%%%%%%%%%%%%% Macros (defrr.tex, revised 06/22/97) %%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}

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

\doit{0}{
% Horizontal alignment of accents: bars, hats, tildes, etc.
\skewchar\fivmi='177 \skewchar\sixmi='177 \skewchar\sevmi='177
\skewchar\egtmi='177 \skewchar\ninmi='177 \skewchar\tenmi='177
\skewchar\elvmi='177 \skewchar\twlmi='177 \skewchar\frtnmi='177
\skewchar\svtnmi='177 \skewchar\twtymi='177
\def\@magscale#1{ scaled \magstep #1}
% For framing we need to put 1 in \doit just below as well as in \framing{1}.
% Framing 
\def\framingfonts#1{
\doit{#1}{\font\twfvmi  = ammi10   \@magscale5 % math italic
\skewchar\twfvmi='177 \skewchar\fivsy='60 \skewchar\sixsy='60
\skewchar\sevsy='60 \skewchar\egtsy='60 \skewchar\ninsy='60
\skewchar\tensy='60 \skewchar\elvsy='60 \skewchar\twlsy='60
\skewchar\frtnsy='60 \skewchar\svtnsy='60 \skewchar\twtysy='60
\font\twfvsy  = amsy10   \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018			% Gothic
\font\sc=font005			% script
\def\Go#1{{\hbox{\go #1}}}	% Gothic for single characters in equations
\def\Sc#1{{\hbox{\sc #1}}}	% script for single characters in equations
\def\Sf#1{{\hbox{\sf #1}}}	% sans serif for single characters in equations
\font\oo=circlew10	      % thick circles (hollow: ` and a-n , solid:  p-z)
\font\ooo=circle10			% thin circles   
\font\ro=manfnt				% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

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

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

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

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

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

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

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

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
	\ialign{\strut \hfil$\displaystyle{##}$&$
	\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
	\everycr{\noalign{\ifdtup \global\dtupfalse
	\vskip-\lineskiplimit \vskip\normallineskiplimit
	\else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous				% eqalignno
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}

% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
}

% Text style parameters
\topmargin=0in				% top margin (less 1") (LaTeX)
\headheight=0in				% height of heading (LaTeX)
\headsep=0in			% separation of heading from body (LaTeX)
\textheight=9in				% height of body (LaTeX)
\footheight=3ex				% height of foot (LaTeX)
\footskip=4ex		% distance between bottoms of body & foot (LaTeX)
\textwidth=6in				% width of body (LaTeX)
\hsize=6in				% " (TeX)
\parskip=\medskipamount			% space between paragraphs (LaTeX)
\lineskip=0pt				% minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em	% space above equation (either)
\belowdisplayskip=1em plus.3em minus.5em	% " below
\abovedisplayshortskip=.5em plus.2em minus.4em	% " above when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em	% " below
\def\baselinestretch{1.2}	% magnification for line spacing (LaTeX)
\thicklines			    % thick straight lines for pictures (LaTeX)

% Title page
\thispagestyle{empty}		    % no heading or foot on title page (LaTeX)
\def\oldheadpic{				% old UM heading
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(349,16)
	\put(325,16){\line(1,0){4}}\put(330,16){\line(1,0){4}}
 \put(340,16){\line(1,0){4}}\put(335,0){\line(1,0){4}}
	\put(340,0){\line(1,0){4}}\put(345,0){\line(1,0){4}}
	\put(329,0){\line(0,1){16}}\put(330,0){\line(0,1){16}}
	\put(339,0){\line(0,1){16}}\put(340,0){\line(0,1){16}}
	\put(344,0){\line(0,1){16}}\put(345,0){\line(0,1){16}}
	\put(329,16){\oval(8,32)[bl]}\put(330,16){\oval(8,32)[br]}
	\put(339,0){\oval(8,32)[tl]}\put(345,0){\oval(8,32)[tr]}
	\end{picture}
	\par
	\thicklines
	\vskip.2in}
\def\border{						% border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics University of
Maryland Elementary Particle Physics}
	\put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
	\end{picture}
	\par\vskip-8mm}
\def\bordero{						% alternate border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
	\put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhdhebaaahjhhdahbahgdedgehgfdiehhgdigicba}
	\put(-20,-241.5){\ooo ababaighefdbfghgeahgdfgafagihdidihiidhiagfedhadbfdecdcdfagdcbhaddhbgfchbgfdacfediacbabab}
	\end{picture}
	\par\vskip-8mm}
\def\headpic{						% UM heading
	\indent
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
	\put(170,16){\line(1,0){4}}
	\put(180,16){\line(1,0){4}}
	\put(175,0){\line(1,0){4}}
	\put(180,0){\line(1,0){4}}
	\put(185,0){\line(1,0){4}}
	\put(169,0){\line(0,1){16}}
	\put(170,0){\line(0,1){16}}
	\put(179,0){\line(0,1){16}}
	\put(180,0){\line(0,1){16}}
	\put(184,0){\line(0,1){16}}
	\put(185,0){\line(0,1){16}}
	\put(169,16){\oval(8,32)[bl]}
	\put(170,16){\oval(8,32)[br]}
	\put(179,0){\oval(8,32)[tl]}
	\put(185,0){\oval(8,32)[tr]}
	\end{picture}
	\par\vskip-6.5mm
	\thicklines}

\def\endtitle{\end{quotation}\newpage}			% end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} \nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\caldsl{{\calD\!\!\! /}}    
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}{\raise -0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}{\raise -0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\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\hep#1{hep-th/{#1}}
\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
\def\cdm{{\Sc D}_{--}} \def\cdp{{\Sc D}_{++}}
\def\Pisl{{\Pi\!\!\!\! /}}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}
%\def\fracmm#1#2{{{#1}\over{#2}}}
%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
	\hbox{$\scriptstyle{#2}$}}}}}			% fraction
\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) } 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}
\def\footnotew#1{\footnote{{\baselineskip 1pt\hsize=7.0in {#1}}}} 
\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
\def\ip{{=\!\!\! \mid}}
%\def\Dot#1{\buildrel{_{_{\hskip 0.01in}\bullet}}\over{#1}}
%\def\Dot#1{\buildrel{_{_{\hskip 0.01in}{_\bullet}}}\over{#1}}
\def\Dot#1{\buildrel{\hskip2.5pt_{\scriptscriptstyle\bullet}}\over{#1}}
\def\dt#1{\Dot{#1}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}} \def\Tr{\,\,{\rm Tr}}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} \def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%% Additional Fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
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\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
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            \textfont1\tenmib \scriptfont1\sevenmib
                               \scriptscriptfont1\fivemib
            \textfont2\tenbsy \scriptfont2\sevenbsy
                               \scriptscriptfont2\fivebsy}
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\def\rlx{\relax\leavevmode}                  
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\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
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\font\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
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\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
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\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
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            \textfont1\tenmib \scriptfont1\sevenmib
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            \textfont2\tenbsy \scriptfont2\sevenbsy
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{\bf Preliminary Version (FOR YOUR EYES ONLY!) \hfill \today} \vskip -0.04in  
}
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{\hbox to\hsize{
%\today
January 1998
\hfill UMDEPP 98--078}}
{\hbox to\hsize{
\doit1{\hfill{hep-th/9802009}} 
}
\vskip -10pt
{\hbox to\hsize{
%\hfill(Revised Version)
}} 

\par 

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\begin{center}
\vglue 0.15in

\baselineskip 18pt 
 
{\large\bf Alternative $\,$Formulation $\,$for} \\  
{\large\bf Duality-Symmetric $\,$Eleven-Dimensional $\,$Supergravity}\\
{\large\bf Coupled $\,$to$\,$ Super $\,$M-5-Brane}$\,$\footnote{This
work is supported in part by NSF grant \# PHY-93-41926.} \\  

\baselineskip 10pt 

\vskip 0.5in

Hitoshi~ N{\small ISHINO}

\vskip 0.08in

{\it Department of Physics} \\[.015in]
{\it University of Maryland} \\[.015in]
{\it College Park, MD 20742-4111, USA} \\[.020in]   
{E-Mail: nishino@nscpmail.physics.umd.edu}

\vskip 2.3in

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

\baselineskip 16pt 

\begin{quotation}

~~~We present an alternative formulation of duality-symmetric 
eleven-dimensional
supergravity with both three-form and six-form gauge fields.  Instead of 
the recently-proposed scalar auxiliary field, we use a simpler lagrangian 
with a non-propagating auxiliary multiplier tensor field with 
eight-indices.  We also complete the superspace formulation in a 
duality-symmetric manner.  An alternative super M-5-brane action 
coupled to this eleven-dimensional background is also presented.  
This formulation bypasses the 
usual obstruction for an invariant lagrangian for a self-dual three-form 
field strength, by  allowing the self-duality only as a solution for field
equations, but not as a necessary condition.  



  

\endtitle

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\vskip 0.1in                                                       
\centerline{\bf 1.~~Introduction}          
         
Eleven-dimensional (11D) supergravity \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}.} 
has been known for a long time to have the field content $~(e\du\m m, 
\psi_\m, A_{\m\n\r})$.  In particular the significance of the 
three-form gauge field 
$~A_{\m\n\r}$~ with its four-form field strength was elucidated, 
when the supermembrane formulation \ref\bst{E.~Bergshoeff, 
E.~Sezgin and P.K.~Townsend, \pl{189}{87}{75}; \ap{185}{88}{330}.} was
established with the consistent couplings to 11D supergravity.  As the
general construction of $~p\-$brane reveals \ref\pbrane{A.~Achucarro,
J.~Evans, P.~Townsend and D.~Wiltshire, \pl{198}{87}{441}.}, 
there may well be an alternative formulation in 
11D that has the Hodge dual seven-form field strength 
$~F_{\m_1\cdots\m_7}$, instead of the four-form field strength 
$~F_{\m\n\r\s}$.  Despite of considerable  
efforts to formulate such 11D supergravity theory using only the seven-form 
field strength, the analysis in \ref\ntvn{H.~Nicolai, P.K.~Townsend 
and P.~van Nieuwenhuizen, Lett.~Nuovo.~Cim.~{\bf 30} (1981) 315; 
R.~D'Auria and P.~Fr\'e, \np{201}{82}{101}.} indicated that there is 
no such a formulation possible in 11D.  This obstruction is also 
reflected in the fact that the lagrangian in \cjs\ has a 
Chern-Simons term containing not only the field strength $~F_{\m\n\r\s}$~ 
but also the gauge field $~A_{\m\n\r}$~ itself, preventing any duality
transformations \ref\nt{H.~Nicolai and P.K.~Townsend, \pl{98}{81}{257}.} 
into the dual field strength $~F_{\m_1\cdots\m_7}$.  However, recent 
development in
M-theory physics \ref\mtheory{{\it For reviews}, M.~Duff, 
\ijmp{11}{96}{5623};
J.H.~Schwarz, Lectures given at the ICTP Spring School (Mar.~1996), 
and TASI
Summer School (June, 1996).}\ref\witten{E.~Witten, {\it Five-Brane 
Effective
Action in M-Theory}, Jour	.~of Geom.~Phys.~{\bf 22} (1997)
103.}\ref\mfive{R.~G\"uven, \pl{276}{92}{49}; O.~Aharony, 
\np{476}{96}{47};
E.~Bergshoeff, M.~de Roo and T.~Ort{\'i}n, \pl{386}{96}{85}; P.~Pasti,
D.~Sorokin and M.~Tonin, \pl{398}{97}{41}; M.~Aganagic, J.~Park, 
C.~Popescu and
J.H.~Schwarz, \np{496}{97}{191}; D.~Sorokin and P.K.~Townsend, 
\pl{412}{97}{265}.}\ref\mfivekappa{I.~Bandos,
K.~Lechner, A.~Nurmagambetov, P.~Pasti, D.~Sorokin and M.~Tonin,
\prl{78}{97}{4332}.}\ref\pss{M.~Perry and J.H.~Schwarz, \np{489}{97}{47};
J.H.~Schwarz, \np{395}{97}{191}; S.P.~de Alwis, {\it Coupling of Branes and
Normalization of Effective Actions in String/M-Theory},
\hep{9705139}.}\ref\nilsson{M.~Cederwall, B.E.W.~Nilsson, and P.~Sundell, 
{\it An Action for the Super-5-Brane in D=11 Supergravity}, 
\hep{9712059}.}\ref\hs{P.S.~Howe and E.~Sezgin, \pl{394}{97}{62}; 
P.S.~Howe,
E.~Sezgin and P.C.~West, \pl{399}{97}{49};   T.~Adawi, M.~Cederwall, 
U.~Gran, M.~Holm and B.E.W.~Nilsson, 
{\it Superembeddings, Non-Linear Supersymmetry and 5-branes},
\hep{9711203}.} suggests a slightly different formulation of such a theory as
the duality-symmetric limit using both the four- and seven-form field 
strengths
at the same time, known as super M-5-brane coupled to electric and magnetic
charges.  Such a formulation should maintain the manifest duality-symmetry
between the four and seven-form field strengths.    

Recently a component formulation for such eleven-dimensional
supergravity theory has been proposed \ref\bbs{I.~Bandos, N.~Berkovitz 
and D.~Sorokin, {\it Duality-Symmetric Eleven-Dimensional Supergravity and 
its Coupling to M-Branes}, \hep{9711055}.}, in which both
the six-form and a three-form gauge fields are present in a
duality-symmetric way.  This formulation has a constraint lagrangian with 
an auxiliary scalar field $~a(x)$,  that yields the duality relation 
between
$~F_{\m\n\r\s}$~ and $~F_{\m_1\cdots\m_7}$.  This mechanism is based on the
constraint lagrangian in \ref\pst{P.~Pasti, D.~Sorokin and M.~Tonin,
\pr{52}{95}{4277}; \ibid{D55}{97}{6292}.} using a gradient of the scalar
auxiliary field forming an unit vector $~v_\m$.    
Even though this formulation uses a single scalar auxiliary field
$~a(x)$~ \bbs, there is much complication for the invariance 
confirmation of the total action due to the non-polynomial and derivative
structure of $~v_\m\equiv (\partial_\m a)\big[\,\big(\partial_\r a\big)^2
\,\big]^{-1/2}$.  This complication is also reflected in the 
`field-dependent'
supersymmetry  algebra $~\{ Q_\a, Q_\b \} = \big(\g^m\big)_{\a\b} \big[\,
P_m -
(\partial_m a) {\cal G} \,\big]$.\footnotew{We point out that this algebra has
a resemblance to the recent formulation in higher-dimensional
supergravity/supersymmetry, F- or S-theories using null-vectors 
\ref\ns{H.~Nishino and E.~Sezgin, 
\pl{388}{96}{569}; H.~Nishino, {\it Supergravity in 10+2 Dimensions as 
Consistent Background for Superstring}, {\it etc.}, \hep{9703214};
\hep{9706148}; \hep{9708064}.}, multi-locality
\ref\stheory{I.~Bars, {\it Algebraic Structure of S-Theory}, 
Talks at 2nd Sakharov Conf.~and Strings '96, \hep{9608061}.}, 
or a gradient of a scalar field \ref\lag{H.~Nishino, {\it Lagrangians and 
Covariant Field Equations for Supersymmetric Yang-Mills Theory in 12D},
\hep{9710141}, to appear in Phys.~Lett.~B.}.} 

In the present paper, we propose an alternative formulation of 
duality-symmetric 
supergravity theory in 11D, that has a much simpler constraint lagrangian,
based on the general technique in 
ref.~\ref\siegelconstr{W.~Siegel, \np{238}{84}{307}.}.  
Our lagrangian has a tensor auxiliary field with eight indices, which
are not totally anti-symmetric.  We will show the simplicity of 
the confirmation of invariance of the total action in our formulation.  
We also re-formulate our system in superspace,
that gives the confirmation of the validity of our theory.  Additionally, 
we present a new but simple super M-5-brane action coupled to our
duality-symmetric 11D supergravity backgrounds.  Our action bypasses the
problem with an invariant lagrangian for a self-dual three-form field
strength \witten, using certain constraint lagrangians.      




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\centerline{\bf 2.~~11D Lagrangian in Component Formulation}

Since our result is simple, it is better for us to give it first, 
leaving the
associated remarks later.  Our field content $~(e\du\m m, \psi_\m, 
A_{\m\n\r};
B_{\m_1\cdots\m_6}, \L^{m_1\cdots m_4n_1\cdots n_4})$~ is almost the 
same as that of Cremmer {\it et al.}~\cjs, except
that we have an additional six-form gauge field
$~B_{\m_1\cdots\m_6}~$\footnotew{We use the symbol $~B$~ instead
of $~A~$ for the six-form gauge field in this paper to distinguish it 
from
the three-form gauge field.} and a tensor auxiliary field
$~\L^{m_1\cdots m_4n_1\cdots n_4}$.  Our total action $~I$~ is simply a 
sum of
three actions $~I_0, ~I_1$~ and $~I_2$, where $~I_0$~ has the original
Cremmer-Julia-Scherk lagrangian \cjs\footnotew{In this section of component
formulation, we use the notation \cjs\ $~(\eta\low{m
n}) = \hbox{diag.}~(+, -, \cdots, -), ~\e^{\,012\cdots 9\, 10} = + 1$.
We use $~{\scst m,~n,~\cdots~=~(0),~(1),~\cdots, ~(10)}$~ for
local Lorentz indices, while $~{\scst \m,~\n,~\cdots~=~0,~1,~\cdots, 
~10}$~ for
curved indices.}, and $~I_1$~ and $~I_2$~ are our deliberately chosen new
actions:   
$$ \li{& I\equiv I_0 + I_1 + I_2~~, ~~~~I_0 \equiv \int d^{11}
     x \,  \Lag_0~~, ~~~~ I_1 \equiv \int d^{11} x \, \Lag_1~~,~~~~ 
     I_2 \equiv \int d^{11} x \, \Lag_2{~~,~~~~~ ~~~~~}   
&(2.1) \cr  
& \Lag_0 \equiv - \frac14 e R(\omega) 
     - i\fracm12 e \Bar\psi_\m \g^{\m\n\r}
     D_\n\big(\frac{\omega+\Hat\omega}2\big) \psi_\r
     -\frac1{48} e F_{\[4\]} F^{\[4\]} 
     + \frac2{(144)^2} \e^{\[3\]\[4\]\[4\]'} A_{\[3\]} F_{\[4\]}
     F_{\[4\]'} \cr       
&~~~~~ ~~ + \frac1{192} e \big[\,\big( \Bar\psi_\m \g^{\m\n\r\s\t\o}
      \psi_\n \big) + 12\big(\Bar\psi{}^\r\g^{\s\t}\psi^\o\big)\, \big] 
     \big(F_{\r\s\t\o} + \Hat F_{\r\s\t\o} \big)  ~~, 
&(2.2) \cr 
& \Lag_1 \equiv +\frac12 e \L^{m_1\cdots m_4 n_1\cdots n_4} 
     \Hat{\cal F}_{m_1\cdots m_4} \Hat{\cal F}_{n_1\cdots n_4} ~~,    
&(2.3) \cr 
& \Lag_2\equiv + \frac12 e \b \Hat{\cal F}_{\[4\]} ^2 
&(2.4\rma) \cr 
& ~~~\,  = + \frac 12 \b e \Hat F_{\[4\]}^2 + \frac1{420} \b e \Hat 
      G_{\[7\]}^2
      - \frac1{7!} \b \e^{\[4\]\[7\]} \Hat F_{\[4\]} \Hat G_{\[7\]}  ~~. 
&(2.4\rmb)  \cr } $$  
The symbol $~{\scst \[n\]}$~ in general denotes the normalized 
anti-symmetric indices, {\it e.g.,} $~F_{\[4\]} F^{\[4\]} \equiv
F_{\m_1\cdots\m_4} F^{\m_1\cdots\m_4}$, in order to save
space.\footnotew{This {\it normalized} $~{\scst\[n\]}$-symbol is
common to all the sections.  In this paper we avoid the usage of 
differential
forms due to drawbacks, when confirming supersymmetric invariance of
actions.}  As usual, other relevant
quantities are such as   $$\Hat F_{\m\n\r\s} \equiv 4\partial_{\[\m}
A_{\n\r\s\]} 
     - 3 \big(\Bar\psi_{\[\m} \g_{\n\r} \psi_{\s\]}\big)~~, ~~~~
     \Hat\omega_{\m r
     s} \equiv \omega_{\m r s} + \fracm i 4 \big(\Bar\psi_\r
    \g\du{\m r s}{\r\s} \psi_\s \big)  ~~,  
\eqno(2.5) $$
with the Lorentz connection $~\omega_{\m r s}$~ containing 
$~\psi\-$torsion with $~\g^{\[5\]}$~ as well as the 
$~\g^{\[1\]}\-$matrices 
\cjs.  All the {\it hatted} fields are supercovariantized 
in component formulation \ref\peter{P.~van
Nieuwenhuizen, \prep{68}{81}{189}.}.   The second action $~I_1$~ is our 
deliberately chosen constraint action with $~\L^{\[4\]\[4\]'}$~ as a
lagrange multiplier.  The $~\L^{\[4\]\[4\]'}$~ is a non-propagating 
multiplier field,  which is {\it not} totally antisymmetric in all 
the eight indices, but
instead with the (anti)symmetry
$$ \L^{m_1m_2 m_3 m_4 n_1 n_2 n_3 n_4} = +  \L^{n_1 n_2 n_3 n_4
     m_1 m_2 m_3 m_4} 
     = - \L^{m_2 m_1 m_3 m_4 n_1 n_2 n_3 n_4} ~~,~~~~ \hbox{\it etc.}  
\eqno(2.6) $$ 
Note also that all the indices in (2.3) are chosen to be local
Lorentz indices, for a technical reason to be mentioned 
later.\footnotew{Whenever the distinction between the
local Lorentz and curved indices are crucial, we avoid the usage of the
symbol $~{\scst \[ n\]}$.}  These features will be 
important, when we confirm the invariance under supersymmetry.  
The $~\Hat{\cal F}_{\[4\]}$~ is defined by 
$$ \li{& \Hat{\cal F}_{m_1\cdots m_4} \equiv 
          \Hat F_{m_1\cdots m_4} - \frac1{7!} 
          \e\low{m_1\cdots m_4}{}^{n_1\cdots n_7} 
          \Hat G_{n_1\cdots n_7} ~~, 
&(2.7) \cr
& G_{\m_1\cdots\m_7} \equiv 7\partial_{\[\m_1} B_{\m_2\cdots\m_7\]} 
     - 35 F_{\[\m_1\cdots\m_4} A_{\m_5\m_6\m_7\]} ~~,  
&(2.8) \cr
&\Hat G_{\m_1\cdots\m_7} \equiv G_{\m_1\cdots\m_7} - i\frac{21}2 
     \big(\Bar\psi_{\[\m_1} \g_{\m_2\cdots\m_6}\psi_{\m_7\]} \big) ~~. 
&(2.9) \cr } $$
Here the field strength $~G_{\[7\]}$~ contains the Chern-Simons form as 
expected
from the consistency in superspace formulation \ref\candiello{A.~Candiello and
K.~Lechner, \np{412}{94}{479}.}.  The lagrangian $~\Lag_1$~ is similar to
those constraint lagrangians in \siegelconstr.  The $~\b$~ in 
$~\Lag_2$~ is an arbitrary real constant subject to conditions
(2.21).  Eq.~(2.4b) for $~\Lag_2$~ is to make 
the $~G_{\[7\]}^2\-$kinetic term explicit.  
Note that the last $~\e\Hat F\Hat G\-$term in (2.4b) is a total divergence 
at the lowest order, so it should be regarded as a trilinear term.  
It is helpful to
remember that $~\Lag_2$~ can be easily obtained by the simple field
redefinition of $~\L\du{\[4\]}{\[4\]'}$~ by a product of
Kronecker's delta (Cf.~(2.15) below).         

Our supersymmetry transformation rule is 
$$ \li{&\d_Qe\du\m m = - i \big(\Bar\e\g^m \psi_\m \big) ~~, 
&(2.10\rma) \cr 
& \d_Q\psi_\m = +D_\m(\Hat\o) \e + i \frac1{144} \big( \g\du\m{\[4\]}
       \Hat F_{\[4\]} - 8 \g^{\[3\]} \Hat F_{\m\[3\]} \big) \e \cr  
& ~~~~~ ~~~~~ - i \frac 1 6 \big( \g_{\m\[4\]} \e \, \L^{\[4\]\[4\]'}
      - 8 \g^{\[3\]}\e\, \L\low{\m\[3\]}{}^{\[4\]'} \big) 
      \Hat{\cal F}_{\[4\]'} \cr 
& ~~~~~ ~~~~~ - i \frac 1 6 \b \big( \g\low\m{}^{\[4\]} \e \, 
       \Hat{\cal F}_{\[4\]} 
       - 8 \g^{\[3\]}\e\, \Hat{\cal F}_{\m\[3\]} \big) 
       \equiv \Hat{\cal D}_\m\e ~~,   
&(2.10\rmb) \cr   
& \d_Q A_{\m\n\r} = + \frac3 2\big(\Bar\e\g_{\[\m\n}\psi_{\r\]} \big) ~~,   
&(2.10\rmc) \cr 
&\d_Q  B_{\m_1\cdots\m_6} = + 3i \big(\Bar\e\g_{\[\m_1\cdots\m_5}
     \psi_{\m_6\]} \big) -20 A_{\[\m_1\m_2\m_3} 
     \big(\d_Q A_{\m_4\m_5\m_6\]}\big) ~~,  
&(2.10\rmd) \cr 
& \d_Q \L^{m_1\cdots m_4 n_1\cdots n_4} = \Big[\, + i\frac32 
       \big(\Bar\e\g^\m\psi_\m\big) 
       \L^{m_1\cdots m_4 n_1\cdots n_4} 
        + 4i \big(\Bar\e\g_r\psi^{\[m_1}\big)
       \L^{m_2 m_3 m_4\] r n_1\cdots n_4}  \cr 
& ~~ -24i \big(\Bar\e\g^{\r\[2\]} \psi^\s \big) 
       \L_{\r\s\[2\]'}{}^{m_1\cdots m_4} \L\du{\[2\]}{\[2\]'n_1\cdots n_4} 
       -96i \big(\Bar\e\g_\r \psi_\s \big) 
       \L_{\r\[3\]}{}^{m_1\cdots m_4} \L\du\s{\[3\] n_1\cdots n_4} \cr 
& ~~ +i\frac 16 \big(\Bar\e\g^{\[4\]\[4\]'\n} 
       \psi_\n \big) \L_{\[4\]}{}^{m_1\cdots m_4} 
       \L_{\[4\]'}{}^{n_1\cdots n_4} - 24i \big(\Bar\e\g^{\r\[2\]\[2\]'} 
       \psi_\r \big) \L\du{\[2\]}{\[2\]''m_1\cdots m_4} \L_{\[2\]'
        \[2\]''}{}^{n_1\cdots n_4} \cr 
& ~~ + 32i \big(\Bar\e\g\du{\[2\]}{\[3\]} 
       \psi_\r \big) \L^{\r\s\[2\] m_1\cdots m_4} \L_{\s\[3\]}{}
       ^{n_1\cdots n_4} 
       + 12i \big(\Bar\e\g^\m \psi_\m\big) 
       \L\du{\[4\]}{m_1\cdots m_4} \L^{\[4\] n_1\cdots n_4} 
       \, \Big] \Big|_{\L \rightarrow \L + \d\cdots\d} \cr 
& ~~ + {\scst(m_i \leftrightarrow  n_i)} {~~.~~~~~ ~~~~~}  
&(2.10\rme) \cr } $$ 
The last manipulation $~{\scst(m_i \leftrightarrow  n_i)}$~ is needed to 
make the r.h.s.~to have the same symmetry as $~\L^{\[4\]\[4\]'}$, while 
the operation $~\big|_{\L \rightarrow \L + \d\cdots\d}$~ is for
the replacement $~\L\du{m_1\cdots m_4}{n_1\cdots n_4}
\rightarrow\L\du{m_1\cdots m_4}{n_1\cdots n_4}+ \b\d\du{\[m_1}{\[n_1}
\cdots\d\du{m_4\]}{n_4\]}$~ of all the $~\L\du{\[4\]}{\[4\]'}$~ in the 
square bracket.  The significance of this operation  will be clarified 
shortly.  The rules (2.10c) and (2.10d) justify the coefficients of 
$~\psi\-$dependent terms
in the supercovariant field strengths given above.  Note also that all the
indices in (2.10e) are local Lorentz indices, but {\it not} curved indices,
because the difference will yield gravitino-linear terms out of elfbein
variations.    

Before confirming the invariance of our action under supersymmetry, 
we first consider the field equations of all the fields.  
The $~\L^{\[4\]\[4\]'}\-$field equation immediately gives 
the duality constraint 
$$\Hat{\cal F}_{m_1\cdots m_4} \equiv \Hat F_{m_1\cdots m_4} 
        - \frac1{7!} \e\low{m_1\cdots m_4}{}^{n_1\cdots n_7} 
        \Hat G_{n_1\cdots n_7} = 0 ~~, 
\eqno(2.11) $$
namely the duality between $~\Hat F_{\[4\]}$~ and $~\Hat G_{\[7\]}$~ 
like \bbs.  This is because the indices $~{\scst \[4\]}$~ and $~{\scst
\[4\]'}$~ in the original $~\L\-$field equation $~\Hat{\cal F}_{\[4\]}
\Hat{\cal F}_{\[4\]'}= 0 $~ are free independent indices,
implying that $~\Hat{\cal F}_{\[4\]}=0$.    
Once eq.~(2.11) is satisfied, we immediately see that the contribution of
both $~I_1$~ and $~I_2$~ to the $~A_{\[3\]}\-$field equation vanishes, 
because it contains one factor of $~\Hat{\cal
F}_{\[4\]}=0$~ by (2.11): 
$$ \Hat D_\m \Hat F\ud\m{\r\s\t} - \frac1{576} e^{-1}
        \e\du{\r\s\t}{\[4\]\[4\]'} \Hat F_{\[4\]} \Hat F_{\[4\]'} = 0 ~~. 
\eqno(2.12) $$  
The same is also true for the gravitino field equation, even though $~\psi_\m$~ 
is involved in $~\Hat{\cal F}_{\[4\]}$:  
$$  i \g^{\m\n\r} \Hat{\cal R}_{\n\r} = 0 ~~,  
\eqno(2.13) $$   
where $~\Hat{\cal R}_{\m\n}\equiv \Hat{\cal D}_\m\psi_\n 
- \Hat{\cal D}_\n\psi_\m$~ is the 
supercovariant field strength of the gravitino in our notation.  Here  
the meaning of {\it hat} is the same one in the r.h.s.~in (2.10b).  
An important feature here is that this gravitino field equation has not 
only the contribution from $~\Lag_0$, but also that from $~\Lag_1$~ 
and $~\Lag_2$~
through the supercovariantized $~\Hat G_{\[7\]}$, proportional to $~\L$~ 
and
$~\Hat{\cal F}$.  This can be confirmed explicitly by taking the variation of
$~\Hat{\cal F}_{\[7\]}$~ with respect to the gravitino.  The
$~B_{\[6\]}\-$field equation is automatically satisfied by (2.11), 
due to the
presence of $~B_{\[6\]}$~ only in $~\Hat{\cal F}_{\[4\]}$.  Now we see that
the elfbein field equation is not affected, because  of the bilinear structure of
$~\Lag_1$~ in  $~\Hat{\cal F}$, as other field equations:  
$$ \Hat R_{\m\n} - \frac13 \big(\Hat F_{\m\[3\]} \Hat
F\du\n{\[3\]} 
     - \frac1{24} g\low{\m\n} \Hat F_{\[4\]}^2 \big) = 0  ~~. 
\eqno(2.14) $$ 
We therefore conclude that the only new effect of $~I_1$~ and $~I_2$~ on 
field equations is simply the duality equation (2.11), 
and none of the field 
equations of the original fields $~e\du\m m,~\psi_\m,~A_{\[3\]}$~ 
are affected.  Moreover, the $~\L\-$field is completely decoupled from any 
field equations in our system.  An important consequence of 
the duality condition (2.11) is that the 
$~B_{\[6\]}\-$field has got dynamical freedom, because of its 
non-vanishing 
divergence due to this duality.  However, note also that 
the superficial kinetic term for $~B_{\[6\]}$~ in $~\Lag_2$~ does 
{\it not}
contribute to the $~B_{\[6\]}\-$field equation, because
there is always the on-shell vanishing factor accompanying the variation.      
Eventually the degrees of freedom for all of these antisymmetric tensor
field stay the same as $~(84 + 84)/2 = 84$.  This feature of degrees of 
freedom is the same as in \bbs.  

We now confirm the invariance of our total action $~I$~ under 
supersymmetry 
dictated by (2.10).  As a universal notation, we distinguish the 
Cremmer {\it et
al.}'s supersymmetry transformation rule $~\d_Q^{(0)}$~ \cjs\ from 
our modified 
terms $~\d_Q^{(1)}$~ in the transformation rule in (2.10) which is
$~\L\-$dependent or $~\b\-$dependent: $~\d_Q =
\d_Q^{(0)} + \d_Q^{(1)}$, {\it except} (2.10e), for which $~\d_Q
\L^{\[4\]\[4\]'} = \d_Q^{(0)}\L^{\[4\]\[4\]'}$, just for convenience of
manipulations.  In other words, $~\d_Q^{(1)}$~ denotes all 
the $~\L\-$dependent and $~\b\-$dependent terms in (2.10).  
We next introduce a technique that drastically simplifies the whole
computation.  Note that $~\Lag_2$~ can be simply obtained from $~\Lag_1$~ 
by the field  redefinition 
$$~\L\du{m_1\cdots m_4}{n_1\cdots n_4}
~\rightarrow~\L\du{m_1\cdots m_4}{n_1\cdots n_4}+ \b\d\du{\[m_1}{\[n_1} 
\cdots \d\du{m_4\]}{n_4\]}~~.       
\eqno(2.15) $$
This implies that to perform the above invariance check, we do not have to
consider $~\Lag_2$~ as an independent lagrangian, but once the invariance 
of  $~I_0+I_1$~ is confirmed for the transformation rule (2.10) with 
$~\b=0$, then we can extrapolate this result to the general case 
$~\b\neq0$, just by the field redefinition (2.15).  

Recalling now $~\d_Q^{(0)}\Lag_0=0$~ \cjs, we can understand the 
invariance of our action under supersymmetry when $~\b=0$~ as   
$$\li{0 
&\!\eqques \d_Q \big( \Lag_0 + \Lag_1 \big) 
         = \big(\d_Q^{(0)} + \d_Q^{(1)}\big) \big(\Lag_0 + \Lag_1 \big) 
         \cr 
& = \d_Q^{(0)} \Lag_0 + \d_Q^{(1)} \big( \Lag_0 + \Lag_1  \big) 
        + \d_Q^{(0)} \Lag_1 \cr 
& = e \big(\d_Q^{(1)}\Bar\psi_\m \big) \Hat{\cal S}^\m + \d_Q^{(0)} 
      \Lag_1 \cr 
& = + i \frac1 6 e \Big(\Bar\e\g_\m{}^{\[4\]} \L\du{\[4\]}{\[4\]'} 
     \Hat{\cal F}_{\[4\]'}    
     + 8 \Bar\e\g^{(3)} \L_{\m\[3\]}{}^{\[4\]} \Hat{\cal F}_{\[4\]} \Big) 
     \big( \Hat{\cal S}^{\m (0)}  + \Hat{\cal S}^{\m(1)} \big)  \cr
& ~~~\, + \frac12 \big[\,\d_Q^{(0)} \big(e \L^{m_1\cdots m_4 n_1\cdots
        n_4}\big)\,\big]
        \Hat{\cal F}_{m_1\cdots m_4} \Hat{\cal F}_{n_1\cdots n_4}  
        + e \L^{m_1\cdots m_4n_1\cdots n_4} 
        \big(\d_Q^{(0)} \Hat{\cal F}_{m_1\cdots m_4} \big) 
        \Hat{\cal F}_{n_1\cdots n_4} {~~.~~~~~ ~~~~~} 
&(2.16) \cr } $$   
Here $~\Hat{\cal S}^\m$~ is for the l.h.s.~of the gravitino field 
equation, while $~\Hat{\cal S}^\m{}^{(1)}$~ denotes the terms in 
$~\Hat{\cal S}^\m$~ coming only from $~I_1$, 
as is easily computed:  
$$\li{&\Hat{\cal S}^\m \equiv e^{-1} \fracm\d{\d\psi_\m} \big(\Lag_0 +
     \Lag_1\big) = - i\frac 1 2 \g^{\m\n\r}\Hat{\cal R}_{\n\r} \equiv 
     \Hat{\cal S}^{\m(0)} + \Hat{\cal S}^\m{}^{(1)} ~~, 
&(2.17) \cr     
&\Hat{\cal S}^\m{}^{(1)} \equiv e^{-1} \fracm\d{\d\psi_\m} \Lag_1 
     = - 6 \g_{\[2\]} \psi_\n 
     \L\du\m{\n\[2\]\[4\]}\Hat{\cal F}_{\[4\]} 
     - \frac12 \g^{\m\n\[4\]} \psi_\n \L\du{\[4\]}{\[4\]'} 
     \Hat{\cal F}_{\[4\]'} ~~. 
&(2.18) \cr } $$
We now easily see that all the terms linear in $~\Hat{\cal S}^{\m(0)}$~ 
cancel themselves by the aid of the identities such as 
$$ \li{&\d_Q^{(0)} \Hat{\cal F}_{m n r s} 
     = -i \frac 16 \big(\Bar\e\g\du{m n r s}t
     \Hat{\cal S}^{\,(0)}_t \big) + i \frac4 3 \big(\Bar\e\g_{\[m n r} 
     \Hat{\cal S}_{s\]}^{\,(0)}\big) \cr 
& ~~~~~ ~~~~~ ~~~~~  - i \big(\Bar\e \g^\m\psi_\m\big) 
     \Hat{\cal F}_{m n r s} + 4i \big(\Bar\e \g_{\[m|} \psi^t \big) 
     \Hat{\cal F}_{t|n r s\]} ~~,  
&(2.19\rma)  \cr
& \d_Q^{(0)} \Hat F_{m n r s} = + 3 \big( \Bar\e\g_{\[m n} 
     \Hat{\cal R}_{r s\]}^{(0)} \big) ~~, 
&(2.19\rmb) \cr 
&\d_Q^{(0)} \Hat G_{m_1\cdots m_7} = - i \frac{21}2 
     \big(\Bar\e\g_{\[ m_1\cdots m_5}\Hat{\cal R}_{m_6 m_7\]}^{(0)} \big) 
     + 7i \big( \Bar\e\g_n\psi_{\[m_1} \big) 
     \Hat{\cal G}_{m_2\cdots m_7\]}{}^n  {~~, ~~~~~ ~~~~~ }   
&(2.19\rmc) \cr  
& \Hat{\cal G}_{m_1\cdots m_7} \equiv +\frac1{4!} \e\low{m_1\cdots m_7}
     {}^{n_1\cdots n_4} \Hat{\cal F}_{n_1\cdots n_4}  ~~. 
&(2.19\rmd) \cr } $$       
We next see that the remaining $~\L\-$dependent terms 
both of the types $~\approx\psi\L\Hat{\cal F}$~ and $~\psi\L^2
\Hat{\cal F}^2$~
cancel themselves, and therefore we establish the supersymmetric 
invariance: $~\d_Q\big(I_0 + I_1 \big)=0$, when $~\b=0$.  
As has been mentioned, since the case of $~\b\neq 0$~ with $~I_2$~  
can be re-obtained by the simple field redefinition (2.15), the total
action $~I_0+ I_1+I_2$~ is also invariant under the transformation rule
(2.10) now with $~\b\neq0$.           
 
The advantage of the algorithm in (2.15) for the invariance check is
that we have to take only the variation 
$~\d_Q^{(0)}$~ by Cremmer {\it et al.}~\cjs.  This considerably  
simplifies the computation, enabling us to fix the transformation rule 
for $~\L^{\[4\]\[4\]'}$.  The presence of the $~\psi\-$linear term in 
(2.19a,c) 
seems to be from the fact that the original Cremmer {\it et al.}'s 
transformation rule $~\d_Q^{(0)}\psi_\m$~ in (2.10b) is not 
duality-symmetric. 

We next consider the on-shell closure of the gauge algebra.   
The only difference of our transformation rule from 
that in \cjs\ is the presence of $~\L\Hat{\cal F}$~ or 
$~\b\Hat{\cal F}\-$terms
in (2.10b), $~\L\-$terms in (2.10e), and (2.10d) itself.   This does not
pose any problem for the closure of gauge algebra, seen as follows.  
First, the
on-shell closure on $~e\du\m m$~ stays the same, because of $~\Hat{\cal
F}_{\[4\]} =0$.  The same is also true for the closure on $~A_{\[3\]}$~ 
and on $~B_{\[6\]}$.  
The least trivial one is the closure on $~\psi_\m$, which
eventually has no problem, because of the key relation 
$~\d_Q\Hat{\cal F}_{m n r
s}=  \d_Q^{(0)}\Hat{\cal F}_{m n r s}$~ equivalent to (2.19a) on-shell.  
Finally the closure on $~\L^{m_1\cdots m_4 n_1\cdots n_4}$~ 
itself has no problem, even though it looks awfully complicated, because 
this auxiliary field is decoupled from any field equations, and in fact, 
it can be gauged away by an extra symmetry mentioned next.    

The absence of $~\L^{\[4\]\[4\]'}$~ from all the field equations suggests 
that this multiplier field might be gauged away as a
non-physical field.  As a matter of fact, we can see the existence
of an extra symmetry \siegelconstr\ for $~\L^{\[4\]\[4\]'}$:    
$$ \li{&\d_\eta\L\du{\m_1\cdots\m_4}{\n_1\cdots\n_4} 
            = \Big[ + \partial_{\[\m_1}
            \eta\du{\m_2\m_3\m_4\]}{\n_1\cdots\n_4} 
            - 2 \d\du{\[\m_1|}{\[\n_1|}\partial_\r
            \eta\low{|\m_2\m_3\m_4\]}{}^{\r|\n_2\n_3\n_4\]} \,\Big] 
            + {\scst (\m_i \leftrightarrow \n_i)} {~~,~~~~~ ~~~~~ ~~~~~}  
&(2.20\rma) \cr 
& \d_\eta A_{\m\n\r} = - \fracm6{24\b-1} \, \eta\du{\m\n\r}{\[4\]} 
     \Hat{\cal F}_{\[4\]} ~~, 
~~~~ \d_\eta B_{\m_1\cdots\m_6} = - \fracm1{24\b} \, \e
     \low{\n\m_1\cdots\m_6}{}^{\[4\]} 
     \eta\low{\[3\]\[4\]} \Hat{\cal F}^{\[3\]\n}  ~~, 
&(2.20\rmb) \cr } $$ 
up to the next order terms.\footnotew{Even though this expression is
only at the lowest order, other higher-order terms can be also fixed which are
skipped in this paper.}  Here $~\eta\du{\m_1\m_2\m_3}{\n_1\cdots\n_4}$~
is an arbitrary space-time dependent parameter, anti-symmetric under
$~{\scst\[\m_1\m_2\m_3\]}$~ and under $~{\scst\[\n_1\cdots\n_4\]}$, but 
with no
other (anti)symmetries.  Since we are interested only in the lowest-order, we
are using the curved indices here.  The last operation in (2.20a) is just 
to make the r.h.s.~have the same symmetry as the l.h.s.  
As the examples in
\siegelconstr, the extra transformations for $~A_{\[3\]}$~ 
and $~B_{\[6\]}$~
vanish on-shell, and more importantly, this symmetry
can gauge away the auxiliary field $~\L^{\[4\]\[4\]'}$, when
$$~\b\neq 0~~, ~~~~\b\neq\frac1{24}~~. 
\eqno(2.21)$$  
Eventually, the only important role
played by $~\L^{\[4\]\[4\]'}$~ is to yield the  constraint (2.11) as 
a multiplier field.  The special case  $~\b=0$~ or 
$~\b={\scst (24)}^{-1}$~ can
be understood as a singular case, when the kinetic term for 
$~A_{\[3\]}$~ or $~B_{\[6\]}$~ disappears.  To see
this more explicitly, consider now the bosonic terms depending only
$~A_{\[3\]}$~ and $~B_{\[6\]}$~ in the total lagrangian  
$$\Lag_{A, B} = +\frac12 \big(\b-\frac1{24} \big) F_{\[4\]}^2 +
     \frac1{420} \b G_{\[7\]}^2 
     + \frac 1{144} \big( \b +\frac1{72} \big) \,\e^{\[3\]\[4\] \[4\]'} 
     A_{\[3\]} F_{\[4\]} F_{\[4\]'} ~~,   
\eqno(2.22) $$
up to non-essential higher-order terms.  
The singular case $~\b={\scst(24)}^{-1}$~ corresponds to the absence of
the kinetic term of $~A_{\[3\]}$, while the other
singular case $~\b=0$~ corresponds to the absence of
$~G_{\[7\]}^2\-$term.  These singular cases $~\b=0,
~\b={\scst (24)}^{-1}$~ do not accommodate the extra symmetry (2.20), and
therefore the $~\L^{\[4\]\[4\]'}\-$field can not be gauged away
as in \siegelconstr.  In other words, the presence of both the 
$~F_{\[4\]}^2$~
and $~G_{\[7\]}^2\-$terms seem to be crucial for our formulation using the
multiplier field, even though these singular cases would give the simplest
lagrangians.  




%\bigskip\bigskip\bigskip

\newpage

\centerline{\bf 3.~~Superspace Formulation}

Once the component formulation has been established, we are ready to 
consider the corresponding superspace formulation, as has been 
almost always the case with supergravity theories.    
Due to the newly introduced six-form gauge field ~$B_{\[6\]}$, we need to 
consider the three independent superspace Bianchi identities (BIs) 
$$ \li{&\nabla_{\[A} T\du{B C)} E - T\du{\[A B|}E T\du{E|C)} D - 
      \frac12 R\du{\[A B| e} f \big({\cal M}\du f e\big)\du{|C)} D 
      \equiv 0 ~~,
&(3.1\rma) \cr  
& \frac1{4!} \nabla_{\[A_1} F_{A_2\cdots A_5)} - \frac1{3!\cdot 2}
      T\du{\[A_1 A_2|} B F_{B|A_3 A_4 A_5)}  \equiv 0 ~~, 
&(3.1\rmb) \cr 
& \frac1{7!} \nabla_{\[A_1} G_{A_2\cdots A_8)} - \frac1{6!\cdot 2}
      T\du{\[A_1 A_2|} B G_{B|A_3\cdots A_8)} 
      +\frac1{(4!)^2} F_{\[A_1\cdots A_4} F_{A_5\cdots A_8)} \equiv 0 ~~. 
&(3.1\rmc) \cr } $$  
We call these Bianchi identities respectively $~(A B C, D)$, $~(A_1\cdots
A_5)$~ and $~(A_1\cdots A_8)\-$types.   In this section of superspace
(and the next section as well), we use the indices 
$~{\scst A,~B,~\cdots}$~ for
the local Lorentz indices in superspace, which can be either bosonic $~{\scst
a,~b,~\cdots}$~ or fermionic $~{\scst \a, ~\b,~\cdots}$.  
The antisymmetrization
symbol in (3.1) is different from the previous section for components, 
because
now we have {\it e.g.,} $~C_{\[A B)} \equiv C_{A B}\pm C_{B A}$~ with no 
factor of $~1/2$.  This notation is common to sections 3 and 4.  
As has been also known \bbs\candiello, the presence of the
Chern-Simons form in (3.1c) is crucial for the lowest engineering 
dimensional BI
at $~d=0$~ {\it via} (3.4) below, also consistent with (2.8).  

From the component result, we can see the relevant superspace 
constraints are  
$$ \li{&T\du{\a\b} c = + i \big(\g^c \big)_{\a\b}~~, ~~~~
     F_{\a\b c d} = + \frac12 \big(\g_{c d} \big)_{\a\b} ~~,  
&(3.2\rma) \cr 
& G_{\a\b c_1\cdots c_5} 
     = - i\frac12 \big( \g_{c_1\cdots c_5} \big)_{\a\b} ~~, 
&(3.2\rmb) \cr
&T\du{\a b}\g = + i \frac1{144} \big( \g\du b{\[4\]} F_{\[4\]} + 
     8 \g^{\[3\]}F_{b\[3\]}  \big) \du\a\g \cr
& ~~~~~ ~~~\,\, - i \frac16 \big( \g\du b{\[4\]} \L\du{\[4\]}{\[4\]'}
              + 8 \g^{\[3\]} \L\du{b\[3\]}{\[4\]'} \big)\du\a\g 
              {\cal F}_{\[4\]'} \cr 
& ~~~~~ ~~~\,\, - i \frac16 \b \big( \g\du b{\[4\]} {\cal F}_{\[4\]}
              + 8 \g^{\[3\]} {\cal F}_{b\[3\]} \big) \du\a\g ~~,   
&(3.2\rmc) \cr
& {\cal F}_{\[4\]} = F_{\[4\]} 
     - \frac1{7!} \e\low{\[4\]}{}^{\[7\]} G_{\[7\]} = 0~~. 
&(3.2\rmd) \cr } $$  
As usual, we do not put any $~{\it hat}$~ on field strengths in superspace,
due to their manifest supercovariance \ref\ggrs{S.J.~Gates Jr., 
M.T.~Grisaru, M.~Ro\v cek and W.~Siegel, {\it
Superspace}, Benjamin/Cummings, Reading, MA, 1983.}.  
As is often with superspace for higher-dimensional supergravity, all the
equations are essentially on-shell \ggrs.  Since dimension $~d=2$~ BI will 
yield the field equation $~{\cal F}_{\[4\]}=0$, as will be seen, 
the presence of the $~{\cal F}\-$terms in (3.2c) should not matter.  
However, inclusion of them
is useful to re-confirm important relationships used in component 
formulation.   
                                                                    
We now analyze these BIs at each engineering dimension.  First
of all, the BIs (3.1a) and (3.1b) are not affected, except for the 
$~{\cal F}\-$dependent terms in $~T\du{\a b}\g$~ 
which we keep as manifest, even though
they vanish on-shell.  Relevantly, the $~(\a\b c d e)\-$type BI at 
$~d=1$~ yields the field equation $~{\cal F}_{\[4\]}=0$.  
The $~(\a b c d e)\-$type BI at $~d=3/2$~ yields 
one of the important relationships:  
$$ \nabla_\a F_{b c d e} = - \frac18 \big(\g_{\[b c|} \big)_{\a\b} 
	T\du{| d e\]}\b ~~, 
\eqno(3.3) $$
which is on-shell equivalent to (2.19a) in component.  All other equations out
of  BIs in (3.1a) and (3.1b) are formally equivalent to the case of
\ref\cf{E.~Cremmer and S.~Ferrara, \pl{91}{80}{61}; L.~Brink and
P.~Howe, \pl{91}{80}{384}.}\cjs.  The only non-trivial confirmation is for
(3.1c).  At $~d=0$~ for the $~(\a\b\g\d c_1\cdots c_4)\-$type BI, the 
following crucial identity is used: 
$$ \big( \g^e \big)_{(\a\b|} \big(\g_{e a b c d} \big)_{|\g\d)}  
\equiv +\frac 18 \big(\g_{\[a b|} \big)_{(\a\b|} 
       \big( \g_{|c d\]} \big)_{\g\d)}~~,   
\eqno(3.4) $$ 
which is confirmed by another identity $~\big(\g^{a b}\big)_{(\a\b|} 
\big(\g_b\big)_{|\g\d)} \equiv 0$.  
It is this identity that requires the presence of the Chern-Simons 
form in the field strength $~G_{\[7\]}$~ \bbs\candiello.  
The next non-trivial confirmation is at $~d=1$~ for $~(\a\b c_1\cdots
c_6)\-$type BI, which consists of three structures of $~\g\-$matrices:
(i) $\g_{c_1\cdots c_6}{}^{\[4\]} F_{\[4\]}$,~ 
(ii) $\g_{\[c_1\cdots c_4|}{}^{\[2\]}
F_{|c_5 c_6\] \[2\]}$,  (iii) $\g_{\[c_1 c_2|} 
F_{|c_3\cdots c_6\]}$, after converting $~F_{\[4\]}$~ into 
$~G_{\[7\]}$~ by (3.2d).  Fortunately, all of these sectors vanish 
by cancellation of the like terms by 
themselves, after the appropriate use of $~\g\-$matrix identities, 
such as $~\g^{\[10\]}\equiv + i \e^{\[10\]m}\g_m$.  At $~d=3/2$, we get 
$$ \nabla_\a G_{b_1\cdots b_7} = - i\frac1{480} 
     \big(\g_{\[b_1\cdots b_5|} \big)_{\a\b} T_{|b_6 b_7\]}{}^\b ~~, 
\eqno(3.5) $$ 
which is easily shown to be on-shell equivalent to (2.19c), and consistent 
with the duality relation (2.11) or (3.2d).    
As usual at $~d=2$, we see that the appearance of $~\theta=0$~ sector 
is consistent with the component field strength $~G_{\[7\]}$~ (2.8) 
with the Chern-Simons form.    




\bigskip\bigskip\bigskip

\centerline{\bf 4.~~Couplings to Super M-5-Brane}   

In this section, we try to couple our 11D supergravity background to 
super M-5-brane.  Our action is in a sense 
simpler than those in \nilsson\ or \mfivekappa\pss\pst, and circumvents 
the usual problem \witten\ for an invariant lagrangian for self-dual field
strength.  

Our fundamental fields in 6D are $~\big(Z^M, g\low{i j}, a_{i j}, 
b_{\[5\]}, \l^{i j k}\subplus, \l, \r^{i j k}\subplus, 
\m_{i j k l m n}^{_{(-)(-)}}, \n_{i j k}
\supminus \big).$\footnotew{Our 6D
notation is $~\big(\eta\low{(i)(j)}\big) 
= \hbox{diag.}\,(+,-,-,-,-,-),~
\e^{\,012\cdots 5} =+1$.  We use $~{\scst (i),~(j),~\cdots~=~
(0),~(1),~\cdots,~(5)}$~ for local Lorentz indices, while $~{\scst
i,~j,~\cdots~=~ 0,~1,~\cdots,~5}$~ for curved indices.}~~Here 
$~Z^M$~ is the 11D
superspace coordinates, $~a_{i j}$~ is antisymmetric field, $~g_{i j}$~ is the
6D metric, while the auxiliary tensor density $~\l^{i j k}\subplus$~ 
and tensor
$~\r^{i j k}\subplus$~ are self-dual with respect to the indices 
$~^{i j k}$. 
The $~\m_{i j k l m n}^{_{(-)(-)}}$~ is a tensor auxiliary field, and is
anti-self-dual with respect to the first three indices $~_{i j k}$, 
as well as to the last  three indices $~_{l m n}$.  The $~\n_{i j k}\supminus$~ 
is a anti-self-dual tensor density auxiliary field.     

Our total action has five parts: the first term 
$~S_{\rm det}$~ with a determinant,  the constraint terms $~S_{\l f}, 
~S_{\l g}, ~S_{\m\r\l}$~ and $~S_{\n\r\l}$:       
$$ \li{ &S\equiv S_{\rm det} + S_{\l f} + S_{\l g} + S_{\m\r\l} 
+ S_{\n\r\l} 
      \equiv \int d^6\s\, \Lag ~~,    
&(4.1) \cr 
& S_{\rm det} \equiv \int d^6\s \, \left[\,
         \det\big( g\low{i j} - \Pi\du i a\Pi_{j a} \big) \, \right]^{1/2} 
         \equiv \int d^6\s \, \Lag_{\rm det} ~~, 
&(4.2) \cr 
& S_{\l f} \equiv\int d^6\s \, \l^{i j k}\subplus f_{i j k}  
         \equiv \int d^6\s \, \Lag_{\l f} ~~, 
&(4.3) \cr  
& S_{\l g} \equiv \int d^6\s \, \left( \frac2{6!} \e^{i_1\cdots i_6} 
      \l g\low{i_1\cdots i_6} + e\l \right) \equiv \int d^6\s \, 
      \Lag_{\l g} ~~,
&(4.4) \cr 
&S_{\m\r\l} \equiv \int d^6\s \, \m_{i j k l m n}^{_{(-)(-)}} 
	\r^{i j k}\subplus \l^{l m n}\subplus      
        \equiv \int d^6\s \, \Lag_{\m\r\l} ~~, 
&(4.5) \cr 
&S_{\n\r\l} \equiv \int d^6\s \, \n_{i j k}\supminus 
	\r^{i j k}\subplus \l       
        \equiv \int d^6\s \, \Lag_{\n\r\l} ~~. 
&(4.6) \cr } $$
The field strengths $~f_{\[3\]}$~ and $~g\low{\[6\]}$~ are  defined by 
\ref\townsend{P.K.~Townsend, \pl{373}{96}{68}.}\ref\cw{M.~Cederwall 
and A.~Westerberg, {\it World-Volume Fields SL(2,Z) and Duality: 
The Type IIB 3-Brane}, 
\hep{9710007}.}\mfivekappa\nilsson
%\ref\cw{M.~Cederwall and A.~Westerberg, \hep{9710007}.}      
$$ \li{&f_{i j k} \equiv \frac12 \partial_{\[i}
            a_{j k\]} - A_{i j k} ~~, ~~~~ 
      A_{i j k} \equiv \Pi\du k C \Pi\du j B \Pi\du i A A_{A B C} ~~,  
&(4.7\rma)  \cr
&g\low{i_1\cdots i_6} \equiv \frac1{5!} \partial_{\[i_1} 
      b_{i_2\cdots i_6\]} 
     - B_{i_1\cdots i_6} + \frac1{48} a_{\[i_1 i_2} 
     F_{i_3\cdots i_6\]}~~.
&(4.7\rmb)  \cr } $$  
The $~\Pi\du i A \equiv \big(\partial_i Z^M\big) E\du M A(Z)$~ are the
pull-backs from 11D superspace to the 6D world-supervolume.  
As usual convention as in (4.7a), the 11D superspace indices $~{\scst
A,~B,~\cdots}$~ can be replaced by the 6D indices $~{\scst i, ~j,
~\cdots}$~ by the use of the pull-backs $~\Pi\du i A$.  
The 6D field strengths $~f$~ and $~g$~ contain the 11D 
superfield potentials like the D-brane couplings \nilsson\townsend\cw.  
Our lagrangian $~\Lag_{\m\r\l}$~ or $~\Lag_{\n\r\l}$~ resembles those in
\siegelconstr, because this lagrangian is also bilinear, but is a product 
of different fields $~\r$~ and $~\l$.  Note that the $~\l^{\[3\]}$~ and
$~\n_{\[3\]}\-$fields are 
tensor densities by definition, and $~f_{i j k}$~ in (4.3) needs
no self-duality projector.  Hence no $~g\low{i j}$~ is involved in
(4.3) and (4.4), {\it except} the $~e\l\-$term in the latter.

Our fermionic symmetry is dictated by the transformation rule 
$$ \li{& \d_\z E^\a \equiv \big(\d_\z Z^M \big) E\du M\a = \k_-^\a \equiv 
      i \big( \g_{\[3\]} \zeta_+\big)^\a \r\subplus^{\[3\]} ~~, 
&(4.8\rma) \cr
&\d_\z E^a \equiv \big(\d_\z Z^M \big) E\du M a = 0 ~~,
     ~~~~ \d_\z \l^{i j k}\subplus = 0 ~~, ~~~~\d_\z \r^{i j k}\subplus 
     = 0 ~~, ~~~~ \d_\z \l = 0 ~~,  
&(4.8\rmb) \cr 
&\d_\z g\low{i j} = \d_\z \big( \Pi\du i a\Pi_{j a} \big) 
     = - i \big( \Bar\k_-\g_{(i} \Pi_{j)} \big) ~~, 
      ~~~~ \d_\z e = - i \big(\Bar\k_- \g^i \Pi_i \big)~~, 
&(4.8\rmc) \cr  
& \d_\z a\low{i j} =\big(\d_\z E^B \big) A_{B i j} ~~,~~~~
     \d_\z b_{i_1\cdots i_5} = (\d_\z E^B) B_{B i_1\cdots i_5} ~~, 
&(4.8\rmd) \cr 
&\d_\z \m_{i j k l m n}^{_{(-)(-)}} = 
    - i \frac1 4\big(\Bar\zeta_+\g\low{i j k}\g^p\g\low{l m n}\Pi_p 
     \big) ~~,
&(4.8\rme) \cr
&\d_\z\n_{i j k}\supminus = - i\frac 16 e
     \big(\z_+\g\low{i j k}\g^{l m n p}\Pi_l \big) f_{m n p}~~. 
&(4.8\rmg) \cr } $$
Here $~e\equiv \det\,(e\du i {(j)}) \equiv {\sqrt {-g}}$~ 
is the determinant of sechsbein in 6D.  The $~\g\-$matrices 
such as $~\g_i$~ is defined by 
$~\g_i \equiv \Pi\du i a \g_a$, satisfying the 6D Clifford algebra
$$ \{\g_i, \g_j \} = 2 g_{i j} I ~~, 
\eqno(4.9) $$ 
under the embedding condition (4.17) below.  We can also specify the
chirality in 6D, defining $~\g\low7 \equiv
\g^{(0)\cdots (5)}$.  The $~\zeta_+$~ is the parameter for our fermionic 
symmetry, and only its positive chirality part is involved 
in our transformation.  
This fermionic symmetry deletes half of the original 32 components 
in the coordinates $~\theta^\m$.        

We first analyze our field equations, starting with that of the 
$~\m\-$field
$$ \r^{i j k}\subplus\l^{l m n}\subplus= 0 ~~. 
\eqno(4.10) $$  
There are two solutions for this field equation: $~\r^{i j k}\subplus=0$,
and/or $~\l^{i j k}\subplus=0$.  However, the former leads to the trivial 
fermionic transformation in (4.8a), to be excluded as a trivial option,
so we concentrate on the latter solution:
$$ \l\subplus^{i j k} = 0 ~~. 
\eqno(4.11) $$
The $~b\-$field equation immediately yields $~\partial_i \l = 0$, 
{\it i.e.},
$~\l={\rm const.}\equiv C$, while the $~\n\-$field equation 
$~\l\r\subplus^{i j k} = 0$~ fixes this constant $~C$~ to be zero:   
$$ \l = 0 ~~.    
\eqno(4.12) $$
Now the $~\r\-$field equation 
$$ \m_{i j k l m n}^{_{(-)(-)}} \l\subplus^{l m n} + \n_{i j k}\supminus 
     \l= 0~~, 
\eqno(4.13) $$
automatically holds under (4.11) and (4.12).  Similarly, the $~a\-$field
equation  
$$ 3 \partial_k \l^{i j k}\subplus - \frac1{24} \e^{i j k_1\cdots k_4}
     F_{k_1\cdots k_4} \l  = 0 ~~,   
\eqno(4.14) $$ 
is also satisfied by (4.11) and (4.12).  The $~\l\-$field equation reads
$$ \frac2{6!} \e^{i_1\cdots i_6} g_{i_1\cdots i_6} + e 
     + \n_{i j k}\supminus \r^{i j k}\subplus = 0 ~~. 
\eqno(4.15) $$
The metric $~g\low{i j}\-$field equation is easy 
to see, because under (4.12) the only contribution is from
$~\Lag_{\rm det}$, whose general variation 
is\footnotew{Note that the undesirable singularity at $~g\low{i j} 
     - \Pi\du i a
\Pi_{j a} = 0$~ in $~\d\Lag_{\rm det}$~ can be easily avoided by 
an alternative lagrangian: 
$~\Lag_{\rm det}' \equiv e^{1-m} \big[
       \det\big(g\low{i j} - \Pi\du i a \Pi_{j a} \big) \big]^{m/2}$~ 
for a real number $~m>2$.} 
$$\d \Lag_{\rm det}= - \frac12 \left[ \, \det\big( g\low{k l} 
- \Pi\du k c
\Pi_{l c} \big)
      \, \right]^{-1/2} \big(g\low{i j} - \Pi\du i a\Pi_{j a}\big) \,  
     \d \big( g^{i j} - \Pi^{i b} \Pi\ud j b \big)~~.   
\eqno(4.16) $$
Thus the metric equation implies the embedding condition
$$ g\low{i j} = \Pi\du i a \Pi_{j a} ~~.
\eqno(4.17) $$  
Under this condition, the $~Z^M\-$field
equation is also satisfied, because under (4.17) the only possible 
contribution to this equation from $~\Lag_{\l f}$~ or $~\Lag_{\l g}$~   
again vanishes under (4.11), {\it e.g.}, 
~$\d\Lag_{\l f}/\d Z^M = \l^{i j k}\subplus \big(\d f_{i j k}
/\d Z^M \big) = 0$.  The only remaining field equation is that 
of $~\l^{\[3\]}$:
$$ f\supminus_{i j k} + \m_{l m n i j k}^{_{(-)(-)}} \r^{l m n}
\subplus = 0 ~~. 
\eqno(4.18) $$
This field equation does {\it not} necessarily implies the 
self-duality $~f\supminus=0$, but it is {\it allowed} as a 
sufficient condition:  
$$ f\supminus_{i j k}=0~~, ~~~~ \m_{i j k l m n}^{_{(-)(-)}} = 0 ~~. 
\eqno(4.19) $$
This feature that the self-duality of $~f$~ is not forced by a 
field equation, but is allowed only as a sufficient condition, 
is expected from the general
argument of M-theory \witten, and also similar to ref.~\nilsson.  
A more generalized self-duality: $~f_{i j
k}\supminus= c f\du{\[i|}{m n} f\du{m n} l f_{l|j k\]}$~ in 
\nilsson\ can be also embedded in (4.18) by $~\r\subplus^{l m n} 
= f\subplus^{l m n}$~ and $~\m\du{i j k}{l m n} = {\scst (}c/6{\scst )} 
f\du{\[i}{\[l m}f\ud{n\]}{j k\]}~$ with the
appropriate duality projection.  From (4.8e) 
it is also clear that the solution (4.19) is not covariant under 
our fermionic symmetry, similarly to \nilsson.  
This is also natural, because
fermionic symmetries can be generally fixed, in such 
a way that unwanted states are eliminated.             

We now study the fermionic invariance of our total action.  
By the help of the 
useful relationship \nilsson 
$$ \li{& \d_\z f_{i j k} = - \big( \d_\z E^B \big) F_{B i j k} 
     = - \frac14 \big(\Bar\k_- \g_{\[i j}\Pi_{k\]} \big) 
     = - i\frac14 \big( \Bar\z_+ \g_{\[3\]} \g_{\[i j} \Pi_{k\]} \big)
     \r^{\[3\]}\subplus ~~,   
&(4.20\rma) \cr 
& \d_\z g_{i_1\cdots i_6} = - i \frac 1{240} 
     \big(\Bar\k_- \g_{\[i_1\cdots
     i_5} \Pi_{i_6\]} \big) + \frac1{24} \big(\Bar\k_- 
      \g_{\[i_1 i_2}\Pi_{i_3} \big) f_{i_4 i_5 i_6\]} ~~, 
&(4.20\rmb) \cr }  $$ 
we easily see that 
$$\li{0\eqques\d_\z \Lag = \, & \d_\z \big(\Lag_{\l f} 
     + \Lag_{\l g} + \Lag_{\m\r\l} + \Lag_{\n\r\l} \big) \cr
= \, & + \l^{i j k}\subplus (-\frac 32) \big(\Bar\k_-\g_{i j} \Pi_k\big) 
     + \l\left[\, i e \big( \Bar\k_-\g_7 \g^i\Pi_i \big) 
     + \frac16 e \big( \Bar\k_- \g_7 \g^{i\[3\]} \Pi_i
     \big) f_{\[3\]} - i e \big( \Bar\k_-\g^i\Pi_i \big)\,\right] \cr 
& - i \frac 14 \big( \Bar\z_+ \g_{\[3\]} \g^i \g_{\[3\]'} \Pi_i\big)  
    \r^{\[3\]}\subplus \l^{\[3\]'}\subplus - i \frac 16 e\l 
      \big( \Bar\z_+\g_{\[3\]} \g^{i\[3\]'} \Pi_i
     \big) f_{\[3\]'} \r\subplus^{\[3\]} \cr 
    = \, & + \frac14 \big(\Bar\k_- \g^k \g_{\[3\]} \Pi_k \big) 
     \l^{\[3\]}\subplus
    +  i \frac 16 e\l \big( \Bar\z_+\g_{\[3\]} \g^{i\[3\]'} \Pi_i
     \big)\r\subplus^{\[3\]} f_{\[3\]'} \cr 
& - i \frac 14 \big( \Bar\z_+ \g_{\[3\]}\g^i \g_{\[3\]'} \Pi_i \big)
\r^{\[3\]}\subplus \l^{\[3\]'}\subplus  
     - i \frac 16 e\l \big( \Bar\z_+\g_{\[3\]} \g^{i\[3\]'} \Pi_i
     \big)\r\subplus^{\[3\]} f_{\[3\]'} \cr  
= \, & + i \frac 14 \big( \Bar\z_+ \g_{\[3\]}\g^i \g_{\[3\]'} \Pi_i \big)
      \r^{\[3\]}\subplus \l^{\[3\]'}\subplus
     - i \frac 14 \big( \Bar\z_+ \g_{\[3\]}\g^i \g_{\[3\]'} \Pi_i \big)
      \r^{\[3\]}\subplus\l^{\[3\]'}\subplus = 0 {~~,~~~~~ ~~~~~}
&(4.21) \cr} $$
based on relations such as $~\g\low7\g_{\[3\]} \l^{\[3\]}\subplus = 
- \g_{\[3\]}\l^{\[3\]}\subplus, ~\Bar\k_- \g\low7 = + \Bar\k_-$, 
{\it etc}. 
 
In our formulation, we have no $~\s\-$model type kinetic term. 
This situation is
similar to ref.~\nilsson, but the system allows a
generalized self-duality for $~f_{i j k}$, as a special case.  
This is natural
in the super M-5-brane formulation, in the sense that the physical 
field is now
$~f_{i j k}$, instead of the $~\s\-$model coordinates $~Z^M$.  It
is interesting that our auxiliary field 
$~\m_{i j k l m n}^{_{(-)(-)}}$~ has the
index symmetries similar to the 11D superspace auxiliary superfield
$~\L^{a_1\cdots a_4 b_1\cdots b_4}$.  



\bigskip\bigskip\bigskip

%\newpage



\centerline{\bf 5.~~Concluding Remarks} 
 
In this paper, we have presented a very simple duality-symmetric local
lagrangian formulation that utilizes only one tensor multiplier field
$~\L^{\[4\]\[4\]'}$.  Compared with the recent paper on a similar 
subject \bbs, our formulation has simpler constraint lagrangians 
whose structure is essentially the same as 
that proposed by Siegel \siegelconstr. The corresponding
superspace formulation with the manifest duality relation between
$~F_{\[4\]}$~ and $~G_{\[7\]}$~ is straightforward.  
Also presented is a new super M-5-brane action, 
allowing the self-duality for the field strength $~f_{i j k}$,
formulated on our 11D duality-symmetric supergravity backgrounds. 

We saw that our new super M-5-brane
action has couplings more natural than the formulation in \nilsson, 
in the sense that the embedding condition of the 6D metric 
in terms of the pull-back comes
out as a field equation.  Our lagrangian is also simpler 
than the formulation
using the unit vector with scalar field \pst\mfivekappa.  
It is also interesting to see 
if the dual version for $~N=1$~ supergravity in 10D \ref\tendual
{S.J.~Gates, Jr.~and H.~Nishino, \pl{157}{85}{157};
\ibid{173}{86}{46}; \ibid{173B}{86}{52}; \np{291}{87}{205}.}   
can be re-obtained by performing double-dimensional reductions
\ref\dhis{M.~Duff, P.S.~Howe, T.~Inami and K.S.~Stelle, \pl{191}{87}{70}.} into
superstrings. 

Note that our invariant lagrangian does {\it not} force the field strength 
$~f_{i j k}$~ to be self-dual as a {\it necessary} condition, but instead, the 
self-duality is allowed as a {\it sufficient} condition.  
Therefore our lagrangian 
bypasses the obstruction for constructing an invariant lagrangian 
for a chiral two-form in 6D \witten\ref\wittenprivate{E.~Witten, 
{\it private
communication}.}.  This obstruction was from various considerations 
leading to the conclusion that a chiral two-form in 6D can not have an 
invariant modular form that is 
needed for an invariant lagrangian \witten\wittenprivate.  
In other words, since our field strength
$~f_{i j k}$~ is not necessarily a self-dual field, we can construct an
invariant lagrangian.  This feature is similar to that in \nilsson.  
    
The method we used in 11D resembles that in \siegelconstr\ with a 
constraint
lagrangian bilinear in the constraint and   linear in the 
multiplier field,
with eight indices in our case.  Such a system prevents the multiplier
field from propagating.  A similar method is also used in our 6D 
super M-5-brane
action.     We have also seen that the existence of the $~F_{\[4\]}^2$~ 
and
$~G_{\[7\]}^2\-$terms are crucial for the multiplier field to be gauged away
by the extra symmetry (2.20) in the standard manner \siegelconstr.  
For this reason we should avoid the singular cases 
$~\b=0,~\b={\scst (24)}^{-1}$.  Despite of the eight-index
auxiliary field $~\L^{\[4\]\[4\]'}$, our action is much simpler than that
in \bbs\ utilizing a scalar field $~a(x)$~ with its gradient 
$~v_\m\equiv\big[\big(\partial_\n a\big)^2\big]^{-1/2} \partial_\m a$~
\pst\ whose {\it non-invariance} under supersymmetry\footnotew{Note that 
the unit vector $~v_\m$~ in \pst\bbs\ is {\it not} invariant under
supersymmetry due to the metric tensor involved in the scalar product 
$~(\partial_\n a)^2\equiv g^{\m\n} (\partial_\m a) (\partial_\n a)$.} 
makes the computation more involved.  

\doit0{
Even though our formulation looks simple, there is an important subtlety
to be emphasized here, has been usually overlooked or ignored
in literature in the past.  For example, the supersymmetric variation of
supercovariant field strengths with curved indices are {\it not} always
`supercovariant' due to gravitino-dependent terms,  caused by the 
variation of
metric tensors.  It is only in terms of local Lorentz indices that these
supercovariant field strengths are free of gravitino-dependent
terms.\footnotew{This point does not seem to be well clarified in \bbs.}    
This feature is most clearly seen in superspace formulation in \ggrs.  
}

Our 11D superspace BIs are satisfied, only if the new 
$~\Hat{\cal F}\-$dependent 
terms vanish.  In other words, these `on-shell vanishing' 
$~\Hat{\cal F}\-$terms
do not satisfy the BIs, in contrast with the usual off-shell formulation
in superspace supergravity, where all the auxiliary-dependent terms also
satisfy the BIs.  In this sense, our superspace constraints are on-shell 
equivalent to those in ref.~\cjs, like the formulation with
scalar auxiliary superfields in \bbs.       

As long as $~\b\neq0,~\b\neq{\scst (24)}^{-1}$~ for the possible 
$~\eta\-$symmetry, there are always kinetic
terms both for $~A_{\[3\]}$~ and $~B_{\[6\]}$, and   
conjugate momenta for these fields exist, 
even though only half of the 
total degrees of freedom are counted as physical ones by the duality 
by the $~\L\-$field equation.  In other words, only the cases 
$~\b\neq 0,~\b\neq{\scst (24)}^{-1}$~ seem to allow simple quantization. 
This feature is more elucidated in our formulation than that with  
scalar field \pst\bbs, and makes our formulation practically more useful.     

Thanks to the simplicity of the system, our formulation provides a good
working ground for studying various aspects of M-theory, such as
D-brane couplings with a two-form field strength, more unified
super M-5-brane couplings with self-dual three-form field strength,
non-perturbative aspects, double-dimensional reduction to supermembrane
\bst\dhis, or relationships with the dual formulation in 10D
\tendual.  

Special acknowledgements are for M.~Cederwall, N.~Berkovits, S.J.~Gates, 
Jr., B.E.W.~Nilsson, and W.~Siegel for important communications.  
  
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