1206.6175.tex
% \count100=1 
% \input defrrr.tex 

%%%%%%%%%%% defrrr.tex %%%%%%%%%%%%%%%%%%% 
\documentstyle[12pt]{article} 

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

\expandafter\ifx\csname amsppt.sty\endcsname\endinput
  \expandafter\def\csname amsppt.sty\endcsname{2.2 (2001/08/07)}\fi

% Underline for text or math 
\catcode`@=11 
\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 letter
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

% Use the following definitions of \pmb, \bfalpha, etc, 
% for `boldfaced' Greek characters. (03/05/99) 
\def\pmb#1{\setbox0=\hbox{${#1}$}%
   \kern-.025em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern.05em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern-.025em\box0 }

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

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

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} 
\raisebox{-.45ex}{$\hskip .48ex { |}$}} 
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} 
\raisebox{.45ex}{$\hskip .48ex { 
|}$}}   
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~} % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}} % D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}} % D'Alembertian
\def\cbo{\Sc [}                            % curly "
\def\pa{\partial}                          % curly d
\def\dell{\bigtriangledown}               % hi ho the dairy-o

\def\pr{\prod}                            % product
\def\iff{\leftrightarrow}                % <-->
\def\conj{{\hbox{\large *}}}             % complex conjugate

\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}} %< or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}} %> or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu %
\llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu %
\llap {$\ddot
        \smile$}}}                           % happy face
\def\dg{\sp\dagger}                         % hermitian 
% conjugate
\def\ddg{\sp\ddagger}                      % double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}                 % superscript (unaligned)
\def\sb#1{{}_{#1}}                              % sub"
\def\oldsl#1{\rlap/#1}                          % poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1} % good slash for lower case

\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}      % " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}    % " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}   % 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}       % 

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

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

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

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

% Text style parameters
\topmargin=0in                          % top margin (less 1") (LaTeX)
\headheight=0in                         % height of heading (LaTeX)
\headsep=0in            % separation of heading from body (LaTeX)
\textheight=9in                         % height of body (LaTeX)
%\footheight=3ex                           % height of foot (LaTeX)
\footheight=15ex                           % height of foot (LaTeX)
\footskip=4ex    % distance between bottoms of body & foot (LaTeX)
\textwidth=6in                          % width of body (LaTeX)
\hsize=6in                              % " (TeX)
\parskip=\medskipamount          % space between paragraphs (LaTeX)
\lineskip=0pt                           % minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em        %space above equation 
%   (either)
\belowdisplayskip=1em plus.3em minus.5em        % " below
\abovedisplayshortskip=.5em plus.2em minus.4em  %"above 
%   when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em  %"below
\def\baselinestretch{1.2}       % magnification for line spacing 
%   (LaTeX)
\thicklines                         % thick straight lines for pictures 
%   (LaTeX)

\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\Biglbracket{\raise0.1pt\hbox{\Big[}{\hskip -4.6pt}\Big[\,}
\def\Bigrbracket{\,\raise0.1pt\hbox{\Big]}{\hskip -4.6pt}\Big]} 

\def\delsl{{{\partial\!\!\! /}}}
\def\nablasl{{{\nabla\!\!\!\!\!{\hskip 1.0pt}/ \,}}}
\def\calDsl{{\calD\!\!\!\! /}\,}    
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}
\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}
\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}
\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}
\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}
\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}
\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\rma{{\rm a}} \def\rmb{{\rm b}} \def\rmc{{\rm c}} 
\def\rmd{{\rm d}} 
\def\rme{{\rm e}} \def\rmf{{\rm f}} \def\rmg{{\rm g}} 
\def\rmh{{\rm h}}  \def\rmi{{\rm i}}  \def\rmj{{\rm j}} 
\def\rmk{{\rm k}}  \def\rml{\ell}  \def\rmm{{\rm m}} 
\def\rmn{{\rm n}} 
\def\Mpl{M_{\rm Pl}}
\def\mw{m\low{\rm W}}
\def\Mgut{M\low{\rm GUT}}
%\def\Re{{\rm Re}\,\,}  \def\Im{{\rm Im}\,\,} 
\def\Re{{\cal R}e\,}  \def\Im{{\cal I}m\,}
\def\tildef{{\tilde f}}
%\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\plpl{{+\!\!\!\!\!{\hskip 0.009in}%
{\raise-1.0pt\hbox{$_+$}}  {\hskip 0.0008in}}} 
\def\mimi{{-\!\!\!\!\!{\hskip 0.009in}%
{\raise-1.0pt\hbox{$_-$}}  {\hskip 0.0008in}}} 

\def\E{{\cal E}}
\def\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.~\& 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}} (19{#2}) #3}
\def\jhep#1#2#3{JHEP {\bf {#1}} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} } 
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}} 
\def\jp#1#2#3{Jour.~Phys.~{\bf {#1}} (19{#2}) {#3}}
\def\ffp#1#2#3{Fort.~f\"ur~Phys.~{\bf {#1}} (19{#2}) {#3}}
%\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
%\def\hepph#1{hep-ph/{#1}} 

%\def\hepth#1{\texttts{hep-th/{#1}}} 
%\def\heplat#1{\texttts{hep-lat/{#1}}} 
%\def\hepph#1{\texttts{hep-ph/{#1}}} 
%\def\texttts#1{\small\texttt{#1}} 
%\def\arxive#1{\texttts{arXiv:{#1}{$\,$}[hep-th]}} 

% In the case the small fonts interfere in \listrefs with \texttt, 
% we have to use  
\def\hepth#1{{hep-th/{#1}}} 
\def\heplat#1{{hep-lat/{#1}}} 
\def\hepph#1{{hep-ph/{#1}}} 
\def\texttts#1{\small\texttt{#1}} 
\def\arXive#1{arXiv:{#1}{$\,$}[hep-th]} 
\def\arxivep#1{arXiv:{#1}{$\,$}[hep-ph]} 

\def\pln#1#2#3{Phys.~Lett.~{\bf {#1}B} (20{#2}) #3} 
\def\npn#1#2#3{Nucl.~Phys.~{\bf B{#1}} (20{#2}) #3}
\def\prln#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (20{#2}) #3}
\def\prn#1#2#3{Phys.~Rev.~{\bf D{#1}} (20{#2}) #3}
\def\cqgn#1#2#3{Class.~\& Quant.~Gr.~{\bf {#1}} (20{#2}) #3} 
\def\cmpn#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (20{#2}) #3} 
\def\jmpn#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (20{#2}) #3} 
\def\apn#1#2#3{Ann.~of Phys.~{\bf {#1}} (20{#2}) #3} 
\def\prepn#1#2#3{Phys.~Rep.~{\bf {#1}C} (20{#2}) #3}
\def\jhepn#1#2#3{JHEP {\bf {#1}} (20{#2}) #3}
\def\ptpn#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (20{#2}) #3}
\def\ijmpn#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (20{#2}) #3}
\def\ncn#1#2#3{Nuovo Cim.~{\bf {#1}} (20{#2}) #3}
\def\ibidn#1#2#3{{\it ibid.}~{\bf {#1}} (20{#2}) #3}
\def\grgn#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (20{#2}) {#3} }
\def\plan#1#2#3{Phys.~Lett.~{\bf A{#1}} (20{#2}) {#3}}
\def\mpln#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (20{#2}) #3} 
\def\zpn#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (20{#2}) {#3}} 
\def\jgtpn#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (20{#2}) {#3}}
\def\rmpn#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (20{#2}) {#3}}
\def\jpn#1#2#3{Jour.~Phys.~{\bf {#1}} (20{#2}) {#3}}
\def\ffpn#1#2#3{Fort.~f\"ur~Phys.~{\bf {#1}} (20{#2}) {#3}}

\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
%\def\uln{{\underline n}} 
\def\uln#1{\underline{#1}}
\def\ulp{{\underline p}} \def\ulq{{\underline q}} 
\def\ulr{{\underline r}}
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\hatm{\hat m}\def\hatn{\hat n}\def\hatr{\hat r}\def\hats{\hat s}
\def\hatt{\hat t}
\def\<<{<\!\!<} \def\>>{>\!\!>} 
%\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
%\def\Check#1{\hbox{\LARGE\v{}}\!\!\!\!\!{#1}} 
\def\Check#1{{\raise-1.0pt\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{\hsize=6.5in {#1}}} 
% The following \footnotes was invented for LaTeX on 07/25/05:  
\def\footnotes#1{{\hskip 1pt}\footnotemark$^)$\footnotetext%
{\hsize=6.5in $^)$~{#1}}} 

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

\begin{document}

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 % Guess what this is for...
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                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
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\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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                               \scriptscriptfont1\fivemib
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%
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% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
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\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
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\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
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\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}
%
%say \footend to put footnotes at end
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\newwrite\rfile
%% We have tampered after #1 in \items which was originally %
% \item and also 
%% the argument of \xdef without [ ].  Also \\ after \items{#1}.
%
% We have to be careful about \ref, when using \label and \eq commands.
\def\ref{[\the\refno]\nref}%
\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi%
\global\advance\refno by1\chardef\wfile=\rfile\immediate%
\write\rfile{\noexpand\Item{#1}\reflabeL{#1\hskip.31in}\pctsign}%
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{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
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\fi\let\next=\writ@line\fi\next\relax}}
\def\striprel@x#1{} \def\em@rk{\hbox{}}
%
\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now 
% unnecessary
%
\def\listrefs{\footatend\vfill\supereject\immediate\closeout%
\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp%
\vfill\eject}\nonfrenchspacing}
%
% The following is the revision of \listrefs to put the list in 
% the same page.
\def\listrefsr{\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}%
\nonfrenchspacing}
% The following is the revision of \listrefs to put the list %
% in the same page with the smaller fonts.
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\font\smallerfonts=cmr9 \font\it=cmti9 \font\bf=cmbx9%
\bigskip{\smallerfonts{% 
\parindent=15pt\escapechar=` \input refs.tmp\vfill\eject}}}
% 
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\baselineskip=12.5pt\centerline{{\bf References}}
\font\smallerfonts=cmr10 \font\it=cmti10 \font\bf=cmbx10%
\bigskip{\smallerfonts{% 
\parindent=18pt\escapechar=` \input refs.tmp\vfill\eject}}}
% 
\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel %
\string#1 is undefined.}%
\nref#1{need to supply reference \string#1.}\fi%
\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
\else#1\count255=1\fi\let\next=\r@fs\fi\next}
%
\def\figures{\centerline{{\bf Figure Captions}}%
\medskip\parindent=40pt%
\def\fig##1##2{\medskip\item{Fig.~##1.  }##2}}
%
% this is ugly, but moore insists
% The following is skipped on 09/14/01 

\newwrite\ffile\global\newcount\figno \global\figno=1
% The following is skipped on 09/14/01.  
\doit0{
\def\fig{fig.~\the\figno\nfig}
\def\nfig#1{\xdef#1{fig.~\the\figno}%
\writedef{#1\leftbracket fig.\noexpand~\the\figno}%
\ifnum\figno=1\immediate\openout\ffile=figs.tmp%
\fi\chardef\wfile=\ffile%
\immediate\write\ffile{\noexpand\medskip\noexpand%
\item{Fig.\ \the\figno. }
\reflabeL{#1\hskip.55in}\pctsign}\global\advance\figno by1\findarg}
%
\def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
\baselineskip14pt\centerline{{\bf Figure Captions}}\nobreak\medskip
\escapechar=` \input figs.tmp\vfill\eject}}
%
\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
\def\figs#1{figs.~\f@gs #1{\hbox{}}}
\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else
\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
%
}

\newwrite\lfile
{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
\def\writedefs{\immediate\openout\lfile=labeldefs.tmp %
\def\writedef##1{%
\immediate\write\lfile{\string\def\string##1\rightbracket}}}
%
\def\writestop{\def\writestoppt%
{\immediate\write\lfile{\string\pageno%
\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
\string\secno\the\secno\string\meqno\the\meqno}% 
\immediate\closeout\lfile}}
%
\def\writestoppt{}\def\writedef#1{}
%
\def\seclab#1{\xdef #1{\the\secno}\writedef{#1\leftbracket#1}%
\wrlabeL{#1=#1}}
\def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
%
\newwrite\tfile \def\writetoca#1{}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
%       use this to write file with table of contents
\def\writetoc{\immediate\openout\tfile=toc.tmp
   \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
   \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}
%       and this lists table of contents on second pass
\def\listtoc{\centerline{\bf Contents}\nobreak%
 \medskip{\baselineskip=12pt
 \parskip=0pt\catcode`\@=11 \input toc.tex \catcode`\@=12 %
 \bigbreak\bigskip}} 
% 
\catcode`\@=12 % at signs are no longer letters
%

% The following is to lift the bottom of the body from page number:
\countdef\pageno=0 \pageno=1
\newtoks\headline \headline={\hfil} 
\newtoks\footline 
 \footline={\bigskip\hss\tenrm\folio\hss}
 %\footline={\hss\tenrm\folio\hss}
\def\folio{\ifnum\pageno<0 \romannumeral-\pageno \else\number\pageno 
 \fi} 

\def\nopagenumbers{\footline={\hfil}} 
\def\advancepageno{\ifnum\pageno<0 \global\advance\pageno by -1 
 \else\global\advance\pageno by 1 \fi} 
\newif\ifraggedbottom

\def\raggedbottom{\topskip10pt plus60pt \raggedbottomtrue}
\def\normalbottom{\topskip10pt \raggedbottomfalse} 

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

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

%\def\circle#1{$\bigcirc{\hskip-9pt}\raise-1pt\hbox{#1}$} 

\def\eqdot{{{\hskip4pt}{\buildrel{\hbox{\LARGE .}} \over =}\,\,{}}} 
\def\eqstar{{\,\,{\buildrel * \over =}\,\,{}}} 
\def\eqques{\,{\buildrel ? \over =}\,{}} 
\def\eqsurface{\,{\buildrel^{\,_{_{_\nabla}}} \over =}\,{}} 

\def\lhs{({\rm LHS})} 
\def\rhs{({\rm RHS})} 
\def\lhsof#1{({\rm LHS~of~({#1})})} 
\def\rhsof#1{({\rm RHS~of~({#1})})} 

\def\binomial#1#2{\left(\,{\buildrel 
{\raise4pt\hbox{$\displaystyle{#1}$}}\over 
{\raise-6pt\hbox{$\displaystyle{#2}$}}}\,\right)} 

\def\Dsl{{}D \!\!\!\! /{\,}} 
\def\doubletilde#1{{}{\buildrel{\mkern1mu_\approx\mkern-1mu}%
\over{#1}}{}}

\def\hata{{\hat a}} \def\hatb{{\hat b}} 
\def\hatc{{\hat c}} \def\hatd{{\hat d}} 
\def\hate{{\hat e}} \def\hatf{{\hat f}} 

\def\circnum#1{{\ooalign%
{\hfil\raise-.12ex\hbox{#1}\hfil\crcr\mathhexbox20D}}}

\def\Christoffel#1#2#3{\Big\{ {\raise-2pt\hbox{${\scst #1}$} 
\atop{\raise4pt\hbox{${\scst#2~ #3}$} }} \Big\} }  

%%%%%%%%%% end of defrrr.tex %%%%%%%%%%%%%

%%%%%%%%%%% End of defrrr.tex %%%%%%%%%%%%
 
\font\smallcmr=cmr6 scaled \magstep2 
\font\smallsmallcmr=cmr5 scaled \magstep 1 
\font\largetitle=cmr17 scaled \magstep1 
\font\LargeLarge=cmr17 scaled \magstep5 
\font\largelarge=cmr12 scaled \magstep0

\def\alephnull{\aleph_0}
\def\sqrtoneovertwopi{\frac1{\sqrt{2\pi}}\,} 
\def\twopi{2\pi} 
\def\sqrttwopi{\sqrt{\twopi}} 

\def\rmA{{\rm A}} \def\rmB{{\rm B}} \def\rmC{{\rm C}} 
\def\HatC{\Hat C} 

%\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}} 

\def\alpr{\a{\hskip 1.2pt}'} 
\def\dim#1{\hbox{dim}\,{#1}} 
% \font\goth = eufm7 scaled \magstep3 
% \font\gothsmall = eufm5 scaled \magstep3 
\def\leftarrowoverdel{{\buildrel\leftarrow\over\partial}} 
\def\rightarrowoverdel{{\buildrel\rightarrow\over%
\partial}} 
\def\ee{{\hskip 0.6pt}e{\hskip 0.6pt}} 

\def\neq{\not=} 
\def\lowlow#1{\hskip0.01in{\raise -7pt%
\hbox{${\hskip1.0pt} \!_{#1}$}}} 
%\def\ZZ{Z\!\!\! Z} 
\def\eqnabla{{{~}\raise-3pt\hbox{${^{^{^{\,\, \nabla}}}}$}{\hskip -12.5pt}={}}} 

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

\font\smallcmr=cmr6 scaled \magstep2 

\def\fracmm#1#2{{{#1}\over{#2}}} 
\def\fracms#1#2{{{\small{#1}}\over{\small{#2}}}} 
\def\low#1{{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}} 
\def\medlow#1{{\raise -1.5pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}

\def\mplanck{M\low{\rm P}} 
\def\mplancktwo{M_{\rm P}^2} 
\def\mplanckthree{M_{\rm P}^3} 
\def\mplanckfour{M_{\rm P}^4} 
\def\mweylon{M\low S}  
\def\mhiggs{M_\medlow H}
\def\mwboson{M \low{\rm W}} 

\def\ip{{=\!\!\! \mid}} 
\def\Lslash{${\rm L}{\!\!\!\! /}\, $} 

\def\leapprox{~\raise 3pt \hbox{$<$} \hskip-9pt \raise -3pt \hbox{$\sim$}~} 
\def\geapprox{~\raise 3pt \hbox{$>$} \hskip-9pt \raise -3pt \hbox{$\sim$}~} 

\def\fR{f (R ) }
\def\FR{F \[ R \]} 
\def\FLaginv{F \[ e^{-1} \Lag_{\rm inv} \]}  
\def\LagSG{\Lag_{\rm SG}} 
\def\Laginv{\Lag_{\rm inv}} 
\def\Lagtot{\Lag_{\rm tot}} 
\def\FprimeLaginv{F\, ' \[e^{-1} \Lag_{\rm inv} \] }   
\def\FdoubleprimeLaginv{F\, '' \[e^{-1} \Lag_{\rm inv} \] }  
\def\Fzeroprime{F\, '\!\!\!_0\,} 


\def\qed{(\hbox{\it Q.E.D.})}

\def\sqrttwo{{\sqrt 2}} 

\def\squarebrackets#1{\left[ \, {#1} \, \right]}  
% The following two are not needed, because of no bothering of putting \,\, 
% \def\braces#1{\left\{ {#1} \, \right\}}  
% \def\parentheses#1{\left( {#1} \right)}  

%: Defs. of Refs. 

%%%%%%%%% Definitions of References %%%%%%%%%%%%%

\def\nrnatmcont{H.~Nishino and S.~Rajpoot, 
{\it `N=1 Non-Abelian Tensor Multiplet in Four Dimensions'}, 
\arXive{1204.1379}, \prn{85}{12}{105017}.}  

\def\bergshoeffsezgincont{	
% `Selfdual Supergravity theories in (2+2)-dimensions', 
E.~Bergshoeff and E.~Sezgin, 
\pl{292}{92}{87}, \hepth{9206101}.}

\def\dwscont{B.~de Wit and H.~Samtleben, 
% `Gauged maximal supergravities and hierarchies of 
% nonabelian vector-tensor systems' 
Fortsch.~Phys.~{\bf 53} (2005) 442, hep-th/0501243.}

\def\dwnscont{B.~de Wit, H.~Nicolai and H.~Samtleben, 
% `Gauged Supergravities, Tensor Hierarchies, and M-Theory' 
JHEP 0802:044,2008, \arXive{0801.1294}.}  

\def\chucont{C.-S.~Chu, 
{\it `A Theory of Non-Abelian Tensor Gauge Field
with Non-Abelian Gauge Symmetry $G \times G$'}, 
\arXive{1108.5131}.}  

\def\sswcont{H.~Samtleben, E.~Sezgin and R.~Wimmer, 
% {\it `(1,0) superconformal models in six dimensions'}, 
\jhepn{12}{11}{062}.}  

\def\atiyahwardcont{M.F.~Atiyah, {\it unpublished}; 
R.S.~Ward, Phil.~Trans.~Roy.~Lond.~{\bf A315} (1985) 451; 
Lect.~Notes in Phys.~{\bf 280} (1987) 106
N.J.~Hitchin, Proc.~Lond.~Math.~Soc.~{\bf 55} (1987) 59;
T.A.~Ivanova and A.D.~Popov, 
% "Some new integrable equations from the self-dual Yang-Mills equations.
Phys.~Lett.~{\bf A205} (1995) 158, 
\hepth{9508129}.} 

\def\belavinetalcont{A.A.~Belavin, A.M.~Polyakov, A.S.~Schwartz and Y.S.~Tyupkin, \pl{59}{75}{85};  
R.S.~Ward, \pl{61}{77}{81}; 
M.F.~Atiyah and R.S.~Ward, \cmp{55}{77}{117}; 
E.F.~Corrigan, D.B.~Fairlie, R.C.~Yates and P.~Goddard, \cmp{58}{78}{223}; 
E.~Witten, \prl{38}{77}{121}.}  

\def\oogurivafacont{H.~Ooguri and C.~Vafa, \mpl{5}{90}{1389};
\np{361}{91}{469}; \ibid{B367}{91}{83}; 
H.~Nishino and S.J.~Gates, \mpl{7}{92}{2543}.}  

\def\siegelcont{W.~Siegel, 
% ``The N=2, (4) String is Self-Dual N=4 Yang-Mills'' 
\pr{46}{92}{3235}, \hepth{9205075}.}  

\def\siegeleightcont{W.~Siegel, \pr{47}{93}{2504}, \hepth{9207043}.} 

\def\ngkcont{S.V.~Ketov, S.J.~Gates, 
Jr.~and H.~Nishino, \pl{307}{93}{323}; 
S.J.~Gates, Jr., H.~Nishino 
and S.V.~Ketov, \pl{307}{93}{331}; 
H.~Nishino, S.J.~Gates, Jr.~and S.V.~Ketov, 
\pl{297}{92}{99}; S.V.~Ketov, H.~Nishino, and 
S.J.~Gates, Jr., \np{393}{93}{149}; 
H.~Nishino, \ijmp{9}{94}{3077}, hep-th/9211042.}  

\def\savvidycont{G.~Savvidy, 
% Extension of the Poincar\'e Group and Non-Abelian Tensor Gauge Fields 
Int.~J.~Mod.~Phys.~{\bf A25} (2010) 5765, \arXive{1006.3005}; 
% Non-Abelian Tensor Gauge Fields
Proc.~Steklov Inst.~Math.~{\bf 272} (2011) 201, 
\arXive{1004.4456}; 
% Particle spectrum of Non-Abelian tensor gauge fields 
Mod.~Phys.~Lett.~{\bf A25} (2010) 1137, \arXive{0909.3859}.}   

\def\stringrelatedcont{S.~Ferrara and M.~Villasante, \pl{186}{87}{85};
P.~Bin\' etruy, G.~Girardi, R.~Grimm, and M.~Muller, \pl{195}{87}{389}; 
S.~Cecotti, S.~Ferrara, and M.~Villasante, Int.~Jour.~Mod.~Phys.
\newline {\bf A2} (1987) 1839; 
M.K.~Gaillard and T.R.~Taylor, \np{381}{92}{577};
V.S.~Kaplunovsky and J.~Louis, \np{444}{95}{191};
P.~Bin\' etruy, F.~Pillon, G.~Girardi and R.~Grimm, \np{477}{96}{175}; 
P.~Bin\' etruy, M.K.~Gaillard and Y.-Y.~Wu, \pl{412}{97}{288}; 
\np{493}{97}{27}493, \ibid{B481}{96}{109};
D.~Lu�st, S.~Theisen and G.~Zoupanos, \np{296}{88}{800}; 
J.~Lauer, D.~L\" ust and S.~Theisen, \np{304}{88}{236}.}  

\def\ngntwocont{S.J.~Gates, Jr.~and H.~Nishino, 
% "Supersymmetric soluble systems embedded in supersymmetric 
% selfdual Yang-Mills theory", 
\pl{299}{93}{255}, \hepth{9210163}.}  

\def\kdvcont{G.~Segal and G.~Wilson, Publ.~Math.~IHES~{\bf 61} (1985) 5; 
V.G.~Drinfeld and V.V.~Sokolov, Sov.~Math.~Dokl.~{\bf 23} (1981) 457; 
Jour.~Sov. Math.~{\bf 30} (1985) 1975.}  

\def\mscont{L.J.~Mason and G.A.J.~Sparling, 
% Nonlinear Schrodinger And Korteweg-De Vries Are Reductions 
% Of Selfdual Yang-Mills.
\pla{137}{89}{29}.}

\def\bdcont{I.~Bakas and D.~Depireux, \mpl{6}{91}{399}.}  

\def\gswcont{M.B.~Green, J.H.~Schwarz and E.~Witten, 
{\it `Superstring Theory'}, Vols.~I \& II, 
Cambridge Univ.~Press (1986).}  

\def\stueckelbergcont{E.C.G.~Stueckelberg, 
Helv.~Phys.~Acta {\bf 11}  (1938) 225;
A.~Proca, J.~Phys.~Radium {\bf 7} (1936) 347; 
D.~Feldman, Z.~Liu and P.~Nath, \prl{97}{86}{021801}; 
{\it For reviews, see, e.g.}, H.~Ruegg and M.~Ruiz-Altaba, \ijmpn{19}{04}{3265}.} 

\def\nrnatensorscont{H.~Nishino and S.~Rajpoot, 
% "Non-Abelian Tensors with Consistent Interactions"  
\prn{72}{05}{085020}, \hepth{0508076}.}  

\def\gswcont{M.B.~Green, J.H.~Schwarz and E.~Witten, 
{\it `Superstring Theory'}, Vols.~I \& II, 
Cambridge Univ.~Press (1986).}  

\def\dbicont{M.~Born and L.~Infeld, Proc.~Roy.~Soc.~Lond.~% 
{\bf A143} (1934) 410; {\it ibid.}~{\bf A144} (1934) 425;
P.A.M.~Dirac, Proc.~Roy.~Soc.~Lond.~{\bf A268} (1962) 57.}% 

\def\gzcont{M.K.~Gaillard and B.~Zumino, 
% "Duality Rotations for Interacting Fields",
\np{193}{81}{221}.}  

\def\bracecont{G.W.~Gibbons and D.A.~Rasheed
% "Electric - Magnetic Duality Rotations in Nonlinear Electrodynamics"
\np{454}{95}{185}, \hepth{9506035};  
D.~Brace, B.~Morariu and B.~Zumino, 
% "Duality Invariant Born-Infeld Theory"
In *Shifman, M.A.~(ed.)~{\it `The many faces of the superworld'}, 
pp.~103-110, \hepth{9905218}; 
M.~Hatsuda, K.~Kamimura and S.~Sekiya, 
% "Electric-Magnetic Duality Invariant Lagrangians" 
Nucl.~Phys.~{\bf 561} (1999) 341; 
P.~Aschieri, \ijmpn{14}{00}{2287}.}  
% "Duality Rotations for Abelian BI Lagrangians"} 

\def\kuzenkocont{S.~Kuzenko and S.~Theisen, 
% "Supersymmetric duality rotations"
\jhepn{03}{00}{034}.}   

\def\pvncont{P.~van Nieuwenhuizen, \prep{68}{81}{189}.}  

\def\schwarzsencont{J.H.~Schwarz and A.~Sen, \np{411}{94}{35}, 
% "Duality symmetric actions" 
\hepth{9304154}.}  

\def\shapereetalcont{A.D.~Shapere, S.~Trivedi and F.~Wilczek, 
% "Dual dilaton dyons"
\mpl{6}{91}{2677};
A.~Sen, 
% "Electric Magnetic Duality in String Theory"
\np{404}{93}{109}.}  
 
\def\aschierietalcont{
P.~Aschieri, D.~Brace, B.~Morariu and B.~Zumino
% "Nonlinear SelfDuality in Even Dimensions"
\npn{574}{00}{551}, \hepth{9909021}.} 
 
\def\nrtencont{H.~Nishino and S.~Rajpoot, 
% "Dual vector multiplet coupled to dual N1 supergravity in 10D"
\prn{71}{05}{085011}.} 

\def\sevenformcont{H.~Nicolai, P.K.~Townsend and P.~van Nieuwenhuizen, 
Lett.~Nuov.~Cim.~{\bf 30} (1981) 315; 
R.~D'Auria and P. Fr\' e, \np{201}{82}{101}.}  

\def\branecont{P.K.~Townsend, 
{\it `p-Brane Democracy'}, hep-th/9507048; 
H.~Nishino, 
% Alternative formulation for duality symmetric 
% eleven-dimensional supergravity coupled to super M five-brane" 
\mpl{14}{99}{977}, \hepth{9802009}.}  

\def\bbscont{I.A.~Bandos, N.~Berkovits and D.P.~Sorokin, 
% "Duality symmetric eleven-dimensional supergravity 
% and its coupling to M-branes" 
\np{522}{98}{214}, \hepth{9711055}.}

\def\nrthreecont{H.~Nishino and S.~Rajpoot, 
% "Duality-symmetric supersymmetric Yang-Mills theory in three dimensions", 
\prn{82}{10}{087701}.}  

\def\ntcont{H.~Nicolai and P.K.~Townsend, \pl{98}{81}{257}.}   

\def\englertwindeycont{F.~Englert and P.~Windey, 
% �Quantization Condition For �t Hooft Monopoles In 
% Compact Simple Lie Groups,� 
\pr{14}{76}{2728}.} 

\def\montonenolivecont{C.~Montonen and D.I.~Olive, 
% �Magnetic Monopoles As Gauge Particles?� 
\pl{72}{77}{117}.} 

\def\olivewittencont{D.I.~Olive and E.~Witten,
% �Supersymmetry Algebras That Include Topological Charges,� 
\pl{78}{78}{97}.} 

\def\osborncont{H.~Osborn, 
% �Topological Charges For N = 4 Supersymmetric Gauge Theories And
% Monopoles Of Spin 1,� 
\pl{83}{79}{321}.} 

\def\cjscont{E.~Cremmer, B.~Julia and J.~Scherk, \pl{76}{78}{409};
E.~Cremmer and B.~Julia, \pl{80}{78}{48}; \np{159}{79}{141}.}

\def\pstcont{P.~Pasti, D.P. Sorokin, M.~Tonin, 
% Note on manifest Lorentz and general coordinate invariance 
% in duality symmetric models"  
\pl{352}{95}{59}, \hepth{9503182}.}     

\def\htwcont{C.~Hull, P.K.~Townsend, \np{438}{95}{109}; 
E.~Witten, \np{443}{95}{85}.}  

\def\tseytlinetalcont{A.A.~Tseytlin, \np{469 }{96}{51}; 
Y.~Igarashi, K.~Itoh and K.~Kamimura, \np{536 }{99}{469}.}  

\def\sstcont{M.B.~Green, J.H.~Schwarz and E.~Witten, 
{\it `Superstring Theory'}, Vols.~I \& II, 
Cambridge Univ.~Press (1986); 
K.~Becker, M.~Becker and J.H.~Schwarz, 
{\it `String Theory and M-Theory:  A Modern Introduction'}, 
Cambridge University Press, 2007.} 

\def\nogocont{M.~Henneaux, V.E.~Lemes, C.A.~Sasaki, S.P.~Sorella, 
O.S.~Ventura and L.C.~Vilar, \pl{410}{97}{195}.}  

\def\ftcont{D.Z.~Freedman, P.K.~Townsend, \np{177}{81}{282};
{\it See also}, V.I.~Ogievetsky and I.V.~Polubarinov, 
Sov.~J.~Nucl.~Phys.~{\bf 4} (1967) 156} 

\def\ksmcont{S.~Krishna, A.~Shukla and R.P.~Malik, {\it `Topologically massive non-Abelian theory: superfield formalism'}, 
\arXive{1008.2649}.} 

\def\topologicalcont{J.~Thierry-Mieg and L.~Baulieu, \np{228}{83}{259}; 
A.H.~Diaz, \pl{203}{88}{408}; 
T.J.~Allen, M.J.~Bowick and A.~Lahiri, \mpl{6}{91}{559};
A.~Lahiri, \pr{55}{97}{5045};
E.~Harikumar, A.~Lahiri and M.~Sivakumar, \prn{63}{01}{10520}.}

\def\ferraracont{G.~Dall'Agata and S.~Ferrara, 
% "Gauged Suprgravity Algebras from Twisted Tori Compactifictions with Fluxes"
\npn{717}{05}{223}, \hepth{0502066}; 
G.~Dall'Agata, R.~D'Auria and S.~Ferrara, 
% "Compactifications on Twisted Tori with Fluxes and Free Differential Algebras"
\pln{619}{05}{149}, \hepth{0503122}; 
R.~D'Auria and S.~Ferrara, 
% "E7 Symmetry and Dual Gauge-Algebra of M-theory on Twisted Seven-Torus"
\npn{732}{06}{389}, \hepth{0504108}; 
R.~D'Auria, S.~Ferrara and M.~Trigiante, 
% "Curvatures and Potential of M-theory in D=4 with Fluxes and Twist"
\jhep{0509}{05}{035}, \hepth{0507225}.}  

\def\nrnacont{H.~Nishino and S.~Rajpoot, 
\hepth{0508076}, \prn{72}{05}{085020}.} 

\def\finncont{K.~Furuta, T.~Inami, H.~Nakajima and M.~Nitta, 
\ptpn{106}{01}{851}, \hepth{0106183}.} 
 
\def\scherkschwarzcont{J.~Scherk and J.H.~Schwarz, \np{153}{79}{61}.}    
 
\def\nepomechiecont{R.I.~Nepomechie, \np{212}{83}{301}.} 

\def\problemnonabeliancont{J.M.~Kunimasa and T. Goto, \ptp{37}{67}{452}; 
A.A.~Slavnov, Theor.~Math.~Phys.~{\bf 10} (1972) 99;
M.J.G.~Veltman, \np{7}{68}{637}; 
A.A.~Slavnov and L.D.~Faddeev, Theor.~Math.
\newline Phys.~{\bf 3} (1970) 312; 
A.I.~Vainshtein and I.B.~Khriplovich, Yad.~Fiz.~13 (1971) 198; 
K.I.~Shizuya, \np{121}{77}{125}; 
Y.N.~Kafiev, \np{201}{82}{341}.    
% D.Z.~Freedman and P.K.~Townsend, \np{177}{81}{282}; 
% D.G.C.~McKeon, Can.~J.~Phys.~{\bf 69} (1991) 1249; 
% J.~Sladkowski, hep-th/9306113.
}  

\def\aflcont{L.~Andrianopoli, S.~Ferrara and M.A.~Lledo, 
% "Axion gauge symmetries and generalized Chern-Simons terms 
% in N=1 supersymmetric theories"
\hepth{0402142}, \jhep{0404}{04}{005};  
R.~D'Auria, S.~Ferrara, M.~Trigiante and S.~Vaula, 
% "Scalar potential for the gauged Heisenberg algebra 
% and a non-polynomial antisymmetric tensor theory"
\pln{610}{05}{270}, \hepth{0412063}.}   

\def\mt{M.~Blau and G.~Thompson,
% Topological Gauge Theories of Antisymmetric Tensor Fields
\ap{205}{91}{130}.}  

\def\hk{M.~Henneaux and B.~Knaepen, 
% All consistent interactions for exterior form gauge fields, 
\pr{56}{97}{6076}, \hepth{9706119}.}

\def\originalcont{S.~Ferrara, B.~Zumino, and J.~Wess, \pl{51}{74}{239}; 
W.~Siegel, \pl{85}{79}{333}; U.~Lindstrom and M.~Ro\v cek, 
\np{222}{83}{285}; 
{\it For reviews of linear multiplet coupled to SG, see, e.g}., P.~Bine�truy, 
G.~Girardi, and R.~Grimm, \prn{343}{01}{255}, {\it and
references therein}.} 

\def\stringrelatedcont{S.~Ferrara and M.~Villasante, \pl{186}{87}{85};
P.~Bin\' etruy, G.~Girardi, R.~Grimm, and M.~Muller, \pl{195}{87}{389}; 
S.~Cecotti, S.~Ferrara, and M.~Villasante, Int.~Jour.~Mod.~Phys.
\newline {\bf A2} (1987) 1839; 
M.K.~Gaillard and T.R.~Taylor, \np{381}{92}{577};
V.S.~Kaplunovsky and J.~Louis, \np{444}{95}{191};
P.~Bin\' etruy, F.~Pillon, G.~Girardi and R.~Grimm, \np{477}{96}{175}; 
P.~Bin\' etruy, M.K.~Gaillard and Y.-Y.~Wu, \pl{412}{97}{288}; 
\np{493}{97}{27}493, \ibid{B481}{96}{109};
D.~Lu�st, S.~Theisen and G.~Zoupanos, \np{296}{88}{800}; 
J.~Lauer, D.~L\" ust and S.~Theisen, \np{304}{88}{236}.}  

\def\threealgebracont{N.~Lamberta and C.~Papageorgakis, 
% Nonabelian (2,0) tensor multiplets and 3-algebras
\jhepn{08}{10}{083}; 
K-W.~Huang and W-H.~Huang, 
% Lie 3-Algebra Non-Abelian (2,0) Theory in Loop Space
arXiv:{1008.3834} \newline [hep-th]; 
S.~Kawamoto, T.~Takimi and D.~Tomino, 
% Branes from a non-Abelian (2,0) tensor multiplet with 3-algebra.
J.~Phys.~{\bf A44} (2011) 325402, \arXive{1103.1223};
% Dp-branes, NS5-branes and U-duality from nonabelian (2,0) theory 
% with Lie 3-algebra. 
Y.~Honma, M.~Ogawa, S.~Shiba, 
\jhepn{1104}{11}{2011}, \arXive{1103.1327};
% The 3-Lie Algebra (2,0) Tensor Multiplet and Equations of Motion 
% on Loop Space.
C.~Papageorgakis and C.~Saemann, \jhep{1105}{11}{099}, 
\arXive{1103.6192}.}  

\def\sgcont{J.~Wess and J.~Bagger, {\it `Superspace and Supergravity'}, 
Princeton University Press (1992).}  

\def\mtheorycont{C.~Hull and P.K.~Townsend,
\np{438}{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'}, Cargese Lectures, 
1997, hep-th/9712004; T.~Banks, W.~Fischler, S.H.~Shenker
and L.~Susskind, \pr{55}{97}{5112}.} 

\def\bbscont{K.~Becker, M.~Becker and J.H.~Schwarz, 
{\it `String Theory and M-Theory:  A Modern Introduction'}, 
Cambridge University Press, 2007.} 

\def\sokatchevcont{E.~Sokatchev, 
% "An Action for N=4 supersymmetric selfdual Yang-Mills theory."
\pr{53}{96}{2062}, \hepth{9509099}.}

\def\romanscont{L.J.~Romans, 
% "Massive N=2a Supergravity in Ten Dimensions", 
\pl{169}{86}{374}.}  

\def\parkescont{A.~Parkes, \pl{286}{92}{265}.}  

\doit0{\def\originalcont{S.~Ferrara, B.~Zumino, and J.~Wess, \pl{51}{74}{239}; 
W.~Siegel, \pl{85}{79}{333}; U.~Lindstrom and M.~Roc\ek, 
% "Scalar tensor duality and N"1, 2 non-linear-models," 
\np{222}{83}{285}.} 
% 
\def\superfieldcont{P.~Binetruy, G.~Girardi and R.~Grimm, 
% "Supergravity Couplings: a Geometric Formulation"
\prepn{343}{01}{255}.}
} % End of \doit 

\def\originalcont{S.~Ferrara, B.~Zumino, and J.~Wess, \pl{51}{74}{239}; 
W.~Siegel, \pl{85}{79}{333}; U.~Lindstrom and M.~Ro\v cek, 
\np{222}{83}{285}.} 
 
\def\superfieldcont{{\it For reviews of linear multiplet coupled to SG, see, e.g}., P.~Bine�truy, 
G.~Girardi, and R.~Grimm, \prn{343}{01}{255}, {\it and references therein}.} 

\def\ggrscont{S.J.~Gates, Jr., M.T.~Grisaru, M.~Ro\v cek and W.~Siegel, 
{\it `Superspace or One Thousand and One Lessons in Supersymmetry'}, 
Front.~Phys.~{\bf 58} (1983) 1-548, hep-th/0108200.} 

\def\typeiibcont{J.H.~Schwarz, \np{226}{83}{269}.} 

\def\selfdualcont{E.~Corrigan, C.~Devchand, D.~Fairie and J.~Nuyts, 
\np{214}{83}{452}; 
R.S.~Ward, \np{236}{84}{381}; 
A.K.~Das, Z.~Khviengia and E.~Sezgin, 
hep-th/9206076, \pl{289}{92}{347}; 
K.~Sfetsos, hep-th/0112117, 
Nucl.~Phys.~{\bf B629} (2002) 417.}% 

\def\octonionscont{M.~G\"unaydin and F.~G\"ursey, 
\jmp{14}{73}{1651}; 
M.~G\"unaydin and C.-H.~Tze, \pl{127}{83}{191}; 
B.~de Wit and H.~Nicolai, \np{231}{84}{506}; 
S.~Fubini and H.~Nicolai, \pl{155}{85}{369};
D.B.~Fairlie and J.~Nuyts, Jour.~Phys.~A: 
Math.~Gen.~{\bf 17} (1984) 2867; 
R.~D\"undarer, F.~G\"ursey and C.-H.~Tze, 
\np{266}{86}{440}.}%

\def\reviewscont{{\it For reviews for reduced holonomies, see, e.g.,}
M.~Atiyah and E.~Witten, Adv.~Theor.~Math.~Phys.~{\bf 6} (2003)1, 
hep-th/0107177; 
M.J.~Duff, {\it `M-Theory on Manifolds of G(2) Holonomy:
The First Twenty Years'}, Talk given at `Supergravity
at 25' (Stony Brook, Dec.~2001), hep-th/0201062; 
L.~Anguelova, C.I.~Lazaroiu, 
hep-th/0204249, JHEP {\bf 0301} (2003) 066; 
{\it and references therein}.}   

\def\gtwocont{B.S.~Acharya and
M.~O'Loughlin, hep-th/9612182, \pr{55}{97}{4521}; 
M.~G\" unaydin and H.~Nicolai,
\pl{351}{95}{169}; hep-th/9502009,
Phys. Lett. {\bf 376B} (1996) 329; 
I. Bakas,
E.G.~Floratos and A.~Kehagias, hep-th/9810042,
Phys. Lett. {\bf 445B} (1998) 69;  E.G.~Floratos and
A.~Kehagias,  hep-th/9802107, Phys. Lett.
{\bf 427B} (1998) 283;  
N.~Hitchin, {\it `Stable forms and
Special Metrics'}, math.DG/0107101; 
M.~Cveti\v c, G.W.~Gibbons, H.~Lu, C.N.~Pope, 
hep-th/0102185, Nucl.~Phys.~{\bf 617} (2001) 151; 
hep-th/0108245, Phys.~Rev.~{\bf D65} (2002) 
106004; B.~Acharya and E.~Witten
{\it `Chiral Fermions from Manifolds of G(2) Holonomy'}, 
hep-th/0109152;  
A.~Brandhuber, hep-th/0112113,
Nucl.~Phys.~{\bf B629} (2002) 393; 
S.~Gukov and J.~Sparks, hep-th/0109025, 
\npn{625}{02}{3}; 
A.~Bilal, J.-P.~Derendinger and 
K.~Sfetsos, \hepth{0111274}, Nucl.~Phys.~{\bf B628} 
(2002) 112.}% 
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%: For Your Eyes only 
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{\bf Preliminary Version (FOR YOUR EYES
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\begin{center} 

%: Title 
{\Large\bf Self-Dual Non-Abelian $\,$N$\,$=$\,$1$\,$ Tensor Multiplet} \\
\vskip 0.05in 
{\Large\bf in $~$D$\,$=$\,$2$\,$+$\,$2$\,$~ Dimensions} \\ 
\vskip 0.1in 
% {\Large\bf } \\ [.3in] 

\baselineskip 9pt 

\vskip 0.21in 

Hitoshi ~N{\smallcmr ISHINO}% 
\footnotes{E-Mail: hnishino@csulb.edu} ~and
~Subhash ~R{\smallcmr AJPOOT}%
\footnotes{E-Mail: rajpoot@csulb.edu} 
\\[.16in]  {\it Department of Physics \& Astronomy}
\\ [.015in] 
{\it California State University} \\ [.015in]  
{\it 1250 Bellflower Boulevard} \\ [.015in]  
{\it Long Beach, CA 90840} \\ [0.02in] 

\vskip 1.7in 

%: Abstract 
{\bf Abstract}\\[.1in]  
\end{center} 
\vskip 0.1in 

\baselineskip 14pt 


~~~We present a self-dual non-Abelian $~N=1$~ supersymmetric 
tensor multiplet in $~D=2+2$~ space-time dimensions.  
Our system has three on-shell multiplets:  
(i) The usual non-Abelian Yang-Mills multiplet $\, (A\du \m I, \l{}^I)$ (ii) 
A non-Abelian tensor multiplet $\,(B\du{\m\n} I , \chi^I, \varphi^I)$,   
and (iii) An extra compensator vector multiplet 
$\, (C\du \m I, \r^I)$.  Here  
the index $~{\scst I}$~ is for the adjoint representation 
of a non-Abelian gauge group.  The duality symmetry relations are $~G\du{\m\n\r} I 
= - \e\du{\m\n\r} \s \, \nabla_\s \varphi^I$, $~F\du{\m\n} I = 
+ (1/2)\, \e\du{\m\n}{\r\s}F\du{\r\s} I$, and $~H\du{\m\n} I = +(1/2)\, \e\du{\m\n}{\r\s}H\du{\r\s} I$, where $~G$~ and $~H$~ are respectively the field strengths of $~B$~ and $~C$.  The 
usual problem with the coupling of the non-Abelian tensor is avoided by 
non-trivial Chern-Simons terms in the field strengths $~G\du{\m\n\r}I$~ and 
$~H\du{\m\n} I$.  For an independent confirmation, we re-formulate 
the component results in superspace.  As applications of embedding 
integrable systems, we show how the $~\calN=2,~r=3$~ and \hbox{$\calN=3, ~r=4$} flows of generalized Korteweg-de Vries equations are embedded into our system.  


\vskip 0.5in  

\baselineskip 8pt 
% PACS 
\leftline{\small PACS:  11.15.-q, 11.30.Pb, 12.60.Jv}  
\vskip 0.03in 
%: Key Words   
\leftline{\small Key \hfil Words:  \hfil Self Dualities, \hfil Duality Symmetry, 
\hfil Non-Abelian Tensor, 
\hfil $\,$N=1$\,$ Supersymmetry,} 
\leftline{\small {\hskip 0.8in} Tensor Multiplet, Consistent Couplings, 
Integrable Systems, KdV Equations.}  

 
% \leftline{\small {\hskip 0.8in} Trace Anomaly, Conformal Anomaly}  

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%: 1. Introduction 

\leftline{\bf 1.~~Introduction}  

Considerable progress has been achieved in constructing theories with 
consistent interactions of non-Abelian tensor fields of 2nd-rank or higher 
%%% 
\ref\dws{\dwscont}%
%%% 
\ref\dwns{\dwnscont}% 
%%% 
\ref\chu{\chucont}%
%%% 
\ref\ssw{\sswcont}.  
The key ingredient is based on the so-called 
`vector-tensor hierarchies' \dws\dwns\chu\ssw, utilizing 
extra Chern-Simons (CS) terms added to the field strengths of non-Abelian tensors.  Another important technique is the engagement of 
generalized Stueckelberg formalism for higher-rank tensors, 
avoiding the usual inconsistency of non-Abelian tensor couplings.  

In \dws, the gauging of five-dimensional (5D) maximal supergravity with $~E_{6(+6)}$~ was generalized in terms of the so-called `vector-tensor hierarchy'.  
The field strength $~\calH_{\m\n\r \, I}$~ 
for a 2nd-rank antisymmetric tensor $~B_{\m\n\, I}$~ is introduced with 
generalized CS terms \dws, such that $~\calH_{\m\n\r \, I}$~ is 
invariant under tensor and vector gauge transformations.  
Subsequently, the relationship of the formulation in \dws\ with M-theory 
%%% 
\ref\mtheory{\mtheorycont}% 
%%% 
\ref\bbs{\bbscont} 
%%% 
was confirmed by representation assignments \dwns. 
Applications to gauged maximal supergravity in 3D 
were also performed with all possible tensor fields \dwns.
It is suggested in \chu\ that the system of the non-Abelian gauge group 
$~G \times G$~ fits nicely to multiple M5-branes with manifest (1,0) supersymmetry.  In \ssw, the original vector-tensor hierarchy was 
simplified further to `minimal vector-tensor hierarchy' in the context of 
conformal $~N=(1,0)$~ supergravity in 6D.  

Motivated by this series of developments \dws\dwns\chu\ssw, 
we have presented in our previous paper 
%%% 
\ref\nrnatm{\nrnatmcont}  
%%%  
an $~N=1$~ supersymmetric formulation of 
non-Abelian tensor in 4D.  
Our formulation is understood as a special case of the so-called 
minimal vector-tensor hierarchy \ssw.  
Our field strengths are the tensor multiplet (TM)  
$~(B\du{\m\n} I, \chi^I, \varphi^I)$,%
\footnotes{We use the indices $~{\scst \m,~\n,~\cdots~=~0,~1,~2,~3}$~ for 
the space-time coordinates.}    
the Yang-Mills vector multiplet (YMVM) $~(A\du \m I, \l^I)$~ and the extra compensating vector multiplet (ECVM) $~(C\du\m I, \r\du\m I)$.  
Following the `vector-tensor hierarchy' \dws\dwns\ssw, we define our field 
strengths by  
\nrnatm
% \footnotes{Our antisymmetrization is normalized as  
% $~X_{\[ a b\]} \equiv X_{a b} - X_{b a}$~ {\it without} 1/2, 
% in order to comply with superspace notation to be engaged later.} 
$$\li{ F\du{\m\n} I & \equiv + 2 \partial_{\[ \m} A\du{\n \]} I 
			+ g f^{I J K} A\du \m J A\du \n K ~~, 
&(1.1\rma) \cr 
G\du{\m\n\r} I & \equiv + 3  D_{\[ \m} B\du{\n\r \] } I 
		- 3  f^{I J K} C\du{\[ \m } J F\du{ \n\r \] } K  ~~, 
&(1.1\rmb) \cr 
H\du{\m\n} I & \equiv + 2 D_{\[ \m } C\du{\n \]} I + g B\du{\m\n} I ~~.
&(1.1\rmc) \cr } $$ 
Relevantly, these field strengths satisfy Bianchi identities
$$ \li{ D_{\[ \m } F\du{ \n\r\] } I & \equiv 0 ~~, 
&(1.2\rma) \cr 
D_{ \[ \m} G\du{ \n\r\s \] } I & \equiv + \fracm 3 2 
		\fIJK F\du{ \[ \m\n} J H\du{ \r\s \]} K ~~, 
&(1.2\rmb) \cr 
D_{ \[\m } H\du{ \n\r \] } I & \equiv + \fracm 13 g \, G\du{\m\n\r} I ~~.  
&(1.2\rmc) \cr } $$ 

Due to the indices $~{\scst \m\n}$~ on $~B\du{\m\n} I$~ or $~{\scst\m}$~ on 
$~C\du\m I$, there should be also proper gauge transformations for 
these fundamental fields.  Let us call them $~\d_\b$~ and $~\d_\g\-$gauge transformations.  In addition to the YM gauge transformation $~\d_\a$, their 
explicit forms are  
$$\li{ \d_\a ( A\du \m I, ~B\du{\m\n} I , ~C\du\m I) 
& = ( + D_a \m^I  , ~ - \fIJK \a^J B\du{\m\n} K , ~- \fIJK \a^J C\du \m K) ~~, 
&(1.3\rma) \cr 
\d_\b ( A\du\m I ,~ B\du{\m\n}I,  ~C\du\m I) 
& = ( 0, ~ + 2 D_{\[ \m} \b\du{ \n \]}  I , ~ - g \b \du \m I) ~~, 
&(1.3\rmb) \cr 
\d_\g ( A\du \m I , ~ B\du{\m\n} I, C\du \m I) 
& = ( 0, ~ - \fIJK F\du{\m\n} J \g^K , ~D_\m \g^I)  ~~. 
&(1.3\rmc) \cr} $$ 
As (1.1c) or (1.3b) shows, $~C\du \m I$~ is a vectorial Stueckelberg field, 
absorbed into the longitudinal component of $~B\du{\m\n} I$.   
Due to the general hierarchy \dws\dwns\ssw, all field strengths are 
covariant under $~\d_\a$~ and invariant under $~\d_\b$~ and $~\d_\g$: 
$$\li{ & \d_\a (F\du{\m\n} I , ~G\du{\m\n\r} I , ~H\du{\m\n} I) 
	= - \fIJK \a^J (F\du{\m\n} K ,~ G\du{\m\n\r} K ,~ H\du{\m\n} K) ~~, 
&(1.4\rma) \cr 
& \d_\b (F\du{\m\n} I  , ~G\du{\m\n\r} I , ~H\du{\m\n} I )  = 0 ~~,
		~~~~ \d_\g (F\du{\m\n} I ,~ G\du{\m\n\r} I , ~H\du{\m\n} I )  = 0 ~~.   
&(1.4\rmb) \cr} $$ 

In the present paper, we apply these developments 
\dws\dwns\chu\ssw\nrnatm\ to 
`self-dual tensor multiplets' in $~2+2$~ dimensions ($D=2+2$).  
The original `self-duality' was implied in terms of Hodge-Poincar\' e duality, 
applied to self-dual Yang-Mills (SDYM) theory
%%% 
\ref\belavinetal{\belavinetalcont}% 
%%% 
\ref\atiyahward{\atiyahwardcont}.  
%%% 
 There are two grounds for the importance of SDYM theory 
\belavinetal\atiyahward.  
First, it has been known that $~N=2$~ superstring 
requires the background YM field be self-dual in $~D=2+2$~ 
space-time dimensions 
%%% 
\ref\oogurivafa{\oogurivafacont}.  
%%% 
Second, SDYM theory seems to be the `master theory' of all (bosonic) 
integrable models in lower dimensions $~1\le D\le 3$ \atiyahward.  
The supersymmetrization of SDYM, {\it i.e.,} self-dual supersymmetric 
Yang-Mills (SDSYM) theory was also accomplished in 1990's
%%%  
\ref\siegel{\siegelcont}%  
%%%   
\ref\ngk{\ngkcont}%  
%%%
\ref\bergshoeffsezgin{\bergshoeffsezgincont}.  
%%% 
In particular, the maximally supersymmetric SDSYM theory in $~D=2+2$~  
is $~N=8$~ case
%%% 
\ref\siegeleight{\siegeleightcont}.  
%%%

From a na\itrema ve viewpoint in the context of SDSYM, 
there appears to be {\it no} strong 
motivation to consider {\it tensor} fields carrying non-Abelian indices.  
Because there are three major objections against such a trial.  
First, the original conjecture \atiyahward\ 
was about SDYM fields, that may generate all the integrable models in lower dimensions.  So an additional tensor field seems redundant.  
Second, even for $~N=1$~ superstring theory
%%%
\ref\gsw{\gswcont}, 
%%% 
a 2-form tensor field 
background should carry {\it no} additional indices, 
so that a non-Abelian tensor seems to be irrelevant.  
Third, even independent of string theory \gsw, it is not interesting enough, 
unless the tensor carries non-trivial indices such as adjoint index with 
non-trivial interactions.  On the other hand, non-Abelian 
tensor couplings to a YM field used to be problematic, 
before the non-Abelian tensor formulations, such as 
\dws\dwns\chu\ssw\nrnatm\ were established.  

Aforementioned three objections, however, are considered obsolete nowadays. 
Definitely, the first objection seems invalid, 
since the duality symmetry between the 3-form field strength $~G_{\m\n\r}$~ 
and the 1-form field strength $~\nabla_\m \varphi$~ of 
a dilaton was predicted as important backgrounds for ~$N=(2,0)$~ heterotic 
$~\s\-$model \oogurivafa.  
The second objection is not strong enough to avoid the discussion of 
non-Abelian tensor with duality and supersymmetry.   
Because even if tensors with additional indices 
may not be directly related to $~N=1$~ \gsw\ or 
$~N=2$~ \oogurivafa\ superstring, 
duality symmetry between a 3-form and 1-form field strengths 
\ngk\ may well be associated with integrable models 
in lower dimensions.  
The third objection has also lost its strong ground, 
because of the above-mentioned 
breakthrough \dws\dwns\chu\ssw\nrnatm.  Moreover, important relationships 
between vector-tensor hierarchy and M-theory \mtheory\ have been 
also established in \dwns.  

Motivated by these viewpoints, especially by the success of the 
supersymmetrization of non-Abelian tensor \nrnatm, 
we give in the present paper the component formulation 
%%% 
\ref\sg{\sgcont}
%%% 
of self-dual non-Abelian tensor multiplet (SDNATM)\footnotes{
The original tensor (or linear) multiplet {\it without} self-duality 
was first formulated in 
%%% 
\ref\original{\originalcont}.  
%%% 
Here we deal with `self-dual' NATM.  
The tensor $~B\du{\m\n} I$~ itself in this multiplet is {\it not} self dual.  However, since the scalar $~\varphi^I$~ and $~B\du{\m\n} I$~ within NATM are dual to each other, we call this multiplet as a `self-dual' NATM.} in $~D=2+2$. 
There are three multiplets in our system: (i) The usual non-Abelian YM vector 
multiplet (VM) $(A\du\m I, \l^I)$, (ii) A SDNATM 
$(B\du{\m\n} I , \chi\du\m I, \varphi^I)$, 
and (iii) An ECVM $~(C\du\m I, \r^I)$.    
Our duality conditions are\footnotes{We use the symbol 
$~\eqstar$~ for an equality associated with dualities, or ans\" atze 
for DRs in section 5.
The derivative $~\nabla_\m \equiv \partial_\m  + g A\du\m I T^I$~ is 
YM non-Abelian group covariant.  The definitions of these field strengths are 
the same as (1.1).  The notation for the $~D=2+2$~ space-time is the same 
as in \ngk, such as $~\g_{\m\n\r\s}= + \e_{\m\n\r\s} \g_5, 
~\e\du{\m\n}{\r\s} \g_{\r\s} = - 2 \g_5 \g_{\m\n}$.}  
$$\li{ G\du{\m\n\r} I \eqstar & \! - \e\du{\m\n\r} \s \nabla_\s \varphi^I ~~, 
&(1.5\rma) \cr 
\nabla_\m \varphi^I \eqstar & \! + \frac 1 6 \e\du\m {\n\r\s} G\du{\r\s\t} I ~~, 
&(1.5\rmb) \cr 
H\du{\m\n} I \eqstar & \! + \frac 12 \e\du{\m\n}{\r\s} H\du{\r\s} I ~~, 
&(1.5\rmc) \cr 
F\du{\m\n} I \eqstar  & \! + \frac 12 \e\du{\m\n}{\r\s} F\du{\r\s} I ~~.   
&(1.5\rmd) \cr } $$ 
Eqs.~(1.5a) and (1.5b) imply the Hodge-Poincar\' e 
duality symmetry between the two 
field strengths $~G\du{\m\n\r} I$~ and $~\nabla_\m \varphi^I$, while 
(1.5c) and (1.5d) are the usual SD for the field strengths $~H$~ and 
$~F$.   
The Abelian case without the adjoint index has been well known for a while 
\ngk.  However, the new ingredient here is that the self-dual TM carrying the 
adjoint index of a non-Abelian gauge group, and we have to 
accomplish the consistent couplings between the tensor field and 
the usual YM gauge field, following the vector-tensor hierarchies 
\dws\dwns\ssw\nrnatm.     

As a general feature of SDYM systems.  
it has been well known that SDYM theory lacks an action, unless 
one breaks Lorentz invariance
%%% 
\ref\parkes{\parkescont}.  
%%%   
This can be easily understood as follows.  If we try to construct the kinetic term 
of a self-dual field strength $~F\du{\m\n} I$, it will be a total divergence:   
$$ \li{ & - \fracm 14 F\du{\m\n} I F^{\m\n\, I} 
	\eqstar - \fracm 1 4 \left( \fracm12 \e\du{\m\n}{\r\s} F\du{\r\s} I \right) 
	F^{\m\n\, I} 
	= - \fracm 18 \e^{\m\n\r\s} F\du{\m\n} I F\du{\r\s} I 
	\eqnabla 0 ~~, ~~~  
&(1.6) \cr} $$ 
where $~\eqnabla$~ is an equality up to a total divergence.  
This is also confirmed by varying $~A\du\m I$~ in (1.6)  
with a zero result, due to the Bianchi identity $~D_{\[\m} F\du{\n\r\]} I
\equiv 0$.  Another typical example is self-dual 5-th rank field strength in the 
so-called type IIB supergravity in 10D 
%%% 
\ref\typeiib{\typeiibcont}.  
%%% 
This property is shared also with the duality symmetric field strengths 
$~G\du{\m\n\r} I$~ and $~D_\m \varphi^I$~ satisfying (1.5a) and (1.5b):  
$$ \li{ - \fracm 1{12} G\du{\m\n\r} I G^{\m\n\r\, I} 
		\eqstar & - \fracm 1{12} ( - \e\du{\m\n\r} \s D_\s\varphi^I) \, 
		G^{\m\n\r\, I} \cr 
= & + \fracm1{12} \e^{\m\n\r\s} G\du{\m\n\r} I D_\s \varphi^I 
		\eqnabla +  \fracm1{12} \e^{\m\n\r\s} \varphi^I 
		D_{\[ \m} G\du{\n\r\s\]} I \cr 
\equiv & + \fracm 1 8 \e^{\m\n\r\s} 
		\fIJK \varphi^I F\du{\m\n} J H\du{\r\s} K ~~,  
&(1.7) \cr} $$ 
where use is made of the Bianchi identity (1.5b).  
Even though the last side of (1.7) is {\it not} vanishing, since it is already at the 
{\it trilinear} interaction, this can {\it no} longer 
regarded as the {\it kinetic} term.  

In order to overcome this general problem with 
SD field strengths, there have been some methods developed, 
such as using harmonic superspace 
%%% 
\ref\sokatchev{\sokatchevcont}.  
%%% 
However, we do not attempt to solve the action problem in this paper, regarding  
it as a separate issue.  So instead of giving an explicit lagrangian, 
we use only the set of field equations.  

As applications of SDNATM, we also show some examples of our system 
generating $~\calN=2,~r=3$~ and $~\calN=3,~r=4$~ 
flows of generalized Korteweg-de Vries (KdV) equations in $~D=1+1$~ 
%%%
\ref\kdv{\kdvcont}.
%%%    
Compared with the case of SDYM system 
%%% 
\ref\ms{\mscont}% 
%%% 
\ref\bd{\bdcont}, 
%%% 
our system is relatively simpler,  
but it still maintains non-trivial feature of embeddings.  
This seems to be the role played by the TM, showing the advantage 
of our SDNATM system over the SDSYM system 
%%% 
\ref\ngntwo{\ngntwocont}. 
%%%   

This paper is organized as follows.  In the next section, we will give first the 
component formulation for SDNATM.  In section 3, we will give the 
superspace re-formulation of the component results.  In section 4, we
mention the difficulty with {\it off-shell} formulation in terms of 
prepotentials and auxiliary fields.  In section 5, we will 
give the embedding of KdV equations in $~D=1+1$, as an important 
application of SDNATM.  In section 6, we point it out that bosonic conditions 
arising in our system after a dimensional reduction (DR) into $~D=1+1$~ 
are equivalent to bosonic equations arising in $~N=2$~
supersymmetric SDSYM theory \ngk.  
The concluding remarks will be given in section 7, with potential 
generalizations to higher space-time dimensions.  


\bigskip\bigskip 

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% \newpage 

%: 2. Component Formulation
\leftline{\bf 2.~~Component Formulation} 
\nobreak  

We first give our results in component language in the most conventional 
notation.  In the next section, we will perform the re-formulation in superspace  
in order to confirm the total consistency.  

The dualities and their supersymmetric partner conditions 
of our system are summarized as (1.5) and the chiralities of fermionic fields
$$ \li{ \g_5 \, ( \l^I , \, \chi^I , \, \r^I)  = & ( - \l^I , \, + \chi^I, \, - \r^I) ~~, 
&(2.1) \cr } $$  
As in the SDSYM case, these chiralities and SD are closely related to 
each other for the SDNATM system.  

The global $~N=1$~ supersymmetry transformation rule is 
$$\li{ \d_Q A\du\m I = & + (\Bar\e\g_\m \l^I) ~~, 
&(2.2\rma) \cr 
\d_Q \l^I = & + \fracm 14 (\g^{\m\n} \e) F\du{\m\n} I ~~, 
&(2.2\rmb) \cr 
\d_Q B\du{\m\n} I = & + (\Bar\e\g_{\m\n} \chi^I) ~~ , 
&(2.2\rmc) \cr 
\d_Q \chi^I = & +  \fracm 1{12} (\g^{\m\n\r} \e) \, G\du{\m\n\r} I 
				- \fracm 12 ( \g^c\e) D_\m \varphi^I 
				+ \fIJK \e \, (\Bar\l^J\r^K) ~~, \
&(2.2\rmd) \cr 
\d_Q\varphi^I = & + (\Bar\e\chi^I)~~,  
&(2.2\rme) \cr 
\d_Q C\du\m I = & + (\Bar\e\g_\m \r^I) ~~, 
&(2.2\rmf) \cr 
\d_Q \r^I = & + \fracm 14 (\g^{\m\n} \e) H\du{\m\n} I ~~.   
&(2.2\rmg) \cr } $$ 
The first and second terms in the r.h.s.~of (2.2d) are the same under the duality 
(1.5a).  The consistency between these rules and the dualities (1.5) or chiralities (2.1) will be confirmed later.  

The field equations for the fermionic fields are 
$$\li{ \Dsl \l^I \eqdot & 0 ~~, 
&(2.3\rma) \cr 
\Dsl \r^I - 2 g \chi^I \eqdot & 0 ~~, 
&(2.3\rmb) \cr 
\Dsl \chi^I - \fracm 14\fIJK (\g^{\m\n} \l^J) H\du{\m\n} K 
		+ \fracm 14 \fIJK (\g^{\m\n} \r^J) F\du{\m\n} K 
				+ g\fIJK \l{}^J \varphi^K \eqdot & 0 ~~.  
&(2.3\rmc) \cr } $$ 

Additional useful transformation rules are 
$$\li{ \d_Q F\du{\m\n} I = & - 2 (\Bar\e \g_{\[\m} D_{\n\]} \l^I) ~~, 
&(2.4\rma) \cr 
\d_Q G\du{\m\n\r} I = & + 3 (\Bar\e\g_{\[\m\n} D_{\r\]} \chi^I) 
				+ 3 \fIJK (\Bar\e\g_{\[ \m} \l^J) H\du{\n\r\]} K 
				+ 3 \fIJK (\Bar\e\g_{\[ \m} \r^J) F\du{\n\r\]} K ~~, 
&(2.4\rmb) \cr 
\d_Q H\du{\m\n} I = & - 2 (\Bar\e\g_{\[ \m} D_{\n\]} \r^I) 
		+ (\Bar\e\g_{\m\n} \chi^I) ~~, 
&(2.4\rmc) \cr } $$ 

The fermionic field equations (2.3) with the chiralities (2.1) 
are consistent with these transformation rules.  For example, we can confirm 
that  
$$\li{&  0 \eqques \d_Q \left(F\du{\m\n} I - \fracm 12 \e\du{\m\n}{\r\s} F\du{\r\s} I \right) 
		= - ( \Bar\e\g_{\m\n} \Dsl \l^I) \eqdot 0 ~~ \qed~~, 
&(2.5\rma) \cr 
&  0 \eqques \d_Q \left( G\du{\m\n\r} I + \e\du{\m\n\r}\s D_\s \varphi^I \right) 
		= + \Big[ \, \Bar\e\g_{\m\n\r} 
		\Big\{ \Dsl\chi^I - \fracm14 \fIJK (\g^{\s\t} \l^J) H\du{\s\t} K \cr 
& {\hskip 1.9in} + \frac14 \fIJK (\g^{\s\t} \r^J) F\du{\s\t} K + g \fIJK \l^J \varphi^K 
		\Big\} \, \Big] \eqdot 0 ~~ \qed~~,  ~~~~~ ~~~~~ ~
&(2.5\rmb) \cr 
&  0 \eqques \d_Q \left( H\du{\m\n} I - \fracm 12 \e\du{\m\n}{\r\s} H\du{\r\s} I \right) 
	= - \left[ \, \Bar\e\g_{\m\n} \left( \Dsl\r^I - 2 g \chi^I \right) \, \right] 
			\eqdot 0 ~~ \qed ~~.    
&(2.5\rmc) \cr } $$ 
In these confirmations, use is made of the $~\g\-$matrix algebra, such as 
$$ \li{ & \e\du{\m\n\r}\s \g_\s = \g_5 \g_{\m\n\r} = - \g_{\m\n\r} \g_5 ~~, ~~~~ 
\{ \g_{\m\n\r} , \, \g^{\s\t} \} = + 12 \d\du{\[\m | } \s \d\du{|\n|} \t \g_{|\r\]} ~~. 
& (2.6) \cr } $$ 
 
\bigskip\bigskip 

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% \newpage 

%: 3. Superspace Re-Formulation
\leftline{\bf 3.~~Superspace Re-Formulation} 
\nobreak  

We have so far presented only component formalism.  Even though 
we have performed cross-confirmations such as (2.5), 
it is still better to have independent confirmation in 
superspace.  To this end, we use the superspace notations, such 
as the indices $~{\scst A ~=~(a,\a,\Dot\a),~B~=~(b,\b,\Dot\b),~\cdots}$~ for the 
superspace coordinates, where $~{\scst a,~b,~\cdots~=~0,~1,~2,~3}$~ 
(or $~{\scst \a,~\b,~\cdots~=~1,~2,~3,~4; ~~ \Dot\a,~\Dot\b,~\cdots~=~\Dot 1,~\Dot 2,~\Dot 3,~\Dot 4}\,$) are for the bosonic (or fermionic) coordinates.  As usual 
the {\it undotted} (or {\it dotted}) indices are for the chiral (or ant-chiral) 
fermions.  Accordingly, our field content in superspace notation is VM $~(A\du a I, 
\Bar\l\du{\Dot\a} I)$, ~SDNATM $~(B\du{a b} I, \chi\du\a I, \varphi^I)$~ and 
ECVM $~(C\du a I, \Bar\r\du{\Dot\a} I) $.  
Our (anti)symmetrizations are such as $~X_{\[ A B ) } \equiv 
X_{A B} - (-1)^{A B} X_{B A}$~ {\it without} the factor of $~1/2$.    

The {\it off-shell} superspace formulation of the original linear multiplet \original\ 
has been systematically studied 
%%% 
\ref\superfield{\superfieldcont}.  
%%% 
In off-shell formulations, the so-called prepotentials drastically simplify the 
total system.  Even though we know that such off-shell 
formulation is much more advantageous than {\it on-shell} formulation,  
we do not have a complete off-shell formulation for {\it non-Abelian} tensor multiplets at the present time, even in the usual 
$~D=3+1$~ space-time \nrnatm.  For this reason, we do not attempt to give 
the off-shell formulation in superspace in this paper.  Instead we use 
the Bianchi identities in superspace, as a guiding principle 
for our {\it on-shell} formulation.  

Based on this principle, following the definitions of the 
$~F,~G$~ and $~H\-$field strengths in the 
component formulation (1.1), our corresponding superspace definitions 
are
% \footnotes{In superspace, we use the antisymmetrization symbols, 
% such as $~X_{\[ A B)} \equiv X_{A B} - (-1)^{A B} X_{B A}$~ {\it without} 
% the factor $~1/2$.}  
$$ \li{ F\du{A B} I \equiv & + \nabla_{\[A} A\du{B)} I 
		- T\du{A B} C A\du C I + g f^{I J K} A\du A J A\du B K  ~~,  
&(3.1\rma) \cr 
G\du{A B C} I \equiv & + \frac 12 \nabla_{\[A} B\du{C D)} I 
			- \frac 12 T\du{\[A B |} D B\du{ D | C)} I  
			- \frac 12 \fIJK C\du{\[ A} J F\du{ B C)} K ~~, 
&(3.1\rmb) \cr 
H\du{A B} I \equiv & + \nabla_{\[ A} C\du{B)} I - T\du{A B} C B\du C I 
		+ g B\du{A B} I ~~. 
&(3.1\rmc) \cr } $$ 
Correspondingly, our superspace 
Bianchi identities (BIds) for these superfield strengths are 
$$\li{ & +\frac 12 \nabla_{\[ A} F\du{ B C)} I  
		- \frac 12 T\du{\[ A B|} D F \du{ D| C)} I \equiv 0 ~~, 
&(3.2\rma) \cr 
& + \frac 16 \nabla_{\[ A} G\du{B C D ) } I 
			- \frac 14 T\du{\[ A B |} E G\du{E | C D)} I 
			- \frac 14 \fIJK F\du{\[ A B} J H\du{C D)} K \equiv 0 ~~, 
&(3.2\rmb) \cr 
& +\frac 12 \nabla_{\[ A} H\du{ B C )} I 
		- \frac 12 T\du{\[ A B |} D H\du{ D | C )} I 
		- g G\du{A B C} I \equiv 0 ~~.   
&(3.2\rmc) \cr } $$ 
These are nothing but the superspace generalization of the component case
(1.2).  These are also parallel to the non-self-dual formulation in 
$~D=3+1$~ \nrnatm.  
Since we have the corresponding non-dual case in $~D=3+1$, even though 
our formulation is {\it on-shell} formulation without prepotentials, the comparison with the $~D=3+1$~ case \nrnatm\ is straightforward.  

There are, however, differences in $~D=2+2$~ about chiralities of spinors, 
compared with $~D=3+1$~ in \nrnatm.    
A special treatment is needed for spinors in $~D=2+2$~\siegel\ngk\bergshoeffsezgin.  The most important feature is that 
{\it dotted} spinors are {\it independent} of {\it un-dotted} spinors.  This 
situation is different from the case of $~D=3+1$~ \nrnatm, where 
{\it dotted} spinors are just complex conjugate to {\it un-dotted} spinors 
%%%
\ref\ggrs{\ggrscont}.
%%%   
This gives certain differences compared with our result in $~D=3+1$~ \nrnatm.

Our superspace constraints at the engineering dimensions $~0 \le d \le 1$~ 
are
$$ \li{ T\du{\a\Dot\b} c = & + (\g^c)_{\a\Dot\b} ~~, ~~~~ 
			G\du{\a\Dot\b c} I = + (\g_c)_{\a\Dot\b} \, \varphi^I ~~, 
&(3.3\rma) \cr 
G\du{\a b c} I = & - (\g_{b c} \chi^I)_\a ~~, ~~~~
		F\du{\a b} I = - (\g_b \Bar\l{}^I)_\a ~~, ~~~~
		H\du{\a b} I =- (\g_b \Bar\r{}^I)_\a ~~, 
&(3.3\rmb) \cr 
\nabla_\a \varphi = & - \chi\du\a I ~~, 
&(3.3\rmc) \cr 
\Bar\nabla_{\Dot\a} \Bar\l\du{\Dot\b} I 
	= & + \frac 14 (\g^{c d})_{\a\b} F\du{a b} I ~~, 
&(3.3\rmd) \cr 
\Bar\nabla_{\Dot\a} \chi\du\b I = & - \frac 1{12} (\g^{c d e})_{\Dot\a\b} G\du{c d e} I 
						- \frac 12 (\g^c)_{\b\Dot\a} \nabla_c \varphi^I 
			 		\eqstar - (\g^c)_{\b\Dot\a} \nabla_c \varphi^I ~~, 
&(3.3\rme) \cr 
\nabla_\a\chi\du\b I = & - C_{\a\b} \fIJK (\Bar\l{}^J \Bar\r^K) ~~, 
&(3.3\rmf) \cr 
\Bar\nabla_{\Dot\a} \Bar\r\du{\Dot\b} I 
= & + \frac 14 (\g^{c d})_{\Dot\a\Dot\b} \, H\du{c d} I 
			+ g C_{\Dot\a\Dot\b} \,  \varphi^I ~~.  
&(3.3\rmg) \cr } $$ 
In (3.3e), the last equality is valid under the duality symmetry (1.5a).  
All other constraints with independent components, such as $~\nabla_\a
\Bar\r\du{\Dot\b} I$~ or $~\Bar\nabla_{\Dot\a} \varphi^I$, {\it etc.}~are all zero. 
In particular, $~\Bar\nabla_{\Dot\a} \varphi^I =0$~ implies that $~\varphi^I$~ is 
a {\it chiral} scalar superfield \ngk.  
These structures are very similar to the TM case in $~D=3+1$~ in \nrnatm.  
The only exceptions are such as the absence of fermionic bilinear terms, 
and coefficients such as those in (2.3a), (2.3d) or (2.3e) 
are half of the corresponding ones in \nrnatm.  These facts are the reflections 
of the chiral nature of our present system.  As usual in superspace, 
the constraints in (3.3) satisfy the BIds at $~0 \le d \le 1$.  

At dimension $~d = 3/2$, BIds (3.2) lead to 
$$ \li {\nabla_\a G\du{b c d} I = & - \frac 12 (\g_{\[ b c} \nabla_{d\]} \chi^I)_\a 
			- \frac 12 \fIJK(\g_{\[b | } \Bar\l^J)_\a H\du{| c d\]} K 
			+ \frac 12 \fIJK(\g_{\[b| } \Bar\r^J)_\a F\du{| c d\]} K~~, ~~~~~ ~~~ 
&(3.4\rma) \cr 
\nabla_\a H\du{b c} I = & + (\g_{\[b } \nabla_{c\]} \Bar\r{}^I)_\a 
				- g (\g_{b c}\chi^I)_\a ~~, ~~~~~ 
			\nabla_\a F\du{b c} I = + (\g_{\[b } \nabla_{c\]} \Bar\l{}^I)_\a ~~, 
&(3.4\rmb) \cr } $$ 
and the fermionic field equations\footnotes{We use the symbol $~\eqdot$~ for 
a field equation, or an ansatz for a solution.}  
$$\li{ (\nablasl\Bar\l^I)_\a & \eqdot 0 ~~, 
&(3.5\rma) \cr 
(\nablasl\Bar\r^I)_\a - 2 g \chi\du\a I & \eqdot 0 ~~, 
&(3.5\rmb) \cr 
(\nablasl\chi^I)_{\Dot\a} 
		- \frac 14 \fIJK (\g^{a b} \Bar\l^J)_{\Dot\a} \, H\du{a b} K
		+ \frac 14 \fIJK (\g^{a b} \Bar\r^J)_{\Dot\a} \, F\du{a b} K 
		+ g \fIJK \Bar\l\du{\Dot\a} J \varphi^K & \eqdot 0 ~~. ~~~~~ ~~~~~ 
&(3.5\rmc) \cr } $$  
Needless to say, these are consistent with the component results 
(2.3) and (2.4).  

Compared with the duality-less case in $~D=3+1$~ \nrnatm, the structures 
in (3.5) have differences as well as similarities.     
The similarity is the parallel structure of the  
constraints (3.3) to \nrnatm.  The difference is that our fermionic field equations
in (3.5) are much simpler, because of {\it chirality} associated with dualities, 
simplifying or deleting certain terms in these field equations.    
Compared with the $~D=3+1$~ case \nrnatm, our 
present system has {\it no} higher-order terms that are skipped in 
\nrnatm.  For example, $~\hbox{(fermion)}^2\-$terms are 
absent in (3.3d) and (3.3g), 
while in $~D=3+1$~ \nrnatm\ corresponding terms are present.  
This is nothing bizarre, considering the fact that each fermion has a  
definite chirality, so that possible terms are limited.  This can be rigorously 
confirmed in superspace than in component language, because Fierz rearrangements are more transparent.  

We next study various self-consistencies of our system.  
First, we can show the consistency of the anticommutators on 
the fermions: 
$$ \li{ \{ \nabla_\a , \Bar\nabla_{\Dot\b} \} \, \Bar\r\du{\Dot\g} I 
		= & + T\du{\a\Dot\b} c \, \nabla_c \Bar\r\du{\Dot\g} I 
		  + \fracm 12 C_{\Dot\b\Dot\g} (\nablasl \Bar\r{}^I - 2 g \chi^I)_\a  
		\eqdot + T\du{\a\Dot\b} c  \, \nabla_c \Bar\r\du{\Dot\g} I ~~, 
& (3.6\rma) \cr 
\{ \nabla_\a , \Bar\nabla_{\Dot\b} \}  \, \chi\du\g I 
= & + T\du{\a\Dot\b} c  \, \nabla_c \chi\du\g I 
			+ C_{\a\g} \Big[ \, \nablasl\chi^I 	
			- \frac 1 4 \fIJK (\g^{a b} \Bar\l{}^J) H\du{a b} K \cr 
& + \frac 1 4 \fIJK (\g^{a b} \Bar\r{}^J) F\du{a b} K	
			+ g \fIJK \Bar\l{}^J \varphi^K \, \Big]_{\Dot\b} 
			\eqdot \! + T\du{\a\Dot\b} c  \, \nabla_c \chi\du\g I  ~~, ~~~~~ ~~~
& (3.6\rmb) \cr 
\{ \nabla_\a , \Bar\nabla_{\Dot\b} \}  \, \Bar\l\du{\Dot\g} I 
= & + T\du{\a\Dot\b} c  \, \nabla_c \Bar\l\du{\Dot\g} I 
				+ \frac 12 C_{\Dot\b\Dot\g} (\nablasl\Bar\l{}^I)_\a
				\eqdot + T\du{\a\Dot\b} c  \, \nabla_c \Bar\l\du{\Dot\g} I  ~~,  
& (3.6\rmc) \cr } $$ 
where use is made of the fermionic field equations (3.5).  

Second, we can also re-obtain SD (1.5) from the fermionic 
field equations (3.5).  For example, we can re-obtain the SD of 
$~F$~ and the $~G$-$\nabla\varphi$~ duality from the $~\Bar\r\-$field equation: 
$$\li{ 0 \eqques & (\g^a)^{\a\Dot\b} \, \Bar\nabla_{\Dot\b}
		\left[ \, (\nablasl \Bar\r^I) - 2 g \chi^I \, \right]_\a  \cr 
\eqstar & - 2 g \left( \nabla^a \varphi^I 
		- \frac 16 \e^{a b c d} G\du{b c d} I \right) 
		+ \nabla_b \left( H^{a b \, I} - \frac 12 \e^{a b c d} H\du{c d} I \right) 
			\eqstar 0 ~~~~~ \qed ~~~~~ ~~
&(3.7) \cr }  $$ 
The symbol $~\eqstar$~ implies that we used the last expression 
of (3.3e).  Eq.$\,$(3.7) holds under (1.5b) and (1.5c).  
		 
Another example is for the self-dualities of $~F$~ and $~H\-$field strengths 
re-obtained from the $~\chi\-$field equation:  
$$\li{ 0 \eqques & (\g^{a b} )^{\a\Dot\b} 
		\Bar\nabla_{\Dot\a} \Big[ \, + (\nablasl\chi^I) 
			- \frac 14 \fIJK (\g^{c d} \Bar\l^J ) H\du{c d} K 
			+ \frac14 \fIJK (\g^{c d} \Bar\r^J ) F\du{c d} K 	
			+ g \fIJK \Bar\l\varphi^K \, \Big]_{\Dot\b} \cr 
\eqstar & \frac 12 \fIJK \left(F\du{ c \[ a |} J 
				- \frac 12 \e\du{c \[a |} {d e} F\du{d e} J \right) H\du{c | b\]} K 
		+ \frac 12 \fIJK F\du{c \[ a|} J \left( H\du{c | b\]} K  
						- \frac 12 \e\du{c | b\]}{d e} H\du{d e} K \right) 
		~~~~~  
&(3.8\rma) \cr 
\eqstar & 0 ~~~~~\qed 
&(3.8\rmb) \cr }  $$  
The symbol $~\eqstar$~ in (3.8a) implies that the last expression of (3.3e) 
is used.  Eq.~(3.8b) holds under the SD on $~F$~ and $~H$, as desired.  



\bigskip\bigskip


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% \newpage 
%: 4. Difficulty with Off-Shell Prepotential Formulation 
% \vbox{ 
\leftline{\bf 4.~~Difficulty with Off-Shell Prepotential Formulation} 
\nobreak 

One may wonder, whether we can use `off-shell' formulation in terms of 
prepotential superfield.  The advantage of off-shell prepotential superfields 
is that we can compare our results with the conventional system with tensor 
(linear) multiplets \original\superfield.  At least in 4D, 
all the prepotentials for our three multiplets for 
{\it Abelian} case have been already known \original\superfield\ggrs\ngk.   

However, there seems to exist some obstruction against such an idea 
for {\it non-Abelian} case.  
The main problem is caused by the following three features.  First, the 
tensor field carries the non-Abelian adjoint index whose superspace formulation 
has never been presented before.  Second, the usual CS-term of the form 
$~F\wedge A - (1/3) A\wedge A \wedge A$~ does {\it not} exist in our 
third-rank field strength (1.2b).  This feature is different from the known 
tensor multiplet in the {\it Abelian} case \superfield\ggrs.  
Third, there are different non-conventional CS-terms in the field strengths 
$~G\du{a b c} I$~ and $~H\du{a b} I$.  
For these reasons, even the usual basic relationship for the scalar superfield 
$~L$:  
$$\li{ & \Biglbracket \nabla_\a , \Bar\nabla_{\Dot\b} \Bigrbracket L 
		= c_1 \, \Big(\s^{c d e}\Big)_{\a\Dot\b} \, 
			\, G\du{c d e}{} + c_2 \tr (W_\a \Bar W_{\Dot\b} ) ~~ 
&(4.1) \cr} $$  
does {\it not} hold.  This is because the $~G\-$term on the right side is 
supposed to carry the adjoint index, while the second $~W\Bar W\-$term 
does {\it not}, due to the trace taken.  

One might think that the already-established `off-shell' prepotential formulation  
\original\superfield\ggrs\ should be applicable to {\it any interactions}.  
However, such an expectation 
is {\it not} valid, because we are dealing with a tensor multiplet with an  
{\it adjoint index}, which is beyond the scope of the conventional prepotential 
formulation for a tensor multiplet as a {\it singlet} of any gauge group.    
This is the reason why even off-shell prepotential formulation for the 
{\it Abelian} tensor multiplet does {\it not} work in the {\it non-Abelian} 
case. 

At the present time, we do not know how to overcome 
obstructions against an {\it off-shell} prepotential superfield formulation. 
The only way we can proceed is to rely on superspace Bianchi identities, 
as we have performed in the previous section, that can guarantee 
the consistency of our component formulation in section 2.  


\bigskip\bigskip


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% \newpage 
%: 5. Generating N=1 and N=2 Flows of Generalized KdV Eqs.
% \vbox{ 
\leftline{\bf 5.~~Generating $~\calN=2$~ and $~\calN=3$~ 
Flows of Generalized KdV Eqs.} 
\nobreak 

As applications of our SDNATM, we give the examples of embedding 
$~\calN=2$\footnotes{We use the symbol ~$~\calN$~ for these flows, 
in order to distinguish them from the number $~N$~ of supersymmetries.} and 
$~\calN=3$~ flows of generalized KdV eqs.  To this end, 
we perform the DR from the original $~D=2+2$~ 
into $~D=1+1$.  
For the original $~D=2+2$, we use the coordinates $~(z,x,y,t)$~ 
with the metric for $~D=2+2$~ \belavinetal: 
$$ \li{ & d s^2 = + 2 d z d x + 2 d y d t ~~.   
& (5.1) \cr} $$ 
The final $~D=1+1$~ has the coordinates $~(x,t)$.  
For simplicity sake, we truncate all the fermionic fields:   
$~\l^I \eqstar \chi^I \eqstar \r^I \eqstar 0$.  
We now see that the SD condition (1.5d) on $~F$~ is  
$$\li{&F_{x t} \eqstar 0~~, 
&(5.2\rma) \cr
&F_{y z} \eqstar 0~~,
&(5.2\rmb) \cr 
& F_{z x} \eqstar F_{t y} ~~,
&(5.2\rmc) \cr } $$ 
with $~\e^{z x y t} = +1$.  
Following the prescription in \belavinetal\ngntwo, 
we regard the YM filed components 
$~A_x$~ and $A_t$~ in $~D=2$~ as {\it pure gauge}:    
$$ A_x \eqstar A_t \eqstar 0 ~~,     
\eqno(5.3) $$    
due to (5.2a).  We also require the independence of all the quantities on the $~y$~
and $~z\-$coordinates: $~\partial_y \eqstar 0,~\partial_z \eqstar 0$, 
so (5.2b) and (5.2c) are equivalent to 
$$\li{& \[ P, B \] \eqstar O ~~, 
&(5.4\rma)  \cr 
&\Dot P + B\, ' \eqstar O ~~, 
&(5.4\rmb)  \cr} $$ 
where $~ P\equiv A_y,~ B\equiv A_z$, and
their {\it prime} and {\it dot} denote respectively the derivatives
$~\partial_x\equiv \partial/\partial x$~ and $~\partial_t\equiv
\partial/\partial t$.   

There are two remarks for the SD condition (1.5c):  First, 
since this SD shares the same index structure with (1.5d), we have the 
conditions parallel to (5.2):   
$$\li{&H_{x t} \eqstar 0~~, ~~~~H_{y z} \eqstar 0~~, ~~~~
			H_{z x} \eqstar H_{t y} ~~.   
&(5.5) \cr } $$  
Second, the $~g B\-$term in the field strength $~H$~ (1.1c) can absorb 
the first gradient terms $~\nabla C$, so that the $~C\-$field has no longer 
a dynamical field as a Stueckelberg field.  So (5.5) is equivalent to  
$$\li{& H_{x t} \eqstar B_{x t} \eqstar 0~~, ~~~~
		H_{y z} \eqstar  B_{y z} \eqstar 0~~, 
				~~~~ H_{z x} \eqstar  B_{z x} 
				\eqstar H_{t y}\eqstar B_{t y} ~~,   
&(5.6) \cr } $$  
where we put $~g=1$~ from now on for simplicity.  

We now perform the DR of the duality (1.5b).  Using also equations above, 
we get 
$$ \li{ \varphi ' \eqstar & + B\, ' _{y t} + \Dot B_{x y} ~~, 
&(5.7\rma) \cr 
\Dot \varphi \eqstar & + B\, '_{t z} + \Dot B_{z x} ~~, 
&(5.7\rmb) \cr 
\[ P, \varphi \] \eqstar & - \[ B, B_{x y} \] - \[ P, B_{z x} \] ~~,  
&(5.7\rmc) \cr 
\[ B, \varphi \] \eqstar & - \[ P, B_{t z} \] - \[  B, B_{y t} \] ~~.   
&(5.7\rmd) \cr } $$ 
For simplicity sake, we impose additional conditions 
$$\li{ & \varphi - B_{z x} \eqstar \varphi - B_{t y} \eqstar 0 ~~, ~~~~
				B_{t z} \eqstar 0 ~~,  
&(5.8) \cr } $$ 
so that (5.7b) and (5.7c) are satisfied.  Eventually, (5.7) is simplified to  
$$\li{ \[ P, X \] + \[ B, Y \] \eqstar & O ~~, 
&(5.9\rma) \cr 
X\, ' - \Dot Y \eqstar & O ~~,  
&(5.9\rmb) \cr } $$ 
where $~X \equiv + \varphi - B_{x z} \eqstar \varphi - B_{y t}, 
~Y\equiv + B_{x y}$.  
After all, the duality conditions in (1.5) are reduced to 
the four equations in (5.4) and (5.9).  

We next give some examples of integrable systems that are generalized by our 
SDNATM system.  As the first example, we show 
that the $~\calN=2, ~r=3$~ flow of the generalized KdV equations \kdv, {\it i.e.,} 
the original KdV equation:\footnotes{We use the symbol $~\eqstar$~ for a 
field equation, or an equality valid upon field equation(s).}    
$$ \li { & 4 \Dot u \eqdot + u ''' + 6 u u' = (u'' + 3 u^2)' 
&(5.10) \cr } $$ 
is embedded into (5.4) and (5.9).  Our ans\"atze for $~P, ~B, X$~ and 
$~Y$~ are
$$\li{ P \eqstar & \pmatrix{ 0 & 0 \cr u & 0} ~~, ~~~~
			B \eqstar - \fracm 14 \pmatrix{ 0 & 0 \cr u''+ 3u^2 & 0} ~~, 
&(5.11\rma) \cr 
X \eqstar & + \fracm 14 \pmatrix{ u'' + 3 u^2 & 0 \cr 0 & 0}~~, ~~~~
		Y \eqstar \pmatrix{u & 0 \cr 0 & 0} ~~.  
&(5.11\rmb) \cr } $$ 
Now eq.~(5.4) is easily satisfied by these $~P$~ and $~B$.  
As for (5.9b), it generates (5.10).  
As for (5.9a), the only non-trivial component is its $21\-$component, 
which also vanishes as $~(1/4) u (u''+3u^2) - (1/4) u (u''+3u^2)=0$.  
Note that this is a non-trivial result, 
because each of the commutators $~\[ P, X\]$~ and 
$~\[ B, Y \]$~ is non-zero.  This is also the reflection of the non-Abelian 
commutators in our SDNATM system, in particular, the non-Abelian 
couplings of TM to YM-field.  

Compared with the SDSYM case \ngntwo, where $~P$~ and $~B$~ were 
just $~1\times 1$~ matrices, our present system is less trivial, because of 
the new SD conditions (5.9).  In our present SDNATM system, 
the $~B$~ and $~P\-$matrices are less trivial $~2\times 2\-$matrices, but still 
the embedding is rather simple.  Also, 
our embedding is relatively simpler, compared with \ms\bd, 
in which a sophisticated $\, H$~ or $~Q\-$matrix is needed.  
Our present SDNATM system is simpler but still non-trivial at the 
same time.  This seems to be the result of the simplification played by the 
new TM, showing the advantage of SDNATM system.     

We next repeat a similar prescription for $~\calN=3, ~r=4$~ flow of generalized 
KdV equations \kdv: 
$$ \li{ 3 \Dot u_2 \eqdot & - u_2^{[4]} + 2 u_3''' - (u_2^2)'' + 4 (u_2 u_3)' ~~, 
&(5.12\rma) \cr 
9 \Dot u_3 \eqdot & - 2 u_2^{[5]} + 3 u_3^{[4]} - 6 u_2 u_2''' 
			-12 u_2' u_2'' - 4 u_2^2 u_2' + 6(u_2 u_3')' + 6(u_3^2)' ~~,  
&(5.12\rmb) \cr } $$ 
where $~{\scst \[n\]}$~ stands for the $~n\-$th derivative by $~\partial_x$.  
These are re-expressed as 
$$\li{ 3 \Dot u_2 \eqdot & \left[ \, - u_2''' + 2 u_3 '' - (u_2)^2 + 4 u_2 u_3 \, \right]' 
		\equiv 3 \left[  \, f(u_2,u_3) \, \right]'  ~,  
&(5.13\rma) \cr
9 \Dot u_3 \eqdot & \left[ \, - 2 u_2^{[4]} + 3 u''' - 3 (u_2')^2 - 6 u_2 u_2''
				- \frac 4 3 u_2^3 + 6 u_2 u_3' + 6(u_3)^2 \, \right]'
				\equiv + 9 \left[  \, g(u_2,u_3) \, \right] '  ~. ~~~~~ ~~~~~ ~~
&(5.13\rmb) \cr}  $$ 

Our ans\" atze for $~P,~B,~X$~ and $~Y$~ are in terms of 
$~4\times 4$~ matrices are 
$$\li{ P \eqstar & \pmatrix{O & O \cr \calU & O} ~, ~~~~
B \eqstar \pmatrix{ O & O \cr - {\cal F} & O} ~, ~~~~ 
X \eqstar \pmatrix{ {\cal F} & O \cr O & O} ~,  ~~~~ 
Y \eqstar \pmatrix{ {\cal U} & O \cr O & O } ~, ~~~~~ ~~~~~ 
&(5.14\rma) \cr
{\cal F} \equiv & \pmatrix{f & 0 \cr g & f} ~~, ~~~~
		{\cal U} \equiv \pmatrix{u_2 & 0 \cr u_3 & u_2} ~~,  
&(5.14\rmb) \cr }  $$ 
where $~f$~ and $~g$~ are given in (5.13), while 
$~\calF$~ and $~\calU$~ are $~2 \times 2$~ matrices.  
We can easily show that all the conditions (5.4) and (5.9) are satisfied 
by these ans\" atze.  In particular, the key relationship is the commutativity 
$~\[ \calU, \calF \] = O$.  
It seems that this kind of patterns can be repeated for higher hierarchies with 
larger $~\calN$~ and $~r$~ for generalized KdV equations \kdv.  

We have seen that the lower flows of generalized KdV equations \kdv\ 
can be embedded into our SDNATM system.  The most important ingredient  
is that the non-Abelian feature of TM is involved into the non-trivial embedding 
of these KdV equations, {\it via} commutators such as $~\[P, X \]$~ 
or $~\[ B, Y \]$.  Even though the presence of the TM seems to 
complicate the system, it simplifies the matrices of $~B$~ and $~P$~ 
compared with SDYM system \ms\bd, where a more complicated $~H\-$matrix is 
needed.  The embedding of KdV equations reveals the advantage 
of our SDNATM system over SDSYM system \siegel\ngk.  



\bigskip\bigskip 

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% \newpage 
%: 6. Relationship with N=2 SDSYM 
% \vbox{ 
\leftline{\bf 6.~~Relationship with $\,$N$\,$=$\,$2$\,$ SDSYM} 
\nobreak 

We can see that our system of SDNATM produces the same set of 
bosonic field equations as those by $~N=2$~ SDSYM theory in $~D=2+2$~
with the field content $~(A\du\m I, \l\du i I, T^I)$, where $~{\scst i~=~1,~2}~$
is the index for $N=2$~ supersymmetry.  Each of $~\l\du 1 I $~ and $~\l\du 2 I$~ 
are Majorana-Weyl spinor with negative chirality, and $~T^I$~ is a 
real scalar in the adjoint representation \ngk.  As shown in \ngk, when the DR into $~D=1+1$~ is performed, the set of bosonic conditions from $~N=2$~ 
SDSYM are (5.4) and  
$$ \li{ & \[ B , \, T\, ' \, \] + \[ P, \, \Dot T \] \eqstar 0 ~~,   
&(6.1) \cr } $$ 

We can show that the condition in (6.1) is equivalent to (5.9) 
arising in our SDNATM.  Let $~U(x,t)$~ be a scalar defined by 
$$ \li{ & U(x,t) \equiv + \int _{t_0}^ t d \t \, X (x, \t) ~~, 
&(6.2) \cr } $$ 
so that 
$$ \li{ & X = \fracmm{\partial U}{\partial t} = \Dot U ~~, ~~~~ 
		X' = \fracmm{\partial^2 U}{\partial x \, \partial t} ~~. 
&(6.3) \cr } $$ 
Integrating (5.9b) over time, we get  
$$ \li{ Y = & + \int_{t_0}^t d \t \, X' (x, \t) 
			= + \int_{t_0}^t d \t \, 
			\fracmm{\partial^2 U(x,\t) }{\partial x \, \partial \t} \cr 
= & + \int_{t_0}^t d \t \, \fracmm{\partial}{\partial\t} 
		\left[ \,  \fracmm{\partial (x,\t) }{\partial x} \, \right] 
		= + \fracmm{\partial U(x,t)}{\partial x }= + U ' ~~~~ 
		\Longrightarrow ~~~~ Y = U ' ~~ . 	
&(6.4) \cr } $$ 
Then (5.9a) is expressed in terms of $~U$~ as 
$$ \li{ & \[ B, U ' \] +\[ P, \Dot U\] \eqstar 0 ~~. 
&(6.5) \cr } $$ 
This is nothing but (6.1) with $~T$~ replaced by $~U$.   
In other words, our $~N=1$~ SDNATM generates the same bosonic 
conditions as $~N=2$~ SDSYM theory \ngk, despite simple supersymmetry 
$~N=1$~  in our system instead of extended $~N=2$~ in \ngk.    

This result is natural, because even though we have only $~N=1$~ 
supersymmetry, since the system of SDNATM is larger than $~N=1$~ SDSYM, 
the enlargement resulted in the equivalence to the {\it enhanced}  
supersymmetry from $~N=1$~ to $~N=2$, when a DR into $~D=1+1$~ is 
performed.       


\bigskip\bigskip 

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% \newpage 
%: 7. Concluding Remarks
% \vbox{ 
\leftline{\bf 7.~~Concluding Remarks} 
\nobreak 

In this paper, following the recent successful formulations 
of non-Abelian tensors \dws\dwns\ssw\nrnatm, we have first presented the component formulation of an $~N=1$~ SDNATM theory with 
non-trivial couplings to YMVM. 
Our system has three multiplets (i) YMVM $\, (A\du \m I, \l{}^I)$, (ii) 
NATM $\,(B\du{\m\n} I , \chi^I, \varphi^I)$, and ECVM $\, (C\du \m I, \r^I)$.   
Similarly to our recent formulation of $~N=1$~ 
TM in $~D=3+1$~ \nrnatm, we need the three multiplets of   
TM, VM and ECVM.  In particular, the ECVM is indispensable for the 
consistent couplings of TM to VM.  The usual YM field strength $~F\du{\m\n} I$, 
and the field strength $~H\du{\m\n} I $~ of the extra compensator vector 
$~C\du\m I$~ should be also self-dual, in order to accomplish the total consistency.  

As independent confirmation, we have also given superspace re-formulation, 
showing the consistency with the component formulation. 
Our superfield formulation is {\it on-shell} formulation based on the 
fundamental superfields VM $~(A\du a I, \Bar\l\du{\Dot\a}I)$,~SDTM $~(B\du{a b} I, \chi\du\a I, \varphi^I)~$ and ECVM $~(C\du a I, \Bar\r\du{\Dot\a}I)$.  
Even though this is {\it on-shell} formulation without prepotentials, 
this is the very first formulation in superspace for a self-dual tensor multiplet.   
This situation is similar to our superspace formulation in \nrnatm\ as the very first superspace formulation for a non-Abelian tensor multiplet.  As for the {\it off-shell} 
formulation, we leave it to future studies, due to non-trivial field strengths 
involved, and the prepotential formulation would be very involved.   

To our knowledge, combining non-Abelian TM with SD, \hbox{$~N=1$} supersymmetry and integrable models has not been 
entertained in the past literature.  We have given not only the component 
formulation, but also superspace re-formulation for the first time, as supporting evidence for the total consistency.  
The successful coupling of a tensor field with the adjoint index of a 
non-Abelian gauge group is based on the extra terms in the field strengths 
$~G$~ and $~H$~ inspired from the recent works \dws\dwns\ssw\nrnatm.  In particular, the extra compensator vector $~C\du\m I$~ in the ECVM 
serves as the Stueckelberg field to be absorbed into the longitudinal 
component of $~B\du{\m\n} I$.  This seems to imply 
that the Stueckelberg mechanism is inevitable for avoiding inconsistency 
by the na\itrema ve couplings of TM.  This feature is common both to 
$~D=3+1$~ \nrnatm\ and $~D=2+2$~ space-time dimensions.  

As applications, we have also given the examples of generalized KdV 
equations for the $~\calN=2,~r=3$~ and $~\calN=3, ~r=4$~ flows.  Our new duality 
symmetry (1.5a) and (1.5b) for the TM provides a set of non-trivial 
conditions (5.9), in addition to those with SDSYM with a pure VM \ngk.  
The embeddings into the $~P$~ and $~B\-$matrices given in \ngntwo\ 
were rather trivial, because they were only 
$~1\times 1\-$matrices, while in our present case, 
the matrices $~P$~ and $~B$~ are at least $~2\times 2\-$matrices.  

Our SDNATM system has 
much simpler embedding configurations, compared with SDSYM theories 
\ngk.  For example, we have seen that our original SDNATM has 
only $~N=1$~ supersymmetry, it generates in $~D=1+1$~ the same 
set of conditions produced by $~N=2$~ SDSYM \ngk.  
Of course, the price to be paid is the introduction of 
the new set of duality symmetry (1.5a) and (1.5b) resulting in (5.9), but it is compensated by the simplification of embedding.  Our configurations are much simpler and more straightforward than \ms\bd, but still non-trivial for generalized KdV equations \kdv.  

The work presented here initiates new directions of research on 
supersymmetric duality symmetry in $~D=2+2$, as well as in higher dimensions.  
To be specific, we can potentially generalize our result beyond 4D  
for {\it generalized} SD 
%%%
\ref\selfdual{\selfdualcont}%
%%% 
\ref\gtwo{\gtwocont}%
%%% 
\ref\reviews{\reviewscont}.  
%%% 
as follows.  
Our SD (1.5) is generalized to 
higher-dimensions in $~D$~ as  
$$\li{ F\du{\m\n}I \eqstar & + \fracm 12 \phi\du{\m\n}{\r\s} F\du{\r\s} I ~~, 
&(7.1\rma) \cr  
G\du{\m\n\r} I \eqstar & \! - \phi\du{\m\n\r} \s \nabla_\s \varphi^I ~~, 
&(7.1\rmb) \cr 
H\du{\m\n} I \eqstar & \! + \frac 12 \phi\du{\m\n}{\r\s} H\du{\r\s} I ~~, 
&(7.1\rmc)  \cr } $$ 
with an appropriate constant $~\phi\du{\m\n}{\r\s} $, {\it e.g.}, 
the octonion structure constant 
%%% 
\ref\octonions{\octonionscont}  
%%% 
in 8D for a reduced holonomy 
$~SO(7) \subset SO(8)$~ \selfdual\gtwo\reviews.   
%%%   
This kind of generalizations especially with non-Abelian 
tensors has become within our reach, after the successful 
formulations of non-Abelian tensors in 4D  
\dws\dwns\ssw\nrnatm.  

Technical details aside, the conceptual lessen we can learn from our present 
result is as follows.  The original Atiyah-Ward conjecture \atiyahward\ 
was that all the lower-dimensional bosonic integrable systems in $~D\le 3$~ 
are generated by SDYM theory in $~D=2+2$.  
In 1990's, this conjecture was further supersymmetrized to SDSYM  
systems \siegel\siegeleight\ngk.  Now it is the next natural step 
to consider the generalization of a SDYM to a SD non-Abelian tensor.  
We can further consider the higher-dimensional 
generalization of SD in 4D to 7D or 8D, based on the so-called 
{\it reduced holonomy} \selfdual\gtwo\reviews, as in (7.1).  
In other words, theories evolve from Abelian groups to non-Abelian groups, from non-supersymmetric to supersymmetric systems, from vectors to tensors, 
and from $~D=4$~ to $~D\ge 5$.  It is clear that our present result has 
historical implication contributing to the past accomplishments 
\selfdual\gtwo\reviews, as well as inducing future applications.  
We also emphasize that the generalization to non-Abelian tensor 
has been made possible, only after the success of NATM in $~D=3+1$~ 
\dws\dwns\ssw\nrnatm.  



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%: Acknowledgement 

\doit1{\bigskip 
We are indebted to the referees of this paper for important suggestions to 
improve the paper. 
This work is supported in part by Department of Energy 
grant \# DE-FG02-10ER41693.  
} 

\doit0{
We are grateful to W.~Siegel for important discussions, and reading 
the manuscript.}

\doit0{
We are grateful to the referees of this paper 
for important suggestions to improve our paper.  
Our research is   
} 

% \bigskip\bigskip\bigskip\bigskip\bigskip

\newpage 

%: References 

% For \listrefrmed we need 
\def\texttts#1{\small\texttt{#1}}

\listrefsrmed


% \listrefsr


\vfill\eject

\end{document} 

%: Worksheet

%%%%%%%%%% Worksheet %%%%%%%%%% 

\doit0{
Even aside from supersymmetry or duality, consistent interactions 
for non-Abelian tensors have been known for a while.  
For example, we have formulated in 
%%% 
\ref\nrnatensors{\nrnatensorscont} 
%%% 
that the introduction of additional higher-rank tensors 
solves the conventional problem with 
non-Abelian tensors.  Another example is generalizing the usual Poincar\' e 
algebra by infinitely many generators as in 
%%% 
\ref\savvidy{\savvidycont}.  
%%% 
However, these works did not accomplish the 
supersymmetric completion of proposed systems.   
It seems extremely difficult to formulate supersymmetric theory with consistent interactions for non-Abelian tensors.  
}

\doit0{We performed in \nrnatm\ the supersymmetrization of non-Abelian tensor {\it without} duality symmetry.  
We have succeeded in formulating a non-Abelian $~N=1$~ 
tensor multiplet (TM) in $~D=3+1$, overcoming the usual problem with 
non-Abelian tensor, following \dws\dwns\chu\ssw.  The main ingredient is the introduction of an extra compensator vector multiplet (ECVM), 
in addition to the non-trivial CS terms in the field strengths of 
$~B$~ and $~C$.  Especially, the extra compensator vector 
$~C\du a I$~ in the ECVM  
plays the role of a Stueckelberg field absorbed into the longitudinal 
component of $~B\du{a b} I$, making the latter massive.  
Encouraged by this result, we apply the same mechanism 
to the {\it self-dual} non-Abelian TM (SDNATM) in $~D=2+2$.  
} 



% \newpage 

%: 2. Preliminaries
\leftline{\bf 2.~~Preliminaries} 
\nobreak  

According to our result in \nrnatm\ based on 
the vector-tensor hierarchies in \dws\dwns\chu\ssw, 
the key ingredient to avoid inconsistencies is to introduce extra CS terms into the field strengths of $~B$~ and $~C$, as shown in (1.3).  In our present 
formulation for SDNATM, the field strengths for $~A\du a I,~B\du{a b} I$~ 
and $~C\du aI$~ are formally the same.\footnotes{Our anti-symmetrization 
rule is {\it e.g.,} $~X_{\[a b\]} \equiv X_{a b} - X_{b a}$~ {\it without} 
the factor $~1/2$.}  
$$\li{ F\du{a b} I \equiv & + \nabla_{\[a} A\du {b\]} I 
			- T\du{a b} c A\du c I 
			+ g f^{I J K} A\du a J A\du b K ~~, 
&(2.1\rma) \cr 
G\du{a b c} I \equiv & + \frac 12 \nabla_{\[ a} B\du{c d\]} I 
			- \frac 12 T\du{\[ a b |} d B\du{ d | c\]} I  
			- \frac 12 \fIJK C\du{\[ a} J F\du{b c\]} K ~~, 
&(2.1\rmb) \cr 
H\du{a b} I \equiv & + \nabla_{\[ a} C\du{b\]} I - T\du{a b} c B\du c I 
		+ g B\du{a b} I ~~, 
&(2.1\rmc) \cr } $$ 
where $~\nabla_a$~ is the usual YM covariant derivative, 
and $~g$~ is the non-Abelian coupling constant with the engineering 
dimension one.\footnotes{Our engineering dimensions are such as 
potential fields $~A\du a I,~B\du{a b} I$~ and $~C\du a I$~ all with 
zero dimensions, while fermions $~\Bar\l\du{\Dot\a}I,~\chi\du\a I$~ and 
$~\Bar\r\du{\Dot\a} I$~ with the dimension 1/2.  In a flat Minkowskian $~D=2+2$, the torsion component $~T\du{a b} c$~ is zero.  The inclusion 
of $~T\du{a b} c$~ in (2.1) and (2.2) is for suggestive purpose 
for superspace Bianchi identities (BI) later.}   
These field strengths, together with the usual YM field strength 
satisfy the Bianchi identities (BIds) 
$$\li{ & + \frac 12 \nabla_{\[ a} F\du{b c\]} I 
			- \frac 12 T\du{\[ a b|} d F\du{ d | c\]} I \equiv 0 ~~, 
&(2.2\rma) \cr 
& + \frac 16 \nabla_{\[ a} G\du{b c d \]} I 
			- \frac 14 T\du{\[ a b |} e G\du{e | c d\]} I 
			- \frac 14 \fIJK F\du{\[ a b} J H\du{c d\]} K \equiv 0 ~~, 
&(2.2\rmb) \cr 
& +\frac 12 \nabla_{\[ a} H\du{ b c)} I 
		- \frac 12 T\du{\[ a b |} d H\du{ d | c )} I 
		- g G\du{a b c} I \equiv 0 ~~.  
&(2.2\rmc) \cr } $$ 
The modified field strengths in (2.1) and the 
$~G$~ and $~H\-$BIds in (2.2) are 
formally the same as the purely bosonic sub-sector of the $~D=3+1$~ case 
\nrnatm.  

We now clarify the gauge transformation properties of these tensors.  
The foundation of these symmetries has been already 
given in \dws\dwns\ssw, but we repeat the relevant features here.  
First, the tensorial transformation associated with $~B$~ is defined on 
all fields by 
$$ \li{ & \d_\b (A\du a I, ~ B\du{a b} I, ~ C\du a I) 
		= ( 0, ~+ \nabla_{\[a } \b\du{b\]} I , ~- g \b\du a I ) ~~, 
&(2.3) \cr } $$ 
As in \dws\dwns\ssw, we need the non-trivial transformation for $~\d_\b C$.
Accordingly, the field strengths are invariant under $~\d_\b$:  
$$\li{ & \d_\b (F\du{a b} I, ~ G\du{a b c} I, ~H\du{a b} I) = (0, ~0, ~0)  ~~. 
&(2.4) \cr } $$ 
The extra CS-terms in (2.1b) and (2.1c) are 
indispensable for the invariance of these field strengths under the 
tensorial transformation $~\d_\b$~ \dws\dwns\ssw\nrnatm.  

Second, there is another gauge symmetry associated with 
$~C\du a I$~ called the $~\d_\g\-$transformation 
defined by\footnotes{Even though we use the 
symbol $~\g$~ which is the same as $~\g\,$-matrices, there will be no 
confusion based on its index $~{\scst I, ~J,~\cdots}$~ and on the context.}  
$$\li{ & \d_\g ( A\du a I, ~C\du a I , ~B\du{a b} I ) 
		= (0,~ + \nabla_a \g^I, ~ - \fIJK F\du{a b}J \g^K) ~~.
&(2.5) \cr } $$ 
As in the case of $~\d_\b$, the non-trivial transformation $~\d_\g B\du{a b} I$~ 
is needed.  Then the invariances follows    
$$\li{ & \d_\g (F\du{a b} I, ~ G\du{a b c} I, ~H\du{a b} I) = (0,~ 0, ~0)~~. 
&(2.6) \cr } $$ 

To summarize, the extra CS-terms in (2.1) are necessary to guarantee the 
invariance of the $~G$~ and $~H\-$field strengths both under $~\d_\b$~ and 
$~\d_\g\-$transformations \dws\dwns\chu\ssw\nrnatm.  It is also crucial that 
the duality symmetries (1.1) become valid, only when all the field 
strengths are invariant both under $~\d_\b$~ and $~\d_\g$.  

The physical meaning of these structures is clear, as in \dws\dwns\ssw\nrnatm.  
The extra compensator vector $~C\du a I$~ is playing a role of Stueckelberg field
%%%
\ref\stueckelberg{\stueckelbergcont}  
%%% 
absorbed into the longitudinal component of $~B\du{a b} I$.  
This is clear from (2.6), because 
the $~C\-$field is gauged away by the $~\d_\b\-$transformation.  
All of these seem to imply that the consistent non-Abelian tensor field 
exists, only if there is a Stueckelberg mechanism served by the 
extra compensator vector 
$~C\du a I$.  This also tells clearly why  
the extra compensator vector $~C\du a I$~ is needed.    



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