~~~} 
&(2.2) \cr } $$
The supercurvature term in (2.1a) is now absent, due to the
lack of manifest local Lorentz covariance.  Needless to say, there is no
Bianchi identity for the supercurvature, such as $~\nabla_{\[A}
R\du{ B C) d}e  - \cdots \equiv 0$, either.  As usual, the 
$~\theta=0$~ sector of $~F_{a b c d}$~ corresponds to the 4-th
rank component field strength in 11D supergravity \cjs\cf.    

More explicitly, the constraints at the mass dimensions 
$~1/2 \le d \le 1$~ for our teleparallel superspace \gnr\ are 
$$\li{& C\du{\a\b} c = + i \big( \g^c \big)_{\a\b}~~, ~~~~ 
          F_{\a\b c d} = + \frac 12 \big( \g\low{c d} \big)_{\a\b} ~~, 
&(2.3\rmb) \cr 
& C\du{\a\b}\g = + \frac14 \big( \g\low{d e} \big)\du{(\a}\g 
       C\du{\b)}{d e} ~~, 
       ~~~~ C\du\a{b c} = - C\du\a{c b} ~~, 
&(2.3\rmc) \cr 
& C\du{\a b} \g = + \frac i{144} \big(\g\du b{d e f g} F_{d e f g} 
     + 8 \g^{d e f} F_{b d e f} \big)\du\a\g 
     - \frac 18 \big( \g^{c d} \big)\du\a\g 
     \big(2C_{b c d} - C_{c d b} \big) ~~.  
&(2.3\rmd) \cr} $$  
We can confirm the satisfaction of Bianchi identities, whose 
details are skipped here, as has been explained in \gnr.   





\bigskip\bigskip\bigskip

% \newpage


\leftline{\bf 3.~~Invariant Lagrangian}  

Since our lagrangian in superspace is simple, we first present it, 
accompanied by notational explanations.  Our 
total action given  as a superspace integral $~I \equiv \int d^{11} x
\,d^{32}
\theta\, 
\Lag$~ is  
$$ \li{ I = \int d^{11} x\, d^{32} \theta\, 
      E^{-1} \calE^{A_1\cdots A_{11} } 
     \Big[ \! & + {\cal G}_{A_{11}A_{10}} \, {\cal F}_{A_9
\cdots A_5} {\cal A}_{A_4\cdots A_1} \cr 
& + {\cal H}_{A_{11}\cdots A_6}   
      {\cal T}\du{A_5 A_4 A_3} B {\cal S}\du{A_2 A_1} C  
      \eta\low{C B} \, \Big] {~~.~~~~~ ~~~~~}   
&(3.1) \cr } $$ 
As the integration measure indicates, our lagrangian is
in the superspace for 11D space-time with 10 bosonic coordinates $~x^m$~
and 32 Majorana coordinates $~\theta^\a$~ \cf.  Our lagrangian is a
generalization of $~B F$~ theory in superspace, in terms of the two
superfield strengths $~{\cal F}_{A B C D E}, ~{\cal T}\du{A B C} D, ~
\calG_{A B}$~ and $~\calH_{A_1\cdots A_6}$~ defined by 
$$ \li{ & {\cal F}_{A B C D E} 
      \equiv \frac1{4!} E_{\[A} {\cal A}_{B C D E ) } 
      - \frac 1{3!\cdot 2} C\du{\[ A B |} F \calA_{F | C D E ) } ~~.  
&(3.2\rma) \cr 
& {\cal T}\du{A B C} D
      \equiv \frac12 E_{\[A} {\cal S}_{B C ) }{}^D 
      - \frac 12 C\du{\[ A B|} E {\cal S}\du{E| C) } D ~~.   
&(3.2\rmb)  \cr
& {\cal G}_{A B}  
     \equiv E_{\[A } {\cal B}_{B ) } 
     - C\du{A B} C {\cal B}_C ~~, 
&(3.2\rmc) \cr 
& {\cal H}_{A_1\cdots A_6} 
     \equiv \frac 1{5!} E_{\[ A_1} {\cal C}_{A_2\cdots A_6 ) }  
     - \frac 1{4!\cdot 2} C\du{\[ A_1 A_2 |} D
     {\cal C}_{D | A_3\cdots A_6)} ~~.   
&(3.2\rmd) \cr } $$ 
In other words, the potential superfields $~{\cal A}_{A B C D}, ~
{\cal S}\du{A B} C, ~ {\cal B}_A$~ and $~{\cal
C}_{A_1\cdots A_5}$~ respectively have the superfield strengths 
$~{\cal F}_{A B C D E},~ {\cal T}\du{A B C} D, ~{\cal G}_{A B}$~ and
$~{\cal H}_{A_1\cdots A_6}$.  The $~E$~ is the superdeterminant 
of the vielbein $~E\du A M$: $~E\equiv \hbox{sdet}\,\big( E\du A M\big)$~
following the notation in superspace \superspace, and this causes the
inverse power of $~E^{-1}$~ in  the integrand in (3.1).  The reason 
we need the products of two pairs of superfield strengths $~{\cal G}
,~ {\calF}$~ and $~\calH,~\cal T$~ will be clarified shortly.    

One of the most important quantities used in our action (3.1) is the
invariant constant tensors $~\calE^{A_1\cdots A_{11}}$~ and 
$~\eta\low{A B}$~ in 11D superspace.  In particular, when all the
indices are bosonic, the former should be totally antisymmetric, while
the latter is totally symmetric, in order our lagrangian (3.1) to be
non-vanishing.  More explicitly they are defined by\footnotew{We mention
that  similar $~\e\-$tensors for superfield actions of Chern-Simons 
type were proposed in 
%%% 
\ref\zupnikpak{B.M.~Zupnik and D.G.~Pak, Russ.~Phys.~J.~{\bf 31}
(1988) 962; Class.~Quant.~Grav.~{\bf 6} (1989) 723.}.}   
%%%  
\vbox{
$$ \li{ & \calE^{A_1\cdots A_{11}} \equiv 
          \frac1{11!} \calE^{\[ A_1\cdots A_{11} ) } \cr 
& ~~~~~ ~~~~~ ~ 
\equiv \cases{ 
\calE^{a_1\cdots a_{11}} 
        \equiv \e^{a_1\cdots a_{11}}~~, \cr
\calE^{ a_1\cdots a_9 \b\g} 
      \equiv \e^{a_1\cdots a_9 b c} \big( \g\low{b c} \big)^{\b\g} ~~, \cr 
\calE^{ a_1\cdots a_7 \b_1\cdots \b_4} 
      \equiv \e^{a_1\cdots a_7 b_1\cdots b_4}
       \big( \g\low{b_1 b_2} \big)^{(\b_1\b_2|} 
       \big( \g\low{b_3 b_4} \big)^{|\b_3\b_4)} ~~, \cr 
\calE^{ a_1\cdots a_5 \b_1\cdots \b_6} 
      \equiv \e^{a_1\cdots a_5 b_1\cdots b_6}
       \big( \g\low{b_1 b_2} \big)^{(\b_1\b_2|} 
       \big( \g\low{b_3 b_4} \big)^{|\b_3\b_4|} 
       \big( \g\low{b_5 b_6} \big)^{|\b_5\b_6)} ~~, \cr 
\calE^{ a_1\cdots a_3 \b_1\cdots \b_8} 
      \equiv \e^{a_1 a_2 a_3 b_1\cdots b_8}
       \big( \g\low{b_1 b_2} \big)^{(\b_1\b_2|} 
       \big( \g\low{b_3 b_4} \big)^{|\b_3\b_4|} 
       \big( \g\low{b_5 b_6} \big)^{|\b_5\b_6|} 
       \big( \g\low{b_7 b_8} \big)^{|\b_7\b_8)} ~~, \cr 
\calE^{ a \b_1\cdots \b_{10}} 
      \equiv \e^{a b_1\cdots b_{10}}
       \big( \g\low{b_1 b_2} \big)^{(\b_1\b_2|} 
       \big( \g\low{b_3 b_4} \big)^{|\b_3\b_4|} 
       \big( \g\low{b_5 b_6} \big)^{|\b_5\b_6|} 
       \big( \g\low{b_7 b_8} \big)^{|\b_7\b_8|} 
       \big( \g\low{b_9 b_{10}} \big)^{|\b_9\b_{10})} {~~, ~} \cr 
0 ~~~~~\hbox{(otherwise)} ~~, \cr } 
&(3.3\rma) \cr
\noalign{~~~ \vskip 0.08in}   
& \eta\low{A B} \equiv \frac12 \eta{\,}\low{( A B\]} \equiv \cases{ 
\eta\low{a b} \equiv \hbox{diag.}~(+, -, \cdots , -) ~~, \cr 
\eta\low{\a \b} \equiv C\low{\a\b} ~~, \cr
0 ~~~~~\hbox{(otherwise)} ~~, \cr  } 
&(3.3\rmb) \cr } $$ 
}
where $~\eta\low{a b}$~ is the usual $~11\times 11$~ symmetric 11D metric
tensor, while $~C_{\a\b}$~ is the usual $~32\times 32$~ antisymmetric
charge-conjugation matrix used as a metric for spinors.  The above
expression for $~\calE^{A_1\cdots A_{11}}$~ is from 10D supergravity
formulation, in particular, what is called `dual formulation' of Type I
supergravity 
%%%
\ref\gn{S.J.~Gates, Jr.~and H.~Nishino, \pl{157}{85}{157};  
\pl{173}{86}{46}; \pl{173}{86}{52}; \np{291}{87}{205}.}.   
%%%
By comparing the usual version and the dual version of Type I
supergravity in superspace \gn, we find that the natural generalization
of the usual 10D $~\e\-$tensor $~\e^{a_1\cdots a_{10}}$~ is given by 
$~\calE^{a_1\cdots a_8 \b\g} \equiv \e^{a_1\cdots a_8 b c} \big(\g\low{b
c} \big)^{\b\g},~ \calE^{a_1\cdots a_6 \b_1 \cdots \b_4} \equiv
\e^{a_1\cdots a_6 b_1\cdots b_4} \big(\g\low{b_1 b_2}\big)^{(\b_1\b_2|}
\big(\g\low{b_3 b_4}\big)^{|\b_3\b_4)} $, and so forth \gn.  In other
words, each pair of bosonic indices in the purely bosonic
$~\e\-$tensor is replaced by a pair of fermionic indices by
contractions with the $~\g\-$matrix $~\big(\g\low{b c} \big)^{\b\g}$. 
There may be some freedom for normalizations for the r.h.s.~of (3.3a),
but this will not matter in our formulation.  More importantly, all of
the components of $~\calE^{A_1\cdots A_{11}}$~ are constant.  

At first glance, the introduction of the $~\calE\-$tensor (3.3a) into
superspace looks unconventional.  However, we point out that this
$~\calE\-$tensor has many analogs in dimensions other than the
above-mentioned 10D case.  For example in supersymmetric Chern-Simons
theories in 3D 
%%%
\ref\ngcs{H.~Nishino and S.J.~Gates, Jr., \ijmp{8}{93}{3371}.},
%%%
it is known that $~N=1$~ Yang-Mills-Chern-Simons theory has an analogous
usage of $~\g\-$matrices in superspace action as if it were a  superspace
$~\calE\-$tensor.  Note also that our invariant tensor 
$~\calE^{A_1\cdots A_{11}}$~ has properties slightly different from  the
purely bosonic case.  First of all, it is {\it not} `maximally' 
antisymmetric in $~11+32=43\-$dimensional superspace.   This is because
in superspace there is no `maximal rank ~$\calE\-$tensor'
used in the same way as for purely bosonic coordinates.  Note that 
$~{\scst \[ A_1\cdots A_r ) } \neq 0$~ for the totally (anti)symmetric
indices even for $~ r\ge 12$~ in 11D, because $~{\scst \[ A_1\cdots A_r
) }$~ is actually `symmetrization' for fermionic coordinates, and
therefore $~{\scst \[ A_1\cdots A_r ) }$~ does not vanish for $~r\ge 12$. 
This is also reflected in the definition of super-determinant $~E \equiv
\hbox{sdet}\,
\big(E\du A M\big) \equiv \exp\big[ \,\str \ln\big( E \du A M\big) \,
\big]$~ with {\it no} reference to the maximal rank (anti)symmetric
$~\calE\-$tensor.  We have to keep this in mind, when performing actual
computation in superspace.    

The fundamental superfields in our theory are the potential
superfields 
$~{\cal A}_{A B C D}, ~{\cal S}\du{A B} C$,   
$~{\cal B}_A, ~ {\cal C}_{A_1\cdots A_5}$, and 
the vielbein $~E\du A M$.  Since $~{\cal F}_{A B C D E}, ~{\cal T}\du{A B
C}D , ~ {\cal G}_{A_1\cdots A_7}$~ and $~{\cal H}_{A_1\cdots A_9}$~ are 
all superfield strengths by definition, they should obey their own 
Bianchi identities.  In fact, it is not difficult to show that 
they satisfy the following Bianchi identities
$$ \li{ & \frac1{5!} E_{\[ A_1 } {\cal F}_{A_2\cdots A_6) } 
      - \frac 1{4!\cdot 2} C\du{\[ A_1 A_2 | } B 
     {\cal F}_{ B | A_3\cdots A_6)}  \equiv 0 ~~,   
&(3.4\rma) \cr 
& \fracm1{3!} E_{\[ A} {\cal T}\du{B C D)} E 
      - \frac 1{2!\cdot 2} C\du{\[ A B|}F 
     {\cal T} \du{F | C D)} E \equiv 0 ~~,  
&(3.4\rmb) \cr 
& \frac 1 2 E_{\[ A } {\cal G}_{B C)}
      - \frac 1 2 C\du{\[ A B | } D 
      {\cal G}_{D | C)}  \equiv 0 ~~, 
&(3.4\rmc) \cr 
& \fracm 1{6!} E_{\[ A_1} {\cal H}_{A_2\cdots A_7)}  
      - \frac 1{5!\cdot 2} C\du{\[ A_1 A_2 |} B 
      {\cal H}_{B | A_3\cdots A_7)} \equiv 0 ~~.  
&(3.4\rmd) } $$
The satisfaction of the Bianchi identity (3.4b) also justifies 
the definition (3.2b) of $~{\cal T}\du{A B C} D$~ which has a
peculiar superscript index $~^D$~ like a free index that stands alone, not
interfering with all other subscript indices $~_{A B C}$, which is 
rather new as superspace formulation.  In other words, this index $~^D$~
behaves like the familiar adjoint representation index in Yang-Mills
theory.   

Our action (3.1) has also gauge invariances, under the transformations 
$$ \li{ & \d_\L {\cal B}_A 
    \equiv E_A \L ~~, 
&(3.5\rma) \cr 
& \d_\L {\cal C}_{A_1\cdots A_5}  
     \equiv \frac1{24} E_{\[A_1} \L_{A_2\cdots A_5)} 
    - \frac1{12} C\du{\[ A_1 A_2 | } B \L_{B| A_3 A_4 A_5)} ~~.  
&(3.5\rmb) \cr }  $$
Here $~\L$'s with different number of indices are 
distinct from each other as arbitrary space-time dependent parameters.
However, as is seen from the pure potential superfields 
$~\calA_{A B C D}$~ and $~\calS\du{A B} C$~ present with no derivatives,
our action (3.1) is {\it not} invariant ($\d_\L ' I\neq 0$) under the
gauge transformations
$$ \li{ & \d_\L '  {\cal A}_{A B C D} 
     = \frac 16 E_{\[A} \L_{B C D)} 
     - \frac 14 C\du{\[A B | } E \L_{E | C D)} ~~, 
&(3.6\rma) \cr 
& \d_\L ' {\cal S}\du{A B} C \equiv 
     E_{\[A} \L\du{B)} C - C\du{A B} D \L \du D C ~~,  
&(3.6\rmb) \cr } $$ 
for the potential superfields $~{\cal A}_{A_1\cdots A_5}$~ and $~{\cal 
S}\du{A B} C$.  We can confirm that $~\d_\L ' I\neq 0$~ more rigorously by
direct computation, with the aid of the Bianchi identities (3.4) and the
definitions (3.2).  

It is worthwhile to mention that our action (3.1) can be 
rewritten after appropriate partial integrations, as  
$$ \li{ I = \int d^{11} x\, d^{32} \theta\, E^{-1} 
     \calE^{A_1\cdots A_{11}} 
\Big[ & \!\! +\frac1{12!} C\du{\[ A_{11} A_{10}|} B \calB_{|B|} 
      {\cal F}_{|A_9\cdots A_5|} {\cal A}_{| A_4\cdots A_1 ) }\cr 
& \!\! + \frac3{12!} C\du{\[A_{11} A_{10} |}B 
     {\cal C}_{|B A_9 \cdots A_6|} 
      {\cal T}\du{|A_5 A_4 A_3|} D {\cal S}\du{|A_2 A_1 )} C 
      \eta\low{C D} \, \Big] {~~ ,  ~~~~~ ~~~}  
&(3.7) \cr } $$ 
where all the 12 indices are totally antisymmetrized in each term.  
For the reason already mentioned, such a total antisymmetrization 
does not lead to a vanishing result, due to the special feature of 
our $~\calE\-$tensor used here, as well as common feature of
superspace \superspace\cf.  Compared with (3.7), the previous expression
(3.1) of our action $~I$~  looks more like a generalized $B F$ theory
\topological, in the sense that each term is a product of a potential
superfield and products of  superfield strengths.  However, the
alternative expression (3.7) is much more  topological in the sense that
the lagrangian in (3.7) changes  as a total divergence under the gauge
transformation (3.5), much like  the case of $B F$ theories or
Chern-Simons theories
\topological.  This feature is  not manifest in (3.1), because superfield
strengths are manifestly  invariant under (3.5).  Note that the index
$~{\scst B}$~  in the second line in (3.7) is under the total
(anti)symmetrizations of 
$12$ indices $~{\scst \[ A_{11} A_{10} B A_9 A_8 \cdots A_1 ) }$.  For
the  reason already mentioned, such an expression with 12 total
(anti)symmetrization does not necessarily vanish in superspace.   

Some readers may wonder, if this type of action starting with the 
trilinear terms such as (3.1) really makes sense.  Because, {\it e.g.,}
for quantizations it is more convenient to start with the bilinear 
terms in the lagrangian, instead of trilinear terms.  This question is 
answered by the other Chern-Simons theories in higher-dimensions
\chamseddine, in which the quantizations can be performed by appropriate
expansions of relevant fields around their non-vanishing v.e.v.'s, so that
the fluctuations for quantized fields will start at the bilinear order
\chamseddine.  In other words, our action (3.1) starting with trilinear
terms suggests some transitions between vacua, as different phases of
this potentially underlying master theory.  For example, we can try 
the v.e.v.'s $~\langle 
\calG_{\hat a \hat b} \rangle = \e_{\hat a\hat b} \neq 0, ~\langle
\calG_{\hat a\bar b} \rangle =0,~\langle \calG_{\bar a\bar b} \rangle =0$~
for the dimensional reduction from 11D into 9D.  Here the indices
$~{\scst \hat a,~\hat b,~\cdots}$~  are for the extra 2D in this
dimensional reduction, while 
$~{\scst \bar a,~\bar b,~\cdots}$~ are for the 9D indices.  By this
reduction, the purely bosonic part in $~\calG \calF\calA\-$term in the
original action (3.1) yields a bilinear Chern-Simons term in 9D:
$~\e^{\bar a_1\cdots \bar a_9}\calF_{\bar a_9\cdots \bar a_5} \calA_{\bar
a_4\cdots \bar a_1}$.  In a similar fashion, we can give nontrivial
v.e.v.'s to other field strengths or covariant superfields that leads us
to many other interesting actions starting with bilinear
terms.\footnotew{We can give such non-vanishing v.e.v.'s, as long as all
the superfield equations (3.12) are satisfied.}  We will give an 
illustrative example of such a dimensional reduction in the next
section.   

The equivalence between the two expressions (3.1) and (3.7) can be 
confirmed by direct computation, including appropriate partial
integrations.  A useful relationship to be used is 
$$ \li{ & \partial_M \big( E^{-1} E\du A M\big) 
     =  C\du{A B} B \equiv E^{-1} C_A ~~. 
&(3.8) \cr } $$ 
Even though we did not write the Grassmann parities explicitly, 
they are to be understood in the standard way.  Namely, when the
nearest neighbor pair of two indices is contracted between northwest
and southeast, there is no cost of sign, while that between northeast and
southwest costs an additional sign.  (We call this `nest form 
contraction'.)  To be more specific, the Grassmann parities in (3.8) can
be explicitly written as 
$$ \li{ & (-1)^{M(A+M)} \partial_M \big( E^{-1} E\du A M\big) 
     = (-1)^B E^{-1} C\du{A B} B \equiv E^{-1} C_A ~~. 
&(3.9) \cr } $$ 
In other words, whenever we have expression like (3.8), we consider 
all the sign changes to put all the indices in the `nest form', {\it
i.e.,} all the contracted indices are in the nearest pairs between
northwest and southwest positions.  As long as we keep this `nest form'
rule in mind, there will arise no ambiguities in signatures, as (3.9)
is uniquely obtained from (3.8).  
In deriving (3.7) from (3.1), we also need the identities 
$$ \li{ & {\cal F}_{\[ A_1\cdots A_5 | } {\cal F}_{| A_6\cdots A_{10} ) } 
    \equiv 0 ~~, ~~~~
     {\cal T}\du{\[ A_1 A_2 A_4 | } E {\cal T}\du{| A_4 A_5 A_6)} F 
     \eta\low{F E} \equiv 0~~. 
&(3.10) \cr } $$  
In the second equation, appropriate Grassmann parities are to be 
understood for the exchange of the indices $~{\scst E}$~ and 
three out of $~{\scst A_1, ~A_2, ~\cdots, ~A_6}$, even though 
they are not explicitly written.  For example, the last equation of (3.10)
implies 
$$ \li{ & \calT\du{\[ A_1 A_2 A_3|} E \, \calT\du{| A_4 A_5 A_6) } F
      \eta\low{F E} 
\equiv \sum_{\buildrel{~6!~{\rm permut.~of}}
      \over{A_1\cdots A_6}} (-1)^{E (A_4 + A_5 + A_6)} 
      \calT\du{A_1 A_2 A_3} E \, \calT\du{A_4 A_5 A_6} F \eta\low{F E}
\cr  & ~~ \, \equiv + (-1)^{E(A_4 + A_5 + A_6)} \calT\du{A_1 A_2 A_3} E 
     \calT\du{A_4 A_5 A_6} F\eta\low{F E} \cr 
& ~~~~~ ~ + 
     (-1)^{E(A_5 + A_6 + A_1) + A_1 (A_2 + A_3 + A_4 + A_5 + A_6) }
     \calT\du{A_2 A_3 A_4} E \, 
     \calT\du{A_5 A_6 A_1} F\eta\low{F E} \cr 
& ~~~~~ ~ + \big[ \, (6!-2)~\hbox{more terms} \, \big] ~~. 
&(3.11) \cr } $$ 
In this paper, we do not write Grassmann parities explicitly, in order
to save considerable amount of space, but they are always to be
understood as in other superspace formulations \superspace\cf.

We now consider the superfield equations from our superspace action in 
(3.1).  Our fundamental superfields are $~{\cal
B}_{A_1\cdots A_4}, ~{\cal S}\du{A B} C, ~{\cal B}_A,~
{\cal C}_{A_1\cdots A_5}$~ and $~E\du A M$~ with no additional 
constraints.  Therefore, their superfield equations are obtained  
as the usual superspace Euler derivatives \superspace:   
$$\li{ \fracm{\d\Lag}{\d A_{A_1\cdots A_4}} 
     = & + \frac 5 {9!\cdot 7!\cdot 2} E^{-1} 
     \calE^{A_1\cdots A_4 B_1\cdots B_7} 
     {\cal G}_{\[ B_7 B_6 |} C\du{|B_5 B_4|} C 
     {\cal A}_{| C B_3 B_2 B_1 ) } \cr 
& - \frac{10}{9! \cdot 4!} E^{-1}  
     \calE^{\[ A_1 A_2 A_3| B_1\cdots B_8} 
     {\cal G}_{B_8 B_7} C\du{B_6 B_5}{|A_4)}           
     {\cal A}_{B_4 \cdots B_1} \eqdot 0 ~~,  
&(3.12\rma) \cr 
\noalign{\vskip 0.15in} 
     \fracm{\d\Lag} {\d {\cal S}\du{A B} C} 
     = & + \frac3{9! \cdot  5! \cdot 2}  
    E^{-1} \calE^{A B D_3\cdots D_{11}} {\cal H}_{\[D_{11} \cdots D_6 |} 
    C\du{| D_5 D_4|} E {\cal S}\du{|E D_3)} F \eta \low{F C} \cr 
& + \frac3{5!} E^{-1} \calE^{\[ A | E_1\cdots E_{10} } 
    {\cal H}_{E_{10}\cdots E_5} C\du{E_4 E_3} {|B)} 
     {\cal S}\du{E_2 E_1} F\eta\low{F C} \eqdot 0 ~~, 
&(3.12\rmb) \cr 
\noalign{\vskip 0.15in} 
\fracm{\d\Lag}{\d {\cal B}_A} 
     = & \frac1{12} E^{-1} \calE^{B_1\cdots B_{11}} 
    C\du{B_{11} B_{10}} A 
     {\cal F}_{B_9\cdots B_5} {\cal A}_{B_4\cdots B_1} \cr 
& - \frac{11}{12!} E^{-1} \calE^{A B_{10} \cdots B_1} 
    C \du{\[ B_{10} B_9|} C {\cal F}_{|C B_8\cdots B_5|} 
    {\cal A}_{|B_4\cdots B_1)} \eqdot 0 ~~, ~~~~
&(3.12\rmc) \cr 
\noalign{\vskip 0.15in} 
\fracm{\d\Lag}{\d {\cal C}_{A_1\cdots A_5}} 
     = & + \frac 1{6!\cdot 4} E^{-1} \calE^{A_1\cdots A_5 B_1\cdots B_6} 
    C\du{\[ B_6 B_5|} C {\cal T}\du{|C B_4 B_3|} D 
     {\cal S} \du{|B_2 B_1)} E \eta\low{E D} \cr 
& - \frac 1{4!\cdot 4} E^{-1} \calE^{\[A_1\cdots A_4| B_1\cdots B_7} 
     C\du{B_7 B_6} {|A_5)} 
    {\cal T}\du{B_5 B_4 B_3} D {\cal S}\du{B_2 B_1} C \eta\low{C D}  
\eqdot 0 {~~, ~~~~~ ~~~~~ ~~}  
&(3.12\rmd) \cr 
\noalign{\vskip 0.15in} 
E \du B M \fracm{\d\Lag}{\d E\du A M} 
   = & - \d\du B A \Lag + 2 E^{-1} E_C X\du B{C A} 
     - 2 E^{-1} C_C X\du B {C A} \cr 
& + 2 E^{-1} C\du{B C} D X\du D{C A}
      - E^{-1} C\du{D C} A X\du B{C D} \eqdot 0 
      {~~, ~~~~~ ~~~~~ ~~}  
&(3.12\rme) \cr } $$ 
where 
$$ \li{ X\du A{B C} \equiv 
& - \calE^{B C D_1\cdots D_9} {\cal B}_A {\cal F}_{D_9\cdots D_5} 
     {\cal A}_{D_4\cdots D_1} 
   - 10 \calE^{B C D_1\cdots D_9} {\cal G}_{D_9 D_8} 
     {\cal A}_{A D_7 D_6 D_5}
    \calA_{D_4\cdots D_1} \cr 
& - 15 \calE^{B C D_1\cdots D_9} {\cal C}_{A D_9\cdots D_6} 
    {\cal T}\du{D_5 D_4 D_3} E \calS\du{D_2 D_1} F \eta\low{F E} \cr 
& - 3 \calE^{B C D_1\cdots D_9} {\cal H}_{D_9\cdots D_4} 
   {\cal S}\du{A D_3} E \calS\du{D_2 D_1} F \eta\low{F E} ~~. 
&(3.13) \cr } $$ 
The symbol $\eqdot$ implies an equality that holds only by the use of 
superfield equation, but not an identity.  In all of the equations in 
(3.12), appropriate Grassmann parities mentioned with (3.11) are to be 
understood, even though they are implicit.  Note that the 
$~{\scst C}\-$index in (3.12a), (3.12c), (3.12d), or the $~{\scst
E}\-$index in (3.12b) are inside of the total (anti)symmetrizations.   
It is no wonder that the
bare potential superfields $~\calA_{A B C D}$~ and $~\calS\du{A B}C$~ with
no derivatives appear in these superfield equations, considering the fact
that our action does not have gauge invariance under (3.6).  We will come
back to this point shortly.  

We are now ready to discuss how these 
superfield equations can embed our teleparallel 11D superspace
supergravity \gnr, characterized by the Bianchi identities (2.1).  First
of all, note that the set of solutions 
$$ \li{ & {\cal T}\du{A B C} D \eqdot 0~~, ~~~~
     {\cal F}_{A B C D E}\eqdot 0~~, 
&(3.14\rma) \cr 
& {\cal B}_A \eqdot 0~~, ~~~~
     {\cal C}_{A_1\cdots A_5} \eqdot 0 ~~,   
&(3.14\rmb) \cr } $$ 
trivially satisfies our superfield equations (3.12) including (3.12e) for
the vielbein, because of $~X\du A{B C}\eqdot 0$~ under (3.14).  At first
glance, this set of solutions looks trivial, leading to no
physical content.  However, the consideration of the identifications 
$$\li{ & {\cal S}\du{A B} C \eqdot C\du{A B} C~~, ~~~~
         {\cal A}_{A B C D} \eqdot F_{A B C D}~~, 
&(3.15) \cr } $$ 
reveals non-trivial nature of our system, because all the
components in the Bianchi identities in 11D superspace (2.1) are 
now satisfied, combined with (3.14), (3.15) and (3.2):  
$$ \li{ & {\cal T}\du{A B C} D 
      \eqdot \frac 12 E_{\[ A} C\du{B C)} D 
     - \frac 12 C\du{\[ A B |} E  C\du{ E| C)} D \equiv 0 ~~, 
&(3.16\rma) \cr 
& {\cal F}_{A B C D E} \eqdot \frac 1{24} E_{\[A} F_{ B C D E ) }
     - \frac 1{12} C\du{\[ A B| } F F_{ F | C D E)} \equiv 0 ~~.   
&(3.16\rmb) \cr } $$ 
In other words, all the teleparallel superspace Bianchi identities \gnr\
for 11D supergravity (2.1) are embedded into the superfield equations 
from our action (3.1), as the simplest but nontrivial solutions.    

One important ingredient here is that there can be other nontrivial 
solutions to our superfield equations in (3.12) other than (3.14), because
there are many indices contracted in (3.12), but not necessarily all the
components of each of the superfield strengths $~{\cal T}\du{A B C} D$~
and $~{\cal F}_{A B C D E}$~ are zero.  This also indicates that our
system of superspace $~B F$~ theory can accommodate more theories than
the ordinary 11D supergravity \cjs\cf, suggesting strongly that our theory
is one of the most natural generalizations of 11D supergravity \cjs\cf, as
a good low energy theory for M-theory \mtheory.   As we have also
mentioned, the feature of our action starting with the trilinear terms
also  suggests some non-trivial vacuum with non-vanishing v.e.v.'s of  
the fundamental superfields, like other Chern-Simons theories in 
higher dimensions \chamseddine.   

We now address the usage of teleparallel superspace in 11D \gnr.  The most
important advantage of teleparallelism \gnr\ is that the supertorsion
Bianchi identity for teleparallel superspace is much more simplified than
the conventional one with the local Lorentz connection $~\phi\du{A b} c$. 
For example, if we had the supercurvature term proportional to $~R\du{\[
A B| c} d \big({\cal M}\du d c\big)\du{|C)} D$~ in the Bianchi identity
(2.1a), then it would look unnatural to embed such a term into another
superfield strength like $~\calT\du{A B C} D$~ in (3.2b).   Or more
importantly, the usual expression of $~\phi\du{A b} c$~ in terms of the
vielbein {\it via} anholonomy coefficients $~C\du{a b} c$~ came out, only
{\it after} solving the Bianchi identity $~\nabla_\a T\du{b c}
d+\cdots\equiv 0$~ at mass dimension $~d = 3/2$.  Therefore, it is more
natural to work directly on the teleparallel superspace \gnr\ from the
outset, where  there is no worry of solving the Bianchi identity at
$~d=3/2$.  From these viewpoints, we consider teleparallel superspace 
\gnr\ is the most natural choice for superspace formulation for our
purpose.   

It is worthwhile to mention that a somewhat analogous trial of using a
third-rank antisymmetric potential field in 11D supergravity
\cjs\ interpreted as a torsion tensor was presented as early as in 1984 
%%% 
\ref\kallosh{R.~Kallosh, \pl{143}{84}{373}.}.  
%%% 
There are differences as well as similarities between our  formulation
and \kallosh.  The similarity is that we are identifying superfield
strengths such as $~F_{A B C D}$~ with the superpotential $~\calA_{A B C
D}$, analogously to the potential $~A_{m n r}$~ embedded into the torsion
$~ T\du{m n}r$~ which is also a field strength \kallosh.  One of the
differences is that our formulation is not just a re-writing of the
conventional 11D supergravity as in \kallosh, but it presents a more
topological generalized $~B F\-$theory as an underlying theory
for 11D supergravity \cjs\cf\ itself.  Additionally, we are dealing with
superspace instead of components formulation \kallosh.  

We now discuss the previously-mentioned 
gauge {\it non}-invariance of our action (3.1) under (3.6).  This is to
guarantee  that our superpotential $~A_{A B C}$~ after the embedding
(3.15) is {\it not} gauged away.  To see this, let us consider a much 
simpler `toy' action 
$$ \li{ & I^\prime = \int d^{11} x\, d^{32} \theta\, 
      E^{-1} \calE^{A_1\cdots A_{11}} \calB_{A_{11}\cdots A_6} 
      \calF_{A_5 \cdots A_1} ~~. 
&(3.17) \cr } $$ 
This action has a manifest gauge invariance under (3.6), and it looks
much simpler than our action $~I$~ in (3.1).  What is wrong with
this action $~I'$~ is that the gauge symmetry (3.6) will gauge away the
embedded potential superfields $~A_{A B C}$~ completely, if we choose  
$$ \li{ & \L_{A B C} = - A_{A B C} ~~. 
&(3.18) \cr } $$ 
In other words, for a system with an action with the gauge symmetry 
(3.6), such embedding as (3.15) becomes just a gauge degree of freedom,
leading to no physical content.  This is a crucial point for our
formulation, because we are embedding  the superfield strength $~F_{A B C
D}$~ into the potential superfield $~\calA_{A B C D}$~ whose superfield
strength $~\calF_{A B C D E}$~ is one hierarchy higher, which can easily
lead to a trivial result, by `cohomological nilpotency'.  Such
obstructions may be the main cause of the delay for the development of
superspace formulations of $B F$ theory \topological\ up until now. 
This also tells us why we needed the products of two pairs of 
superfield strengths $~\cal G,~\calF$~ and $~\calH,~\calT$~ in (3.1) 
as a generalization of $B F$ theory.  It further tells us 
how difficult it is to `embed' such a simple looking set of 
Bianchi identities as (2.1) into some superfield equations obtained under
an action principle.  

This sort of gauge-non-invariance $~\d_\L' I \neq 0$~ under (3.6) seems
rather awkward at a first glance, because there do not seem to be so many
other examples in physics.  However, we emphasize that this is what is
happening rather often in supergravity theories.  First of all, we have
to remind ourselves that our system is not locally Lorentz covariant from
the outset.  This can be explicitly confirmed by the variation of 
the vielbein: 
$$ \li{ & \d_{\rm L} E\du A M = L\du A B E\du B M ~~, ~~~~
      \d_{\rm L} E \du M A = - E\du M B L\du B A ~~, 
&(3.19) \cr } $$ 
with the infinitesimal local Lorentz parameter $~L^{A B} = L^{A B} (Z) =
- (-1)^{A B} L^{B A} \equiv \Big( L^{a b} , L^{\a\b} \equiv (1/4) (\g\du
c d)^{\a\b} L\du d c\Big)$.  Accordingly, the anholonomy coefficients 
$~C\du{A B} C$~ acquires an inhomogeneous term at the end: 
$$ \li{ & \d_{\rm L} C\du{A B} C = L\du{\[A |} D C \du{D|B)} C 
      - C\du{A B} D L\du D C + E_{\[A} L\du{B)} C ~~,   
&(3.20) \cr } $$ 
and therefore our action (3.7) lacks local Lorentz invariance under
(3.19).  Analogously to this local Lorentz symmetry, 
the local symmetry for our solution $~A_{A B C}$~ in (3.15):
$$ \li{ & \d_\L'' A_{A B C} = \frac 12 E_{\[A} \, \L''_{B C)} 
     -\frac 12 C\du{\[A B|} D \L''_{D | C)} ~~ , 
&(3.21) \cr } $$ 
is not manifest in our original action, while this symmetry is  realized
within a particular set of solutions at the superfield equation level. 
This phenomenon is not new, but has been well-known in supergravity
theories, such as local $~SU(8)$~ symmetry in $~N=8$~ supergravity in 4D
\cjs\ related to U-duality.  To put it differently, our starting action
does not have to possess all the local symmetries needed at the end, but
those symmetries such as (3.19) or (3.21) are realized only among
their particular solutions.  In fact, once we have obtained the 11D
supergravity as a solution to the superfield equations (3.12), the local
symmetry (3.21) for $~A_{A B C}$~ is recovered as in usual 11D
supergravity \cf.  Once we have accepted this viewpoint, it is becomes
natural that our system lacks certain symmetries or invariances such as
(3.6a) at the level of action or lagrangian.  






% \bigskip\bigskip\bigskip


\newpage






% \newpage

\leftline{\bf 4.~~Concluding Remarks}

In this paper, we have presented a simple lagrangian formulation of 
$~B F$~ theory \topological\ in superspace \superspace\ that can
accommodate all the necessary Bianchi identities for teleparallel 11D
superspace supergravity \gnr.  We have two expressions (3.1) and (3.7) for
our action $~I$.  The latter is much more `topological' than its
alternative expression (3.1), in the sense that the lagrangian changes as
a total divergence in superspace under gauge transformations.   We have
seen that the 11D superspace Bianchi identities are realized as one of
the simplest but nontrivial solutions to our superfield equations from
our superspace lagrangian.  This also suggests that the superfield
equations for our lagrangian allow more solutions than those for 11D
supergravity \cf, and therefore, our topological theory can be one of the
most natural generalization of the conventional 11D supergravity \cjs\cf. 
Our action (3.1) starts with the trilinear order terms, instead of the
usual bilinear ones, suggesting the transition of vacuum from the
original one where superfields are expanded.  This feature is also common
to other Chern-Simons theories in higher dimensions \chamseddine.  

It has been generally expected that M-theory \mtheory\ will reveal itself
as a topological or geometrical theory, such as Chern-Simons theory
\achu\chamseddine\green, $~B F$~ theory 
\topological, or topological theory \linsmolin\ like Poincar\' e gravity
can be realized as a more geometrical conformal gravity theory, when the
dimensionful gravitational coupling becomes negligible, or at higher
energy, or at quantum level.  In this paper, we have presented a
generalized $~B F$~ theory as a topological/geometrical generalization of
11D supergravity \cf, relying on formulation in teleparallel superspace
\gnr.  Our action (3.1) starting with trilinear terms indicates   
possible nontrivial v.e.v.'s yielding bilinear terms.  This suggests the
existence of many different phases of our theory with nontrivial vacuum
structures.   
 
To our knowledge, we stress that such a formulation of `generalized'
topological $B F$ theory \topological\ in 11D superspace \cf\ has never
been presented.  There seem to have been three main obstructions against
such formulations in  superspace in the past.  First, the supercurvature
term in the $~T\-$Bianchi identity \superspace\ prohibits the simple
embedding of the supertorsion tensor $~T\du{A B} C$~ into a higher-rank
tensor.  We have overcome this obstruction by getting rid of the
supercurvature term in (2.1a), adopting teleparallel superspace \gnr. 
Second, superspace formulation \superspace\ seemed to lack the appropriate
invariant constant $~\E\-$tensor for fermionic coordinates, as a simple
analog of $~\e\-$tensor for topological gauge theories only with the
bosonic coordinates.  We have overcome this obstruction by the use of
$~\calE\-$tensor (3.3a) acquired from experience in 3D Chern-Simons theory
in superspace \ngcs\ and the dual Type I supergravity \gn, as the most
natural higher-dimensional application.  Third, as has been mentioned at
the end of the last section, the `cohomological nilpotency' invalidates
the idea of such embeddings as (3.15), due to the gauge symmetry that
gauge away physical degrees of freedom.  This obstruction has been
overcome by forbidding the troublesome gauge symmetry (3.6) by the use of
Chern-Simons factor as in (3.1).  In other words, our system lacks the
gauge invariance under such a troublesome gauge symmetry (3.6).  
     
We stress that the usage of Lorentz non-covariant formulation presented 
in this paper is nothing peculiar or eccentric, but based on
the recent developments related M-theory.  This is not only
motivated by higher dimensional supergravity theories in 
%%%
\ref\hdsg{H.~Nishino, \pl{428}{98}{85}, hep-th/9703214; 
\pl{437}{98}{303-314}, hep-th/9706148; 
\np{542}{99}{217-261}, hep-th/9807199.}
%%%  
for F-theory 
%%%
\ref\ftheory{C.~Vafa, \np{469}{96}{403}.} 
%%% 
or S-theory
%%%
\ref\stheory{I.~Bars, \pr{55}{97}{2373}.}, 
%%% 
but also by other results related to M-theory \mtheory\ or
superstrings in 
$~D\le 11$.  For example, in ref.~% 
%%% 
\ref\blo{E.~Bergshoeff, Y.~Lozano and T.~Ortin, 
\np{518}{98}{363}, \hepth{9712115}.} 
%%% 
a new supergravity formulation with background with Killing isometry has 
been presented in 11D.  Even though the exact terminology
`teleparallelism' is not used in the paper, such a formulation in 11D
lacks manifest local Lorentz covariance within 11D, due to the Killing
isometry.  Another example is found in a recent paper 
%%% 
\ref\bffhp{M.~Blan, J.~Figueroa-O'Farrill, C.~Hull and G.~Papadopoulos,
{\it ``A New  Maximally Supersymmetric Background of Type IIA
Superstring''}, \hepth{0110242}.}   in which some new maximally
supersymmetric backgrounds are found for  Type IIA superstring.  Despite
the fact that  the phrase `Lorentz non-covariance' was not used in
\bffhp, their result shows the importance of maximally supersymmetric
backgrounds which lacks  the full 10D local Lorentz covariance.  These
are just simple  examples in this new direction associated with M-theory
\mtheory, and the importance of exploring Lorentz non-covariant
formulation is now clear.  We emphasize that 
the result in this paper is based on a profound supporting evidence for
Lorentz non-covariant teleparallelism in general supergravity/superstring
theories, such as recent developments associated with M-theory \mtheory.  

In our paper, we have identified supertorsions (anholonomy coefficients)
$~C\du {A B} C$~ with the supertensor $~{\cal S}\du{A B} C$.  Even 
though this kind of identification seems unusual or artificial at first
sight, we  stress that there have been analogous methods since 1980's. 
One good example can be found in a paper     
%%% 
\ref\gnz{M.T.~Grisaru, H.~Nishino and D.~Zanon, \pl{206}{88}{625}; 
\np{314}{89}{363}.}  
%%% 
on $~N=1$~ superspace in 10D.  It has been demonstrated in \gnz\ that the
consistent embedding of the supertorsion 
$~T\du{a b} c$~ into a third-rank superfield strength $~H_{a b c}$~ is
possible by the relationship $~H_{a b c} = - (1/2) T_{a b c}$.  It is
to be stressed that our approach has provided a generalization of such an
embedding to the case of 11D supergravity \cjs\cf.  

It seems that the teleparallelism formulation \gnr\ is universally
possible  in any space-time dimensions, once the conventional Lorentz
covariant formulation in superspace \superspace\ is established for any 
supergravity 
%%% 
\ref\nahm{W.~Nahm, \np{135}{78}{149}; 
{\it `Supergravity in Diverse Dimensions'}, 
{\it eds}.~A.~Salam and E.~Sezgin, Vols.~{\bf 1} and {\bf 2},
North-Holland/World Scientific (1989).}.  
%%%
However, in this paper we have taken advantage of 
11D teleparallel superspace \gnr\ in which there are only two 
Bianchi identities needed for supergravity.  It is this simplicity of 
superspace that enabled us to formulate our $~B F$~ theory that can 
embed 11D supergravity theory \cjs\cf.   

Our system has another noteworthy aspect about local symmetries.  We have
seen that not only local Lorentz symmetry, but also local symmetry for
$~A_{A B C}$, that are not present at the action level, can be recovered
at the level of superfield equations.  We stress that this phenomenon is
not new at all, but has been observed in many different contexts in
supergravity theories in the past, such as the local $~SU(8)$~ symmetry
for $~N=8$~ supergravity in 4D \cjs, that can be realized only among the
field equations, but not at the action or lagrangian level.  

Our result in this paper has opened a wide avenue for applications of
this formulations to other supergravities in diverse dimensions in $~1\le
D\le 10$~ \superspace\nahm.  For example, our formulation provides an
action principle even for certain supergravity theories, such as Type
IIB  in 10D, that forbid conventional action formulations.  Another
interesting study will be on the dimensional reduction of our action
into $~1\le D \le 10$~ that will generate not only all the other known
supergravity theories in these lower dimensions \nahm, but also
unknown ones.  

Even though our topological superfield theory lagrangian (3.1) looks 
simple, it is not merely a single term of $~B F$~ theory, but is an
intricate combination of two terms.  Neither is it in component language,
but it is in terms of superspace language \superspace\ with manifest
supersymmetry.  It can be understood as a `generalized' topological $B F$
theory or Chern-Simons theory \topological\ in the sense that the
lagrangian in (3.7) changes as a total divergence under certain gauge
transformation.  It has space-time supersymmetry instead of `fake'
supersymmetry of supergroups used in Chern-Simons supergravity
formulations \achu\chamseddine\green.  It is not formulated in ordinary
superspace \superspace, but it is more naturally formulated in peculiar
teleparallel superspace \gnr.  We find that our formulation is so simple
and elaborate at the same time that this theory might well be the genuine
fundamental underlying theory of M-theory \mtheory\ as the low energy
limit, accommodating the conventional 11D supergravity \cjs\cf.       


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\doit1{
We are grateful to the referee of this paper who gave important 
suggestions to improve the original manuscript.   
} 

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