hep-th0107202.tex
\count100=1  
\documentstyle[12pt]{article}

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

\doit0{
% Horizontal alignment of accents: bars, hats, tildes, etc.
\skewchar\fivmi='177 \skewchar\sixmi='177 \skewchar\sevmi='177
\skewchar\egtmi='177 \skewchar\ninmi='177 \skewchar\tenmi='177
\skewchar\elvmi='177 \skewchar\twlmi='177 \skewchar\frtnmi='177
\skewchar\svtnmi='177 \skewchar\twtymi='177
\def\@magscale#1{ scaled \magstep #1}
}

% For framing, put 1 in \doit below as well as in \framing{1}.     
\doit{0}{
% Framing 
\def\framingfonts#1{
\doit{#1}{\font\twfvmi  = ammi10   \@magscale5 % math italic
\skewchar\twfvmi='177 \skewchar\fivsy='60 \skewchar\sixsy='60
\skewchar\sevsy='60 \skewchar\egtsy='60 \skewchar\ninsy='60
\skewchar\tensy='60 \skewchar\elvsy='60 \skewchar\twlsy='60
\skewchar\frtnsy='60 \skewchar\svtnsy='60 \skewchar\twtysy='60
\font\twfvsy  = amsy10   \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018                        % Gothic
\font\sc=font005                        % script
\def\Go#1{{\hbox{\go #1}}}      % Gothic for single characters 
%  in equations
\def\Sc#1{{\hbox{\sc #1}}}      % script for single characters 
%  in equations
\def\Sf#1{{\hbox{\sf #1}}}      % sans serif for single characters 
%  in eqs
\font\oo=circlew10        % thick circles (hollow: ` and a-n , 
%  solid: p-z)
\font\ooo=circle10                      % thin circles   
\font\ro=manfnt                         % font with rope
\def\kcl{{\hbox{\ro 6}}}                % left-handed rope
\def\kcr{{\hbox{\ro 7}}}                % right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}        % top end for 
%  left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}        % " right
\def\kbl{{\hbox{\ro \char'136}}}        % " bottom left
\def\kbr{{\hbox{\ro \char'137}}}        % " right
}}  
}

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

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


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

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

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

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

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

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

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

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

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

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

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

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

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
        \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)

% Title page
% \thispagestyle{empty}               % no heading or foot on title page 
%   (LaTeX)
\def\oldheadpic{                                % old UM heading
        \setlength{\unitlength}{.4mm}
        \thinlines
        \par
        \begin{picture}(349,16)
        \put(325,16){\line(1,0){4}}\put(330,16){\line(1,0){4}}
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        \end{picture}
        \par
        \thicklines
        \vskip.2in}
\def\border{                                            % border
        \setlength{\unitlength}{1mm}
        \newcount\xco
        \newcount\yco
        \xco=-24
        \yco=12
        \begin{picture}(140,0)
        \put(\xco,\yco){$\ktl$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcl$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbl$}}
        \xco=158
        \yco=12
        \put(\xco,\yco){$\ktr$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcr$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
        \put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
        \end{picture}
        \par\vskip-8mm}
\def\bordero{               % alternate border
        \setlength{\unitlength}{1mm}
        \newcount\xco
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        \xco=-24
        \yco=12
        \begin{picture}(140,0)
        \put(\xco,\yco){$\ktl$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcl$}
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        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbl$}}
        \xco=158
        \yco=12
        \put(\xco,\yco){$\ktr$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcr$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbr$}}
        \put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhd%
hebaaahjhhdahbahgdedgehgfdiehhgdigicba}
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        \par\vskip-8mm}
\def\headpic{                                           % UM heading
        \indent
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        \par
        \begin{picture}(29,16)
        \put(165,16){\line(1,0){4}}
        \put(170,16){\line(1,0){4}}
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        \thicklines}

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

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\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\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\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}
{\raise-0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}
{\raise-0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\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\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\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}}

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

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

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

\begin{document}

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

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

\def\inbar{\vrule height1.5ex width.4pt depth0pt}
\def\sinbar{\vrule height1ex width.35pt depth0pt}
\def\ssinbar{\vrule height.7ex width.3pt depth0pt}
\font\cmss=cmss10
\font\cmsss=cmss10 at 7pt
% \def\ZZ{\rlx\leavevmode
%             \ifmmode\mathchoice
%                    {\hbox{\cmss Z\kern-.4em Z}}
%                    {\hbox{\cmss Z\kern-.4em Z}}
%                    {\lower.9pt\hbox{\cmsss Z\kern-.36em Z}}
%                    {\lower1.2pt\hbox{\cmsss Z\kern-.36em Z}}
%               \else{\cmss Z\kern-.4em Z}\fi}
\def\ZZ{{}Z {\hskip -6.7pt} Z{}} 
\def\Ik{\rlx{\rm I\kern-.18em k}}  % Yes, I know. This ain't capital.
\def\IC{\rlx\leavevmode
             \ifmmode\mathchoice
                    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
                    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
                    {\hbox{\kern.28em\sinbar\kern-.25em{\rm C}}}
                    {\hbox{\kern.25em\ssinbar\kern-.22em{\rm C}}}
             \else{\hbox{\kern.3em\inbar\kern-.3em{\rm C}}}\fi}
\def\IP{\rlx{\rm I\kern-.18em P}}
\def\IR{\rlx{\rm I\kern-.18em R}}
\def\IN{\rlx{\rm I\kern-.20em N}}
\def\Ione{\rlx{\rm 1\kern-2.7pt l}}

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

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


\def\almostshipout#1{\if L\l@r \count1=1 \message{[\the\count0.%
\the\count1]}
      \global\setbox\leftpage=#1 \global\let\l@r=R
 \else \count1=2
  \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
      \hbox to\fullhsize{\box\leftpage\hfil#1}}  
   \global\let\l@r=L\fi} \fi


% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string%
\string##1}}}
\nolabels
%
% tagged sec numbers
\global\newcount\secno \global\secno=0
\global\newcount\meqno \global\meqno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi %if desired
\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1
\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1%
\message{(\secsym\the\subsecno.%
 #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
%
\def\appendix#1#2{\global\meqno=1\global\subsecno=0%
\xdef\secsym{\hbox{#1.}}
\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
%
%       \eqn\label{a+b=c}       gives displayed equation, numbered
%                               consecutively within sections.
%     \eqnn and \eqna define labels in advance (of eqalign?)
%
\def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1\wrlabeL#1}
\def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
\writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
\global\advance\meqno by1\wrlabeL{#1$\{\}$}}
\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}
%
%                        footnotes
\newskip\footskip\footskip8pt plus 1pt minus 1pt 
% \footskip sets footnote baselineskip 
\def\footnotefont{\ninepoint}
\def\f@t#1{\footnotefont #1\@foot}
\def\f@@t{\baselineskip\footskip\bgroup\footnotefont\aftergroup%
\@foot\let\next}
\setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt} %
\global\newcount\ftno \global\ftno=0
\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}
%
%say \footend to put footnotes at end
%will cause problems if \ref used inside \foot, %
%instead use \nref before
\newwrite\ftfile
\def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
$^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
\immediate\write\ftfile{\noexpand\smallskip%
\noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
\def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
\centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}
\def\footatend{}
%
%     \ref\label{text}
% generates a number, assigns it to \label, generates an entry.
% To list the refs on a separate page,  \listrefs
%
\global\newcount\refno \global\refno=1
\newwrite\rfile
%% We have tampered after #1 in \items which was originally %
% \item and also 
%% the argument of \xdef without [ ].  Also \\ after \items{#1}.
%
% We have to be careful about \ref, when using \label and \eq commands.
\def\ref{[\the\refno]\nref}%
\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi%
\global\advance\refno by1\chardef\wfile=\rfile\immediate%
\write\rfile{\noexpand\Item{#1}\reflabeL{#1\hskip.31in}\pctsign}%
\findarg\hskip10.0pt}%  
%       horrible hack to sidestep tex \write limitation
\def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
\gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
\edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup%
\else\ifx\next\empty%
\else\immediate\write\wfile{\the\toks0}%
\fi\let\next=\writ@line\fi\next\relax}}
\def\striprel@x#1{} \def\em@rk{\hbox{}}
%
\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now 
% unnecessary
%
\def\listrefs{\footatend\vfill\supereject\immediate\closeout%
\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp%
\vfill\eject}\nonfrenchspacing}
%
% The following is the revision of \listrefs to put the list in 
% the same page.
\def\listrefsr{\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}%
\nonfrenchspacing}
% The following is the revision of \listrefs to put the list %
% in the same page with the smaller fonts.
\def\listrefsrsmall{\immediate\closeout\rfile\writestoppt
\baselineskip=11pt\centerline{{\bf References}}
\font\smallreffonts=cmr9 \font\it=cmti9 \font\bf=cmbx9%
\bigskip{ {\smallreffonts% 
\parindent=15pt\escapechar=` \input refs.tmp\vfill\eject}}}
%
\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel %
\string#1 is undefined.}%
\nref#1{need to supply reference \string#1.}\fi%
\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
\else#1\count255=1\fi\let\next=\r@fs\fi\next}
%
\def\figures{\centerline{{\bf Figure Captions}}%
\medskip\parindent=40pt%
\def\fig##1##2{\medskip\item{Fig.~##1.  }##2}}
%
% this is ugly, but moore insists
% 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
%


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

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

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

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

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

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


\font\smallcmr=cmr6 scaled \magstep2 
\font\largecmrseventeen=cmr17 scaled \magstep1 
\font\LargeLarge=cmr17 scaled \magstep5 

\def\alephnull{~$\large{\aleph_0}\,$~} 
\def\alephnulllarge{${\displaystyle\aleph_0}$~}  
\def\Bo{\bo{\hskip 0.03in}} 
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) }
\def\.{.$\,$} 
\def\alpr{\a{\hskip 1.2pt}'} 

\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}

\def\eqques{{}{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}{}}

\def\Christoffel#1#2#3{\left\{ {\buildrel #1 
\over{\scst #2 #3}} \right\} } 
\def\Christoffelsmall#1#2#3{\big\{ {\buildrel #1 
\over{\scst #2 #3}} \big\} } 

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

\font\smallcmr=cmr6 scaled \magstep2 

\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\ul{\underline} 
\def\un{\underline} 
\def\-{{\hskip 1.5pt}\hbox{-}} 
\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\ip{{=\!\!\! \mid}} 
\def\Lslash{${\rm L}{\!\!\!\! /}\, $}
\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\eqstar{~{\buildrel * \over =}~}
%\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~}
%\def\eqdot{{}{\on{\hbox\Dot}=}{}}
%\def\eqques{{}{\on{\hbox{\large ?}}=}{}} 
\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~{}}
\def\eqques{~{\buildrel\hbox{\large?} \over =}~{}}

\def\Sabcd{S^{a b c d}} 
\def\Gaualpha{\,\big(\G^a u \big)^\a} 
\def\Gbubeta{\,\big(\G^b u \big)^\b} 
\def\Gaubeta{\,\big(\G^a u \big)^\b} 
\def\Gbualpha{\,\big(\G^b u \big)^\a} 

\def\framing#1{\doit{#1} 
{\framingfonts{#1} 
\border\headpic 
}} 

\def\Check#1{\raise0pt\hbox{$\buildrel{\raise0pt\hbox{$_{_{_\smile}}$}}%
\over{\raise1pt\hbox{${#1}$}}$}}
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{\Large\bf Unimodular~ Supergravity}
%\footnote{This work is supported in part 
% by NSF grant \# PHY-93-41926.} 
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\baselineskip 9pt 

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



\vskip 2.3in 

{\bf Abstract}\\[.1in]  
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\baselineskip 15.5pt 

~~~We present the locally supersymmetric formulation of unimodular
gravity theory in $~D~ (1\le D \le 11)$~ dimensions,  namely
supergravity theory with the metric tensor whose determinant is
constrained to be unity.  In such a formulation, the usual  fine-tuning
of cosmological constant is no longer needed, but its value is
understood as an initial condition.  Moreover, the zero-ness of the
cosmological constant is concluded as the most probable configuration,
based on the effective vacuum functional.  We also show that the
closure of supersymmetry gauge algebra is consistent with  the
unimodular condition on the metric. 


\vskip 0.7in

\leftline{\small PACS: ~04.65, ~02.40.H, ~02.40.K, ~02.40.M}
\vskip -0.05in
\leftline{\small Key Words: Supergravity, Cosmological Constant, 
Unimodular Gravity}
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%\leftline{\small {\hskip 0.8in} Teleparallelism}

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\centerline{\bf 1.~~Introduction} 


Ever since Einstein's `blunder' 
%%% 
\ref\einstein{A.~Einstein, Sitzungsber, Berlin (1917).},
%%% 
how to understand the zero or extremely small cosmological constant 
without fine-tuning has been a long-standing problem both 
at the classical and quantum levels 
%%% 
\ref\weinberg{{\it For reviews, see, e.g.,} S.~Weinberg, \rmp{61}{89}{1}.}.  
%%%     
Lagrangian formulation of general
relativity admits it, no known symmetry forbids it, and up until
recently, it was not even required empirically.  Recent Type Ia supernova
observation 
%%% 
\ref\supernove{A.G.~Riess et al., Astron.~J.~{\bf 116} (1998) 1009; 
P.M.~Garnavich et al., Astrophys.~J.~{\bf 509} (1998) 74; 
S.~Perlmutter 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 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, 
{\it `The Beginning of the end of the Anthropic Principle'}, 
DAMTP-2000-03, 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.  Here we address 
only the `first' cosmological constant problem.  

In a certain formulation, the cosmological constant problem can be
understood  as an `initial condition' instead of extremely small number
adjusted by hand as an artificial `fine-tuning'.   Such a theory is
called `unimodular gravity' theory,  in which the determinant of the
metric tensor is constrained to be  unity, originally developed in 
%%% 
\ref\unimodular{A.~Einstein, Sitzungsber, d.~Preuss. Akad.~d.~Wissench., Pt.~1, 433 (1919). English translation in
H.A.~Lorentz, A.~Einstein {\it et al}., 
{\it The Principle of Relativity}, Dover, New York (1952); 
J.~Anderson, D.~Finkelstein, Amer.~J.~Phys.~{\bf 39/8} (1971) 901; 
J.J.~van der Bij, H.~van Dam and Y.J.~Ng, Physica {\bf A116} 
(1982) 307.}% 
%%% 
\ref\wilczeketal{F.~Wilczek, \pr{104}{84}{111}; 
A.~Zee, in {\it `Proceedings of the Twentieth Annual Orbis Scientiae
on High Energy Physics'}, 1985 edited by S.L.~Mintz and A.~Perlmutter
(Plenum, New York, 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, \ibid{40}{89}{2598}; 
M.~Henneaux and C.~Teitelboim, Phys.~Lett.~{\bf B222} 
(1989) 195.}.   
%%%
Motivated by the development of a possible solution to the cosmological
constant based on baby universe with wormholes
%%%
\ref\wormholes{T.~Banks, \np{309}{88}{493}; S.~Coleman,
\ibid{B310}{88}{643}; S.~Giddings and Strominger, \ibid{B321}{89}{481};
S.~Adler, \prl{62}{89}{373}; I.~Klebanov, L.~Susskind and T.~Banks, 
\np{317}{89}{665}.},  
%%%
the authors in 
%%%
\ref\ngdam{Y.J.~Ng and H.~van Dam, \prl{65}{90}{1972};
Jour.~Math.~Phys.~{\bf 32} (1991) 1337.} 
%%% 
computed the effective vacuum functional in unimodular 
gravity theory as 
$$ Z = \int d \m (\L) \,\, \hbox{exp}  
      \left( \frac{3\pi} {G \L} \right)~~, 
\eqno(1.1) $$  
where $~G$~ is the Newton's constant, and $~d \m(\L)$~ is the
path-integral  measure for the `scalar field' $~\L(x)$.  Even though
$~\L$~  is initially a scalar field, it is constrained to be space-time
independent  by a lagrange multiplier field.   Similarly to wormhole
models \wormholes, this $~Z$~ has a singularity at $~G^2
\L=0$, and therefore the most probable configuration is the one with 
the vanishing cosmological constant $~\L=0$.  

Therefore, unimodular 
gravity theory provides not only the interpretation of the 
cosmological constant as an initial condition, but also the reason 
why the cosmological constant should be zero.  Unimodular gravity theory
can also provide an alternative solution to the strong CP problem 
%%% 
\ref\fnd{P.H.~Frampton, Y.J.~Ng and H.~van Dam,
\jmp{33}{92}{3881}.}.    
%%%

Considering these developments in unimodular gravity theory, as a
possible solution to the cosmological constant problem in gravity
physics and also to other fine-tuning problems in particle physics, it
seems imperative to consider its supersymmetric generalization, namely
to construct unimodular supergravity with local supersymmetry.  In this
paper, we take the first step toward this direction, namely we present
a unimodular supergravity theory with the unit determinant of the
metric tensor.  We present the lagrangian formulation of unimodular
supergravity, in which the metric has a unit determinant as a field
equation aided by lagrange multiplier fields.  We confirm the closure
of the gauge algebra, based on the universal notation in superspace 
%%% 
\ref\ggrs{S.J.~Gates, Jr., M.T.~Grisaru, M.~Rocek and W.~Siegel, {\it
`Superspace'}, Benjamin/Cummings, Reading, MA, 1983.}, 
%%%
in any arbitrary space-time dimensions $~D~(1\le D \le 11)$~ 
in which ordinary supergravity theory exists
%%% 
\ref\nahm{W.~Nahm, \np{135}{78}{149}.}.      
%%%
We use the superspace Bianchi identities \ggrs\ in order to show the  
closure of the gauge algebra holds universally, as long 
as the ordinary supergravity theory is possible.     

Some readers may develop a question about the real necessity of
unimodular formulations for supergravity theories:     
Since the cosmological constant vanishes so long as supersymmetry is
not broken, why do we need to impose the unimodular condition to avoid
the usual fine-tuning of the cosmological constant?  To answer this 
question, we mention examples of non-vanishing cosmological
constants in certain supergravity theories, such as massive
ten-dimensional ($D=10$) type IIA supergravity 
%%% 
\ref\romans{L.J.~Romans, \pl{169}{86}{374}.}
%%% 
which has a free parameter proportional to the cosmological constant,
even though the action is invariant under proper supersymmetry
transformations.  In this paper, we consider the fact that certain
supergravity theories have non-zero cosmological constant as a free
parameter, but is still invariant under its proper supersymmetry
transformations.  In other words, we take the standpoint
that a unimodular formulation for the cosmological
constant is still important even in a supergravity theory.  




\bigskip\bigskip\bigskip

%\vfill\eject

\centerline{\bf 2.~~Dimension-Independent Formulation of 
Unimodular Gravity}

We start with the review of lagrangian formulation of
non-supersymmetric unimodular gravity in arbitrary space-time
dimensions $~D$.  We then try to formulate unimodular supergravity in
an arbitrary $~D$, as long as ordinary superspace formulation 
\ggrs\ is possible \nahm.    

As the basic principle, we start with the condition that the
determinant of the metric tensor is a constant.  However, just for
simplicity, we  choose this constant to be unity.  Then the question is
how one can  impose such a condition on the ordinary Einstein's general
relativity, hopefully from a constraint lagrangian.  Suppose the total
lagrangian $~\Lag$~ is composed of three terms $~~\Lag_R$, $~\Lag_{\rm
M}$~ and $~\Lag_\L$, where $~\Lag_R$~ is the standard Hilbert
lagrangian, $~\Lag_{\rm M}$~ is a general matter lagrangian whose
details are not crucial here, and $~\Lag_\L$~ is the constraint
lagrangian introduced  in order to  fix the determinant of the metric
tensor:\footnotew{We are using the signature $~\big(\eta_{m n}\big) =
\hbox{diag}.\,(+,-,-,\cdots,-)$.  The  inverse power for the
determinant $~e$~ in $~\Lag_{\rm C}$~ is  due to the definition of
$~e\equiv \det\big(e\du a m \big)$~  complying with the notation in
\ggrs.  Accordingly, the scalar curvature $~R(e)\equiv R(e,\phi(e))$~ 
complies with the definition of the Lorentz connection $~\phi\du{m a}
b$~ in \ggrs, and the Hilbert lagrangian has the opposite sign to the
most common notation 
%%%
\ref\mtw{{\it See, e.g.,} C.W.~Misner, K.S.~Thorne and J.A.~Wheeler, {\it
`Gravitation'}~ (W.H.~Freeman and Company, NY, 1972); 
S.~Weinberg, {\it `Gravitation and Cosmology'} (John Wiley \& Sons, 
NY, 1971); L.D.~Landau, E.M.~Lifshitz, {\it `The Classical Theory of Fields'}, 
4th English ed., Pergamon Press (1975).}.}           
%%% 
$$\li{ \Lag & \equiv  \Lag_R + \Lag_{\rm M} + \Lag_\L 
      = + \frac14 e^{-1} R(e) + \Lag_{\rm M} + \L (e^{-1} -1) ~~.  
&(2.1) \cr } $$ 
The $~\L\equiv \L(x)$~ is a real scalar auxiliary field, and $~ e\equiv
\det\big( e \du a m \big)$.  Here we are using the notation in \ggrs\ 
generalized to $~D\-$dimensional space-time \nahm%
%%%
\ref\kt{T.~Kugo and P.K.~Townsend, \np{211}{83}{157}.},  
%%% 
namely the indices $~{\scst m,~n,~,\cdots~=~0,~1,~\cdots,~D-1}$~ are for
curved coordinates, while $~{\scst a,~b,~\cdots~=~0,~1,~\cdots, ~D-1}$~
are for local Lorentz coordinates.  In general, the matter lagrangian 
$~\Lag_{\rm M}$~ contains fermionic fields with the vielbeins $~e\du a
m$.  The $~R(e)$~ is the scalar curvature in terms of the Levi-Civita
connection $~{\scst\Christoffelsmall\r\m\n}$~ with no torsion, and all 
the possible torsion terms are separated in $~\Lag_{\rm M}$.  The
gravitational field equation is 
$$\li{ & e \fracm{\d\Lag}{\d e\du m a} 
    = - \frac 12 \big[\, R \du a m (e) - \frac12 e\du a m R(e) 
       - \k T\du a m - 2 e\du a m \L \, \big] \eqdot 0 ~~, 
&(2.2) \cr } $$ 
where $~\k$~ is the gravitational coupling, and the symbol $~\eqdot$~ 
represents a field equation, distinguished from algebraic identities. 
The $~T\du a m$~ is the usual energy-momentum tensor $~\k T\du a m  
\equiv +  2 e \d\Lag_{\rm M} / \d e\du m a $.  
Note that the scalar field $~\L\equiv\L(x)$~ enters 
in the gravitational field equation (2.2), 
as if it {\it were} the cosmological `constant'.       
Now taking the trace of (2.2) yields
$$ \li{ & \L \eqdot \frac1{2D} \Big[ \, \frac{2-D} 2 \, R(e) 
     - \k T \, \Big]~~, 
&(2.3) \cr } $$
where, as usual, $~T\equiv T\du m m $.  Using (2.3) back in 
(2.2), we get
$$ \li{ & R_{m n}(e) - \frac1D g_{m n} R(e) 
     \eqdot \k\big(T_{m n} - \frac1D g_{m n} T \big)~~.  
&(2.4) \cr } $$
Needless to say, this expression covers the familiar case of $~D=4$~ in 
\weinberg. Now the usual technique is to take the covariant divergence of
(2.4) to get 
$$ \li{ & \nabla_n \big[\, R\du m n (e) - \frac1D \d\du m n R(e) 
	-\k T\du m n
     + \k \frac1D \d\du m n T  \, \big] \cr 
& = \frac12 \nabla_m R(e) - \frac1D \nabla_m R(e) - \k \nabla_n T\du m n  
     + \k \frac1D \nabla_m T \cr
& \eqdot - \frac1D \nabla_m \Big[\, \frac{2-D} 2 R(e) - \k T \,\Big]  \cr  
& \eqdot - 2 \nabla_m \L \eqdot 0 ~~.  
&(2.5) \cr} $$
Here $~\nabla_m$~ has the Christoffel connection  
$~{\scst\Christoffelsmall r m n}$~ only in terms of the vielbein with
no torsion.  As usual, use is made of the Bianchi identity and the
matter field equation  
$$\li {& \nabla_n R\du m n(e)  \equiv + \frac12 \nabla_m R(e)~~, 
& (2.6\rma) \cr 
& \nabla_n T\du m n \eqdot 0 ~~,  
& (2.6\rmb) \cr } $$
together with the form for $~\L$~ in (2.3).  
Eq.~(2.6b) is the energy-momentum conservation, which is valid  as long
as $~\Lag_{\rm M}$~ is invariant under general coordinate 
transformations.  Eq.~(2.5) implies nothing other than the constancy of
$~\L$, and therefore the $~\L\-$term in (2.2) can be regarded as the
cosmological constant in the gravitational field equation.  In other
words, in this formulation, the value of the cosmological constant is
understood as an `initial condition' instead of `fine-tuning' performed
by hand \weinberg\wilczeketal\unimodular.  Finally, the $~\L\-$field 
equation $~\d\Lag / \d \L \eqdot 0$~ yields the unimodular condition 
$$ \li{& e \eqdot 1~~.  
&(2.7) \cr } $$
Thus we have a lagrangian formulation in which the unimodular condition
is automatically derived from the total lagrangian.   


\bigskip\bigskip\bigskip

%\vfill\eject






\centerline{\bf 3.~~Dimension-Independent Unimodular Supergravity}

We now generalize this to supergravity in dimensions $~D~(1\le D\le
11)$~ \nahm, as long as it allows an invariant lagrangian in superspace
formulation \ggrs.    

Let $~\Lag$~ be the total lagrangian composed of the usual 
supergravity and matter multiplets part $~\Lag_0$~ which is 
locally invariant up to a total divergence, and the constraint lagrangian 
$~\Lag_{\rm C}$: 
$$\li{ & \Lag\equiv \Lag_0 + \Lag_{\rm C}~~, 
&(3.1\rma) \cr 
& \Lag_{\rm C} \equiv \Lag_\L + \Lag_\r \equiv \L (e^{-1} - 1) 
     + e^{-1} \r^{\un\a} \big[ \, i  \big( \g^a\psi_a \big)_{\un\a} 
      - T\du{\un\a b} b \, \big]  ~~.    
&(3.1\rmb) \cr } $$ 
The spinorial auxiliary field $~\r^{\un\a}$~ is needed as a
`superpartner' of $~\L$.  Here the {\it underlined} spinorial indices $~{\scst \un\a,~\un\b,~
\cdots}$~ include all the possible internal indices, such as those for 
$~Sp(1),~SO(N)$~ or dottedness for chiralities, {\it etc.}~\nahm\kt.   
In our notation, the multiplication by the $~\g\-$matrices satisfies  
$~\big(\g^a\psi_a \big)_{\un\b} = \big( \g^a \big)\du{\un\b}{\un\g} 
\psi_{a\un\g} = -\big( \g^a \big)_{\un\b\un\g} 
\psi\du a{\un\g}$.\footnotew{There may well be some subtlety about  the
multiplication of the $~\g\,$-matrices, depending on $~D$,  in which
the spinorial metric is not the antisymmetric  charge-conjugation
matrix $~C_{\un\a\un\b}$~ but just the Kronecker's delta
$~\d_{\un\a\un\b}$~ \kt.   Even though we do not go into the details of 
such a subtlety in this paper, our results will be general for 
$~^\forall D$.}      The $~T\du{\un\a b} c $~ in (3.1) is a
supertorsion component in superspace appearing in the general
supersymmetry transformation of the vielbein      
$$ \li { & \d_Q e\du a m = + i \big( \Bar\e \g^m \psi_a \big) 
         - \e^{\un\g} \, T\du{\un\g a} b e\du b m ~~,     
&(3.2) \cr } $$
as is derived easily from eq.~(5.6.28) in \ggrs, with the  universal
constraint $~T\du{\un\a\un\b} c = i \big(\g^c \big)_{\un\a\un\b}~$ in
superspace.  (Cf.~(4.1) below).  For $~T\du{\a b} b$~ in (3.1)  or
(3.2), we take the $~\theta=0$~ sector as is usually expressed by  the
symbol $~|$~ \ggrs, but we omit this symbol consistently throughout in
this paper.  In most formulations of supergravity in diverse dimensions 
%%%
\ref\salamsezgin{{\it `Supergravity in Diverse Dimensions'}, 
{\it eds}.~A.~Salam and E.~Sezgin, Vols.~{\bf 1} and {\bf 2},
North-Holland/World Scientific (1989).},  
%%%
the second term in (3.2) is usually absent, but we keep this term 
just for completeness.  The $~T\du{\a b} c$~ can be generally 
composed of fundamental superfields, but its detailed structure 
is not crucial here.    

Needless to say, $~\Lag_0$~ contains the Hilbert lagrangian $~+(1/4)
R(e)$, so that the general structure (2.1) is still valid even for
supergravity lagrangian (3.1), when all the fields (including
$~\r^{\un\a}$) other than the vielbein are separated from $~\Lag_0 +
\Lag_\r$~ and included into the `matter' lagrangian $~\Lag_{\rm M}$. 
This is also the reason why we have put the factor $~e^{-1}$~ in 
$~\Lag_\r$~ for its regular behavior as a spinor under the
general coordinate transformations.  

The supersymmetry transformation rules for
the new fields $~\L$~ and $~\r^{\un\a}$~ are   
$$ \li{ & \d_Q\L = - e \, \d_Q e^{-1} 
     = \e^{\un\a} \big[ \, i\big( \g^a\psi_a\big)_{\un\a}
     - T\du{\un\a b} b \, \big] \L~~, 
&(3.3\rma) \cr 
& \d_Q \r^{\un\a} = e \e^{\un\a} \L - e\r^{\un\a}\, \d_Q e^{-1} 
				=	e \e^{\un\a} \L + \r^{\un\a} \e^{\un\b} \, 
     \big[ \, i\big( \g^a\psi_a\big)_{\un\b}
     - T\du{\un\b c} c \, \big] ~~,    
&(3.3\rmb) \cr } $$
while other component fields in $~\Lag_0$~ transform in the usual way.  
Some remarks are to be made here.  First, we note that our supersymmetry
transformation rule must be also constrained, such that the condition
$~e \eqdot 1$~ is satisfied.  Second, using (3.3), we get the
supersymmetry transformation of the condition $~e^{-1} \eqdot 1$~ 
(2.7) as  
$$ \li{ & e \d_Q e^{-1} = - \e^{\un\a} 
    \big[ \, i\big( \g^a\psi_a\big)_{\un\a}
     - T\du{\un\a c} c \, \big] \eqdot 0~~~\Longrightarrow~~~
i \big(\g^a\psi_a \big)_{\un\a} 
    - T\du{\un\a b} b \eqdot 0 ~~. 
&(3.4) \cr } $$ 
Third, the supersymmetry transformation of (3.4) itself should
also vanish: 
$$ \li{ & \d_Q \big[ \, i \big(\g^a\psi_a \big)_{\un\a} 
         - T\du{\un\a b} b \, \big] \eqdot 0 ~~.  
&(3.5) \cr } $$ 
The explicit form of this can be seen in (4.2).   
In this sense, our supersymmetry transformation rule is `semi-on-shell',
namely all the conditions related to the unimodular condition $~e
\eqdot 1$~ to be respected by the `constrained' supersymmetry
transformation.  The word `semi' is used, because we do not use
field equations other than those related to the unimodular condition 
(2.7). 

We now confirm the invariance of the total lagrangian 
$~\Lag_0$~ under supersymmetry $~\d_Q$:    
$$\li{ \d_Q & \big( \Lag_0 + \Lag_{\rm C}\big) 
       = \d_Q \Lag_{\rm C} \cr  
\eqdot & - e \big( \d_Q e^{-1} \big) 
    \L \big(e^{-1} -1 \big) 
     + \L \d_Q e^{-1} \cr 
& + \big( \d_Q e^{-1} \big) \r^{\un\a} 
    \big[ \, i \big(\g^a\psi_a \big)_{\un\a} 
         - T\du{\un\a b} b \, \big] 
     + e^{-1} \big[ \, e\e^{\un\a} \L 
     - e \r^{\un\a} \big( \d_Q e^{-1} \big) \, \big] 
     \big[ \, i \big(\g^a\psi_a \big)_{\un\a} 
         - T\du{\un\a b} b \, \big] {~~~~~ ~~~~~} \cr 
= \, & + e \big(\d_Q e^{-1} \big) \L 
     + \L \big[ \, i \big(\g^a\psi_a \big)_{\un\a} 
         - T\du{\un\a b} b \, \big] = 0 ~~,
&(3.6) \cr } $$ 
where we have used (3.5) and the fact that 
$~\d_Q\Lag_0 =0$~ up to a total divergence is taken for granted. 
Our result is universal and applicable to any supergravity theory that
allows a superspace formulation \ggrs\ and lagrangian formulation
\salamsezgin.   

As for the field equations in our unimodular supergravity, these are
exactly parallel to (2.2) - (2.7) for the non-supersymmetric case.  This
is because,  as was also mentioned, the structure of the total lagrangian
(3.1) with local supersymmetry is exactly the same as that in (2.1),
when all the gravitino-dependent terms and other fields (including
$~\r^{\un\a}$) in $~\Lag_0$~ and $~\Lag_\r$~ are separated and
collected into $~\Lag_{\rm M}$.  In particular, this $~\Lag_{\rm M}$~ is
to have the regular invariance under general coordinate transformations,
in order to have the energy-momentum conservation (2.6b).  For example,
the factor $~e^{-1}$~ in $~\Lag_\r$~ becomes important, 
because if this factor were absent, then $~\r^{\un\a}$~ would have
to transform as a spinor `density' instead of a regular spinor, which
would modify the energy-momentum conservation (2.6b).  Eventually, the
field equations (2.2) - (2.7) are still valid, yielding the same
condition of $~\L \eqdot \hbox{const.}$~with $~e \eqdot 1$.  Hence we
emphasize that our total lagrangian (3.1) is valid as the locally
supersymmetric unimodular supergravity lagrangian,  and is applicable to
any supergravity theory with lagrangian formulation \nahm\salamsezgin\ 
in arbitrary dimensions $~D~ (1\le D \le 11)$.   


\bigskip\bigskip\bigskip

% \vfill\eject



\centerline{\bf 4.~~Closure of Gauge Algebra in Unimodular Supergravity}

This section contains the most non-trivial part of our 
formulation.  We confirm the closure of gauge algebra on our
supersymmetric unimodular conditions (2.7), (3.4) and (3.5).  Note
that since the result in this section is algebraic, it is more general than
the lagrangian formulation of the previous section, as long as a given
supergravity theory allows superspace formulation \ggrs.    

The supersymmetry transformation of (2.7) yielded an additional condition
(3.4).  We first review the derivation of (3.4) based on the general
formulae (5.6.28) in \ggrs:  First, we get    
$$\li{ \d_Q e\du a m 
& = - \e^{\un\b} T\du{\un\b a} b e\du b m 
     -\e^{\un\b} \psi\du a{\un\g} T\du{\un\g\un\b} b e\du b m 
     = - \e^{\un\g} \big[ + i \big(\g^m \big)_{\un\g\un\d} 
     \psi\du a {\un\d} + T\du{\un\g a} b e\du b m \, \big] ~~.   
&(4.1) \cr} $$
Next, multiplying this by $~e^{-1} e\du m a $, we get $~\d_Q e^{-1}$~ 
in (3.4).  The supersymmetry transformation of (3.5) in turn yields the 
additional condition
$$ \li{ i \big( \g^a D_a (\Hat\phi) \e \big)_{\un\b} 
& + \e^{\un\g} \nabla_{\un\g} T\du{\un\b c} c \cr  
& + i \big(\g^a \big)_{\un\b\un\g} \Big[ \, \e^{\un\d} T\du{\un\d a}{\un\g} 
        + \e^{\un\d} T\du{\un\d a}b \psi\du b{\un\g} 
        + \e^{\un\d} \psi\du a{\un\e} T\du{\un\e\un\d}{\un\g} 
     + \e^{\un\d} \psi\du a{\un\e} T\du{\un\e\un\d} b 
     \psi\du b{\un\g} \, \Big] \eqdot 0 {~.~~~~~ ~~~}  
&(4.2) \cr } $$
The Lorentz connection $~\Hat\phi$~ contains 
what is called the $~\psi\-$torsion 
%%% 
\ref\pvn{P.~van Nieuwenhuizen, \prep{68}{81}{189}.} 
%%%
as well as other torsion components {\it via} $~T\du{a b} c$: 
$\Hat\phi\du{m a} b \equiv \Hat\phi\du{m a} b (e,\psi,T)$.  
These will be given explicitly in (4.9).    
Eq.~(4.2) is easily confirmed by the expressions for 
$~\d_Q \psi\du a{\un \g}$~ given in \ggrs\ and $~\d_Q T\du{\un\b c} c
= - \e^{\un\g} \nabla_{\un\g} T\du{\un\b c} c$, as    
$$\li{ 0 \eqdot \, & \d_Q \big[ \, i\big(\g^a\psi_a \big)_{\un\b} 
     - T\du{\un\b c} c \, \big]  
     = - i \big(\g^a\big)_{\un\b\un\g}
     \big(\d_Q\psi\du a{\un\g} \big) 
     - \d_Q T\du{\un\b c} c \cr 
= \, & - i\big(\g^a \big)_{\un\b\un\g} 
    \Big[\,  D_a (\hat\phi) \e^{\un\g} 
    - \e^{\un\d} T\du{\un\d a}{\un\g} 
    - \e^{\un\d} T\du{\un\d a} b \psi\du b{\un\g}  
	-\e^{\un\d} \psi\du a {\un\e} T\du{\un\e\un\d}{\un\e} 
    T\du{\un\e\un\d}{\un\g} - \e^{\un\d} \psi\du a{\un\e}
    T\du{\un\e\un\d} b \psi\du b{\un\g} \, \Big] \cr 
& + \e^{\un\g} \nabla_{\un\g} T\du{\un\b c} c ~~.  
&(4.3) \cr }$$
The condition (4.2) dictates the space-time dependence
of the parameter $~\e^{\un\a}$.       

On the other hand, under translations, the determinant 
$~e^{-1}$~ transforms as
$$ \li{ & \d_P e^{-1} = - \xi^m \partial_m e^{-1} 
     - e^{-1} \partial_m \xi^m 
     = -\partial_m \big(e^{-1} \xi^m \big) ~~,  
&(4.4) \cr } $$ 
where $~\xi^m$~ is the parameter for the translation.  
The non-trivial confirmation now is to see whether the 
parameter $~\xi^m$~ satisfies the condition 
$$ \li{&\partial_m \big( e^{-1} \xi^m \big) \eqdot 0 ~~,  
&(4.5) \cr } $$
to be consistent with our first condition (2.7).  In particular, the
parameter $~\xi^m$~ is to be identified with that arising from the
commutator of two supersymmetries $~\[ \d_Q(\e_1) , 
\d_Q (\e_2) \] = \d_P (\xi^m)$~ \ggrs:  
$$ \li{ & \xi^a \equiv i \big(\Bar\e_1\g^a\e_2 \big) ~~.   
&(4.6) \cr } $$
Eq.~(4.5) is further rewritten {\it via} (4.6) as 
$$\li{ & i \big[\, D_m(\Hat\phi) \big( e^{-1} e\du a m\big) \, \big]
     \big(\Bar\e_1\g^a\e_2 \big)  
     + \big[\, i e^{-1} \big(\Bar\e_1\g^a D_a (\Hat \phi) \e_2  \big) 
     - {\scst (1 ~ \leftrightarrow ~ 2)} \, \big]  \eqdot 0 ~~. 
&(4.7) \cr } $$
The first term here can be further rewritten in terms of 
supercovariant anholonomy coefficients \ggrs:  
$$  \li{ C\du{a b}c & \equiv \big( e\du a n \partial_n e\du b m 
     - e\du b n \partial_n e\du a m \big) e\du m c 
     + i \big(\Bar\psi_a \g^c \psi_b \big) 
     - \psi\du{\[ a |}{\un\a} T\du{\un\a |b \]} c 
     + T\du{a b} c \cr 
&\equiv C\du{a b}c (e) + C\du{a b}c (\psi,T) {~~,~~~~~ ~~~~~} 
&(4.8) \cr } $$ 
related to $~\Hat \phi_{a b c} \equiv
\Hat\phi_{a b c}(e,\psi)$~ as
$$\li{ & \Hat\phi_{a b c} = + \frac12 \big( C_{a b c} 
      - C_{a c b}  + C_{c b a} \big) ~~.  
&(4.9) \cr } $$
The derivative factor in the first term in (4.7) can be 
re-expressed as 
$$ \li{ e D_m (\Hat\phi) \big( e^{-1} e\du a m  \big) 
& = -  C\du{a b}b (\psi , T) \cr 
& = - i \big( \Bar\psi_a \g^b \psi_b\big) 
     + \psi\du a{\un\a} T\du{\un\a b} b 
     - \psi\du b{\un\a} T\du{\un\a a} b - T\du{a b} b \cr 
& = - \Bar\psi\du a {\un\b} 
       \big[ \, i \big( \g^b \psi_b \big)_{\un\b} -T\du{\un\b b} b\, \big] 
     - \psi\du b{\un\a} T\du{\un\a a} b - T\du{a b} b \cr 
&  \eqdot -\psi\du b{\un\a} T\du{\un\a b} b - T\du{a b} b ~~.  
&(4.10) \cr } $$
Here the first term in the penultimate line has vanished 
due to our condition (4.4).  Now using (4.10) in (4.7), we can 
confirm (4.5) {\it via} (4.7) as 
$$\li{ 0 \eqques & \big( \psi\du b{\un\a} T\du{\un\a b} b \xi^a  
    - T\du{a b} b \xi^a \big)
    + \big[ \, i\big(\Bar\e_1\g^a D_a (\Hat\phi)\e_2 \big)   
    -{\scst (1~\leftrightarrow~ 2)} \, \big] \cr 
\eqdot & -i \psi \du b{\un\a} T\du{\un\a a} b
     \big(\Bar\e_1\g^a\e_2\big)
     - i T\du{a b} b \big( \Bar\e_1 \g^a\e_2 \big) \cr 
& + \e_1^{\un\b} \e_2^{\un\d} 
    \Big[ - i \big(\g^a\big)_{(\un\b|\un\g} 
     T\du{|\un\d) a}{\un\g} 
     - i \big( \g^a \big)_{(\un\b|\un\g} T\du{|\un\d) a} b
      \psi\du b{\un\g} \cr 
& ~~~~~ ~~~~~ ~~~~~ ~~ - i\big( \g^a \big)_{(\un\b|\un\g} 
     \psi\du a{\un\e} T\du{\un\e|\un\d)}{\un\g} 
       - i \big(\g^a \big)_{(\un\b|\un\g} 
     T\du{\un\e|\un\d)} b \psi\du b{\un\g}
     -\nabla_{(\un\b} T\du{\un\d ) c} c  \,\Big]~~. 
&(4.11) \cr } $$  
Here use is also made of the condition (4.2) to get rid of the derivative
term of $~\e$.  The symbol $~\eqques$~ is used, because the equality 
under question has yet to be confirmed.  

In order to simplify the term $~\nabla_{\un\b} T\du{\un\d c} c$~  in
(4.11), we next use the supertorsion Bianchi-identity   
$$ \li{ \nabla_{( \un\b} T\du{\un\d ) c} c 
    + \nabla_c T\du{\un\b\un\d} c
     & - T\du{\un\b\un\d} d T\du{d c} c 
     - T\du{\un\b\un\d}{\un\e}  T\du{\un\e c} c \cr 
&  - T\du{c (\un\b|} d T\du{d | \un\d ) } c 
     - T\du{c (\un\b|}{\un\e} T\du{\un\e |\un\d)} c 
    - R\du{\un\b\un\d c} c \equiv 0 ~~.       
&(4.12) \cr } $$
Due to the universal constraint $~T\du{\un\a\un\b} c = i \big(\g^c
\big)_{\un\a\un\b}$~ \ggrs, the second term in the first line vanishes. 
Because of (anti)symmetry of indices, the first term in the second line
and the last term also vanish.  Eventually we get  
$$ \li{& \nabla_{(\un\b} T\du{\un\d) c} c 
     \equiv T\du{\un\b\un\d}{\un\e} T\du{\un\e c} c
	+ i \big( \g^c \big)_{(\un\b| \un\e} 
       T\du{c | \un\d)} {\un\e} 
      + i\big( \g^d \big)_{\un\b\un\d} T\du{d c} c~~,    
&(4.13)  \cr } $$
which, after the substitution into 
the last term in (4.11), simplifies the latter as 
$$\li{  0 \eqques & \! - i \psi\du b{\un\a} 
     T\du{\un\a a} b \big(\Bar\e_1\g^a\e_2 \big) 
    - i T\du{a b} b \big(\Bar\e_1\g^a\e_2 \big) \cr 
& + \e_1^{\un\b} \e_2^{\un\d} \,  
	\big[ - i \big(\g^a \big)_{ ( \un\b | \un\g} 
     T\du{|\un\d) a}{\un\g} 
     - i \big( \g^a \big)_{(\b | \un\g} 
     T\du{|\un\d ) a} b \psi\du b{\un\g}  \cr  
& ~~~~~ ~~~~~ ~~  - i\big( \g^a \big)_{(\un\b |\un\g} 
     \psi\du a{\un\e} T\du{\un\e|\un\d)}{\un\g} 
     + \big( \g^a \big)_{(\un\b|\un\g} 
     \psi\du a{\un\e} \big(\g^b \big)_{\un\e| \un\d) } 
     \psi\du b{\un\g}  \cr 
& ~~~~~ ~~~~~ ~~ - T\du{\un\b\un\d}{\un\d} T\du{\un\e c} c  
     - i T\du{c( \un\b|}{\un\e} \big( \g^c \big)_{\un\e | \un\d)} 
     - i \big( \g^b \big)_{\un\b\un\d} T\du{d c} c \, \big]  
&(4.14\rma) \cr
\eqdot & \e_1^{\un\b} \e_2^{\un\d} \,  
     \big[ + i \big( \g^a \big)_{\un\b\un\d} \psi\du b{\un\g} 
     T\du{\un\g a} b 
     - i \big( \g^a\big)_{(\un\b|\un\g} T\du{|\un\d ) a}b 
     \psi\du b{\un\g} \cr 
& ~~~~~ ~~~~~ - i \big( \g^a \big)_{(\un\b|\un\g} 
     \psi\du a{\un\e} T\du{\un\e|\un\d)}{\un\g} 
     - i T\du{\un\b\un\d} {\un\e} \big( \g^c\psi_c \big)_{\un\e} \,\big]
&(4.14\rmb) \cr  
     \eqdot & \frac12 \e_1^{\un\a} \e_2^{\un\b} \, \psi\du b{\un\g} \,  
     \big[ \, i\big( \g^a \big)_{(\un\a\un\b} T\du{\un\g) a} b 
     - i \big( \g^b \big)_{(\un\a|\un\d} T\du{|\un\b\un\g)}{\un\d} 
     \, \big] ~~,       
&(4.14\rmc) \cr } $$
after various cancellation among like terms.  In (4.14a), the last
term in the first line and the last  term in the last line cancel each
other, so do the first term in the second line and the second terms in
the last line.  The last term in the third line vanishes due to the
antisymmetry $~\big( \g^a\psi_b \big)_{ ( \un\b | } \big( \g^b\psi_a
\big) _{|\un\d)} \equiv 0$.  From (4.14b) to (4.14c), use is also
made of the condition (4.4)  replacing $~T\du{\un\e c}  c$~ by
$~i\big(\g^c\psi_c \big)_{\un\e}$.  Now our last task is to show that
(4.14c) vanishes.  Fortunately, this can be easily
done by the use of another supertorsion Bianchi-identity
$$\li{ & \frac12 \nabla_{(\un\a} T\du{\un\b\un\g)} b 
      - \frac 12 T\du{(\un\a\un\b|} d T\du{d|\un\g)} b 
    - \frac 12 T\du{(\un\a\un\b|}{\un\d} T\du{\un\d|\un\g)} b
     - \frac 12 R\du{(\un\a\un\b\un\g) } b \equiv 0 ~~, 
&(4.15) \cr } $$
where the first and last terms vanish.  Eq.~(4.15) implies that 
(4.14c) vanishes identically, and therefore (4.5) vanishes as desired, 
{\it via} (4.11) and (4.7).  This concludes our proof of (4.5)
for the parameter (4.2).   
 
Before concluding this section, we briefly consider the closure of 
supersymmetry on our new fields $~\L$~ and $~\r^{\un\a}$.  First, notice
that the {\it on-shell} closure on these fields is easier to handle than
the lagrangian invariance, because here we can use field equations.  
Second, as has been mentioned, once the {\it semi-on-shell} condition
(3.4) is considered, both of the fields $~\L$~ and $~\r^{\un\a}$~ do not
transform.  In other words, the commutator $~\[ \d_Q(\e_1) , \d_Q(\e_2)
\] $~ vanishes, when acting on both of these fields.  At first sight,
this sounds puzzling, because this also means the absence of translation
generated on both of these fields.  However, note that the field $~\L$~
is to be a constant after all, so that its translation is required to
vanish.   As for the field $~\r^{\un\a}$, it can be completely gauged
away by an appropriate local supersymmetry in (3.3b), when $~\L 
\eqdot \hbox{const}$.  In other words, any transformation of
$~\r^{\un\d}$, including the usual translation, can be re-absorbed into
a new supersymmetry parameter.  Therefore the vanishing of the
commutator $~\[ \d_Q(\e_1) , \d_Q(\e_2) \]$~ on both of these new
fields poses no problem for closure of supersymmetry.  




\bigskip\bigskip\bigskip

% \vfill\eject



\centerline{\bf 5.~~$N=1$~ Unimodular Supergravity in $~D=4$~ 
as An Example}

Once we have established our general formulation of unimodular 
supergravity, it is easier to look into some explicit examples.  
Here we give an example of old minimal supergravity 
%%%
\ref\oldminimal{K.S.~Stelle and P.C.~West, \pl{74}{78}{330};
S.~Ferrara and P.~van Nieuwenhuizen, \pl{74}{78}{333}.}  
%%% 
in $~D=4$. 

Complying with the superspace notation so far, we give the 
superspace constraints for supertorsions and supercurvatures of 
$~D=4, \, N=1$~ old minimal supergravity \oldminimal\ 
for the component field content $~(e\du a m, \psi\du a{\un\a}, 
S, P, A_m)$:
$$\li{ & T\du{\un\a\un\b} c = + i \big( \g^c \big)_{\un\a\un\b} ~~,  
&(5.1\rma)  \cr 
& T\du{\un\a b}{\un\g} 
    = - \frac i 6 \big(\g_b \big)\du{\un\a}{\un\g} \, S 
      - \frac 1 6 \big(\g_5 \g_b \big)\du{\un\a}{\un\g} \, P
      - \frac i 3 \big( \g_5\g_b\g^c \big)\du{\un\a}{\un\g} \, A_c ~~, 
&(5.1\rmb) \cr 
& \nabla_{\un\a} S 
    = + \frac 1 2 \big(\g^{a b}\big)_{\un\a\un\g} \, 
					T\du{a b}{\un\g}
	= +\frac i 2 \big( \g^m{\cal R}_m \big)_{\un\a} ~~, 
&(5.1\rmc) \cr 
& \nabla_{\un\a} P 
    = - \frac i 2 \big(\g_5 \g^{a b}\big)_{\un\a\un\g} \,  
					T\du{a b}{\un\g}  
	= +\frac 1 2 \big( \g_5 \g^m{\cal R}_m \big)_{\un\a} ~~, 
&(5.1\rmd) \cr 
& \nabla_{\un\a} A_b 
    = + \frac 3 4 \big(\g_5\g\du b{c d} \big) \du{\un\a}{\un\g} \, 
					    T\du{c d}{\un\g}
      - \frac 1 2 \big(\g_5 \g_b\g^{c d} \big)_{\un\a\un\g}\,  
					    T\du{c d}{\un\g} 
= +\frac {3i} 2 \big(\g_5 {\cal R}_b \big)_{\un\a} 
	-\frac i 2 \big(\g_5 \g_b \g^c {\cal R}_c \big)_{\un\a} 
{~~, ~~~~~ ~~~~~} 
&(5.1\rme) \cr 
& T\du{a b} c = + \frac 2 3 \e\du{a b}{c d} \, A_d~~, 
&(5.1\rmf) \cr } $$ 
with $~{\cal R}_{a\,\un\a} \equiv -(i/2) \big(\g\du a{b c} \big)_{\un\a\un\b}
T\du{b c}{\un\b}$, 
while all other remaining supertorsion components $~T\du{\un\a b} c, ~
T\du{\un\a\un\b}{\un\g}$~ between the dimensionality $~0 \le d \le 1$~
are zero.  Here the {\it underlined} spinorial indices are for 
the four-component spinors: $~{\scst\un\a~=~1,~2,~3,~4}$. 
The component invariant lagrangian $~\Lag_0$~ of 
supergravity corresponding to (3.1a) \oldminimal\ 
and $~\Lag_{\rm C}$~ of 
(3.1b) are  
$$ \li{ \Lag_0 + \Lag_{\rm C}= & + \frac 1 4 e^{-1} R (e, \phi(e))  
     - \frac 1 2 \e^{m n r s}\Bar\psi_m \g_5 \g_n D_r(\phi(e)) \psi_s
     - \frac1 6 e^{-1} \big( S^2 + P^2 - A_m^2 \big) \cr 
& + \L (e^{-1} - 1) 
     + i e^{-1} \r^{\un\a} \big( \g^m\psi_m \big)_{\un\a} ~~,       
&(5.2) \cr} $$ 
due to the absence of $~T\du{\un\a b} c$.  
The supersymmetry transformation
rules corresponding equations to (3.2), (3.3a) and (3.3b) are
simple, because  we simply drop the last terms with $~T\du{\un\a b} c$. 
By adding also the transformations of $~\psi\du a{\un\a}, ~S,~P$~ 
and $~A_a$, we complete the supersymmetry transformation rule as 
$$ \li{ & \d_Q e\du a m = + i \big( \Bar\e \g^m \psi_a \big) ~~,     
&(5.3\rma) \cr  
&\d_Q \psi\du m{\un\a} = D_m(\phi(e,\psi))\e^{\un\a} 
	   - \frac i 6 \big(\g_m \e \big)^{\un\a} S 
	   + \frac 1 6 \big(\g_5 \g_m \e \big)^{\un\a} P 
	   + \frac i 3 \big( \g_5 \e \big)^{\un\a} A_m 
    - \frac i 6 \big(\g_5 \g\du m n \e \big)^{\un\a} A_n 
    {~, ~~~~~ ~~~~~} 
&(5.3\rmb) \cr 
&\d_Q S = -\frac i 2 \big( \Bar\e \g^m{\cal R}_m \big)~~, ~~~~
	~~~~ \d_Q P  = -\frac 1 2 \big( \Bar\e \g_5\g^m{\cal R}_m \big)~~, 
	\cr 
& \d_Q A_m = - \frac{3i} 2 \big( \Bar\e \g_5 {\cal R}_m \big) 
	+ \frac i 2\big( \Bar\e \g_5 \g_m\g^n {\cal R}_n \big) ~~, 
&(5.3\rmc) \cr 
& \d_Q\L = + i\big( \Bar \e \g^m \psi_m\big) ~~, 
&(5.3\rmd) \cr 
& \d_Q \r^{\un\a} = + e \e^{\un\a} \L + i e \r^{\un\a} \big( \Bar\e
     \g^m\psi_m \big)  ~~.    
&(5.3\rme) \cr } $$ 
The previous invariance confirmation (3.6) for our total lagrangian
$~\Lag \equiv \Lag_0 + \Lag_{\rm C}$~ is performed in exactly the same
way here, and also the closure on all the fields as in section 4 as
well.  

Note that even though we are using here the `off-shell' formulation  of
$~D=4,\, N=1$~ supergravity with the old minimal multiplet \oldminimal, 
the closure of gauge algebra related to the unimodular condition, {\it
i.e.,} those equations in section 5, are `semi on-shell'.  This seems 
inevitable, as long as we impose the unimodular condition from outside,
even if it is implied by `auxiliary' multiplier fields $~\L$~ and
$~\r^{\un\a}$~ at the lagrangian level.  
\doit0
{In other words, the unimodular
condition associated with the new fields $~\L$~ and $~\r^{\un\a}$~ is
intrinsically `on-shell' condition, that can not be made `off-shell' by
appropriate auxiliary fields.
}  


\bigskip\bigskip\bigskip 


\centerline{\bf 6.~~$N=1$~ Unimodular 
Supergravity in $~D=11$~ as Another Example} 

As another instructive and useful application, we look at $~N=1$~
supergravity in $~D=11$.  There is a slight difference in this system 
compared with the previous $~D=4$~ case.   

In our unimodular supergravity formulation, as some careful readers may
have noticed, we have seen that all the supertorsions/supercurvatures
constraints in superspace have not been modified, but there are
additional constraints on fields such as (2.7) and (3.4), or
constraints on the supersymmetry parameter (4.2).  In 
component language, this is equivalent to the fact that all the
transformation rules for the original fields, such as
$~e\du a m$~ and $~\psi\du a{\un\a}$~ are not modified formally, but
these fields are more constrained than before by the constraints (2.7)
and (3.4), {\it etc.}  The only new transformation rule is for the new
fields $~\L$~ and $~\r^{\un\a}$.  Therefore considering  superspace
Bianchi identities \ggrs%
%%%
\ref\cf{E.~Cremmer and S.~Ferrara,
\pl{91}{80}{61}; L.~Brink and P.~Howe, \pl{91}{80}{384}.}, 
%%%
there will be no `modifications' for the original field equations for
the  original fields.\footnotew{We use here the words `constraints'
distinguished from `modifications', because all the original form of 
field equations are formally maintained.}  The only new ingredient is
the constraints (2.7) or (3.4) on the original fields, together with the
constraint (4.2) on the supersymmetry parameter.   

Considering these points, it is now clear that in the case of $~N=1$~  
supergravity in $~D=11$, there will be no cosmological constant
possible.  This is because all the original field equations including
also the gravitational one are maintained, allowing  no cosmological
constant.  To be more specific, the gravitational field equation
implied by the original Bianchi identities \ggrs\cf 
$$ \li{ & R_{m n} 
     = - \frac 1 3 \big( F_{m r s t} F\du n{r s t} 
     -  \frac1{12} g_{m n} F_{r s t u} F^{r s t u} \big) ~~, 
&(6.1) \cr } $$ 
stays the exactly the same even in the unimodular case with no
cosmological constant.  The unimodular condition $~e \eqdot 1$~ of
(2.7) follows from the $~\L\-$field equation $~\d\Lag / \d \L \eqdot0$~ 
consistently with supersymmetry, while (6.1) forces not only $~\L
\eqdot\hbox{const.}$~ but also $~\L \eqdot \hbox{const.} \eqdot 0$. 
Namely, we get $~\L$~ to be zero exactly, maintaining the original field
equations consistent under supersymmetry.  

From this viewpoint, our formulation of unimodular supergravity is 
more meaningful, when the value of cosmological constant is not
determined by local supersymmetry itself, such as in $~1\le D \le 10$.



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% \newpage



\centerline{\bf 7.~~Effects of Superpartner Fields to Vacuum Functional}

The vacuum functional (1.1) in unimodular gravity \ngdam\ had been 
derived, ignoring any contributions to $~Z$~ by matter fields other
than graviton.  However, in our unimodular supergravity, there are
superpartner matter fields, such as gravitino or other bosonic as well
as fermionic fields.  Therefore, it is reasonable to ask about their
possible contributions to the vacuum functional.  To answer this
question, we point out that the derivation of (1.1) is based on the
assumption that the effects of `backgrounds' (but {\it not}  
fluctuations) of matter fields are negligible 
%%% 
\ref\baumhawking{E.~Baum, \pl{133}{84}{185}; 
S.W.~Hawking, \ibid{134B}{84}{403}.}.
%%% 
As a matter of fact, the vacuum functional (1.1) is derived from 
$$ \li{ Z & = \int d \m (\L) \, \, \exp\big[ - S_\L (\Bar g_{\m\n}, 
     \Bar \phi) \big]  \cr 
& = \int d \m (\L) \, \, \exp\big[ - S_\L (\Bar g_{\m\n}, 
     0) \big]  \cr 
& = \int d \m (\L) \, \, \exp  
      \left( \frac{3\pi} {G \L} \right)~~,  
&(7.1) \cr } $$  
where $~S_\L (\Bar g_{\m\n} , 0 ) = - {3\pi}/(\L G)$~ is the  action for
a four-sphere background metric $~\Bar g_{\m\n}$.  In other words, it is
the `background' values $~\Bar \phi$~ (distinguished from their 
original values $~\phi$) of all the matter fields  that might contribute
to $~Z$.  In the non-supersymmetric case \ngdam, it is assumed that 
$~\Bar\phi$~ are all vanishing, and therefore, the second line in (7.1)
follows.  In our present paper, we rely on the same assumption for
superpartner fields in our unimodular supergravity in a given
$~D\-$dimensional space-time, such as gravitino, fermionic matter or
higher-rank bosonic fields.  Since the purpose of this paper is to
consider the vanishing of a cosmological constant within the given
$~D\-$dimensional space-time, instead of  
compactifications with Freund-Rubin type background values for
bosonic fields, our assumption here seems quite legitimate.  To conclude,
the presence of superpartner fields in unimodular supergravity theory
does not upset the good feature with vacuum functional inherent in
unimodular non-supersymmetric gravity.         



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\centerline{\bf 8.~~Concluding Remarks}

In this paper, we have shown that the unimodular supergravity  theory
can be formulated in any space-time dimensions $~D~ (1\le D\le 11)$, in
which ordinary supergravity theory exists \nahm, based on the universal
notation in superspace \ggrs.  We have presented the  supergravity
lagrangian that generates the unimodular determinant  condition as a
field equation of a multiplier field.   We have confirmed the invariance
of our lagrangian under local supersymmetry up to total divergence.   
We have seen that the non-trivial closure of the gauge algebra  is
confirmed with the help of Bianchi identities in superspace, in  a highly
sophisticated but universal way applicable to any supergravity theory
\ggrs\salamsezgin, independent of the  space-time dimensions $~D ~(1\le
D\le 11)$.      

In section 2, we have presented a lagrangian formulation, assuming that
the  basic supergravity theory allows a lagrangian in $~D~(1\le D \le
11)$.  Therefore those supergravity theories allowing no lagrangian
formulations, such as type IIA supergravity in 10D are excluded in
section 2.  However, armed with the algebraic closure confirmed in
section 4, we can  also include those supergravity theories even
without lagrangian formulations.  This is another advantage of our
analysis of gauge algebra in section 4.   

As for the possible contributions by superpartner matter fields to the
vacuum functional (1.1), we have understood that the usual assumption
for the background values (but not their fluctuations) of matter
fields to be vanishing, and will not affect the vacuum functional (7.1). 
This is reasonable for fermionic fields such as gravitino, and other
bosonic fields as well, within a given $~D\-$dimensional space-time, in
which a conventional supergravity theory can be formulated.  Therefore,
the good feature of unimodular non-supersymmetric gravity has
been inherited to our unimodular supergravity.    

Some readers may be wondering about other quantum behaviour of 
unimodular supergravity from a general viewpoint.  For example, one
might think that the condition of unit determinant of the metric would
introduce unphysical degrees of freedom, as it is equivalent to a
nontrivial gauge condition, and therefore unitarity of the theory is
doubtful.   Furthermore, at the loop level new divergences might appear 
as densities of arbitrary weight.  Such a worry, however, is not
appropriate.  This is because we have established supergravity 
formulation of unimodular gravity, which is compatible with superstring
whose quantum behaviour is much better, or supposed to be finite to all
orders, compared with general relativity.  In this context, we
re-emphasize the importance of investigating supergravity formulation
of unimodular gravity.  It is not just pure curiosity that we need to
establish supergravity formulation for unimodular gravity, but it
is only after supergravity formulation is established, that we have
a good control of the quantum behaviour of unimodular gravity theory.  

The success of the universal formulation of unimodular supergravity 
indicates that the concept of unit determinant for 
metric tensor has fundamental significance, compatible with 
supersymmetry.  In other words, the necessity of such a formulation 
in order to understand the vanishing cosmological constant as an initial
condition instead of fine-tuning by hand, is compatible also with 
supersymmetry, which is another important concept in particle physics.   

We believe that our results presented in this paper will provide a  good
working ground for future study of unimodular supergravity/supersymmetry
models that may provide solutions for other `fine-tuning'
problems both in gravity and particle physics.   
  

\bigskip\bigskip


We are grateful to J.~Gates, Jr.~for helpful discussions.
Special acknowledgment is due to M.~Luty for initial collaboration 
on the subject, and for suggesting to include the explicit 
examples of $~D=4$~ and $~D=11$~ unimodular supergravity theories.    
Final acknowledgement is due to the referee of this paper who helped us
for clarifying important aspects of unimodular supergravity.  


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\end{document} 