hep-th0404088.tex
%%%%% 0-F.F.S.new.tex %%%%%%%% 
\count100=1 

\documentstyle[12pt]{article}

\def\doit#1#2{\ifcase#1\or#2\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 letters
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

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

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

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

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

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

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

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

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

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

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

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
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        \else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous      % eqalignno
        \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{$\sp{#1)}$}
}

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%   (either)
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%   (LaTeX)
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%   (LaTeX)

% Title page
% \thispagestyle{empty}               % no heading or foot on title page 
%   (LaTeX)
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        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
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Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
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\def\endtitle{\end{quotation}\newpage}  % end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\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\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\calDsl{{\calD\!\!\!\! /}}
%\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}
%{\raise-0.03em\hbox{$_+$}}
%{\hskip 0.05pt}}{}{\hskip0.03em}}} 
%\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}
%{\raise-0.05em\hbox{$_-$}}
%{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\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.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\jhep#1#2#3{Jour.~High Energy Phys.~{\bf {#1}} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} }
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}}
%\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
%\def\hepph#1{hep-ph/{#1}} 
\def\hepth#1{\texttt{hep-th/{#1}}} 
\def\heplat#1{\texttt{hep-lat/{#1}}} 
\def\hepph#1{\texttt{hep-ph/{#1}}} 
\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.~and 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{Jour.~High Energy Phys.~{\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\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}
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%%%%%%%%% Worksheet %%%%%%%%%%%%%
% \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \thispagestyle{empty}

\doit0{
{\bf Preliminary Version (FOR YOUR EYES
ONLY!)\hfill\today
} \\[-0.25in] 
%{\hfill\today} 
\\[-0.3in]  
}

{\hbox to\hsize{\hfill
hep-th/0404088}} 
\vskip -0.06in 
{\hbox to\hsize{\hfill CSULB--PA--04--3}} 
\vskip -0.18in 
\hfill 
{(Revised Version)} 
\\ 

\begin{center} 

\vskip 0.03in 

{\Large\bf Hodge Duality and Cosmological Constant} 
\\  
%{\Large\bf Vector Multiplet in 3D}% 
{\hskip 0.01in}%
%\footnote{Work supported 
%in part by NSF Grant \# 0308246}
\\    [.1in] 

\baselineskip 9pt 

\vskip 0.36in 

Hitoshi ~N{\smallcmr ISHINO}%
\footnote{E-Mail: hnishino@csulb.edu}~  and 
~Subhash ~R{\smallcmr AJPOOT}%
\footnote{E-Mail: rajpoot@csulb.edu} 
%~and ~ 
%Bradley G{\smallcmr OLD} 
\\[.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 1.7 in 

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

\vskip 0.1in 

\baselineskip 14pt 


~~~We present a simple mechanism to eliminate cosmological constants
both in supersymmetric and non-supersymmetric theories.  This
mechanism is based on the Hodge (Poincar\'e) duality between a
$\,0\-$form and $\, D\-$form field strengths in $\, D\-$dimensional
space-time.  The new key ingredient is the introduction of an extra
Chern-Simons term into the  
$\, D\-$form field strength $~H$~ dual to the $\,0\-$form field strength. 
Our formulation can be also made consistent with supersymmetry. 
Typical applications to  four-dimensional $~N=1$~ supergravity and to
ten-dimensional type IIA  supergravity are given.  The success  of our
formulation for both supersymmetric and non-supersymmetric systems
strongly indicates the validity of our mechanism even after
supersymmetry breakings at the classical level.  
Our mechanism may well be applicable to quantized systems, 
at least for supersymmetric cases with 
fundamental D-brane actions available.  


\vskip 0.55in

\leftline{\small PACS: 98.80.Hw, 04.65.+e, 12.60.Jv, 04.20.Gz} 
\vskip -0.00in 
\leftline{\small Key Words:  Cosmological Constant, 
General Relativity, Supergravity, Supersymmetry,} 
\vskip -0.03in 
\leftline{{\hskip 0.8in} \small Four-Dimensions, Ten-Dimensions}   
\vskip -0.05in 
\vfill\eject 

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\footskip=0.36in 
\def\baselinestretch{0.8} 
%\footheight=1in 
%\bottomfraction=.25

\pageno=2 


\leftline{\bf 1.~~Introduction}  

The smallness or zero-ness 
of the cosmological constant to the accuracy 
of $~10^{-120}$~ in the dimension of $~\hbox{(mass)}^4$~ has
been mystifying theoretical particle physicists since its inception  
%%%
\ref\weinberg{{\it For classic and recent reviews, see e.g.,} 
S.~Weinberg, \rmp{61}{89}{1}; 
E.~Witten, {\it `The Cosmological Constant from the 
Viewpoint of String Theory'}, Lecture at DM2000, 
hep-ph/0002297;  S.M.~Carroll, Living Rev.~Rel.~{\bf 4} (2001) 1,
astro-ph/0004075.}. 
%%% 
The problem is not whether we {\it can} have a model with a zero
or extremely small cosmological  constant, but that we {\it must}
always perform `artificial' fine-tunings with the accuracy of
$~10^{-120}$~ to remove the cosmological constant both at
classical and quantum levels.  To date, there seems to be no
satisfactory mechanism for its explanation, either in fundamental
high energy theories including superstring theory 
%%% 
\ref\gsw{M.~Green, J.H.~Schwarz and E.~Witten, {\it
`Superstring Theory'}, Vols.~{\bf I} and {\bf II}, 
Cambridge University Press (1987).}, 
%%%
M-theory 
%%% 
\ref\mtheory{C.~Hull and P.K.~Townsend,
\np{438}{95}{109}, \hepth{9410167}; 
E.~Witten, \np{443}{95}{85}, \hepth{9503124}; 
P.K.~Townsend, {\it `M-theory from its Superalgebra'},
\hepth{9712004}; T.~Banks, W.~Fischler, S.H.~Shenker
and L.~Susskind, \pr{55}{97}{5112}, \hepth{9610043}.},  
%%% 
or in cosmological models \weinberg.  
Supergravity theory with local supersymmetry 
%%%
\ref\pvn{P.~van Nieuwenhuizen, \prep{68}{81}{189}.} 
%%%
was once expected to provide a nice mechanism of fine-tuning the
cosmological constant to be exactly zero.  
However, it turned out that supergravity theory also
needs  an artificial fine-tuning  
%%%
\ref\dz{S.~Deser and B.~Zumino, \prl{38}{77}{1433}.}%
%%% 
\ref\matter{E.~Cremmer, S.~Ferrara, L.~Girardello 
and A.~van Proeyen, \np{212}{83}{413}.}, 
%%% 
even though the quantum divergences in such theories  
are better controlled by supersymmetry.  
The dilaton scale invariance common in supergravity 
originated from string tree level invariance
%%% 
\ref\wittendilaton{E.~Witten, \pl{155}{85}{151}.} 
%%% 
was also expected to play an important role.  
But the questions are whether such a tree-level symmetry is 
maintained at higher-loops, and how to avoid a massless dilaton
\gsw.  Some attempts have also been made in 
quantum cosmology, based on baby universes or wormholes  
%%%
\ref\wormholes{S.W.~Hawking, \pl{195}{87}{337}; \pr{37}{88}{904}; 
G.~Lavrelashvili, V.~Rubakov and P.~Tinyakov, 
JETP Lett.~{\bf 46} (1987) 167;
S.~Giddings and A.~Strominger, \np{306}{88}{867}; 
\ibid{B307}{88}{854}; S.~Coleman, \np{307}{88}{867}.}, 
%%%
but there seems to be the difficulty of 
forbidding the creation of undesirably `big' baby universes
%%%
\ref\klebanov{T.~Banks, I.~Klebanov 
and L.~Susskind, \np{317}{89}{665};
I.~Klebanov and L.~Susskind, {\it `DPF Conf.~1988'}, Talk 
at Div.~of Particles and Fields of APS, Storrs, CT, 1988, p.~786.}. 

Ever since Einstein's blunder 
%%% 
\ref\einstein{A.~Einstein, Sizungsber, d.~Preuss, 
Akad.~d.~Wissensch., Pt.~1, 433 (1919), English translation:
H.A.~Lorentz, A.~Einstein {\it et al.}, {\it `The Principle of Relativity'}, 
Dover, NY (1952).},
%%% 
the cosmological constant has been an 
enigma in general relativity.  Lagrangian formulation of general
relativity admits it, while no known symmetry forbids it, and up until
recently, it was not required empirically.   Recent Type Ia supernova
observation 
%%% 
\ref\supernove{A.G.~Riess {\it et al.}, 
Astron.~J.~{\bf 116} (1998) 1009; 
P.M.~Garnavich {\it et al.}, Astrophys.~J.~{\bf 509} (1998) 74; 
S.~Perlmutter {\it et al.}, Astrophys.~J.~{\bf 517} (1999) 565.}   
%%% 
provides evidence that the universe is accelerating at a  greater
rate now than in the past, and implies a non-zero cosmological
constant $~\L\neq 0$.  An interesting implication of this is that
energy density $~\O_\L$~ associated with non-zero $~\L$~ is of
the same order of  magnitude as the matter density of the
universe, giving rise to the so-called second cosmological problem. 
This has led to a flurry of activity  explaining the two cosmological
problems, and involves the anthropic  principle \weinberg% 
%%% 
\ref\kane{G.L.~Kane, M.J.~Perry, A.N.~Zytkow, 
New Astron.~{\bf 7} (2002) 45, astro-ph/0001197 },
%%% 
quintessence 
%%%
\ref\quintessence{P.J.E.~Peebles and B.~Ratra, 
Astrophys.~J.~{\bf 325} 
(1988) L17; B.~Ratra and P.J.E.~Peebles: \pr{37}{88}{3406}; 
C.~Wetterich, Nucl.~Phys.~{\bf B302} (1988) 668.},  
%%%
new interactions, extra dimensions, phase  transitions, and
space-time fluctuations.  However, more data are  required
before definite conclusions can be drawn.   In our present work,
we will assume $~\L=0$, and present a dynamical mechanism that 
uniquely determines the value $~\L=0$.  

Understanding the zero-ness or smallness of the cosmological constant 
has been also attempted in terms of 
`unimodular gravity' formulation \einstein%
%%% 
\ref\unigr{J.~Anderson, D.~Finkelstein, Amer.~J.~Phys.~{\bf 39} (8) (1971)
901; 
J.J.~van der Bij, H.~van Dam,  Y.J.~Ng, Physica {\bf A116} (1982) 307; 
F.~Wilczek, \pr{104}{84}{111};
A.~Zee, in S.L.~Mintz and A.~Perlmutter (Eds.), {\it `Proceedings of 
the Twentieth Annual Orbis Scientiae on High Energy Physics'}, 
Plenum, NY (1985); 
W.~Buchm\"uller and N.~Dragon, \pl{207}{88}{292};
W.G.~Unruh, \pr{40}{89}{1048};
W.G.~Unruh and R.M.~Wald, \pr{40}{89}{2598};
M.~Henneux and C.~Teitelboim, \pl{222}{89}{481}.} 
%%% 
or its supersymmetric version `unimodular supergravity'
%%% 
\ref\unisg{H.~Nishino and S.~Rajpoot, \hepth{0107202}, 
\pln{528}{02}{259}.}.   
In these formulations, the zero-ness or smallness of  the
cosmological constant $~\L$~ is less serious, because the value of
$~\L$~ is no longer something put in by hand, but is regarded 
as an `initial condition'.  However, we still have  to input such a
fine-tuned initial condition at the beginning, based on a  special
constraint $~\hbox{det}~(e\du\m m) = 1$~ which seems rather
{\it ad hoc}, unfounded or artificial.  

One important development in the 1990's is the discovery of 
dualities playing crucial roles in string physics,  
such as S-, T- and U-dualities 
%%% 
\ref\dualities{{\it See, e.g.,}  
C.~Vafa and E.~Witten, \np{431}{94}{3}, \hepth{9408074}; 
\ibid{B447}{95}{261}, \hepth{9505053};  
E.~Alvarez, L.~Alvarez-Gaume and Y.~Lozano, 
Nucl.~Phys.~Proc.~Suppl.~{\bf 41} (1995) 1, \hepth{9410237}; 
J.H.~Schwarz, Nucl.~Phys.~Proc.~\newline  
Suppl.~{\bf 55B} (1997) 1, \hepth{9607201};  
A.~Giveon, M.~Porrati and E.~Rabinovici, 
\prep{244}{94}{77}, \hepth{9401139};   
N.A.~Obers, B.~Pioline, \prep{318}{99}{113}, \hepth{9809039}; 
S.~Forste, Fort.~Phys.~{\bf 50} (2002) 221, \hepth{0110055}; 
{\it and references therein}.}. 
%%% 
Dualities are similar to symmetries, but they have 
distinct features, such as describing different phases 
of coupling constants.  It is a general conviction that 
the cosmological constant problem may be solved by 
invoking symmetries or something similar.  Therefore  
it is a natural step  
to consider certain dualities as a trial solution 
to this problem.  

For example, a scenario of supersymmetry breaking 
maintaining the zero cosmological constant in 
three-dimensions (3D) with conical singularities and 
strong/weak duality 
%%% 
\ref\wittensw{E.~Witten, \np{443}{95}{85}, \hepth{9503124}.}.  
%%% 
was proposed by Witten
%%% 
\ref\witten{E.~Witten, \ijmp{10}{95}{1247}, \hepth{9409111}; 
\mpl{10}{95}{2153}, \hepth{9506101}.}.  
%%%  
Such a conical space-time has a deficit angle, lifting the
degeneracy between bosons and fermions.  
An explicit model in 3D has been presented based on 3D,
$N=2$~ supergravity, and mechanisms applicable to 4D based on
strong/weak coupling duality have been also proposed
%%% 
\witten\ref\bbsnishino{K.~Becker, M.~Becker and
A.~Strominger, \pr{51}{95}{6603}; H.~Nishino, \pl{370}{96}{65}.}.  
%%% 
However, it is not clear how such duality with 3D can help for
building realistic models with required Lorentz invariance in 
4D.   

The duality we rely on in our present paper is the so-called Hodge
(Poincar\'e) duality, {\it i.e.,} the relationship between the
$~r\-$form  and $~(D-r)\-$form field strengths in $~D$~
space-time  dimensions.   
Typical examples are  
$~\partial_\m \varphi = (1/3!) e^{-1} \e\du\m{\n\r\s}G_{\n\r\s}$~
between the 1-form $~d \varphi$~  and the 3-form 
$~G \equiv 3 d B$~ field strengths in 4D, 
or 3-form vs.~7-form field strengths in 10D  
%%% 
\ref\ng{J.~Gates, Jr.~and H.~Nishino, 
\pl{157}{85}{157}; \ibid{173B}{86}{46 \& 52}; 
\np{282}{87}{1}; \ibid{B291}{87}{205}; H.~Nishino and S.J.~Gates, Jr., 
\pl{189}{87}{45}; H.~Nishino, \pl{258}{91}{104}.}.  
%%% 

The Hodge (Poincar\'e) duality used in this paper is 
between a $~0\-$form and a $~D\-$form in 
$~D$~ space-time dimensions.  
The `$0\-$form field  strength' is understood better in mathematical
terms:  Let $~\a$~ be a `$0\-$form field strength'.  Since it is a
field  strength, it must satisfy a `Bianchi identity' 
$~d \a =0$.  As $~\a$~ is a $~0\-$form, the scalar field
$~\a$~ is  nothing but a constant.  Even though this fact 
itself seems to have poor content, its significance will be 
gradually elucidated in this paper.  Let the cosmological constant term  
in a lagrangian in $~D\-$dimensions be  
$$ \li{ & {\sqrt{-g}} \, \L \equiv \frac 12 a \, m^2 \, e~~, 
&(1.1) \cr } $$ 
where $~a$~ is a non-zero numerical constant.    
We emphasize here that $~\L$~ can contains not only 
`classical' cosmological constant, but also `quantum' 
cosmological constant. 
We use this notation (1.1), in order 
to follow the common usage in supergravity theories in 
diverse dimensions
%%% 
\ref\ss{{\it `Supergravity in 
Diverse Dimensions'}, {\it eds.}~A.~Salam and E.~Sezgin  
(North-Holland/World-Scientific, 1989).}.  
%%% 
We next regard $~m$~ as such a `$0\-$form field strength'.  
Now we invoke the duality relationship between 
a $~0\-$form $~m$~ and a maximal-rank 
$~D\-$form field strength $~H$~ to be  
$$ \li{ & m = \fracm 1{D!} e^{-1} \e^{\m_1\cdots\m\low D} 
      H_{\m_1\cdots\m\low D} ~~,  
&(1.2) \cr } $$ 
where $~H = D d C +\,$`Chern-Simons', where $~C$~ is 
a $~(D-1)\-$form potential.  Even though these points are similar to 
the duality transformation dictated 
%%% 
\ref\nt{H.~Nicolai and P.K.~Townsend, \pl{98}{81}{257}.},   
%%% 
there is an essential difference.  In our system, the 
existence of the Chern-Simons term in the field strength $~H$~ 
plays an important role, in addition to the conventional 
duality transformation \nt.  

Introducing maximal-rank field strengths is not a new 
idea.  For example, the compactifications 
%%% 
\ref\fr{P.G.O.~Freund and M.A.~Rubin, \pl{97}{80}{233}.} 
%%% 
of 11D supergravity 
%%% 
\ref\cjs{E.~Cremmer, B.~Julia and J.~Scherk,  \pl{76}{78}{409}.}
%%% 
or $~N=8$~ supergravity in 4D
%%% 
\ref\cj{E.~Cremmer and  B.~Julia, \np{159}{79}{141}.}   
%%% 
have motivated the introduction of  4-form field strength 
$~F_{\m\n\r\s}$~  
%%% 
\ref\ant{A.~Aurilia, H.~Nicolai and P.K.~Townsend, 
\np{176}{80}{509}.}.  
%%% 
However, our formulation differs from these 
previous ones by the additional `Chern-Simons term' 
within the maximal-rank field strength $~H$.  

In this paper, we will show that maintaining this duality with 
the $~H\-$field strength including the extra Chern-Simons term at the 
lagrangian level excludes the possible non-zero cosmological 
constant at the field equation level.  
This mechanism works for both supersymmetric and 
non-supersymmetric systems.  As interesting applications, we 
present as examples $~N=1$~ supergravity in 4D \dz\ and 
massive type IIA supergravity in 10D 
%%% 
\ref\romans{L.~Romans, \pl{169}{86}{374}.}%
%%% 
\ref\typeiibrane{
%E.~Bergshoeff, M.~Green, 
%G.~Papadopoulos and P.K.~Townsend, \hepth{9511079};
E.~Bergshoeff, M.~de Roo, M.~Green, 
G.~Papadopoulos and P.K.~Townsend,
\np{470}{96}{113}, \hepth{9601150}.}%  
%%% 
\ref\supereight{H.~Nishino, \hepth{9901027}, \pl{457}{99}{51}.}.    
%%% 




\bigskip\bigskip\medskip 

% \newpage 



\leftline{\bf 2.~General Mechanism in Arbitrary 
$D\-$Dimensions}   

Consider a general $~D\-$dimensional space-time, and 
let $~\Lag_0$~ be any lagrangian with the Hilbert 
action term: $-(1/4) e R$.  Separate the cosmological 
constant term $~e\L$~ from $~\Lag_0$, as\footnotew{We use 
the metric $~(\eta_{m n}) = \hbox{diag.}~(+, -, -, \cdots, -)$~ 
in this paper.}  
$$\li{ &\Lag_1 \equiv \Lag_0 + e\L 
     \equiv \Lag_0 + \frac 12 a m^2 e ~~, 
&(2.1) \cr } $$
where $~a m^2/2 \equiv \L$~ as in (1.1).   
We now perform the duality transformation from the 
zero-form $~m$~ to the $~D\-$form field strength $~H$.  
To this end, we first rewrite $~m$~ as the scalar field $~M(x)$, 
and then add the constraint lagrangian dictating the constancy 
of $~M$~ \nt:
$$\li{ & \Lag_{\rm tot} \equiv \Lag_0 + \frac 12 a e M^2  
     + \Lag_{\rm c} ~~, 
&(2.2\rma) \cr 
&\Lag_{\rm c} \equiv 
     -  \frac1{D!} \, a \, \e^{\m_1\cdots\m\low D}  \, M \, 
     H_{\m_1\cdots\m\low D} ~~,  
&(2.2\rmb) \cr 
& H_{\m_1\cdots\m\low D} \equiv 
     D \, \partial_{\[\m_1} C_{\m_2\cdots\m\low D\] } 
    + M \, K_{\m_1\cdots\m\low D}~~. 
&(2.2\rmc) \cr} $$ 
Compared with \nt, the last term in (2.2c) is 
a`generalized' Chern-Simons term.  
The $~K$~ is a $~D\-$form potential field whose $~(D+1)\-$form
field strength is `formally' defined, but it does not actually 
exist in $~x\-$space-time, due to its over-rank $~D+1>D$.  
The $~M K\-$term in (2.2c) is a 
generalized Chern-Simons term for the following reasons.  
First, it is a product of a `0-form' field strength $~M$~ and a $~D\-$form 
potential $~K$.  Second, the exterior derivative 
of (2.2c) formally gives a product of a $~0\-$form 
and a $~(D+1)\-$form `field strengths':
$$ \li{ & (D+1) d H = M L~~, 
&(2.3) \cr } $$ 
where $~L \equiv (D+1) d K$~ is the $~(D+1)\-$form field strength:  
$~L_{\m_1\cdots\m\low{D+1}} 
\equiv (D+1) \partial_{\[\m_1} 
K_{\m_2 \cdots \m\low{D+1}\]}$.  

The Chern-Simons term in (2.2c) is motivated 
by the superspace formulation \supereight\ of massive 
Type IIA supergravity \typeiibrane\romans.   
In this sense, the Chern-Simons term in (2.2c) is not artificially 
put in by hand, but has been well-motivated in terms of local 
supersymmetry in superspace \typeiibrane\supereight.    

The $~C\-$field equation is now 
$$ \li{ & \partial_\m M = 0 ~~\Longrightarrow ~~M = \hbox{const.}~~, 
&(2.4) \cr } $$ 
while the $~M\-$field equation yields\footnotew{The index  
$~{\scst \[n\]}$~ stands for totally antisymmetrized 
$~n$~ indices in order to save space, {\it e.g.,} $~\e^{\[D\]} K_{\[D\]} 
\equiv\e^{\m_1\cdots\m\low D}  K_{\m_1\cdots\m\low D}$.}   
$$\li{ & M  \eqdot + \frac 1{D!} e^{-1} \e^{\[D\]} H_{\[D\]} 
    + \frac 1{D!} e^{-1} \e^{\[D\]} M K_{\[D\]} ~~.
&(2.5) \cr } $$ 
Our new feature is that the $~K\-$field equation yields the 
zero cosmological constant 
$$\li{ & M \eqdot 0 ~~\Longrightarrow ~~ \L \eqdot 0 ~~.
&(2.6) \cr} $$ 
Therefore, the last term in (2.5) also vanishes, while yielding the 
desirable duality $~M  \eqdot + (1/D!)
e^{-1} \e^{\[D\]} H_{\[D\]}$.  

We are using the Hodge duality with the extra Chern-Simons 
term $~M K$~ in $~H~$ as the important guiding principle.  
There might be many other mechanisms 
yielding the same effect to remove the cosmological constant.  
For example, the $~C\-$field in (2.2c) can be absorbed into the field
redefinition of the $~K\-$field.  By the same token, a lagrangian term 
$~a m\Tilde C$~ with a scalar density $~\Tilde K$~ dual to $~K$~ 
can set $~m=0$.  Furthermore, an extreme method is
just to put $~\L=0$~ `by hand' from the outset.  However,
these mechanisms are not well founded by any principles, geometrical
formulations, or dynamical field equation to yield the desired results.  
In this context, we re-emphasize the importance of 
$~0\-$form $~D\-$form Hodge duality with the extra Chern-Simons 
term $~M K$~ in the field strength $~H$.  

As is easily seen, our mechanism is based on a 
very general form of cosmological constant in (2.1).  
Therefore, our formulation is also applicable 
a system with spontaneous symmetry breakings, 
such as $~SU(2)_{\rm W} \times U(1)_Y\rightarrow U(1)_{\rm em}$.  
Moreover, our mechanism 
may well be compatible with quantized systems, as long as 
the cosmological constant 
$~\L$~ includes also quantum effects with loop corrections.        
We will be back to this point shortly.  


\bigskip\bigskip\bigskip

% \newpage 

\leftline{\bf 3.~~Application to 4D Supergravity} 

As the most important application of our mechanism, 
we use it for $~N=1$~ supergravity in 4D.  For simplicity, 
we adopt the formulation of massive gravitino with a 
cosmological constant \dz, with no matter multiplet.    
We start with the lagrangian 
$$ \li{ \Lag_{\rm 4D} 
     = \, & - \frac 14 e R 
     - \frac i 2 e (\Bar\psi_\m \g^{\m\n\r} D_\n \psi_\r ) 
       - \frac 12 M (\Bar\psi_\m\g^{\m\n} \psi_\n ) 
       + \frac3 2 e M^2 
       -  \frac 1 8 \, \e^{\m\n\r\s} M  H_{\m\n\r\s}   
      {~~, ~~~~~ ~~~} 
&(3.1) \cr } $$ 
where for $~D=4$~ and $~a=3$~ in (2.2):  
$$ \li{ & H_{\m\n\r\s} 
     \equiv 4 \partial_{\[\m}  C_{\n\r\s\]}  + M K_{\m\n\r\s} ~~.
&(3.2) \cr } $$ 
All the terms in (3.1) except the last one are exactly the same as 
the original $~N=1$~ supergravity lagrangian with a cosmological 
constant \dz\ with the mass parameter $~m$~ replaced by $~M$.  

The action $~I_{\rm 4D} \equiv \int d^4 x \, \Lag_{\rm 4D}$~ 
is invariant under $~N=1$~ supersymmetry 
$$ \li{ & \d_Q e\du\m m = -i  (\Bar\e\g^m\psi_\m) ~~, ~~~~  
      \d_Q\psi_\m = D_\m \e + \frac i 2 M \g_\m \e ~~, \cr 
&\d_Q C_{\m\n\r} = + i (\Bar \e \g\low 5\g_{\[\m \n} \psi_{\r\]} )  ~~,  
     ~~~~ \d_Q K_{\m\n\r\s} = 0 ~~,  ~~~~ \d_Q M = 0 ~~.    
&(3.3) \cr } $$ 
The transformation $~\d_Q C$~ is rather easily fixed, in such a  way that all
the $~\partial M\-$dependent terms are cancelled  by $~\d_Q	 C$~ {\it via}
the $~M H\-$term.  The fact that the 
$~M\-$field is invariant under supersymmetry poses no problem,  due to
its constancy.  Similarly, as will be seen,  all the effect of the $~K\-$field   
disappears from the whole field equations justifies the invariance  of
$~K$~ under supersymmetry.    

We can verify the closure of supersymmetry, 
as in the conventional system.  In particular, the closure on 
the $~C\-$field is easily verified, up to a local gauge transformation 
$~\d_\l C_{\m\n\r\s} = 4 \partial_{\[\m} \l_{\n\r\s\]} $~ \nt\typeiibrane. 

The new feature of our system compared with \nt\ is the 
existence of the extra Chern-Simons term with the $~K\-$field.  
In fact, the $~K\-$field equation yields 
$$ \li { & M \eqdot 0 ~~\Longrightarrow ~~ \L \eqdot 0 ~~. 
&(3.4) \cr } $$ 
Once this is satisfied, the $~C\-$field equation 
$$ \li{ & M \eqdot \hbox{const.}~~
&(3.5) \cr } $$ 
is automatically satisfied.  
Accordingly, the $~M\-$field equation is simply 
$$ \li{& M 
      \eqdot + \frac 1{24} e^{-1} \e^{\m\n\r\s} \Hat H_{\m\n\r\s}  ~~, 
&(3.5) \cr } $$ 
where $~\Hat H$~ is the usual supercovariantized field strength 
\pvn.  All the fermion bilinear terms in (3.4) are completely absorbed into 
the supercovariantization of the field strength $~H$, which also 
reconfirms the transformation rule $~\d_Q C$.  
Note that we skipped the $~M K\-$term like that in (2.5), because 
of the vanishing of  the $~M\-$field itself.  Needless to say, due to 
the vanishing $~M\-$field, each side of (3.5) vanish on-shell.  
The on-shell duality relationship (3.5) is also consistent with 
supersymmetry, just as in the usual duality transformation \nt.      

If we do not include the Chern-Simons term $~M K$~ in $~H$, 
and apply the usual duality transformation in \nt\  
to the supergravity system in 4D \dz, 
the scalar field $~M$~ in the cosmological constant term 
$~3 e M^2 /2$~ 
can take any arbitrary value, as in \nt\typeiibrane.  In our 
formulation, however, the dynamical $~K\-$field equation {\it uniquely} 
fixes $~\L \eqdot 0$.    

We have thus seen that our mechanism is 
applicable to supergravity  in 4D, consistently with local supersymmetry.  
Applications to supergravity with more matter multiplets, 
even with supersymmetry breakings {\it via} super Higgs effects \matter\ 
can be accomplished without further essential problems
%%% 
\ref\nrnext{H.~Nishino and S.~Rajpoot, {\it in preparation}.}.  



\bigskip\bigskip\bigskip

% \newpage 

\leftline{\bf 4.~~Application to 10D Massive Type IIA Supergravity} 

As another interesting application, we look into the case of 
massive type IIA supergravity in 10D \romans\ 
which is obtained from massless type IIA supergravity 
%%% 
\ref\typeiia{F.~Gianni and M.~Pernici, \pr{30}{84}{325}; 
M.~Huq and M.~Namazie, \cqg{2}{85}{293}; 
C.~Campbell and P.~West, \np{243}{84}{112}.}.  
%%% 
A cosmological constant term in 
Massive type IIA supergravity \romans\ is actually identified 
with a dilaton potential term proportional to $~m^2 e e^{-5 \varphi}$.  
We show that our mechanism constrains the mass parameter $~m$~ to 
be zero, uniquely yielding a zero cosmological constant.  
Most importantly, this can be done consistently with
local $~N=(1,1)$~ supersymmetry in 10D. 

One caveat is that we need the massive Type IIA supergravity 
lagrangian with the smooth limit of $~m\rightarrow 0$.  
In fact, the lagrangian in \romans\ does not do the 
job, because the field strength $~F_{\m\n}$~ has been absorbed into 
the $~B_{\m\n}\-$field, and the limit $~m\rightarrow 0$~ is 
not smooth.  For this reason, we need to reconstruct the 
original lagrangian where the field strength $~F_{\m\n}$~ is 
present.  

Keeping this caveat in mind, we apply our mechanism 
to Type IIA supergravity in 10D \typeiia\romans.  
The field content of our system is $~(e\du\m m , \psi_\m, 
A_{\m\n\r} , B_{\m\n} , A_\m, \chi, \varphi, M, C_{\m_1\cdots\m_9}, 
K_{\m_1\cdots \m_{10}})$, where only $~M, ~C$~ and $~K$~ 
are our new fields.  Our lagrangian is now fixed to be  
$$ \li{ \Lag_{\rm 10D} = \, &  
- \frac 14 e R - \frac i 2 e ( \bar\psi_\m\g^{\m\n\r} D_\n\psi_\r) 
      +  \frac i2 e (\Bar\chi\g^\m D_\m \chi) 
     + \frac 12 e (\partial_\m  \varphi)^2 
     - \frac 1{48} e e^{-\varphi} (\fprime{\[4\]})^2 
   + \frac 1{12} e e^{2\varphi} (G_{\[3\]})^2 \cr 
&  -\frac 14 e e^{-3\varphi} (\fprime{\m\n})^2 
     + \frac 1{96} e e^{-\varphi/2} 
     \Big[\, (\Bar\psi{}^\m \g_{\[\m|} \g^{\[4\]} \g_{|\n\]} \psi^\n) 
     - \frac i{\sqrt 2} (\Bar\psi_\m\g^{\[4\]} \g^\m\chi) 
    + \frac 3 4 (\Bar\chi\g^{\[4\]} \chi) \, \Big] 
      \fprime{\[4\]} ~  \cr 
& + \frac1{24} e e^\varphi \Big[\, i (\Bar\psi{}^\m 
     \g\low{11} \g_{\[\m|} \g^{\[3\]} \g_{|\n\]} \psi^\n) 
     - {\sqrt2} (\Bar\psi_\m \g^{\[3\]} \g^\m\chi) \, \Big] G_{\[3\]} 
     - \frac 1{\sqrt 2} e (\Bar\psi_\m\g\low{11} \g^\n\g^\m \chi) 
     \partial_\n\varphi \cr 
& + \frac 18 e e^{-3\varphi/2} \Big[\, 
     (\Bar\psi{}^\m \g\low{11} \g_{\[\m|} \g^{\r\s} 
    \g_{|\n\]} \psi^\n)  
     - \frac{3i}{\sqrt2} (\Bar\psi_\m\g^{\r\s} \g^\m\chi)  
     + \frac 5 4 (\Bar\chi\g\low{11}\g^{\r\s} \chi) \, \Big]  \fprime{\r\s} \cr 
& +  \frac1{1152} \,\e^{\, \m\n\r\s\t\l\o\psi\varphi\chi} 
    \Big( F_{\m\n\r\s} F_{\t\l\o\psi} 
     B_{\varphi\chi} +  12 F_{\m\n\r\s} F_{\t\l} 
     B_{\o\psi} B_{\varphi\chi} 
     + 48 F_{\m\n} F_{\r\s} 
     B_{\t\l} B_{\o\psi} B_{\varphi\chi} \cr 
& {\hskip 0.5in}  + 4  M F_{\m\n\r\s} B_{\l\t}B_{\o\psi} 
     B_{\varphi\chi} + 36 M F_{\m\n} B_{\r\s} B_{\t\l} 
     B_{\o\psi} B_{\varphi\chi} 
     + \frac{36} 5 M^2 B_{\m\n} B_{\r\s} B_{\t\l} B_{\o\psi} 
    B_{\varphi\chi} \Big) \cr
& + \frac 1 8 e M e^{-5\varphi/2} (\Bar\psi_\m\g^{\m\n}\psi_\n) 
     - \frac{5i}{8{\sqrt2}} e M e^{-5\varphi/2} (\Bar\psi_\m\g 
      \low{11} \g^\m \chi)
     - \frac{21}{32} e M e^{-5\varphi/2} (\Bar\chi\chi) \cr 
&  - \frac 1 8 e e^{-5\varphi} M^2 
     + \frac 1{4(10!)} \e^{\[10\]} M H_{\[10\]} ~~. 
&(4.1) \cr } $$ 
Our $~\g\low{11}$~ satisfies $~(\g\low{11})^2 = + I$. 
The conventional field strengths are defined by 
\romans\typeiibrane
$$ \li{ & \fprime{\m\n} \equiv F_{\m\n} 
      + M B_{\m\n} ~~, ~~~~
     F_{\m\n} \equiv 2\partial_{\[\m} A_{\n\]} ~~, ~~~~  
     G_{\m\n\r} \equiv 3 \partial_{\[\m} B_{\n\r\]} ~~, \cr 
& F\du{\m\n\r\s}{ \!\!\!\!\! \!\!\!\!\!\! ' ~~~} \equiv 
     F_{\m\n\r\s}  + 12 B_{\[\m\n} \fprime{\r\s\]} 
      - 6 M B_{\[ \m\n} B_{\r\s \]} ~~, 
     ~~~~ F_{\m\n\r\s} \equiv 4 \partial_{\[\m} A_{\n\r\s\]} ~~, 
&(4.2) \cr } $$ 
while $~a = -1/4$~ in (2.2), and our $~H_{\[10\]} $~ 
contains Chern-Simons $~M K\-$term as in (2.2) for $~D=10$: 
$$ \li{ & H_{\m_1\cdots \m_{10}} \equiv 
     10 \partial_{\[\m_1} C_{\m_2\cdots \m_{10} \] } 
    + M K_{\m_1\cdots\m_{10}} ~~. 
&(4.3) \cr } $$ 

Our action $~I_{\rm 10D} \equiv \int d^{10} x\, \Lag_{\rm 10D}$~ is 
invariant under supersymmetry
$$ \li{ \d_Q e\du\m m  = \, & -i  (\Bar\e\g^m \psi_\m) ~~, ~~~~
     \d_Q\varphi = - \frac 1{\sqrt2} (\Bar\e\g\low{11}\chi) ~~, 
&(4.4\rma) \cr 
\d_Q \psi_\m = \, & + D_\m (\Hat\o) \e 
     - \frac i {32} e^{-3\varphi/2} 
     \g\low{11} (\g\du\m{\r\s} -14\d\du\m\r \g^\s ) 
     \e \fprime{\r\s} \cr 
& - \frac 1{48} e^\varphi \g\low{11} (\g\du\m{\n\r\s} 
         - 9 \d\du\m\n \g^{\r\s} )\e G_{\n\r\s} \cr 
& + \frac i {128} e^{-\varphi/2} (\g\du\m{\n\[3\]} 
      - \frac{20}3 \d\du\m\n \g^{\[3\]} ) \e 
     F\du{\n\[3\]}{\!\!\!\!\!\!\!\!\!'}  ~~
          - \frac i {32} M e^{-5\varphi/2} \g_\m \e ~~, 
&(4.4\rmb) \cr 
\d_Q A_{\m\n\r} = \, & + \frac 3 2 e^{\varphi/2}  
     ( \Bar\e\g_{\[\m\n} \psi_{\r\]} )  
      +  \frac i {4\sqrt2} e^{\varphi/2} (\Bar\e\g\low{11} 
     \g_{\m\n\r}\chi) - 6 B_{\[\m\n | } (\d_Q A_{|\r\]} ) ~~, 
&(4.4\rmc) \cr 
\d_Q A_\m = \, & + \frac 12 e^{3\varphi/2} 
     (\Bar\e\g\low{11} \psi_\m)
      + \frac{3i} {4\sqrt2}e^{3\varphi/2} (\Bar\e\g_\m\chi) ~~, 
&(4.4\rmd) \cr 
\d_Q B_{\m\n} = \, &  - i  e^{-\varphi} (\Bar\e\g\low{11}
     \g_{\[\m}\psi_{\n\]} ) 
      + \frac 1{2\sqrt2} e^{-\varphi} (\Bar\e \g_{\m\n}\chi) ~~, 
&(4.4\rme) \cr 
\d_Q\chi = \, & +\frac i {\sqrt2} \g\low{11} 
     \g^\m \e \partial_\m\varphi
     + \frac 3{8\sqrt2}  e^{-3\varphi/2} \g^{\r\s} \e \fprime{\r\s} \cr 
&  + \frac i {12\sqrt2} e^\varphi \g^{\[3\]} \e G_{\[3\]} 
     + \frac 1{96\sqrt2} e^{-\varphi/2} \g\low{11} \g^{\[4\]} \e 
     \fprime{\[4\]} 
     + \frac5{8\sqrt2} M e^{-5\varphi/2} \e ~~,   
&(4.4\rmf) \cr 
\d_Q C_{\m_1\cdots\m_9} = \, &  - 9 e^{-5 \varphi/2} 
     ( \Bar\e \g\low{11} \g_{\[\m_1\cdots\m_8} \psi_{\m_9 \] } ) 
     + \frac{5i}{2\sqrt2} e^{-5 \varphi/2} (\Bar\e 
     \g_{\m_1\cdots\m_9} \chi) ~~, ~~~~ 
&(4.4\rmg) \cr 
   \d_Q K_{\m_1\cdots\m_{10}} = \, & 0~~,  ~~~~\d_Q M = 0 ~~, 
   { ~~~~~ ~~~}  
&(4.4{\rm h}) \cr } $$ 
The transformation rule for $~C$~ is fixed,  
such that the new terms compared with \romans\ 
are invariant under supersymmetry up to a total divergence.  
Despite the presence of the dilaton exponents, our 
mechanism works smoothly also in this massive type IIA 
supergravity in 10D.  

The closure of supersymmetry can be also easily verified, 
as in the conventional system \typeiia\romans.  
In particular, the closure on the $~C\-$field is 
fixed in such a way that all the $~\partial M\-$dependent 
terms can be cancelled by $~\d_Q C$~ \nt\typeiibrane.  

The $~K\-$field equation yields
$$ \li{ & M \eqdot 0~~\Longrightarrow ~~ \L\eqdot 0 ~~, 
&(4.8) \cr } $$ 
which automatically satisfies the $~C\-$field equation 
$$\li{ & M \eqdot \hbox{const.}~~
&(4.9) \cr } $$
which is automatically satisfied by (4.8).  
The $~M\-$field equation yields the desirable duality relationship 

\newpage
$$ \li{ & M \eqdot \frac 1{10!} e^{-1} e^{5\varphi} 
     \e^{\[10\]} \Hat H_{\[10\]} - \frac{21}8 e^{5\varphi/2} 
    (\Bar\chi\chi) ~~, 
&(4.10) \cr } $$ 
which is also invariant under supersymmetry.    
Note the presence of the $~(\Bar\chi\chi)\-$term in 
the duality relationship (4.10) that can be traced back to the 
involvement of the exponential function $~e^{5\varphi} $~ 
with $~\Hat H$, which generates the $~\chi H\-$term, 
necessitating the $~(\Bar\chi\chi)\-$term.  
Our mechanism results in a system of type IIA supergravity 
with zero dilaton potential or zero cosmological constant.   

Note that the consequence $~M\eqdot 0$~ is not artificially 
put in by hand, but required by the $~K\-$field equation.  
Out of an innumerable number of values of $~M$~ keeping 
the supersymmetric invariance of the action, the 
$~K\-$field equation picks up the unique value $~M\eqdot 0$.    
In this sense, the vanishing of the cosmological constant is 
required by the dynamics of our system, but not artificially 
installed by hand.  

Even though the basic mechanism is parallel to the previous 
4D case, the 10D case has slight differences.  First, 
we have regarded the dilaton potential term as the 
`cosmological constant' term.  Second, we need the 
special dilaton dependence in exponents, 
such as $~e^{5\varphi}$~ in the duality (4.7).  
while this slightly complicates the 
system, its presence does not affect the essential structure 
of our mechanism.  

The success of our mechanism applied to 
type IIA supergravity in 10D \typeiia\romans\ 
is not a coincidence, but in a sense is a natural  
result.  This is because the original introduction of 
the maximal-rank field strength \typeiibrane\ was motivated by 
the study of massive type IIA supergravity \romans\ 
which was further re-formulated in superspace \supereight.   
Note also that our present formulation here differs from that in 
\typeiibrane\ due to the generalized Chern-Simons term 
$~M K$~ introduced into the maximal field strength $~H$~ 
which was motivated by superspace formulation \supereight.  

The 10-th rank field $~K$~ is 
not {\it ad hoc} or artificial, but is clarified by the super ninebrane 
action in 10D formulated on 10D super world-volume 
\supereight:\footnotew{We use $~{\scst \un\a, ~\un\b,~\cdots~=~
1,~\cdots,~32}$~ for fermionic, while $~{\scst a,~b,~\cdots~=~
0,~1,~\cdots,~9}$~ for bosonic coordinates.}  
$$ \li{ & I_{p=9} \equiv \int d^{10} \s 
    \left[ \, + \frac 1 2{\sqrt {-g}} \, g^{i j} \Pi\du i a \Pi_{j a} 
    - 4 {\sqrt{-g}} 
    + \frac 1{10!} \e^{i_1\cdots i_{10}} \, \Pi\du{i_1}{A_1} \cdots
       \Pi\du{i_{10}}{A_{10}} K_{A_{10} \cdots A_1} \right] 
     ~~, ~~~~~ ~~~ 
&(4.11) \cr} $$ 
where $~K_{A_1\cdots A_{10}}$~ is the 10-th rank potential 
superfield, while $~{\scst i, ~j, ~\cdots~=~0,~1,~\cdots,~9}$~ 
are the coordinates for the 10D super world-volume.  The set of 
constraints to be used is so-called `$\b\-$function favored 
constraints' (BFFC) to simplify the action above \supereight.  
This action is invariant under the fermionic $~\k\-$symmetry:
$$ \li{ & \d_\k E^{\un\a} = \frac 12 
      (I+ \G) \ud{\un\a}{\un\b} \k^{\un\b} ~~, ~~~~
      \d_\k E^a = 0 ~~, \cr 
& \G\du{\un\a}{\un\b} = \frac 1{10!{\sqrt{-g}}} \e^{i_1\cdots i_{10}}
     \Pi\du{i_1}{a_1} \cdots \Pi\du{i_{10}}{a_{10}}
     (\s_{a_{10}\cdots a_1}) \du{\un\a}{\un\b} ~~,  
&(4.12)\cr } $$ 

\newpage

\noindent
with $~\d_\k E^A\equiv (\d_\k Z^M)E\du M A$, and $~\G^2 \equiv I$~ 
\supereight.  The BFFC background constraints are given by (2.10) 
in \supereight, satisfying the Bianchi identities (2.1) - (2.6) in 
\supereight\footnotew{The superfields $~M,~ N,~
C$~ and $~H$~ in \supereight\ correspond respectively to 
$~C, ~H,~K$~ and $~L$~ in this present paper.}.  
These are all consistent with the fermionic $~\k\-$invariance of the 
action (4.11), in particular with the field strength \supereight
$$\li{ & H_{A_1\cdots A_{10}}  \equiv 10 
      \nabla_{\[ A_1} C_{A_2\cdots A_{10})}  + M K_{A_1\cdots A_{10}}~~.
&(4.13) \cr } $$ 
which is nothing but the superspace generalization of  (4.3).  

This feature seems to be 
common to arbitrary supergravity theories, {\it i.e.,}  
the dimensionality of super world-volume 
$~D$~ of a $~p = (D-1)\-$brane coinciding 
with that of the target space-time ~$D$~ \supereight.  
This gives an independent justification of the generalized 
Chern-Simons term $~M K$~ into the field 
strength $~H$, in particular, consistently with D-branes
%%% 
\ref\dbranes{P.K.~Townsend, {\it `P-Brane Democracy'}, in the
proceedings of March 1995 PASCOS/Johns Hopkins Conference, 
hep-th/9507048; J.~Polchinski, \prl{75}{95}{184}; 
J.~Dai, R.G.~Leigh and J.~Polchinski, \mpl{4}{89}{2073}.} 
%%% 
possibly at quantum levels.  

As a matter of fact,  
according to past experiences in supergravity and superstring 
theories, important geometrical features in superspace at classical level
are   well-preserved and consistent at quantum levels, unless there is
certain anomaly.  Since we have the D-brane action supporting the
consistency of our backgrounds with our $~M K\-$type Chern-Simons 
term, we have a strong foundation to believe the  quantum consistency of
our  mechanism.  
  


\bigskip\bigskip\bigskip

% \newpage 

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

In this Letter, we have presented a new mechanism based on 
Hodge duality with a Chern-simons term yielding a zero 
cosmological constant.  The key prescription is summarized as 
\vskip 0.00in
\Item{(1)} Regard $~m$~ in the cosmological constant 
$~\L\equiv a\, m^2/2$~ as a `0-form field strength', replacing  
$~m$~ by an $~x\-$dependent scalar field $~M(x)$.  
\vskip -0.05in 
\Item{(2)} Define the $~D\-$form field strength $~H = D d C + M K$~ 
with the Chern-Simons term $~M K $.  
\vskip -0.05in 
\Item{(3)} Add a constraint lagrangian $~\Lag_c \equiv 
- (a/D!)\e^{\[D\]} M H_{\[D\]}$.  
\vskip -0.05in 
\Item{(4)} The $~C\-$field equation yields $~M\eqdot\hbox{const.}$  
\vskip -0.05in 
\Item{(5)} The $~K\-$field equation yields $~M\eqdot 0 ~\Longrightarrow
~\L =0$, {\it i.e.,} the zero cosmological constant.  
\vskip -0.05in
\Item{(6)} The $~M\-$field equation yields the 
duality $~M \eqdot (1/D!) e^{-1} \e^{\[ D\]} H_{\[ D\]}$.  The difference 
from the usual duality transformation \nt\ from $~0\-$ to $~D\-$forms 
\typeiibrane\ is that both sides vanish due to $~M\eqdot 0$.    

\newpage

We have also seen that our mechanism is applicable
also to  supergravity theories in diverse dimensions  \ss\ consistently with
local supersymmetry.  As explicit applications, we have seen the examples
of  4D supergravity \dz\ and 10D massive type IIA supergravity
\typeiia\romans.   Interestingly, the action invariances under 
supersymmetry allow $~^\forall\L \equiv a M^2$, if there is no  extra term
$\approx M K$~ in the field strength $~H$.   It is the $~K\-$field equation
that forces ~$M$~ to   vanish, consistently with local supersymmetry.   The
vanishing cosmological constant is required by the dynamics  of our
system, but is not `artificially' installed by hand.  

Even though the consistency with 
local supersymmetry looks remarkable, it is 
a natural consequence.  In fact, the introduction 
of  the $\,M\-$field \typeiibrane\ was originally 
motivated by massive type IIA supergravity in 10D 
\romans, and in particular, the maximal-rank field $~K$~
was introduced in superspace \supereight.  It is our present 
mechanism with the particular lagrangian that relates 
this Chern-Simons term to the 
vanishing cosmological constant as dynamical $~K\-$field 
equation in component formulation.  This is because the 
equation $~M\eqdot 0$~ is a dynamical equation which 
is always consistent with supersymmetry, superspace or 
fermionic $~\k\-$symmetry, but is produced only by a 
component lagrangian term such as $~M \wedge H\-$term~ in (4.1).      

Our mechanism is based on the non-physical fields 
$~M, ~C, ~K$, which do not 
affect any physics other than the cosmological constant $~\L$~ 
itself.  In other words, the exclusion of $~\L$~ is 
naturally accomplished by such non-physical fields.  Note also that 
the introduction of the Chern-Simons term $~M K$~ in the 
maximal-rank field strength $~H= D d C + M K$~ inspired by 
superspace formulation \supereight\ is the key ingredient 
that has not been presented in component 
formulation in the past, not to mention the context of 
cosmological constant problem.     

Even though there might be many other mechanisms 
yielding the same effect $~M\eqdot 0$, such as using only a scalar 
density field, those formulations lack underlying principles 
or symmetries, as our 0-form $~D\-$form 
Hodge duality that necessitates our mechanism with the 
new Chern-Simons $~M K\-$term in the field strength $~H$.     

Even though we have discussed only classical systems, there are 
a few reasons to believe that our mechanism works in quantized systems, 
at least for supersymmetric cases with fundamental D-brane actions 
\typeiibrane.  The first question may be the consistency of duality 
transformation \nt\ at quantum levels.   This is not too difficult to
answer, because the original duality \nt\ is applied to massive type IIA
D-brane formulation \typeiibrane\ for the R-R higher-rank
fields of Type IIA superstring.  In particular, the existence of
$~9\-$form potential is not artificial manipulation, but is well-founded by
super-eightbrane formulation \typeiibrane\ which is supposed to be
consistent at the quantum level.  

A subtler question is the justification of
the extra Chern-Simons term $~M K$~ in the field strength $~H$. 
As is shown in the last section, supporting evidence is 
that the maximal $D\-$rank 
superfield is related to $~p=(D-1)\-$brane action \supereight\ 
compatible with general D-brane formulations \dbranes\ 
and $~\k\-$invariance.   
Therefore it is natural to expect 
that our $~M K\-$type Chern-Simons term in $~H$~
is respected at each level of quantum loop computations.
In fact, according to our past experiences in supergravity or superstring,  
any geometrical feature in classical superspace backgrounds for
extended objects is supposed to be maintained at quantum levels, 
unless there is anomaly to upset it.  Since our $~M K\-$type 
Chern-Simons form has a geometrical foundation, 
we have a good reason to believe its 
quantum consistency of our mechanism for supersymmetric cases
with D-brane actions.  Of course, 
however, we lose such grounds for non-supersymmetric systems with
no D-brane action available.   For this reason, quantum consistency may
fail for non-supersymmetric  cases with no fundamental D-brane
actions.\footnotew{If  $~\Lambda$~ contains also quantum effects, 
our mechanism  in section 2
may still be valid at quantum levels.  However, we do not  get into the
details of this kind for non-supersymmetric systems in  this paper, due to
the more basic question with non-renormalizability. }  

Compared with unimodular gravity \unigr\ or unimodular supergravity 
\unisg, our mechanism is superior.  First,   
in the former \unigr\unisg,  
the cosmological constant is regarded as an `initial 
condition' instead of a parameter fixed by hand.  In our 
formulation, the value $~\L =0$~ comes out as the unique solutions to 
$~K\-$field equations.  In other words, the dynamics of our system
determines the special value $~M\eqdot 0$~ out of 
$~^\forall M$~ that are 
allowed for action invariance.   
Second, our 0-form $D\-$form Hodge duality relationships  
such as (1.2) are motivated mathematically, instead 
of the particular constraint $~\hbox{det} (e\du\m m) = 1$~ 
which looks rather {\it ad hoc} and artificial.   

The fact that our formulation is successful both for 
supersymmetric and non-supersymmetric systems 
strongly suggests the validity of our mechanism 
even after supersymmetry breakings, at least at classical levels.  
This is one of the most important features of our mechanism.
We re-emphasize that the examples given in this paper cover
only a small subset of all the systems that our mechanism can 
be applied to.  By following the simple steps (1) through (6) at 
the beginning of this section, one can immediately apply our mechanism 
to gravity or supergravity theories in diverse dimensions.  
Our mechanism may well be consistent also at 
quantum levels for supersymmetric backgrounds, 
if consistent D-brane actions are available.  


\bigskip\bigskip
% \newpage

\doit1{We are grateful to W.~Siegel 
for important discussions.  
This work is supported in part by NSF Grant \# 0308246.}


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

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%%%%%%%%%%%%%


$~F\du{\m_1\m_2} I_1~, ~\cdots, ~
F\du{\m\low{2M-3}\m\low{2M-2}}{I\low{M-1}} ~, ~
\newline F\du{y\m\low{2M-1}} {I\low M}$, except for the
last  factor 
$~ F\du{y\m\low{2M-1}}{I\low M} = \partial_y 
A\du{\m\low{2M-1}}{I\low M}$, like  



\ngaleph{H.~Nishino and S.J.~Gates, Jr.,
hep-th/9606090, \np{480}{96}{573-588}.  


%%% 
\ref\aaortin{{\it See, e.g.,} N.~Alonso-Alberca and T.~Ortin,
\npn{651}{03}{263}, \hepth{0210011}.}. 
%%% 

In other words, we have given
the supersymmetric non-Abelian CS 
theory for an arbitrary gauge group $~G$~ with 
arbitrarily many extended supersymmetries, which is
the most general compared with similar theories in the
past \ngscs\ngaleph.  

%%%
\ref\siegel{W.~Siegel, \pl{128}{83}{397}.}%
%%%

%%% 
\ref\bns{E.~Bergshoeff, H.~Nishino and
E.~Sezgin, \pl{166}{86}{141};  G.~Atkinson,
U.~Chattopadhyay and S.J.~Gates, Jr.,
\ap{168}{86}{387}.}.   
%%% 

%%%%%%%%%%%%%%% 

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2. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76 (1978) 409

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6. Y. Lozano, Eleven Dimensions from the Massive D-2-brane,
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%%%%%%

1. J. Polchinski, `Dirichlet-Branes and Ramond-Ramond
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2. G. Gibbons, M.B. Green and M.J. Perry, in preparation.

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%%%%%%

M.~Dine, P.~Huet and N.~Seiberg, \np{322}{89}{301};


%%%%%%%
However, this will be equivalent to the formulation above,   
because the $~m\-$field equation yields 
$~N_{\[D\]} + m C_{\[D\]} \eqdot 0$~ 
under the same $~\O\-$field 
equation (2.2), while $~M_{\[D-1\]}\-$field equation requires 
$~m(x)$~ to be a constant $~m$.  This $~m$~ should 
eventually vanish, since the $~C\-$field equation further implies 
that $~m \eqdot 0$, consistently with the $~m\-$field 
equation above and (2.2).  Such a formulation 
has the slight advantage in that both $~m(x)$~ and $~N_{\[ D\]}$~ 
are treated as fields, and the final equation $~m \eqdot 0$~ 
comes out as a field equation, instead of a constant 
restricted to be zero.  However, since $~m$~ is a `0-form field 
strength', it becomes nothing but a constant at some stage.  
In any case, these two formulations 
are basically equivalent, despite the minor difference.  


%%%%%%%%%%%%%%

