\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 equations
\font\oo=circlew10	      % thick circles (hollow: ` and a-n , solid:  p-z)
\font\ooo=circle10			% thin circles   
\font\ro=manfnt				% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

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

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


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

% 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}}
 \put(340,16){\line(1,0){4}}\put(335,0){\line(1,0){4}}
	\put(340,0){\line(1,0){4}}\put(345,0){\line(1,0){4}}
	\put(329,0){\line(0,1){16}}\put(330,0){\line(0,1){16}}
	\put(339,0){\line(0,1){16}}\put(340,0){\line(0,1){16}}
	\put(344,0){\line(0,1){16}}\put(345,0){\line(0,1){16}}
	\put(329,16){\oval(8,32)[bl]}\put(330,16){\oval(8,32)[br]}
	\put(339,0){\oval(8,32)[tl]}\put(345,0){\oval(8,32)[tr]}
	\end{picture}
	\par
	\thicklines
	\vskip.2in}
\def\border{						% border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics University of
Maryland Elementary Particle Physics}
	\put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
	\end{picture}
	\par\vskip-8mm}
\def\bordero{						% alternate border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
	\put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhdhebaaahjhhdahbahgdedgehgfdiehhgdigicba}
	\put(-20,-241.5){\ooo ababaighefdbfghgeahgdfgafagihdidihiidhiagfedhadbfdecdcdfagdcbhaddhbgfchbgfdacfediacbabab}
	\end{picture}
	\par\vskip-8mm}
\def\headpic{						% UM heading
	\indent
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
	\put(170,16){\line(1,0){4}}
	\put(180,16){\line(1,0){4}}
	\put(175,0){\line(1,0){4}}
	\put(180,0){\line(1,0){4}}
	\put(185,0){\line(1,0){4}}
	\put(169,0){\line(0,1){16}}
	\put(170,0){\line(0,1){16}}
	\put(179,0){\line(0,1){16}}
	\put(180,0){\line(0,1){16}}
	\put(184,0){\line(0,1){16}}
	\put(185,0){\line(0,1){16}}
	\put(169,16){\oval(8,32)[bl]}
	\put(170,16){\oval(8,32)[br]}
	\put(179,0){\oval(8,32)[tl]}
	\put(185,0){\oval(8,32)[tr]}
	\end{picture}
	\par\vskip-6.5mm
	\thicklines}

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

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} \nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\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:
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% \baselineskip 16pt\oddsidemargin=0.03in 
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\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}}
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\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}

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\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
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\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
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\def\ref{[\the\refno]\nref}%
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\def\striprel@x#1{} \def\em@rk{\hbox{}}
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\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
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%
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\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
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\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
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\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}}}
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\def\writestop{\def\writestoppt{\immediate\write\lfile{\string\pageno%
\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
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\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
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%
\catcode`\@=12 % at signs are no longer letters
%

\doit0{
\def\footnote#1{\let\@sf=\empty 
 \ifhmode\edef\@sf{\spacefactor=\the\spacefactor}\/\fi
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\def\vfootnote#1{\insert\footins\bgroup
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\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} 
}

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%     \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}%
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%\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf
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%\interlinepenalty100 \let\par=\endgraf
%     \leftskip=0pt \rightskip=0pt
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%     \smallskip\Item{#1}\bgroup\strut\aftergroup\@foot\let\next}
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\newtoks\footline 
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\def\nopagenumbers{\footline={\hfil}} 
\def\advancepageno{\ifnum\pageno<0 \global\advance\pageno by -1 
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\newif\ifraggedbottom
\def\raggedbottom{\topskip10pt plus60pt \raggedbottomtrue}
\def\normalbottom{\topskip10pt \raggedbottomfalse} 

\def\on#1#2{{\buildrel{\mkern2.5mu#1\mkern-2.5mu}\over{#2}}}
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\def\Dot#1{\dt{#1}}

\count100=1 

\font\smallcmr=cmr6 scaled \magstep2 
\font\largetitle=cmr17 scaled \magstep1 

\font\LargeLarge=cmr17 scaled \magstep5 

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\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\dim#1{\hbox{dim}\,{#1}} 
\def\ZZ{Z\!\!\! Z} 

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

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

\def\Christoffel#1#2#3{\left\{ {\buildrel #1 
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\def\hepth#1{hep-th/{#1}} 

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\font\smallcmr=cmr6 scaled \magstep2 

\def\plpl{{+\!\!\!\!\!{\hskip 0.009in}{\raise -1.0pt\hbox{$_+$}} 
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\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\eqdot{~{\buildrel{\normalsize\cdot} \over =}~} 
\def\eqstar{~{\buildrel * \over =}~} 

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

\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~} 
\def\Check#1{\raise0pt\hbox{$\buildrel{\raise0pt\hbox{$_{_{_\smile}}$}}% 
\over{\raise1pt\hbox{${#1}$}}$}} 

% If we need the framing in the cover page, put 1 after the following 
% \framing-command, and put 0 otherwise: 
\framing{0} 

\doit1{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0105138}}\par 
{\hbox to\hsize{\hfill UMDEPP 01--053}}\par 
\vskip -0.1in 
\hfill 
\\ 

\begin{center} 

\vskip -0.6in 
\vskip 0.272in 

{\bf\largetitle Alternative ~N$\,$=$\,$2~ Supergravity ~in ~Singular} 
\\[0.026in] 
{\bf\largetitle Five ~Dimensions 
~with ~Matter/Gauge Couplings}\footnote{This 
work is supported in part by NSF grant \# PHY-98-02551.} \\[.1in]  

\baselineskip 9pt 

\vskip 0.26in 

Hitoshi ~N{\smallcmr ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
\\[.16in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland} \\ [.015in] 
{\it College Park, MD 20742-4111} \\[.20in]   

and \\[0.2in] 

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 0.8in 

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

\baselineskip 14.5pt 

~~~We present an extended study of our previous work on an alternative
five-dimensional $~N=2$~ supergravity theory that has a single
antisymmetric tensor and a dilaton as a part of supergravity multiplet.  
The new fields are natural Neveu-Schwarz massless fields in superstring
theory.  Our total matter multiplets include $~n$~ copies of vector
multiplets forming the sigma-model coset space $~SO(n,1)/SO(n)$,  and
$~n'$~ copies of hypermultiplets forming the quaternionic K\"ahler manifold
$~Sp(n',1)/Sp(n')\times Sp(1)$.  We complete the couplings of matter
multiplets to supergravity with the gauged group of the type $~SO(2)
\times Sp(n') \times Sp(1) \times H \times  [\,U(1)\,]^{n-p+1}$~ for an
arbitrary gauge group $~H$~ with $~p\equiv \dim H + 1$, and the isotropy
group $~Sp(n') \times Sp(1)$~ of the coset $~Sp(n',1) / Sp(n') \times
Sp(1)$~ formed by the hypermultiplets.  We also describe the 
generalization to singular 5D space-time as in the conventional formulation


\vskip 0.35in

\leftline{PACS: 04.50.+h, 04.65.+e, 11.30.Pb}
\vskip -0.05in
\leftline{Key Words: Supergravity, Five-Dimensions, Singular
Space-Time, M-Theory}

\vfill\eject 

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\centerline{\bf 1.~~Introduction} 

The importance of supergravity in 5D space-time manifests itself in many 
contexts, such as the supersymmetrization 
%%%
\ref\abn{R.~Altendorfer, J.~Bagger and D.~Nemeschansky, 
{\it `Supersymmetric Randall-Sundrum Scenario'}, \hepth{0003117}; 
T.~Gherghetta and A.~Pomarol, Nucl.~Phys.~{\bf B586} (2000) 141, 
hep-ph/0003129; 
N.~Alonso-Alberca, P.~Meessen and T.~Ortin, Phys.~Lett.~{\bf 482B} 
(2000) 400, \hepth{0003248}; 
A.~Falkowski, Z.~Lalak, S.~Pokorski, Phys.~Lett.~{\bf 491B} (2000) 172, 
\hepth{0004093}; {\it `Four-Dimensional Supergravities
from Five-Dimensional Brane Worlds'}, \hepth{0102145}; 
M.~Zucker, {\it `Supersymmetric Brane World Scenarios 
from Off-Shell Supergravity'}, \hepth{0009083}.}%  
%%%
\ref\bkvp{E.~Bergshoeff, R.~Kallosh and A. Van Proyen, JHEP
{\bf 0010} (2000) 033, \hepth{0007044}.}  
%%%  
of Randall-Sundrum type brane-world scenario  
%%% 
\ref\rs{L.~Randall and R.~Sundrum, \prl{83}{99}{3370};
\ibid{83}{99}{4690}; 
J.~Lykken and L.~Randall, JHEP {\bf 0006} (2000) 014.},    
%%% 
namely, gauged supergravity in 
singular 5D space-time.  In ref.~\bkvp, the introduction of a 4th-rank
antisymmetric tensor $~A_{\m\n\r\s}$~ made it easier to handle supergravity
in such a singular space-time with the orbifold-type singularity
$~S^1/\ZZ_2$.  Another important aspect of 5D supergravity is related to
what is called holographic anti-de-Sitter and superconformal field theory 
(AdS/SCF) correspondence, 
namely the conjecture that the large $~N$~ limit of 
$~SU(N)$~ superconformal field theories in 4D are dual equivalent to 
supergravity on AdS space-time in 5D
%%% 
\ref\holography{G.~'t Hooft, {\it `Dimensional Reduction in Quantum 
Gravity'}, in Salam Festschrift, 
A.~Aly, J.~Ellis and S.~Randjbar-Daemi, {\it eds}., 
World Scientific, Singapore (1993), 
gr-qc/9310026; L.~Susskind, \jmp{36}{95}{6377};
L.~Susskind and E.~Witten, {\it `The Holographic Bound in Anti-de Sitter
Space'}, \hepth{9805114}.}%
%%% 
\ref\maldacena{J.M.~Maldacena, \atmp{2}{98}{231}; 
E.~Witten, Adv.~Theor.~Math.~Phys.~{\bf 2} (1998) 253; \newline 
S.~Gubser, I.R.~Klebanov and A.M.~Polyakov, \pl{428}{98}{105}.}.   
%%% 
In both of these aspects of 
5D supergravity, the presence of the 5D cosmological constant, 
{\it via} the gauging of the $~N=2$~ 
automorphism group $~SL(2,\IR) 
= Sp(1)$~ (or its $~SO(2)$~ subgroup) 
plays a crucial role.   

The conventional on-shell formulation of $~N=2$~ supergravity in 5D 
was initiated in 
%%%
\ref\gst{M.~G\"unaydin, G.~Sierra and P.K.~Townsend,
Phys.~Lett.~{\bf 133B} (1983) 72; 
\ibid{144B}{84}{41}; \prl{53}{84}{332}; \np{242}{84}{244};
\ibid{B253}{85}{573}.} 
%%%
in which an arbitrary number of vector multiplets is
coupled to supergravity, and generalized further in 
%%% 
\ref\gz{M.~G\"unaydin and M.~Zagermann, Phys.~Rev.~{\bf D62}
(2000) 044028, \hepth{0002228}; Nucl.~Phys.~{\bf B572} (2000) 
131, \hepth{9912027}.}%
%%%
\ref\gzrecent{M.~G\"unaydin and M.~Zagermann, 
{\it `Gauging the Full R-Symmetry Group in 
Five-Dimensional $\,{\scst\large N=2}\,$ Yang-Mills/Einstein/Tensor
Supergravity'}, \hepth{0004117}.}%  
%%% 
\ref\cda{A.~Ceresole and G.~Dall'Agata, 
Nucl.~Phys.~{\bf B585} (2000) 143, \hepth{0004111}.}.      
%%%  
However, in these formulations \gst\gz\gzrecent\cda, the dilaton field as
one of the important NS fields does not have manifest dilaton scale
invariance.  Moreover, an additional complication is that the tensor fields
in \gst\gz\gzrecent\cda\ appear in symplectic pairs, obeying the
`self-duality' condition in odd space-time dimensions, and therefore the
single antisymmetric tensor field $~B_{\m\n}$~ as another important NS
field 
%%%
\ref\gsw{M.B.~Green, J.H.~Schwarz and E.~Witten, {\it `Superstring
Theory'}, Vols.~{\bf I} and {\bf II}, 
Cambridge University Press (1987).}       
%%%
is mixed up with  other tensor fields.  In order to overcome these 
drawbacks in these on-shell formulations \gst\gz\gzrecent\cda, we may try
an off-shell formulation 
%%% 
as an alternative, but such a formulation lacks the 
manifest ~$\s\-$model geometry formed by scalars, which is `hidden' at
the off-shell level before eliminating auxiliary fields.  This is similar
to the 4D case of K\"ahler manifold structure in on-shell $~N=1$~
supergravity 
%%%
\ref\cjfgvn{E.~Cremmer, B.~Julia, J.~Scherk, S.~Ferrara, L.~Girardello,
and P.~van Nieuwenhuizen, \np{147}{79}{105};
E.~Cremmer, S.~Ferrara, L.~Girardello and A.~van Proyen, 
\np{212}{83}{413}.}  
%%%
which is hidden in the off-shell formulation. 

In our previous paper
%%%
\ref\nr{H.~Nishino and S.~Rajpoot, 
Phys.~Lett.~{\bf 502B} (2001) 246.},  
%%%
we have proposed an alternative on-shell $~N=2$~ supergravity
multiplet in 5D, which has an irreducible field content larger
than the conventional one \gst\gzrecent\cda\ 
including an antisymmetric tensor and a dilaton fields that are 
Neveu-Schwarz (NS) massless fields in superstring theory \gsw.  
Our supergravity multiplet has the field content 
$~(e\du\m m, \psi\du\m A, B_{\m\n}, 
\chi^A, A_\m, \s)$~ with ~12+12~ on-shell degrees of freedom, where the
f\"unfbein $~e\du\m m$, the gravitini
$~\psi\du\m A$, and the graviphoton ~$A_\m$~ are the same as the conventional 
$~N=2$~ supergravity \gst\gz\cda\gzrecent, while an antisymmetric tensor 
$~B_{\m\n}$, a dilatino $~\chi^A$, and a dilaton $~\s$~ are our new
field content.  Among these, the antisymmetric
tensor $~B_{\m\n}$~ and the dilaton $~\s$~ are natural NS
massless fields in superstring theory \gsw.

In the present paper, we will continue the study of our
alternative on-shell $~N=2$~ supergravity \nr\ coupled to $~n$~ copies of
vector multiplets and $~n'$~ copies of hypermultiplets.  In our
formulation, the dilaton and the antisymmetric fields as the important NS
fields are treated separately from other scalars.  Our $~n$~ scalars
$~\varphi^\a$~ form the coordinates of the
$~\s\-$model coset $~SO(n,1)/ SO(n)$, while the $~4n'$~ scalars
$~\phi^{\un\a}$~ form  the coordinates of the $~\s\-$model coset
$~Sp(n',1)/Sp(n')\times Sp(1)$~ \nr.  We present the general gaugings of 
our system in the presence of hypermultiplets that were not given in our 
previous paper \nr, and consider our supergravity in singular 5D
space-time, as the supersymmetrization of Randall-Sundrum brane-world
scenario \rs, following the prescription of \bkvp\ for dealing with 
the orbifold-type $~S^1/\ZZ_2$~ singularity.  

This paper is organized as follows:  In section 2, we review
our $~N=2$~ alternative supergravity as a notational preparation before
gaugings.  In section 3, we give the general treatment for the gauging of
an arbitrary non-Abelian gauge group that has nothing to do with the coset
$~Sp(n',1) / Sp(n')
\times Sp(1)$.  Section 4 is devoted to our main focus in the present
paper, namely, to show how to gauge the automorphism group
$~Sp(1) = SL(2,\IR)$~ of $~N=2$~ supersymmetry, or more generally the
whole isotropy group $~Sp(n')\times Sp(1)$~ of the coset $~Sp(n',1)/
Sp(n') \times Sp(1)$, in the presence of hypermultiplets, which was not
accomplished in our previous paper \nr.  As a by-product, we will   give
the most general case of gauging of the total group 
$~SO(2) \times Sp(n') \times Sp(1) \times H \times  [\,U(1)\,]^{n-p+1}$~ for
an arbitrary gauge group $~H$~ with $~p\equiv \dim H + 1$, and the
isotropy group $~Sp(n') \times Sp(1)$~ of the coset $~Sp(n',1) / Sp(n')
\times Sp(1)$~ formed by the hypermultiplets.  Section 5 is for the
formulation of our alternative supergravity in a singular 5D space-time,
with the orbifold-type singularity $~S^1/\ZZ_2$, 
{\it i.e.}, the supersymmetrization \abn\bkvp\ 
of Randall-Sundrum brane-world scenario \rs.  
Section 6 is for our conclusion, while the important notations 
and conventions are given in the Appendix.   


\bigskip\bigskip\bigskip

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\centerline{\bf 2.~~Coupling of Vector Multiplets and Hypermultiplets}
\centerline{\bf to 5D, $~N=2$~ Supergravity}

We start with reviewing the couplings of 5D, $N=2$~ supergravity to
vector multiplets and hypermultiplets \nr, before general  non-Abelian
gaugings.   The field content of the multiplet of supergravity is 
$~(e\du\m m, \psi\du\m A, A\du\m I, B_{\m\n}, \chi^A, \varphi^\a,  \l^{a
A}, \phi^{\un\a}, \psi^{\un a})$~ with $~12 + 12$~ on-shell  degrees of
freedom \nr.  Here $~{\scst\m,~\n,~\cdots}$~ are for the  curved world
indices, while $~{\scst m,~n,~\cdots}$~  are local Lorentz with the
metric  $~\big(\eta\low{m n}\big) = \hbox{diag.}~(-,+,+,+,+)$,  $~e\du\m
m$~ is the f\"unfbein, $~\psi\du\m A$~  is the gravitino with $~{\scst
A~=~1,~2}$~ for $~{\bf 2}\-$representation of  the automorphism group
$~Sp(1)= SL(2,\IR)$~ for the $~N=2$~ supersymmetry.  The
raising/lowering of the indices $~{\scst A,~B,~\cdots}$~ is performed by
the $~Sp(1)$~ metric $~\e_{A B},~\e^{A B}$, and therefore special  
attention is needed for superscript/subscript of these indices,  in
particular, their inner products.   As in \nr, we use here
$~Sp(1)=SL(2,\IR)$~ notation instead of $~SU(2)$ as the automorphism
group, in order to make all the bosonic fields manifestly real, just for
simplicity.   The vectors $~A\du\m I~ {\scst (I~=~0,~1,~2,~\cdots,~n)}$~
form the $~({\bf n+1})\-$representation 
of $~SO(n,1)$~ in the coset $~SO(n,1) / SO(n)$~ 
%%%
\ref\gns{S.J.~Gates, Jr., H.~Nishino and E.~Sezgin, \cqg{3}{86}{21}.}%  
%%%
\ref\agst{M.~Awada, P.K.~Townsend, M.~G\"unaydin and G.~Sierra, 
\cqg{2}{85}{801}.}.  
%%% 
The $~\varphi^\a~ {\scst (\a~=~1,~2,~\cdots,~n)}$~ are 
the $~n\-$dimensional
$~\s\-$model coordinates of the coset $~SO(n,1) / SO(n)$,
$~\l^{a A}~{\scst (a~=~1,~2,~ \cdots,~n)}$~ are in the 
$~{\bf n}\-$representation of $~SO(n)$, $~\phi^{\un\a} ~{\scst
(\un\a~=~1,~2,~\cdots,~4n')}$~ are the $~4n'\-$dimensional coordinates of
the quaternionic K\"ahler manifold  $~Sp(n',1) / Sp(n') \times Sp(1)$, and
$~\psi^{\un a}~{\scst (\un a~=~1,~2, ~\cdots,~2n')}$~ are in the $~{\bf
2n'}\-$representation of $~Sp(n')$.  As described in \nr, this is  the
combination of our multiplet of supergravity $~(e\du\m m,
\psi\du\m A, A_\m, B_{\m\n}, \chi^A, \s)$,  $~n$~ copies of the vector
multiplets $~(C_\m, \l^A, \varphi)$, and $~4n'$~  copies of the
hypermultiplets $~(\phi^{\un\a}, \psi^{\un a})$.   In particular, the
graviphoton $~A_\m$~ is identified with the zero-th  component $~A\du\m
0$, while the ~$n$~ copies of the vector field $~C_\m$~ from the vector
multiplets renamed as $~A\du\m 1 ,~A\du\m 2 ,~\cdots,~ A\du\m n$,
combined into the unified notation   $~A\du\m I~ {\scst
(I~=~0,~1,~2,~\cdots,~n)}$.  Since the indices $~{\scst I, ~J,~\cdots}$~
are with the indefinite metric 
$~\big(\eta\low{I J}\big) = \hbox{diag.}~(-,+,+,\cdots ,+)$, we make the
raising/lowering of these indices explicit.  Note that our multiplet of
supergravity is distinct from the conventional one
$~(e\du\m m, \psi_\m{}^A, A_\m)$~ \gst, in which only the f\"unfbein,
gravitino and the graviphoton form the irreducible field content.  

The geometrical relationships associated with the coset $~ 
SO(n,1) / SO(n)$~ are conveniently listed up as \gns\nr
$$ \li{ & \[ H_{a b} , H_{c d} \] = \d_{b c} H_{a d} 
	- \d_{a c} H_{b d} + \d_{a d} H_{b c} - \d_{b d} H_{a c}~~, 
&(2.1\rma) \cr 
&\[ H_{a b} , K_c \] = \d_{b c} K_a - \d_{a c} K_b ~~, ~~~~
    \[ K_a , K_b \]  = + 2 H_{a b} ~~. 
&(2.1\rmb) \cr 
& L\du A I \partial_\a L\du I B 
     = \frac 12 A\du \a{a b} \big(H_{a b} \big) \du A B 
	+ V\du \a a \big( K_a\big)\du A B ~~, ~~~~ A\du{\a b} c 
     = L\du b I \partial_\a L\du I c ~~,    
&(2.1\rmc) \cr 
& \big( H_{a b} \big) \du c d 
    = \d_{a c} \d\du b d - \d_{b c} \d\du a d~~, ~~~~ 
    \big( K_a \big)_{b (0)} = \big( K_a \big)_{(0) b} 
    = - {\sqrt 2} \d_{a b} ~~,   
&(2.1\rmd) \cr
& L\du I A L\du A J = \d\du I J~~, ~~~~ 
	L\du A I L\du I B = \d\du A B~~,   
&(2.1\rme) \cr 
& L_I \equiv L\du I{(0)}~~, ~~~~L^I  \equiv L\du {(0)} I~~,  
     ~~~~ L_I L^I = +1 ~~, ~~~~L\du a I L_I \equiv 0 ~~, ~~~~
     L\du I a L^I  \equiv 0 {~~, ~~~~~ ~~~~~}   
&(2.1\rmf) \cr 
& L_{I J} \equiv \eta\low{A B} L\du I A L\du J B = - L_I L_J 
+ L\du I a L_{J a} 
&(2.1{\rm g}) \cr 
& L_{I J} L^J = - L_I~~,~~~~ L_{I J} L\du a J = + L_{I a}~~, 
&(2.1{\rm h}) \cr 
& D_\a L_I = \partial_\a L_I = - {\sqrt2} L\du I a V_{\a a}
	~~,~~~~ D_\a L\du I a = - {\sqrt2} L_I V\du \a a~~,  
 ~~~~ \partial_\a L_{I J} = 0 ~~, 
&(2.1{\rm i}) \cr 
& \[ D_\a, D_\b \] L\du I a = - 2
	\big( V\du\a a V\du \b b - V\du\b a V\du \a b \big) L_{I b} ~~,  
&(2.1{\rm j}) \cr 
& R\du {\a\b}{a b} = - 2
	\big( V\du\a a V\du \b b - V\du\b a V\du \a b \big) ~~, 
     ~~~~ R = -2 n(n-1) \le 0 ~~,   
&(2.1{\rm k}) \cr 
&D_\a X_a \equiv \partial_\a X_a + A\du{\a a} b X_b~~,
&(2.1\ell) \cr } $$ 
which are self-explanatory exactly in the same notation as in \nr.  
The Cartan decomposition of the $~SO(n,1)$~ Lie algebra is 
dictated by the $~SO(n)$~ generators $~H_{a b}$~ and the coset generators 
$~K_a$, satisfying (2.1).  The indices $~{\scst 
a,~b,~\cdots~=~(1),~(2),~\cdots,~(n)}$~ are  for the vectorial
representation of $~SO(n)$.  The indices $~{\scst
A,~B,~\cdots~=~((0),a),~((0),b),~
\cdots~=~(0),~(1),~(2),~\cdots,~(n)}$~ 
are for the local coordinates on $~Sp(n',1)/Sp(n')\times 
Sp(1)$.\footnotew{The indices  $~{\scst A,~B,~\cdots}$~ used both for
the $~{\bf 2}\,$-representations and for these local Lorentz coordinates
are {\it not} to be confused each other, as long as we keep track of the
context they are used.}  In other words, $~{\scst
A,~B,~\cdots~=~((0),a), ~((0),b),~\cdots}$~ are the
$~(n+1)\-$dimensional extension of the original $~n\-$dimensional
indices $~{\scst a,~b,~\cdots}$. The indices $~{\scst
I,~J,~\cdots~=~0,~1,~\cdots,~n}$~ are for the curved coordinates, while
$~{\scst \a,~\b,~\cdots~ = ~1,~2,~\cdots,~n}$~ are for the coordinates
on $~Sp(n',1)/Sp(n')\times Sp(1)$.  The raising/lowering of the
indices $~{\scst A,~B,~\cdots}$~ is performed by the metric tensor
$~\big(\eta\low{A B} \big) = \hbox{diag.}~(-,+,+,\cdots,+)$.  The
Maurer-Cartan form made of $~L\du I A$~ decomposes as in (2.1c). 
Eq.~(2.1d) gives the explicit components of $~H$'s and $~K$'s, while
(2.1e) - (2.1i) are relevant orthonormality relations.  Eq.~(2.1i) -
(2.1$\ell$)  are for the $~SO(n)$~ covariant derivative $~D_\a$.   

As for the geometry related to the 
quaternionic K\"ahler manifold $~Sp(n',1) / Sp(n')\times Sp(1)$~
we start with the representative $~L^{\un a \un\a}$, which satisfies the 
Maurer-Cartan form for the coset $~Sp(n',1) / Sp(n')\times Sp(1)$~ 
%%%
\ref\bw{J.~Bagger and E.~Witten, \np{222}{83}{1}.}%  
%%%
\ref\ns{H.~Nishino and E.~Sezgin, \pl{144}{84}{187}; 
\np{278}{86}{353}; \ibid{B505}{97}{497}.}%
%%%
\cda\gz\gzrecent\nr:
$$\li{& L^{-1} \partial_{\un\a} L = A\du{\un\a} i T^i 
	+ A\du{\un\a}{\un I} T_{\un I} + V\du{\un\a}{\un a A}
	K_{\un a A} ~~, 
&(2.2\rma) \cr
& g_{\un\a\un\b} V\du{\un a A}{\un\a} V\du{\un b B}{\un\b}  
	= \e\low{\un a \un b} \e\low{A B}~~, ~~~~
	V_{\un\a\un a A} V\du{\un\b}{\un a B} 
	= + \frac 12 g_{\un\a\un\b} \d\du A B
	  - \frac 12 F\du{\un\a\un\b} i \big(T^i\big)\du A B ~~, 
&(2.2\rmb)  \cr 
& J_{\un\a\un\b}{}^i = -  \big(T^i\big)\du A B 
	\big( V_{\un\a \un a B} V \du{\un\b}{\un a A}  
	- V_{\un\b \un a B} V \du{\un\a}{\un a A} \big) 
     = - \frac12 F\du{\un\a\un\b} i  ~~,  
&(2.2\rmc) \cr } $$
where $~T_{\un I} ~~ {\scst ({\un I}~=~1,~2,~\cdots,~n'(n'+1)/2)}$~ are 
the generators of $~Sp(n')$, $~T^i ~~{\scst (i~=~1,~2,~3)}$~ 
are the generators of $~Sp(1)$, while $~K_{\un a A}$~ are the 
coset generators of $~Sp(n',1) / Sp(n')\times Sp(1)$~ \bw\ns.  
All of these equations involve the vielbein $~V\du{\un\a}{\un a
A}$~  for this quaternionic K\"ahler manifold.    
 
With all other details of geometry skipped, our lagrangian 
before gaugings is \nr 
$$\li{ e^{-1} \Lag_0  = & - \frac14 R 
	- \frac12 \big(\Bar\psi_\m\g^{\m\n\r}D_\n\psi_\r \big)
	- \frac1{12} e^{-4\s} G_{\m\n\r}^2 
	- \frac1 4 e^{-2\s} \big( L\du I a L_{J a} + L_I L_J \big) 
	F\du{\m\n}I F^{\m\n\, J}  \cr 
& - \frac12 \big(\Bar\l{}^a\g^\m D_\m \l_a \big) 
	- \frac12 g\low{\a\b} 
	\big( \partial_\m \varphi^\a \big) 
	\big( \partial^\m \varphi^\b \big)
	- \frac 3 4 \big(\partial_\m\s\big)^2 
	- \frac12\big( \Bar\chi\g^\m D_\m\chi\big) \cr 
& - \frac12 g\low{\un\a\un\b} \big(\partial_\m\phi^{\un\a} \big)
	\big(\partial^\m\phi^{\un\b} \big) 
	- \frac12 \big( \Bar\psi{}^{\un a} 
	\g^\m D_\m\psi_{\un a} \big) \cr 
& + \frac i{\sqrt2} V\du\a a \big( \Bar\psi_\m\g^\n\g^\m \l_a \big) 
	\partial_\n\varphi^\a 
	+ \frac{{\sqrt3}i}2 
	\big( \Bar\psi_\m\g^\n\g^\m \chi \big)\partial_\n\s 
	+i V\du{\un\a}{\un a A} 
	\big( \Bar\psi_\m \g^\n\g^\m \psi_{\un a} \big) 
	\partial_\n \phi^{\un\a} \cr 
& - \frac i{4\sqrt2} e^{-\s} \big( \Bar\psi_\m \g^{\m\n\r\s} \psi_\n 
	+ 2\Bar\psi{}^\r \psi^\s \big) L_I F\du{\r\s} I \
 + \frac i{6\sqrt3} e^{-2\s} 
	\big(\Bar\psi_\m\g^{\r\s\t} \g^\m\chi\big) G_{\r\s\t} \cr 
& - \frac1{24} e^{-2\s} \big( \Bar\psi_\m \g^{\m\n\r\s\t} \psi_\n 
	- 6 \Bar\psi{}^\r\g^\s\psi^\t \big) G_{\r\s\t} 
	-\frac5{72} e^{-2\s} 
	\big( \Bar\chi\g^{\m\n\r} \chi\big) G_{\m\n\r} \cr
& - \frac1{2\sqrt2} e^{-\s} \big(\Bar\psi_\m\g^{\r\s} \g^\m \l_a \big) 
	L\du I a F\du{\r\s} I
	- \frac1{2\sqrt6} e^{-\s} 
	\big(\Bar\psi_\m \g^{\r\s}\g^\m \chi \big) L_I F\du{\r\s} I 
	- \frac i{12\sqrt2} e^{-\s} 
	\big( \Bar\chi \g^{\m\n} \chi \big) L_I F_{\m\n}{}^I  \cr 
& + \frac i{4\sqrt2} e^{-\s} \big( \Bar\l{}^a \g^{\m\n} \l_a \big) 
	L_I F\du{\m\n} I  + \frac1{24} e^{-2\s} 
	\big(\Bar\l{}^a \g^{\m\n\r}\l_a \big) G_{\m\n\r} 
	+ \frac1{24} e^{-2 \s} \big( \Bar\psi{}^{\un a} 
	\g^{\m\n\r} \psi_{\un a} \big) G_{\m\n\r} \cr  
& + \frac i{\sqrt6} e^{-2\s} \big(\Bar\chi\g^{\m\n}\l_a \big) 
	L\du I a F\du{\m\n} I 
	- \frac1{4\sqrt2} e^{-\s} \big(\Bar\psi{}^{\un a} 
	\g^{\m\n} \psi_{\un a} \big) 
	L_I F\du{\m\n} I {~~, ~~~~~ ~~~~~} 
&(2.3) \cr }$$ 
yielding an invariant action $~S_0$~ under supersymmetry
$$\li{ & \d_Q e\du\m m = + \big(\Bar\e\g^m\psi_\m\big) ~~, ~~~~
	\d_Q \s = + \frac i{\sqrt3} \big(\Bar\e\chi\big) ~~, ~~~~ \cr
& \d_Q \psi\du\m A = + D_\m \e^A + \frac i{6\sqrt2} e^{-\s}    
	\big(\g\du\m{\r\s} - 4 \d\du\m\r \g^\s \big)  
	\e^A L_I F\du{\r\s} I 
	+ \frac1{18} e^{-2\s} \big( \g\du\m {\r\s\t} 
	- \frac3 2\d\du\m\r\g^{\s\t} \big) \e^A G_{\r\s\t} ~~, \cr  
& \d_Q A\du\m I = - \frac i{\sqrt2} e^\s L^I \big( \Bar\e\psi_\m\big) 
	+ \frac1{\sqrt6} e^\s L^I \big(\Bar\e\g_\m \chi\big) 
	+ \frac1{\sqrt2} e^\s \big( \Bar\e\g_\m \l^a\big) L\du a I 
	~~, \cr 
& \d_Q B_{\m\n} = + e^{2\s} \big( \Bar\e\g_{\[\m} \psi_{\n\]} \big) 
	+ \frac i{\sqrt3} e^{2\s} \big( \Bar\e\g_{\m\n}\chi\big) 
	- 2L_{I J} A\du{\[\m|} I \big( \d_Q A\du{|\n\]} J \big) ~~, \cr 
& \d_Q \chi^A = - \frac1{2\sqrt6} e^{-\s} \g^{\m\n} \e^A L_I 
	F\du{\m\n} I 
	    + \frac i{6\sqrt3} e^{-2\s} \g^{\m\n\r} \e^A G_{\m\n\r} 
	    - \frac{{\sqrt3}i}2 \g^\m \e^A\partial_\m \s ~~,~~~~~  \cr 
& \d_Q \varphi^\a 
	= +\frac i{\sqrt2} V\du a \a \big(\Bar\e\l^a \big) ~~, \cr 
& \d_Q \l^{a A} = - \frac1{2\sqrt2} e^{-\s} 
	\g^{\m\n}\e^A L\du I a F\du{\m\n} I  
	- \frac i{\sqrt2} \g^\m \e^A V\du\a a 
	\partial_\m \varphi^\a ~~, \cr 
& \d_Q \phi^{\un\a} = + i V\du{\un a A} {\un\a} 
	\big( \Bar\e^A\psi^{\un a}\big)  ~~, \cr 
& \d_Q \psi^{\un a} 
   = - i V\du{\un\a}{\un a A} \g^\m \e\low A 
   \partial_\m \phi^{\un\a} ~~,    
&(2.4) \cr } $$ 
up to quartic fermion (or quadratic fermion) terms in the 
lagrangian (or transformation rules).  
Here we have omitted the $~Sp(1)$~ indices $~{\scst A,~B,~\cdots}$~ in
the $~Sp(1)\-$invariant products, {\it e.g.,}
$~\big(\Bar\e\g^m\psi_\m\big)  \equiv \big(\Bar\e{}^A\g^m\psi_{\m
A}\big)$.  

As in the usual dilaton couplings in supergravity 
%%%
\ref\ss{{\it `Supergravity in Diverse Dimensions'}, 
eds.~A.~Salam and E.~Sezgin, North-Holland, World Scientific.}, 
%%%
the antisymmetric field $~B_{\m\n}$~ and the vectors $~A_\m{}^I$~ are
scaled,  when the dilaton $~\s$~ is shifted by a constant value: 
$$ \li{ & \s \rightarrow \s + c~~, ~~~~B_{\m\n} 
   \rightarrow e^{2c} B_{\m\n} 	~~, ~~~~A_\m{}^I \rightarrow 
     e^c A_\m{}^I ~~, 
&(2.5) \cr } $$ 
where $~c$~ is an arbitrary constant parameter.  
This global symmetry controls the various exponential couplings of 
$~\s$~ in the lagrangian (2.3).   

The various covariant derivatives and the field strength $~G_{\m\n\r}$~ 
in these equations are given by 
$$\li{ & D_\m \e^A \equiv 
   D_\m (\o) \e^A  
   + \big(\partial_\m \phi^{\un\a} \big) A\du{\un\a} i 
   \big( T^i \, \e \big)^A ~~,  \cr  
& D_{\[\m} \psi\du{\n\]} A \equiv 
	     D_{\[\m |} (\o) \psi\du{|\n\]} A  
     + \big(\partial_{\[\m |} \phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \, \psi_{|\n\]} \big)^A  ~~, \cr 
& D_\m \chi^A \equiv D_\m (\o) \chi^A  
     + \big(\partial_\m\phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \, \chi \big)^A  ~~, \cr 
& D_\m \l^{a A} \equiv D_\m (\o) \l^{a A}  
     + \big(\partial_\m\varphi^\a \big) A\du\a{a b} 
     \l\du b A  
     + \big(\partial_\m\phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \l^a \big)^A  ~~, \cr 
& D_\m \psi^{\un a} \equiv D_\m (\o) \psi^{\un a}  
     + \big(\partial_\m\phi^{\un\a} \big) A\du{\un\a} {\un I} 
     \big( T^{\un I} \, \psi \big)^{\un a} ~~, \cr 
& G_{\m\n\r} \equiv 3\partial_{\[\m} B_{\n\r\]} 
	- 3 L_{I J} F\du{\[\m\n} I A\du{\r\]} J ~~,  
&(2.6) \cr } $$  
where $~A\du\a{a b}$~ is the composite $~SO(n)$~ connection on 
the coset $~SO(n,1) / SO(n)$, while $~A\du\a{\un I}$~ and  
$~A\du\a i$~ are respectively the composite connections  
of $~Sp(n')$~ and $~Sp(1)$~ in $~Sp(n',1) / Sp(n') \times Sp(1)$.  
The action of the generators $~T^i$~ or $~T^I$~ is such as 
$~\big(T^i \e\big)^A \equiv \big(T^i\big)^{A B}\e\low B$~ or 
$~\big(T^{\un I} \psi\big)^{\un a} \equiv \big(T^{\un I}\big)^{\un a
\un b}\psi_{\un b}$.   
Since we are not concerned with the quadratic fermionic terms in the 
transformation rule (2.4), the Lorentz connection $~\o\du\m{r s}$~ 
contains the usual unholonomy coefficients just made of the f\"unfbeins.  

Compared with the conventional formulations \gst, there is a similarity 
as well as basic difference.  The similarity is that  our
tensor field $~B_{\m\n}$~ can be dualized into a vector field $~B_\m$~ 
by a duality transformation  so that the final field content will be
$~(e\du\m m, \psi\du\m A, A_\m, B_\m, \chi^A, \s)$. From this viewpoint,
our system (2.1) is `dual equivalent' to the conventional formulation
with only one vector multiplet, in particular the dilaton field plays
the coordinate of $~SO(1,1)$, as usual in superstring theory.  However,
the caveat at this stage is that even though such a duality
transformation is possible even after  coupling vector multiplets, the
resulting $~\s\-$model structure  is qualitatively different from that
given in the conventional  formulations \gst\gz\cda\gzrecent, as has
been also explained in our previous paper \nr.   

As has been also stressed in \nr, the antisymmetric field $~B_{\m\n}$~ 
and the dilaton $~\s$~ are the natural NS massless fields in
superstring \gsw\ or M-theory 
%%%
\ref\mtheory{{\it For reviews of M-theory, see, e.g.,} 
W.~Taylor IV, {\it `The M(atrix) Model of M-Theory'}, Lectures for NATO
school {\it `Quantum Geometry'} (Iceland 1999), \hepth{0002016};
T.~Banks,  {\it `TASI Lecture Note on Matrix Theory'}, 
\hepth{9911068}; {\it and references therein}.}.  
%%%          
Therefore, it is more
natural to have a supergravity with these fields in the point field
theory limit.  Another advantage of introducing an antisymmetric tensor 
$~B_{\m\n}$~ is associated with the recent development 
of non-commutative geometry 
in which the tensor $~B_{\m\n}$~ develops certain non-trivial constant 
value.  We stress the fact 
that our supergravity multiplet contains the NS fields $~B_{\m\n}$~
and $~\s~$ as irreducible component fields, indicating that our 
supergravity is a more natural point field theory limit of 
superstring theory \gsw\ or M-theory \mtheory\ than the conventional one
\gst\gz\cda\gzrecent.  



\bigskip\bigskip\bigskip 

% \vfill\eject


\centerline{\bf 3.~~Non-Abelian Gauging of Subgroup of $~SO(n,1)$} 

We next establish general non-Abelian gaugings in the presence of
vector multiplets and hypermultiplets.  In this section, we consider
the case that the gauged non-Abelian group $~G$~ has nothing to do with 
isotropy groups $~Sp(n')\times Sp(1)$~ in the coset
$~Sp(n',1)/Sp(n')$, but is just any other independent Lie group, which may 
be needed for more practical model building.  
Since all the $~n$~ copies of vectors in the vector multiplets together
with the graviphoton in the multiplet of supergravity form the 
$~({\bf n+1})\-$representation of $~SO(n,1)$~ in the coset 
$~SO(n,1)/SO(n)$, we need special care for such non-Abelian gaugings.
Such non-Abelian gauging has been performed in the conventional 
formulation \gst, as well as recent works in \gz\cda\gzrecent, and in
other dimensions such as in 7D 
%%% 
\ref\bks{E.~Bergshoeff, I.G.~Koh and E.~Sezgin, \pr{32}{85}{1353}.}. 
%%% 
In the formulation below, we will mainly follow the notation in \bks,
in which the coset space formed by the scalars in the vector  multiplets
is $~SO(n,3) / SO(n) \times SO(3)$.  This is slightly different from our
coset $~SO(n,1)/SO(n)$, but we still can take advantage of the
similarity between them.  

First of all, the non-Abelian gauge group $~G$~ should be the subgroup of 
$~SO(n,1)$, and at the same time $~\dim G = n+1 $~ should be
satisfied, due to the coset structure to be maintained. 
Second, the structure constant $~f\du{I J} K$~ should satisfy
the relationship \bks
$$ \li{ & f_{I J K}\equiv f\du{I J} L L_{L K} 
    = f_{\[I J K\] }  ~~.
&(3.1) \cr } $$ 
where $~L_{I J}$~ is the indefinite metric on $~SO(n,1) / SO(n)$~ as in
section 2.  This condition is satisfied when this indefinite 
metric $~L_{I J}$~ is identified with the Cartan-Killing metric
$~\eta\low{I J}$~ of $~G$.  To be specific, we can have 
$~G = SO(2)\times H\times \big[\, U(1)\, \big]^{n-p+1}$, so that
$~\dim H = p-1, ~\dim G= n+1$, and   
$$\li{ & \big(L_{I J}\big) \equiv \big(\eta\low{I J} \big) 
     = \hbox{diag.}\,\big( -1, ~\eta\low{I'J'}\, , \,~ +1, +1, 
      \cdots, +1 \big)~~,     
&(3.2) \cr } $$ 
where $~{\scst I',~J'~=~1,~2,~\cdots,~p-1}$~ with $~1\le p \le n+1$. 
Also  in (3.2), the first $~-1$~ is the Cartan-Killing metric of
$~SO(2)$~ for the $~0\-$th direction in the $~(n+1)\-$dimensions, 
$~\eta\low{I'J'}$~ is that of $~H$, while the last $~(+1, +1,
\cdots, +1)$~ are the metrics for the Abelian factor groups $~\big[\,
U(1)\, \big]^{n-p+1}$.  In the special case of $~p=n+1$, there is no
$~U(1)$~ factor group.  This situation is similar to that in \bks.  

For such a gauge group $~G$, we introduce the minimal coupling with 
the coupling constant $~\Bar g$.  Typically, we have \bks 
$$ \li{ & P_{\m a} \equiv L\du a I \partial_\m L 
     = - {\sqrt 2} V_{\a a} \partial_\m\varphi^\a \cr 
& ~~~~~ ~~~~~ \longrightarrow~~
     {\cal P}_{\m a} \equiv L\du a I 
     \big( \partial_\m L_I + \Bar g f\du{I J}K A\du\m J L_K \big) 
     \equiv L\du a I {\cal D}_\m L_I 
     \equiv -{\sqrt 2} V_{\a a}{\cal D}_\m \varphi^\a 
     {~~, ~~~~~ ~~~~~}
&(3.3\rma) \cr 
& A\du{\m a} b \equiv \big( \partial_\m\varphi^\a\big) A\du{\a a} b \cr  
& ~~~~~ ~~~~~ \longrightarrow~~ 
     {\cal A}\du{\m a} b \equiv A\du{\m a}b 
     + \Bar g f\du{I J} K A\du\m I L\du a J L\du K b~~,
&(3.3\rmb) \cr 
&D_\m \l^{a A} \equiv D_\m (\o) \l^{a A}  
     + \big(\partial_\m\varphi^\a \big) A\du\a{a b} \l\du b A  
     + \big(\partial_\m\phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \l^a \big)^A  \cr 
& ~~~~~ ~~~~~ \longrightarrow~~ {\cal D}_\m \l^{a A} \equiv 
     D_\m (\o) \l^{a A}  
     + {\cal A}\du\m {a b} \l\du b A  
     + \big(\partial_\m\phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \l^a \big)^A  ~~,  
&(3.3\rmc) \cr 
& G_{\m\n\r} ~~ \longrightarrow ~~ 
	~~ G_{\m\n\r} \equiv 3 \Big( 
	\partial_{\[\m} B_{\n\r\]} - L_{I J} F\du{\[ \m\n}I A\du{\r\]} J
	+ \frac 1 3 \Bar g f\du{I J} K A\du\m I A\du\n J A_{\r K} \Big)
	~~. 
&(3.3\rmd)  \cr  } $$
Eq.~(3.3a) is none other than the standard minimal non-Abelian 
coupling for the adjoint index $~{\scst I}$.  Needless to say, the
structure constants $~f\du{I J} K$~ with the indices $~{\scst I,~J,~K}$~
for any of the $~U(1)$~ factor groups or $~SO(2)$~ are supposed to vanish. 
So effectively, only the indices $~{\scst I',~J',~K'}$~ on 
$~f\du{I J}K$~ remain.  If we rewrite (3.3a) as 
$$ \li{ & {\cal P}\du\m\a \equiv V\du a\a {\cal P}\du\m a 
     \equiv -{\sqrt 2} \big( \partial_\m \varphi^\a 
     - \Bar g A\du\m I \xi\ud \a I \big) 
     \equiv - {\sqrt 2} {\cal D}_\m \varphi^\a ~~, 
&(3.4) \cr } $$ 
then its comparison with (3.3a) implies that 
$$ \li{ & \xi\ud \a I = 
    -\frac 1 {\sqrt 2} f\du{I J} K V^{a\a} L\du a J L_K ~~, 
&(3.5) \cr } $$
with the Killing vectors $~\xi\ud \a I$~ in the directions of 
the gauged group $~G$. 
Eq.~(3.3c) has a new term for the  non-Abelian
coupling.  Relevantly, we have 
$$ \li{ & {\cal D}_\m L_I = {\cal P} \du\m a L_{I a} ~~, ~~~~
     {\cal D} _\m L\du I a = {\cal P} \du\m a  L_I~~, ~~~~
{\cal D}_\m L^I = - {\cal P}\du\m a L\du a I ~~, ~~~~
{\cal D}_\m L\du a I = - {\cal P}_{\m a} L^I {~~. ~~~~~ ~~~~~}
&(3.6) \cr } $$ 
By defining
$$ \li{ & C_{a b} \equiv f\du{I J} K L\du a I L\du b J L_K
    = -{\sqrt 2} \xi\ud \b I V_{\b b} L\du a I = -  C_{b a} ~~, 
&(3.7) \cr } $$ 
we have the important relationship
$$ \li{ &{\cal D}_{\[\m} {\cal P}_{\n\] a}
    = + \frac 12 \Bar g F\du{\m\n} I L\du I b C_{a b}~~, 
&(3.8) \cr } $$
by the use of another identity
$$ \li{ & L\du I b C_{a b} = - f\du{I J}K L\du a J L_K~~, 
&(3.9) \cr } $$ 
confirmed by (2.1).
As has been already mentioned, in expressions like (3.7) - (3.9), 
the structure constants $~f\du{I J} K$~ in any directions 
of the $~U(1)$'s or $SO(2)$~ 
are supposed to vanish, not to mention any other `mixed' directions of 
different groups.  The same is also true for the index $~{\scst I}$~ in 
the last term in (3.3d), in which any irrelevant component gives the 
vanishing of $~A\du\m I$.  These geometrical structures are parallel
to the 7D case in \bks.  

Note that in this non-Abelian gauging, the gaugini $~\l^{a A}$~ are 
{\it not} in the adjoint representation, as opposed to the usual vector 
multiplets in higher dimensions \ss, such as that in 10D with the gaugino
in the adjoint representation.  This is in a sense not surprising, because
the gaugino fields should form the $~{\bf n}\-$representation instead of
the $~({\bf n+1})\-$representation of $~SO(n,1)$, and therefore their 
range of indices should differ from that of the vector fields.  
This situation in 5D is similar to the original work in \gst, or also in
\cda\bks.  

We next introduce the Killing 
vectors $\Hat \xi^{\un\a \hat I}$~ for the direction of the gauged
group $~G$.  In order to fix an invariant action, we also follow the
result in \bks, and we can  postulate an additional term needed in the
lagrangian \bks 
$$ \li{ & e^{-1} \Lag_{\Bar g} = + i a \Bar g e^\s 
     C_{a b} \big(\Bar\l{}^a \l^b \big) ~~,  
&(3.10) \cr } $$ 
for the non-Abelian gauging, while putting {\it no
explicit}$\,$\footnotew{The word `explicit' here implies any 
$~\Bar g$-dependent term other than those hidden in the covariant
derivatives  such as (3.3).}  
$~\Bar g\-$dependent terms in the transformation rules.  Now the
variation of
$~\Lag_{\Bar g}$~ generates only two sorts of terms, when fermionic
cubic terms are ignored: (i) $~\Bar g \l F\-$terms and (ii) $~\Bar g \l
{\cal  D}\varphi\-$terms.  For the term in (i), the variation of 
the $~\psi_\m \l {\cal D} \varphi~$ Noether-term is the only
counter-contribution,  while for the term in (ii), the kinetic term of
$~\varphi$~ is the only contribution to cancel.  Both of these two
sectors yield the same condition $~a = + 2^{-3/2}$~ consistently.  

Armed with these preliminaries, we are ready to give the lagrangian 
$$\li{ e^{-1} \Lag_1  = & - \frac14 R  
	- \frac12 \big(\Bar\psi_\m\g^{\m\n\r} D_\n\psi_\r \big)
	- \frac1{12} e^{-4\s} G_{\m\n\r}^2 
	- \frac1 4 e^{-2\s} \big( L\du I a L_{J a} + L_I L_J \big) 
	F\du{\m\n}I F^{\m\n\, J}  \cr 
& - \frac12 \big(\Bar\l{}^a\g^\m {\cal D}_\m \l_a \big) 
	- \frac12 g\low{\a\b} 
	\big( {\cal D}_\m \varphi^\a \big) 
	\big( {\cal D}^\m \varphi^\b \big) 
	- \frac 3 4 \big(\partial_\m\s\big)^2 
	- \frac12\big( \Bar\chi\g^\m {\cal D}_\m\chi\big) \cr 
& - \frac12 g\low{\un\a\un\b} \big(\partial_\m\phi^{\un\a} \big) 
	\big(\partial^\m\phi^{\un\b} \big) 
	- \frac12 \big( \Bar\psi{}^{\un a} 
	\g^\m D_\m\psi_{\un a} \big) \cr 
& + \frac i{\sqrt2} V\du\a a \big( \Bar\psi_\m\g^\n\g^\m \l_a \big) 
	{\cal D}_\n\varphi^\a + \frac{{\sqrt3}i}2 
	\big( \Bar\psi_\m\g^\n\g^\m \chi \big)\partial_\n\s 
	+i V\du{\un\a}{\un a A} 
	\big( \Bar\psi_\m \g^\n\g^\m \psi_{\un a} \big) 
	\partial_\n \phi^{\un\a} \cr 
& - \frac i{4\sqrt2} e^{-\s} \big( \Bar\psi_\m \g^{\m\n\r\s} \psi_\n 
	+ 2\Bar\psi{}^\r \psi^\s \big) L_I F\du{\r\s} I 
 + \frac i{6\sqrt3} e^{-2\s} 
	\big(\Bar\psi_\m\g^{\r\s\t} \g^\m\chi\big) G_{\r\s\t} \cr 
& - \frac1{24} e^{-2\s} \big( \Bar\psi_\m \g^{\m\n\r\s\t} \psi_\n 
	- 6 \Bar\psi{}^\r\g^\s\psi^\t \big) G_{\r\s\t} 
	-\frac5{72} e^{-2\s} 
	\big( \Bar\chi\g^{\m\n\r} \chi\big) G_{\m\n\r} \cr
& - \frac1{2\sqrt2} e^{-\s} \big(\Bar\psi_\m\g^{\r\s} \g^\m \l_a \big) 
	L\du I a F\du{\r\s} I
	- \frac1{2\sqrt6} e^{-\s} 
	\big(\Bar\psi_\m \g^{\r\s}\g^\m \chi \big) L_I F\du{\r\s} I 
	- \frac i{12\sqrt2} e^{-\s} 
	\big( \Bar\chi \g^{\m\n} \chi \big) L_I F_{\m\n}{}^I  \cr 
& + \frac i{4\sqrt2} e^{-\s} \big( \Bar\l{}^a \g^{\m\n} \l_a \big) 
	L_I F\du{\m\n} I  + \frac1{24} e^{-2\s} 
	\big(\Bar\l{}^a \g^{\m\n\r}\l_a \big) G_{\m\n\r} 
	+ \frac1{24} e^{-2 \s} \big( \Bar\psi{}^{\un a} 
	\g^{\m\n\r} \psi_{\un a} \big) G_{\m\n\r} \cr  
& + \frac i{\sqrt6} e^{-2\s} \big(\Bar\chi\g^{\m\n}\l_a \big) 
	L\du I a F\du{\m\n} I 
	- \frac1{4\sqrt2} e^{-\s} \big(\Bar\psi{}^{\un a} 
	\g^{\m\n} \psi_{\un a} \big) 
	L_I F\du{\m\n} I \cr 
& + \frac i{2\sqrt2} \Bar g e^\s C_{a b} \big( \Bar\l{}^a \l^b\big) 
     ~~,~~~~~ 
&(3.11) \cr }$$ 
with all of the $~{\cal D}_\m$~ and $~G_{\m\n\r}$~ in (3.3), 
yielding an invariant action $~S_1$~ under supersymmetry
$$\li{ & \d_Q e\du\m m = + \big(\Bar\e\g^m\psi_\m\big) ~~, ~~~~
	\d_Q \s = + \frac i{\sqrt3} \big(\Bar\e\chi\big) ~~, ~~~~ \cr
& \d_Q \psi\du\m A = + D_\m \e^A + \frac i{6\sqrt2} e^{-\s}    
	\big(\g\du\m{\r\s} - 4 \d\du\m\r \g^\s \big)  
	\e^A L_I F\du{\r\s} I 
	+ \frac1{18} e^{-2\s} \big( \g\du\m {\r\s\t} 
	- \frac3 2\d\du\m\r\g^{\s\t} \big) \e^A G_{\r\s\t} ~~, \cr  
& \d_Q A\du\m I = - \frac i{\sqrt2} e^\s L^I \big( \Bar\e\psi_\m\big) 
	+ \frac1{\sqrt6} e^\s L^I \big(\Bar\e\g_\m \chi\big) 
	+ \frac1{\sqrt2} e^\s \big( \Bar\e\g_\m \l^a\big) L\du a I 
	~~, \cr 
& \d_Q B_{\m\n} = + e^{2\s} \big( \Bar\e\g_{\[\m} \psi_{\n\]} \big) 
	+ \frac i{\sqrt3} e^{2\s} \big( \Bar\e\g_{\m\n}\chi\big) 
	- 2L_{I J} A\du{\[\m|} I \big( \d_Q A\du{|\n\]} J \big) ~~, \cr 
& \d_Q \chi^A = - \frac1{2\sqrt6} e^{-\s} \g^{\m\n} \e^A L_I 
	F\du{\m\n} I 
	    + \frac i{6\sqrt3} e^{-2\s} \g^{\m\n\r} \e^A G_{\m\n\r} 
	    - \frac{{\sqrt3}i}2 \g^\m \e^A\partial_\m \s  ~~ , \cr 
& \d_Q \varphi^\a 
	= +\frac i{\sqrt2} V\du a \a \big(\Bar\e\l^a \big) ~~, \cr 
& \d_Q \l^{a A} = - \frac1{2\sqrt2} e^{-\s} 
	\g^{\m\n}\e^A L\du I a F\du{\m\n} I  
	- \frac i{\sqrt2} \g^\m \e^A V\du\a a 
	{\cal D}_\m \varphi^\a ~~, \cr 
& \d_Q \phi^{\un\a} = + i V\du{\un a A} {\un\a} 
  \big( \Bar\e^A\psi^{\un a}\big)  ~~, \cr 
& \d_Q \psi^{\un a} 
   = - i V\du{\un\a}{\un a A} \g^\m \e\low A \partial_\m \phi^{\un\a}
   ~~,     
&(3.12) \cr } $$ 
Note that there is no need of any explicitly $~\Bar g\-$dependent terms in
the  transformation rule.  There is no potential term generated in this
gauging, which is similar to the conventional $~N=2$~ theories in 5D
\gz\cda\gzrecent.  Compared with \bks, since our vector fields do not
carry extra $~Sp(1)$~ indices, no scalar potential term is generated. 
 
Analogous to (2.5), we have the scaling invariance of $~\Lag_1$~ 
when the coupling constant $~\Bar g$~ transforms as
$$\li{ & \Bar g \rightarrow e^{-c} \Bar g~~, 
&(3.13) \cr } $$
when the fields transform as in (2.5).   



 

\bigskip\bigskip\bigskip



\centerline{\bf 4.~~$Sp(n')\times Sp(1)\-$Gauging}  

In our previous paper \nr, we studied the gauging of  $~SO(2)$~ which is
the subgroup of $~Sp(1)$~ in the isotropy groups $~Sp(n') \times Sp(1)$~
in the coset $~Sp(n',1) /Sp(n')\times Sp(1)$.  Most of the geometric
relationships related to the coset $~SO(n,1) / SO(n)$~ are parallel to
the $~SO(2)\-$gauging \nr, so we give important relations in such a way
that the comparison with \nr\ is easy to make.  

Our total gauged group in this section is $~G = SO(2) \times Sp(n') \times 
Sp(1) \times \Bar H \times [\,U(1)\,]^{n-p+1}$, which is a special case of the 
previous section.  In fact, the first $~SO(2)$~ is for the 
$~{\scst I~=~0}\-$direction for the indices $~{\scst I ~= ~0,~
1,~\cdots, ~n+1}$, and the groups 
$~Sp(n')\times Sp(1)$~ are regarded as a special case of $~H
\equiv Sp(n')\times Sp(1) \times \Bar H$~ for the group $~H$~ in the
last section, and an  arbitrary gauge group $~\Bar H$~ with
$~\dim{\Bar H} = p - n'(2n'+1) - 4$, such that the previous condition
$~\dim H = p -1$~ is maintained.  Since the dimension $~p$~ is still 
arbitrary, we have enough freedom for choosing the group $~\Bar H$~ 
for a large enough dimension of $~n$.  

Accordingly, we arrange our index convention as follows.  
Among the indices 
$~{\scst I,~J,~\cdots~=~0,~1,~2,}$
\newline $~{\scst \cdots,~n}$~ for
the total $~n+1$~ copies of vector fields, we use the indices $~{\scst
\un I,~ \un J,~\cdots =~1, ~2,~\cdots, ~ n'(2n'+1)}$~ for the adjoint
representation of $~Sp(n')$, and combine them with $~{\scst i,~j,~
\cdots~=~1,~2,~3}$~ for that of $~Sp(1)$, in terms of the combined
indices $~{\scst\hat I~\equiv~(\un I, i),~ \hat J~\equiv~(\un J, 
j),~\cdots}$~ for the gauged groups  $~Sp(n')\times Sp(1)$.  
For the adjoint indices for $~\Bar H$, we use the {\it barred} ones: 
$~{\scst \Bar I, ~\Bar J, ~\cdots~=~1,~2,~\cdots,~p-1}$.  As for the remaining 
product groups $~SO(2)\times [\,U(1)\,]^{n-p+1}$, we do not need 
particular indices in this section, so we do not specify the indices for 
these groups.  Compared with the indices $~{\scst
I,~J,~\cdots}$, all these indices $~{\scst \hat I, ~\hat J,
~\cdots;~ \un I,~\un J,~\cdots; ~i,~j,~\cdots;~ \Bar I,~\Bar J,~\cdots}$~ 
do not need distinctions of their raising/lowering due to their
positive definite metrics.  Therefore their contractions are
given as superscripts like $~A^{\un I} B^{\un I}$.     

In our previous paper \gst, the $~SO(2)\-$gauging was performed by
introducing  the constant vectors $~V^I$, with the coupling constant
$~g$.  In our present case of $~Sp(n')\times Sp(1)\-$gauging, this
$~SO(2)$~ group is enlarged to $~Sp(n')\times Sp(1)$.  
In this section, we use the coupling constant $~g$~ for $~Sp(1)$, 
$~g'$~ for $~Sp(n')$, and $~\Bar g$~ for $~\Bar H$.  
Accordingly, all the combination of $~g V_I A\du\m I \xi^{\un\a} $~ in \nr\ will
be replaced by $~g A\du\m i \xi^{\un\a i} + g' A\du\m{\un I}
\xi^{\un\a\un I}$, where $~\xi^{\un\a\un I}$~ and $~\xi^{\un\a i}$~ are
the Killing vectors for the gauged groups $~Sp(n')\times Sp(1)$~ in the
coset $~Sp(n',1) / Sp(n') \times Sp(1)$.  

Accordingly, the covariant derivatives on $~Sp(1)$~ non-invariant 
fermions acquire the $~Sp(1)$~ minimal couplings in addition to the 
$~D_\m$'s or $~\partial_\m\varphi^\a$~ in section 2 as 
$$\li{ D_\m \e^A & ~~\longrightarrow ~~ 
   {\cal D}_\m \e^A \equiv 
   D_\m (\o) \e^A 
   + \big({\cal D}_\m \phi^{\un\a} \big) A\du{\un\a} i 
   \big( T^i \, \e \big)^A + g A \du\m i 
   \big( T^i \, \e \big)^A ~~, 
&(4.1\rma) \cr   
D_{\[\m} \psi\du{\n\]} A & ~~\longrightarrow ~~ 
    {\cal D}_{\[\m} \psi\du{\n\]} A \equiv 
     D_{\[\m |} (\o) \psi\du{|\n\]} A  
     + \big({\cal D}_{\[\m |} \phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \psi_{|\n\]} \big)^A + g A \du{\[\m|} i \big( T^i
     \psi_{|\n\]} \big)^A  ~, ~~~~~ ~~~~~  
&(4.1\rmb) \cr  
D_\m \chi^A & ~~\longrightarrow ~~
	{\cal D}_\m \chi^A \equiv 
  D_\m (\o) \chi^A  
     + \big({\cal D}_\m\phi^{\un\a} \big) A\du{\un\a} i 
     \big( T^i \chi \big)^A   
     + g A \du\m i \big( T^i \chi \big)^A  ~~, 
&(4.1\rmc) \cr  
D_\m \l^{a A} & ~~\longrightarrow ~~ 
	{\cal D}_\m \l^{a A} \equiv  
	D_\m (\o) \l^{a A}  
     + {\cal A} \du\m {a b} \l\du b A  
     + \big({\cal D}_\m\phi^{\un\a} \big) 
    A\du{\un\a} i \big( T^i \l^a \big)^A 
	+ g A \du\m i \big( T^i \l^a \big)^A ~~,   
&(4.1\rmd) \cr  
\partial_\m \varphi^\a & ~~ \longrightarrow ~~
     {\cal D}_\m \varphi^\a \equiv \partial_\m \varphi^\a 
      - A\du\m I \Hat \xi\ud\a I~~, 
&(4.1\rme) \cr 
D_\m \psi^{\un a} & ~~\longrightarrow ~~ 
	{\cal D}_\m \psi^{\un a} \equiv D_\m (\o) \psi^{\un a}  
     + \big({\cal D}_\m\phi^{\un\a} \big) A\du{\un\a} {\un I} 
     \big( T^{\un I} \, \psi \big)^{\un a} 
     + g' A\du\m {\un I} \big( T^{\un I} \psi\big)^{\un a} ~~, 
&(4.1\rmf) \cr  
\partial_\m \phi^{\un\a} & ~~\longrightarrow ~~ 
	{\cal D}_\m \phi^{\un\a} \equiv \partial_\m \phi^{\un\a} 
     - g' A\du\m{\un I} \xi^{\un\a\un I} 
     - g A\du\m i \xi^{\un\a i} \equiv \partial_\m \phi^{\un\a} 
     - A\du\m{\hat I} \Hat\xi^{\un\a\hat I} ~~, 
&(4.1{\rm g}) \cr } $$ 
with the generalized Killing vectors
$$ \li{ & \Hat\xi\ud{\un\a} I  \equiv
     \cases{ ~ \Hat\xi^{\un\a \hat I} \equiv 
     \cases{ \Hat\xi^{\un\a\un I} \equiv g'\xi^{\un\a \un I} ~~~
      &\hbox{(for $~Sp(n')$)} ~~,  \cr 
      \Hat\xi^{\un\a i} \equiv 
     g\xi^{\un\a i} ~~~ & \hbox{(for $~Sp(1)$)} ~~, \cr }
\cr ~~~~~ \cr 
~ 0 ~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~\,  \hbox{(otherwise)} ~~.  \cr
}   
&(4.2) \cr } $$ 
For example, an expression like $~\Hat\xi\ud{\un\a} I
L^I$~ actually means $~\Hat\xi\ud{\un\a} I L^I
\equiv \Hat\xi^{\un\a\hat I} L^{\hat I} 
\equiv g' \xi^{\un\a\un I} L^{\un I} + g \xi^{\un\a i} L^i$.  The
absence of the component $~\Hat \xi^{\un\a \Bar J}$~ is understood from 
the fact that the group $~\Bar H$~ has nothing to do with the 
coset $~Sp(n',1)/ Sp(n') \times Sp(1)$.  In
(4.1), all the terms other than explicit
$~g\-$terms are just the previous covariant derivatives in (2.6) in which
$~\partial_\m\phi^{\un\a}$~ is  replaced by $~{\cal D}_\m\phi^{\un\a}$,
and the matrices $~T^{\hat I}$~ are the anti-hermitian generator of 
$~Sp(n') \times Sp(1)$, as its index $~{\scst \hat I}$~ reveals.  This
structure is similar to the models in \gst\gz\cda\gzrecent.  

The covariance of the derivatives in (4.1) are 
confirmed by considering the transformations of these fields under
the gauged groups $~G\equiv SO(2) \times Sp(n')\times Sp(1) \times \Bar H 
\times [\,U(1)\,]^{n-p+1}$, such as 
$$ \li{ &\d_{\rm G} \phi^{\un\a}  
     = + \a^I \Hat\xi\ud{\un\a} I ~~, ~~~~
     \d_{\rm G} A\du\m I = \partial_\m \a^I 
     + \Hat f\du{J K} I A\du\m J \a^K  
     \equiv D_\m \a^I ~~, 
&(4.3\rma)  \cr 
& \d_G \Hat \xi^{\un\a}{}_I  = \a^J \Hat\xi\ud{\un\b} I 
     \big( \partial_{\un\b} \Hat\xi\ud{\un\a} J \big)
     - \Hat f\du{I J} K \a^J \Hat\xi\ud{\un\a} K ~~, 
&(4.3\rmb) \cr 
& \d_G \big( {\cal D}_\m\phi^{\un\a} \big) 
     = \a^I \big({\cal D}_\m\phi^{\un\a} \big) 
     \big( \partial_{\un\b} \Hat\xi\ud{\un\a} I \big) ~~, 
&(4.3\rmc) \cr } $$ 
for the local parameters $~\a^I$~ for the 
gauged groups in $~G$, and the structure constants 
$$ \li{ & \Hat f\du{I J} K 
    \equiv \cases{ ~\Hat f^{\hat I\hat J\hat K} = 
    \cases{~\Hat f^{\un I \un J \un K} 
      = ~g' f^{\un I\un J \un K} 
      	{\hskip 0.2in}
        \hbox{(for $~Sp(n')$)} ~~, \cr 
     ~\Hat f^{i j k} 
      = ~g\e^{i j k} 
	{\hskip 0.43in}\hbox{(for $~Sp(1)$)} ~~, \cr}\cr 
     ~\Hat f^{\Bar I \Bar J \Bar K} 
      = ~\Bar g f^{\Bar I\Bar J\Bar K} 
      {\hskip 1.0in} 
	\hbox{(for $~\Bar H$)}~~, \cr  
     ~0 {\hskip 1.97in} 
     \hbox{(otherwise)} ~~, \cr } 
&(4.4\rma) \cr 
& \Hat f_{I J K} \equiv \Hat f\du{I J} L L_{L K} = \Hat f_{\[I J K\]} ~~, 
&(4.4\rmb) \cr } $$ 
for the  respective structure constants for
$~Sp(n'), ~Sp(1)$~ and $~\Bar H$~ in the  combined notation. 
Since the $~SO(2)$~ group in the negative metric $~0\-$th direction is
Abelian, it does not enter (4.4), and therefore we do not need
to distinguish the super/subscripts on the r.h.s.~of (4.4a).  

The field strength (3.3d) should be also modified by all the 
non-Abelian couplings:
$$ \li{ & G_{\m\n\r} \equiv 
	3 \Big( \partial_{\[\m} B_{\n\r\]} 
	- L_{I J} F\du{\[ \m\n} I A\du{\r\]} J 
	+ \frac 1 3 \Hat f_{I J K} A\du\m I A\du \n J A\du \r K \Big) 
	~~. 
&(4.5) \cr } $$
The commutator of two covariant derivatives acting on $~\e^A$~ 
provides certain important geometric quantity in our system:
$$ \li{ \[ {\cal D}_\m , {\cal D}_\n \] \e\low A  
     = \, & -\frac14 R\du{\m\n}{m n} \g_{m n} \e\low A 
      + \big({\cal D}_{\[\m|} \phi^{\un\a}  \big) 
     \big({\cal D}_{|\n\] } \phi^{\un\b}  \big) F\du{\un\a\un\b}i 
     \big( T^i \e\big)_A  
     - F\du{\m\n}{\hat I} \Hat C^{i\hat I} \big( T^i \e \big)_A 
     {~~, ~~~~~ ~~~} 
&(4.6) \cr } $$ 
where the function $~\Hat C^{i\hat I}$~ is defined by 
$$ \li{ & \Hat C^{i J} \equiv 
     \cases{ ~\Hat C^{i \hat J} 
~= ~\cases{ \Hat C^{i \un J}  \equiv g' C^{i\un J} 
     \equiv g' A\du{\un\a} i \xi^{\un\a\un J} 
     ~~~ & \hbox{(for ~$~Sp(n')$)} ~~, \cr 
      \Hat C^{i j} \equiv g C^{i j} 
      \equiv g\big( A\du{\un\a} i \xi^{\un\a j} 
     - \d^{i j} \big)  ~~~ & \hbox{(for ~$~Sp(1)$)} ~~ , \cr } \cr
~~~\cr  
~0 ~~~~~ (\hbox{otherwise}) ~~, \cr }   
&(4.7) \cr } $$  
which is analogous to the $~N=2$~ case in 6D \ns, or our combination
$~\Hat C^{i\hat J} T^i$~ is an analog of $~P\du{I i} j$~ in the notation
in \cda.  The component $~\Hat C^{i j}$~ in (4.7) implies that all 
the terms with $~g T^2 V_I $~ in \nr\ should be replaced by $~- 
T^i \Hat C\ud i I$, when we gauge $~G\equiv SO(2) \times Sp(n') \times
Sp(1)\times\Bar H \times [\, U(1)\,]^{n-p+1}$~ instead of
$~SO(2)$~ in \nr.  Some illustrative examples of the replacements of the
terms  in \nr\ are given by  
$$ \li{ + g T^2 V_I ~~ & \longrightarrow ~~
       - T^i \Hat C^i{}_I ~~, 
&(4.8\rma) \cr 
+ g\xi^{\un\a} V_I ~~ & \longrightarrow ~~ 
        + \Hat \xi\ud{\un\a} I ~~, 
&(4.8\rmb) \cr 
+ \frac i{2\sqrt 2}g  e^\s \big( \Bar\psi_\m 
     \g^{\m\n} T_2 \psi_\n\big) V_I L^I  ~~ & \longrightarrow ~~
     - \frac i{2\sqrt2} e^\s \big( \Bar\psi_\m 
     \g^{\m\n} T_i \psi_\n\big) \Hat C\ud i I L^I ~~,  
&(4.8\rmc) \cr
- \frac 18 g^2 e^{2\s} V_I V_J L^{I J} ~~ &
      \longrightarrow~~
     - \frac18 \Hat C\ud i I \Hat C\ud i J L^{I J} ~~. 
&(4.8\rmd) \cr } $$ 
Needless to say, when the gauged group is truncated from $~G\equiv SO(2)
\times Sp(n') \times Sp(1)\times\Bar H \times [\, U(1)\,]^{n-p+1}$~ back
into $~SO(2)$~ with the $~V_I$'s as in \nr, then all the r.h.s.~in (4.8)
go back to their l.h.s.  This can provide a good confirmation at
various stages of computations, in particular the invariance check of
total action under supersymmetry.  Due to the indefinite metric
involved, special care is needed for  the contraction of the $~{\scst
I}\-$indices here, while   the ups/downs of the index $~{\scst i}$~ does
not matter.      Relevantly, we can define the covariant derivative on
$~\Hat C^{i\hat I}$~ as 
$$ \li{ & D_{\un\a} \Hat C^{i\hat J} 
     \equiv \partial_{\un\a} \Hat C^{i\hat J} 
     + \e^{i j k} A\du{\un\a} j \Hat C^{k\hat J} ~~,  
&(4.9) \cr } $$
so that 
$$ \li{ {\cal D}_\m \Hat C^{i\hat I} & \equiv 
     \partial_\m \Hat C^{i\hat I} + g\e^{i j k} A\du\m j \Hat C^{k\hat I} 
	+ \Hat f^{\hat I \hat J \hat K} A\du\m{\hat J} 
	\Hat C^{i \hat K} 
     + \e^{i j k} \big( {\cal D}_\m \phi^{\un\a} \big) 
     A\du{\un\a} j \Hat C^{k\hat I} \cr 
& = \big({\cal D}_\m\phi^{\un\a} \big) 
      \big(D_{\un\a} \Hat C^{i\hat I} \big)  {~~. ~~~~~}
&(4.10) \cr } $$ 
To confirm the last equality, we need the relationship
$$ \li{ & \Hat\xi^{\un\a \hat I} \partial_{\un\a} 
      \Hat C^{i \hat J} 
	= \Hat f^{\hat I \hat J\hat K} \Hat C^{i \hat K}
	+ \Hat f^{\hat I i k} \Hat C^{k \hat J} ~~, 
&(4.11) \cr } $$
derived from the Lie derivatives
$$\li{ & {\cal L}_{\Hat \xi^{\hat I}}\Hat \xi^{\un \a \hat J} \equiv 
	\Hat \xi^{\un\b \hat I} \partial_{\un\b} \Hat \xi^{\un\a \hat J} 
	- \Hat \xi^{\un\b \hat J} \partial_{\un\b} \Hat \xi^{\un\a \hat I} 
	= \Hat f^{\hat I \hat J \hat K} \Hat \xi^{\un\a \hat K} ~~, \cr  
& {\cal L}_{\Hat \xi^{\hat I}} A\du{\un \a} i \equiv
	\Hat \xi^{\un\b \hat I} \partial_{\un\b} A\du{\un\a} i 
	- \big( \partial_{\un\a} \Hat \xi^{\un\b \hat I} \big)
	A\du{\un\b} i = \Hat f^{\hat I i j } A\du{\un\a} j ~~.
&(4.12) \cr } $$
Note that as in the 6D case in \ns, there is no term with $~\Hat 
f^{\hat J\hat K\hat L} A\du{\un\a}{\hat K}$~ needed in (4.9), in order 
to be consistent with supersymmetry of the action.  Other
important corollaries with these $~\Hat C$'s are such as 
$$ \li{ & D_{\un\a} \Hat C^{i\hat I} 
      = \Hat\xi^{\un\b\hat I} F\du{\un\a\un\b} i ~~,  ~~~~ 
     \Hat\xi^{\un\a\hat I} D_{\un\a} \Hat C^{i\hat I} \equiv 0 ~~,  
&(4.13) \cr } $$ 
which have parallel structures as in the 6D case \ns.  

The most crucial relationship involving $~\Hat C^{i J}$~ in our system is
the  constraint required by the supersymmetric invariance of the total 
action, needed for the consistency between the two cosets
$~SO(n,1)/SO(n)$~ and $~Sp(n',1)  /Sp(n') \times Sp(1)$:  
$$ \li{ & F\du{\un\a\un\b}i \Hat\xi^{\un\a}{}_I \Hat\xi^{\un\b}{}_J 
     - \e^{i j k} \Hat C\ud j I \Hat C\ud k J 
     - \Hat f\du{I J} K \Hat C\ud i K  = 0 ~~.  
&(4.14) \cr } $$ 
This constraint is required by the cancellation of $~\l\-$linear terms 
with the structure $~\big(\Bar\e T^i \l\big) $~ with one $~T^i\-$generator
sandwiched.  This constraint is also analogous to eq.~(3.15) in 
\cda, or to eqs.~(2.21) - (2.24) in \gzrecent.  
The necessity of such a constraint is natural from the fact that 
the vector fields $~A\du\m I$~ in our system are {\it both} in the 
$~({\bf n+1})\-$representation of $~SO(n,1)$~ {\it and} the adjoint 
representations of the gauged groups in $~G$~ at the same time.  And 
therefore their mutual consistency, in particular, under supersymmetry 
requires such a constraint.  It is taken for granted that 
in (4.14), there are many trivially vanishing components for each
terms depending on the combination of  the adjoint indices.  For example,
according to (4.4), the structure constants $~\Hat f\du{I J} K$~ vanishes
identically for any $~U(1)\-$directions, or for any `mixed' directions 
of different gauge groups.  However, note that the first term 
in (4.14) does not automatically vanish for such `mixed' directions.  
Our previous $~SO(2)$~ gauging in \nr\ also satisfies (4.14) trivially, 
because the last two terms vanish, while $~\Hat \xi\ud{\un\a} I 
\rightarrow g \xi^{\un\a} V_I$~ makes the first term vanish, too.     

The tensor $~C_{a b}$~ in (3.7) is also redefined 
in terms of $~\Hat f\du{I J} K$~ by 
$$\li{ & \Hat C_{a b} \equiv \Hat f\du{I J} K L\du a I L\du b J  L_K 
	= - \Hat C_{b a} ~~.   
&(4.15) \cr } $$

With these preliminaries, we now give our lagrangian\footnotew{We 
mention the errors in signatures of terms in our previous paper \nr.  The
sign errors in the $~g$-linear lagrangian (4.2) and
$~g$-linear transformation rule (4.3) in \nr\ are now corrected in (4.16)
and (4.17).}    
$$\li{ e^{-1} \Lag_2  = & - \frac14 R  
	- \frac12 \big(\Bar\psi_\m\g^{\m\n\r}{\cal D}_\n\psi_\r \big)
	- \frac1{12} e^{-4\s} G_{\m\n\r}^2 
	- \frac1 4 e^{-2\s} \big( L\du I a L_{J a} + L_I L_J \big) 
	F\du{\m\n}I F^{\m\n\, J}  \cr 
& - \frac12 \big(\Bar\l{}^a\g^\m {\cal D}_\m \l_a \big) 
	- \frac12 g\low{\a\b} 
	\big( {\cal D}_\m \varphi^\a \big) 
	\big( {\cal D}^\m \varphi^\b \big) 
	- \frac 3 4 \big(\partial_\m\s\big)^2 
	- \frac12\big( \Bar\chi\g^\m {\cal D}_\m\chi\big) \cr 
& - \frac12 g\low{\un\a\un\b} \big({\cal D}_\m\phi^{\un\a} \big) 
	\big({\cal D}^\m\phi^{\un\b} \big) 
	- \frac12 \big( \Bar\psi{}^{\un a} 
	\g^\m {\cal D}_\m\psi_{\un a} \big) \cr 
& + \frac i{\sqrt2} V\du\a a \big( \Bar\psi_\m\g^\n\g^\m \l_a \big) 
	{\cal D}_\n\varphi^\a + \frac{{\sqrt3}i}2 
	\big( \Bar\psi_\m\g^\n\g^\m \chi \big)\partial_\n\s 
	+i V\du{\un\a}{\un a A} 
	\big( \Bar\psi_\m \g^\n\g^\m \psi_{\un a} \big) 
	{\cal D}_\n \phi^{\un\a} \cr 
& - \frac i{4\sqrt2} e^{-\s} \big( \Bar\psi_\m \g^{\m\n\r\s} \psi_\n 
	+ 2\Bar\psi{}^\r \psi^\s \big) L_I F\du{\r\s} I 
 + \frac i{6\sqrt3} e^{-2\s} 
	\big(\Bar\psi_\m\g^{\r\s\t} \g^\m\chi\big) G_{\r\s\t} \cr 
& - \frac1{24} e^{-2\s} \big( \Bar\psi_\m \g^{\m\n\r\s\t} \psi_\n 
	- 6 \Bar\psi{}^\r\g^\s\psi^\t \big) G_{\r\s\t} 
	-\frac5{72} e^{-2\s} 
	\big( \Bar\chi\g^{\m\n\r} \chi\big) G_{\m\n\r} \cr
& - \frac1{2\sqrt2} e^{-\s} \big(\Bar\psi_\m\g^{\r\s} \g^\m \l_a \big) 
	L\du I a F\du{\r\s} I
	- \frac1{2\sqrt6} e^{-\s} 
	\big(\Bar\psi_\m \g^{\r\s}\g^\m \chi \big) L_I F\du{\r\s} I 
	- \frac i{12\sqrt2} e^{-\s} 
	\big( \Bar\chi \g^{\m\n} \chi \big) L_I F_{\m\n}{}^I  \cr 
& + \frac i{4\sqrt2} e^{-\s} \big( \Bar\l{}^a \g^{\m\n} \l_a \big) 
	L_I F\du{\m\n} I  + \frac1{24} e^{-2\s} 
	\big(\Bar\l{}^a \g^{\m\n\r}\l_a \big) G_{\m\n\r} 
	+ \frac1{24} e^{-2 \s} \big( \Bar\psi{}^{\un a} 
	\g^{\m\n\r} \psi_{\un a} \big) G_{\m\n\r} \cr  
& + \frac i{\sqrt6} e^{-2\s} \big(\Bar\chi\g^{\m\n}\l_a \big) 
	L\du I a F\du{\m\n} I 
	- \frac1{4\sqrt2} e^{-\s} \big(\Bar\psi{}^{\un a} 
	\g^{\m\n} \psi_{\un a} \big) 
	L_I F\du{\m\n} I \cr 
& + \Big[ - \frac i{2\sqrt2} e^\s 
      \big(\Bar\psi_\m\g^{\m\n} T^i \psi_\n \big) \Hat C\ud i I L^I 
     + \frac1{\sqrt2} e^\s \big( \Bar\psi_\m \g^\m T^i \l^a \big) 
      \Hat C\ud i I L\du a I \cr 
& ~~~~~ + \frac i{2\sqrt2} e^\s \big( \Bar\l^a T^i \l_a \big)
     \Hat C\ud i I L^I + \frac{2i}{\sqrt 6} e^\s 
     \big( \Bar\chi T^i\l^a\big) \Hat C\ud i I L\du a I \cr 
& ~~~~~  - \frac i{6\sqrt2} e^\s \big(\Bar\chi T^i \chi \big)  
     \Hat C\ud i I L^I + \frac 1{\sqrt6} e^\s 
     \big(\Bar\psi_\m \g^\m T^i \chi\big) \Hat C\ud i I L^I \cr 
& ~~~~~  + {\sqrt 2} i e^\s \big(\Bar\psi{}^{\un a} \l^{a A} \big)
     V_{\un\a \un a A} \Hat\xi\ud{\un\a} I L\du a I 
     + \frac 1{\sqrt2} e^\s 
     \big( \Bar\psi\du\m A \g^\m \psi^{\un a} \big) V_{\un\a \un a A}
     \Hat\xi\ud {\un\a} I L^I \cr 
& ~~~~~  + \frac{2i}{\sqrt6} e^\s \big(\Bar\chi{}^A \psi^{\un a} \big) 
     V_{\un\a \un a A} \Hat\xi\ud{\un\a} I L^I 
     + \frac i{4\sqrt2} e^\s 
     \big( \Bar\psi{}^{\un a} \psi^{\un b}\big) V\du{\un a} {B\un\a} 
     V_{\un\b\un b B} \big(D_{\un\a} \Hat\xi\ud{\un\b} I \big) 
     L^I 
     + \frac i{2\sqrt2} e^\s \Hat C_{a b} \big( \Bar\l{}^a\l^b \big) 
     \, \Big] \cr 
& + \Big[ - \frac18 e^{2\s} \Hat C^{i I} \Hat C^{i J}  L_{I J} 
	- \frac14 e^{2\s} \Hat\xi\du{\un\a}I \Hat\xi^{\un\a J} 
	 L_I L_J \, \Big]  ~~,~~~~~ 
&(4.16) \cr }$$ 
where the penultimate pair of the square brackets $~[~~]$~ is for the 
terms at $~{\cal O}(\Tilde g)$, while the last pair is for the 
terms at $~{\cal O}(\Tilde g^2)$, where $~\Tilde g$~ is any minimal
coupling constant for gaugings among $~g,~g'$~ or $~\Bar g$.  This is because 
$~\Hat C^{i I},~\Hat C_{a b}$~ and $~\Hat \xi^{\un\a I} $~ are all 
at $~{\cal O} (\Tilde g)$.   
Our lagrangian (4.16) yields an action $~S_2$~ invariant under
supersymmetry 
$$\li{ & \d_Q e\du\m m = + \big(\Bar\e\g^m\psi_\m\big) ~~, ~~~~
	\d_Q \s = + \frac i{\sqrt3} \big(\Bar\e\chi\big) ~~, ~~~~ \cr
& \d_Q \psi\du\m A = + {\cal D}_\m \e^A + \frac i{6\sqrt2} e^{-\s}    
	\big(\g\du\m{\r\s} - 4 \d\du\m\r \g^\s \big)  
	\e^A L_I F\du{\r\s} I 
	+ \frac1{18} e^{-2\s} \big( \g\du\m {\r\s\t} 
	- \frac3 2\d\du\m\r\g^{\s\t} \big) \e^A G_{\r\s\t} \cr 
& ~~~~~ ~~~~~ ~~ - \frac i{3\sqrt2} e^\s \big(\g_\m T^i\e \big)^A 
     \Hat C\ud i I L^I ~~, \cr  
& \d_Q A\du\m I = - \frac i{\sqrt2} e^\s L^I \big( \Bar\e\psi_\m\big) 
	+ \frac1{\sqrt6} e^\s L^I \big(\Bar\e\g_\m \chi\big) 
	+ \frac1{\sqrt2} e^\s \big( \Bar\e\g_\m \l^a\big) L\du a I 
	~~, \cr 
& \d_Q B_{\m\n} = + e^{2\s} \big( \Bar\e\g_{\[\m} \psi_{\n\]} \big) 
	+ \frac i{\sqrt3} e^{2\s} \big( \Bar\e\g_{\m\n}\chi\big) 
	- 2L_{I J} A\du{\[\m|} I \big( \d_Q A\du{|\n\]} J \big) ~~, \cr 
& \d_Q \chi^A = - \frac1{2\sqrt6} e^{-\s} \g^{\m\n} \e^A L_I 
	F\du{\m\n} I 
	    + \frac i{6\sqrt3} e^{-2\s} \g^{\m\n\r} \e^A G_{\m\n\r} 
	    - \frac{{\sqrt3}i}2 \g^\m \e^A\partial_\m \s  
     + \frac1{\sqrt6} e^\s \big( T^i \e \big)^A \Hat C\ud i I L^I  
      ~~ , \cr 
& \d_Q \varphi^\a 
	= +\frac i{\sqrt2} V\du a \a \big(\Bar\e\l^a \big) ~~, \cr 
& \d_Q \l^{a A} = - \frac1{2\sqrt2} e^{-\s} 
	\g^{\m\n}\e^A L\du I a F\du{\m\n} I  
	- \frac i{\sqrt2} \g^\m \e^A V\du\a a 
	\partial_\m \varphi^\a + \frac 1{\sqrt2} e^\s \big( T^i \e\big)^A 
     \Hat C\ud i I L^{a I} ~~, \cr 
& \d_Q \phi^{\un\a} = + i V\du{\un a A} {\un\a} 
  \big( \Bar\e^A\psi^{\un a}\big)  ~~, \cr 
& \d_Q \psi^{\un a} 
   = - i V\du{\un\a}{\un a A} \g^\m \e\low A {\cal D}_\m \phi^{\un\a} 
     - \frac1{\sqrt 2} e^\s \e\low B V\du{\un\b}{\un a B} 
     \Hat\xi\ud{\un\b} I L^I ~~,    
&(4.17) \cr } $$ 

Similarly to the $~SO(2)\-$gauging \nr, the potential term is positive 
definite, {\it except for} the term with $~L_I\equiv L\du I {(0)}$~ in
$~L_{I J} =  L_{a I} L\du J a - L_I L_J$~ \gst\gz\cda\gzrecent\nr: 
$$ \li{ & V_{\rm pot} 
     = + \frac18 e^{2\s} \Hat C^{i I} \Hat C^{i J}  L_{I J} 
	+ \frac14 e^{2\s} \Hat \xi\du{\un\a}I \Hat \xi^{\un\a J} L_I L_J ~~.   
&(4.18) \cr } $$ 

As in section 3, our lagrangian $~\Lag_2$~ has the scaling invariance 
when $~g,~ g'$~ and $~\Bar g$~ transform like 
$$ g \rightarrow e^{-c} g ~~, ~~~~g' \rightarrow e^{-c} g'~~  , ~~~~
          \Bar g \rightarrow e^{-c} \Bar g~~, ~~~~ $$ 
$$ \Hat C^{i I} \rightarrow e^{-c} \Hat C^{i I}~~, ~~~~
	\Hat \xi^{\un\a I} \rightarrow e^{-c} \Hat\xi^{\un\a I}
	~~, ~~~~ \Hat C_{a b} \rightarrow e^{-c} \Hat C_{a b} ~~, 
\eqno(4.19) $$ 
in addition to (2.5).  

Similarly to the 6D case with the $~Sp(n')\times Sp(1)\-$gauging \ns, 
any subgroup of these gauge groups can be also gauged  consistently with
supersymmetry, even though the details of its process are skipped here. 
In such a case, the indices $~{\scst \un I, ~\un J,~\cdots}$~ and
$~{\scst i, ~j,~k}$~ are to be replaced by the corresponding indices of
such gauged subgroups.  In particular, for the $~SO(2)$~ subgroup gauging
out of the $~Sp(1)$~ above, only the second direction $~{\scst 2}$~ out
of the original indices $~{\scst i,~j,~k}$~ is relevant, so that we can
use the notation such as $~T_2$~ as in \nr.  As the
$~SO(2)\-$gauging described in \nr\ or in \gst\gz\cda\gzrecent, 
we can combine the products of $~U(1)$~ groups, by introducing the 
constant couplings $~V_I~~{\scst (I, ~J,~\cdots~=~0,~1,~2,~\cdots,~n)}$,
as a slight generalization of the single $~SO(2)$~ subgroup gauging.  In
any of these cases, our results above are formally valid, and only the
interpretation or the range of indices are changed.  





\bigskip\bigskip\bigskip

% \vfill\eject 

\centerline{\bf 5.~~Alternative $~N=2$~ Supergravity in Singular
5D Space-Time}

As has been developed in 
\abn\bkvp\ 
for the conventional $~N=2$~ supergravity \gst\gz\cda, 
we can generalize our alternative $~N=2$~ supergravity 
into singular 5D space-time, as supersymmetrization 
of Randall-Sundrum brane-world scenario \rs.   

As in our previous paper \nr, we follow the prescription in \bkvp\ 
designed for the case of Abelian $~SO(2)$~ gauging for the singular 
5D space-time with the orbifold-type singularity of $~S^1/\ZZ_2$.  
However, since our present total gauged group is non-Abelian: $~G 
= SO(2) \times Sp(n') \times Sp(1) \times \Bar H\times [\,
U(1)\,]^{n-p+1}$ which is much bigger than just $~SO(2)$, we need 
special care when applying the method in \bkvp.  

We start with fixing the bulk 5D space-time action $~S_{\rm bulk}~$
before considering the singularity.  
Mimicking the Abelian case \bkvp, we first replace the original
$~Sp(1)\-$gauging coupling constant $~g$~ {\it everywhere} in $~\Lag_2$~ 
by a space-time-dependent real scalar field $~G(x)$, and then introduce a
fourth-rank antisymmetric tensor potential 
$~A_{\m\n\r\s}$, with a  new term in the lagrangian \bkvp 
$$ \li{ & S_{A G} \equiv \int d^5 x \, 
	\Lag_{A G} \equiv \int d^5 x \, \Big(
	\frac1{24} \e^{\m\n\r\s\t} A_{\m\n\r\s} 
	\partial_\t G \Big) ~~.
&(5.1) \cr } $$ 
The reason we replace only $~g$~ by $~G(x)$~ is that this coupling is for 
the $~Sp(1)$~ group that can contain the $~SO(2)$~ subgroup in 
our previous case \nr\ which is analogous to the Abelian group in \bkvp. 
The scalar field $~G(x)$~ has inherited the scale transformation property
from the coupling constant $~g\rightarrow
e^{-c} g $~ under the scaling transformation (2.5).  Accordingly,   
for the action $~S_{A G}$~ to be also invariant under this scale 
transformation, $~A_{\m\n\r\s}$~ should be also rescaled as 
$$ \li{ &  G \rightarrow e^{-c} G~~, ~~~~
A_{\m\n\r\s} \rightarrow e^c A_{\m\n\r\s}~~. 
&(5.2) \cr } $$
when other fields and constants are transforming like (2.5) and (4.19)
{\it except for} $~g$~ now replaced by ~$G$.  

The total 5D bulk action is now $~S_{\rm bulk} 
\equiv  S_2 + S_{A G} \equiv \int d^5 x \, \big( \Lag_2
+ \Lag_{A G}\big)$.  Here $~S_2$~ is no longer invariant under supersymmetry, 
but has terms proportional to $~\partial_\m G$, which is supposed to be
cancelled by the variation of $~S_{A G}$~ \abn\bkvp.  
There are eight sectors contributing to such $~\partial_\m G\-$dependent
terms out of $~\Lag_2$~ after the replacement $~g\rightarrow G(x)$:  
(i) ~gravitino kinetic term, ~~(ii) $~\big( \Bar\psi_\m\l\big) 
{\cal D}\varphi\-$Noether term, ~~ (iii) ~$\big( \Bar\psi_\m\chi\big) 
G_{\n\r\s}\-$Noether term, ~~ (iv) 
$~\big( \Bar\psi_\m\psi_\n\big) G_{\r\s\t}\-$Noether term, ~~
(v) ~$ \big( \Bar\psi_\m\psi_\n\big) \Hat C\-$term, ~ 
(vi) ~$ \big( \Bar\psi_\m\l\big) \Hat C\-$term, ~ 
(vii) ~$ \big( \Bar\psi_\m\chi\big) \Hat C\-$term, ~ 
(viii) ~$ \big( \Bar\psi_\m\psi^{\un a}\big) \Hat C\-$term.  The 
terms (i) - (iv) contribute, when a derivative $~{\cal D}_\m$~ hits 
either some covariant derivatives or field strengths, after a partial
integration of the contribution 
$~\d_Q\psi_\m \approx {\cal D}_\m\e$, while (v) - (viii) terms 
contribute, when the derivative hits {\it hatted} quantities $~\Hat
C^{i J}$~ or $~\Hat\xi\ud{\un\a} I$.  All of these terms containing 
the derivative $~\partial_\m G$, are therefore cancelled by the
appropriate supersymmetry transformation $~\d_Q A_{\m\n\r\s}$~ in 
$~\d_Q\Lag_{A G}$.  If we restrict ourselves to the case
of linear order supersymmetry transformation of $~\d_Q
A_{\m\n\r\s}$,\footnotew{The word `linear' here does not include 
the quantities $~C^{i J}$~ or $~\xi^{\un\a}{}_I$.  This is 
because in the reduced case of Abelian ~$SO(2)$-gauging, $~C^{i j}$~ 
is reduced to be a constant, while since $~C^{i j}$~ has 
the part $~A\du{\un\a} i \xi^{\un\a j}$, and therefore it is more
convenient to  regard the $~\xi$'s as the same order as the $~C$'s
itself.} then we easily see that only (v) - (viii) terms contribute.  For
example, (iii) term contributes the cubic combination $~\e^{i j k} A\du\n
i A\du \r j A\du \s k$~ coming from the Chern-Simons term in
$~G_{\m\n\r}$, and  such terms are omitted from now on.  After these
considerations, we get 
$$ \li {\d_Q \Big( \Lag_2 \Big|_{g\rightarrow G} \Big) 
= \frac1{24} \e^{\m\n\r\s\t} \Big[ \! 
& + 2{\sqrt 2} e^\s \big(\Bar\e\g_{\[\m\n\r |} T^i \psi_{|\s\]} \big)
     C^{i j} L^j  
     + \frac i{\sqrt 2} e^\s \big(\Bar\e\g_{\m\n\r\s} T^i \l^a\big)
      C^{i j} L_a{}^j  \cr  
& + \frac i{\sqrt 6} 
     e^\s \big(\Bar\e\g_{\m\n\r\s} T^i \chi \big)
      C^{i j} L^j  
     + \frac i{\sqrt 2} e^\s \big(\Bar\e\g_{\m\n\r\s} \psi^{\un a} \big)
      V_{\un\a {\un a} A} \xi^{\un\a i} L^i \, \Big] 
      \partial _\t G {~~, ~~~~~ ~~~}  
&(5.3) \cr } $$  
up to cubic or higher order terms.  
Note that there are {\it no hats} on $~C^{i j}$~ and $~\xi^{\un\a i}$ 
here, and $~L^i$~ is the $~{\scst I~=~i}$~ component of $~L^I$.  
Eq.~(5.3) is supposed to be cancelled by the new
supersymmetry transformation rule $~\d_Q A_{\m\n\r\s}$~ in $~\d_Q\Lag_{A
G}$:\footnotew{The previously-mentioned correction of sign errors in
(4.2) in \nr\ leads to the corrected signs for the first and third terms
both in (5.3) and (5.4).}   
$$ \li{ \d_Q A_{\m\n\r\s} = & - 2{\sqrt 2} e^\s \big(\Bar\e\g_{\[\m\n\r |}
T^i \psi_{|\s\]} \big)
      C^{i j} L^j  
     - \frac i{\sqrt 2} e^\s \big(\Bar\e\g_{\m\n\r\s} T^i \l^a\big)
      C^{i j} L_a{}^j  \cr  
& - \frac i{\sqrt 6} e^\s \big(\Bar\e\g_{\m\n\r\s} T^i \chi \big)
      C^{i j} L^j  
     - \frac i{\sqrt 2} e^\s \big(\Bar\e\g_{\m\n\r\s} \psi^{\un a} \big)
      V_{\un\a {\un a} A} \xi^{\un\a i} L^i  + \hbox{(quadratic terms)} 
      {~. ~~~~~ ~~~} 
&(5.4) \cr } $$ 
The fact that the $~C$'s and $~\xi$'s here have {\it no hats} 
is consistent with the scaling property (5.2).  
As the standard first step of this prescription \bkvp, we require
$~\d_Q G = 0$, so that there is no other contribution from $~\d_Q
\Lag_{A G}$.  Our previous result \nr\ can be recovered easily by
truncating $~\psi^{\un a} \rightarrow 0$~ and reducing $~C^{2 2} = - 1,
~C^{i j}~\hbox{(otherwise)} = 0$.  Now our action $~S_{\rm bulk}$~ is
invariant under (5.4), $~\d_Q G=0$~ and (4.17) with $~g\rightarrow
G$.  

Since we are now dealing with the prescription in \bkvp\ originally
designed for Abelian gauging without hypermultiplets, 
applied to our non-Abelian gauging also with hypermultiplets, 
it is better to confirm the closure of supersymmetry on the field
$~A_{\m\n\r\s}$~ by the commutator $~\[\d_Q(\e_1), \d_Q(\e_2) \]
= \d_P \big(\Bar\e_2\g_m \e_1\big)$~ acting on $~A_{\m\n\r\s}$, 
where $~\d_P (\eta_m)$~
implies the  translation operator.  In what follows, we confirm this
closure up to quadratic field level.  The linear terms in this commutator 
are composed of six sectors: ~$~F_{\m\n}, ~G_{\m\n\r}, ~\partial\s ,~
\partial\varphi,~\partial\phi\-$linear sectors, and $~\Hat C^2$~ or 
$~\Hat\xi^2\-$terms.  Here, the first four sectors
work with no problem, while the $~\partial\phi\-$linear sector needs
special care.  To be more specific, we get 
$$ \li{ \[\d_Q(\e_1), \d_Q(\e_2) \] A_{\m\n\r\s} \Big|_{\partial\phi}  
      = & - e^{-1} e^\s \e\du{\m\n\r\s}\t 
     \Big[ \, \frac12 \big( \Bar\e_1\e_2\big) L^i  \, \Big] 
     \Hat\xi\du{\un\a} i {\cal D}_\t\phi^{\un\a} \cr 
& - 2 i e^\s \big( \Bar\e_2 \g_{\[\m\n\r|} T^i \e_1 \big) 
	F\du{\un\a\un\b} i \xi^{\un\a j} L^j {\cal D}_{|\s\]} 
     \phi^{\un\b} ~~, 
&(5.5) \cr} $$ 
where the last term can be interpreted just as the usual desirable gauge 
transformation of the type $~\partial_{\[\m} \L_{\n\r\s\]}$~ up to 
quadratic terms, while the first term needs special care.  This term 
is actually interpreted as an $~Sp(1)$~ gauge transformation of 
$~A_{\m\n\r\s}$.  Even though this seems slightly bizarre at first 
sight, it can be easily understood, once we notice that the
$~\phi^{\un\a}\-$kinetic term is {\it no longer} $~Sp(1)$~ invariant
after the replacement $~g\rightarrow G(x)$.  In fact, after this 
replacement, (4.3c) is to be modified as 
$$ \li{ & \d_G \big( {\cal D}_\m\phi^{\un\a} \big) 
     = \a^I \big({\cal D}_\m\phi^{\un\a} \big) 
     \big( \partial_{\un\b} \Hat\xi\ud{\un\a} I \big) 
       + \a^i \xi^{\un\a i} \partial_\m G ~~, 
&(5.6) \cr } $$ 
with the new effect of $~\partial G$, while all other equations in 
(4.3) are `formally' intact. This results in the non-trivial 
contribution of the $~\phi\-$kinetic term under the gauge 
transformation $~\d_G$: 
$$ \li{ & \d_G \Big[ - \frac12 e g^{\m\n} g_{\un\a\un\b} 
     \big({\cal D}_\m \phi^{\un\a} \big)  
     \big({\cal D}_\n \phi^{\un\b} \big)  \, \Big]   
      = \frac1{24} \e^{\m\n\r\s\t} 
     \Big[\, e^{-1} \e\du{\m\n\r\s}\o 
     \, \a^i \xi\du{\un\a} i \,  
      \big( {\cal D}_\o \phi^{\un\a} \big)  \, \Big] 
					\partial_\t G ~~, 
&(5.7) \cr} $$  
It is now clear that this contribution can be cancelled by an 
extra transformation $~\d_G A_{\m\n\r\s}$~ {\it via} $~\Lag_{A G}$, 
such that 
$$ \li{ & \d_G A_{\m\n\r\s} = - e^{-1} \e\du{\m\n\r\s}\t \,  
     \a^i \xi\du{\un\a} i  \, {\cal D}_\t \phi^{\un\a} ~~.   
&(5.8) \cr  } $$
In other words, when we identify $~\a^i \equiv (1/2) \big(\Bar\e_2\e_1 \big)
L^i$~ in (5.5), then the first term in (5.5) is absorbed into the 
~$Sp(1)$~ gauge transformation.   

Even though the result that the tensor potential field
$~A_{\m\n\r\s}$~ is transforming under the gauge group $~Sp(1)$~ seems
unnatural at first glance, this is nothing new in supergravity. 
In fact, in \bkvp\ it was pointed out that the original action $~S_2$~ is 
no longer $~R\-$invariant, {\it i.e.,} in our case $~Sp(1)$~ 
{\it non}-invariant producing a quadratic terms in fermions after the
replacement 
$~g\rightarrow G(x)$.\footnotew{These terms did not matter in our treatment, 
because we are looking into only linear terms in the transformation rule 
of $~\d_Q A_{\m\n\r\s}$.}   
Analogous situation can be found in Green-Schwarz 
mechanism in anomaly cancellation in the usual formulation 
%%% 
\ref\gs{M.~Green and J.H.~Schwarz, \pl{149}{84}{117}.}% 
%%% 
\gsw\ or in the dual formulation 
%%%
\ref\ngdual{S.J.~Gates, Jr.~and H.~Nishino, \pl{157}{85}{157}.}, 
%%% 
in which the tensor field $~B_{\m\n}$~ or $~M_{\m_1\cdots
\m_6}$~ transforms under Lorentz as well as gauge transformation, as the
zero-slope limit effect of superstring theory.  

Going back to our closure question, the only left over sector is 
the $~\Hat C^2$~ and $~\Hat\xi^2\-$terms which turn out to be 
$$ \li{ & \[\d_Q(\e_1), \d_Q(\e_2) \] A_{\m\n\r\s} \Big|_{\Hat C^2}
    = \frac14 \e_{\m\n\r\s\t} e^{2\s} \eta^\t 
    \big( C^{i j} \Hat C\ud i K L^{j K} 
     + 2 \xi\du{\un\a} i \Hat\xi\ud{\un\a} J L^i L^J \big) ~~, 
&(5.9) \cr } $$
where $~\eta^\t \equiv \big( \Bar\e_2\g^\t \e_1\big)$.  
If this system is analogous to the Abelian case \bkvp\nr, these two terms
are supposed to be proportional to $~\eta^\t F_{\t\m\n\r\s}$, upon 
the use of $~G\-$field equation,  
for the field strength of the potential $~A_{\m\r\s\t}$: 
$~F_{\m\n\r\s\t} \equiv 5 \partial_{\[\m} A_{\n\r\s\t\]}$.  
In fact, the $~G\-$field equation is easily obtained as 
$$ \li{ & F_{\m\n\r\s\t} = \frac 14 e^{-1} e^{2\s} \e_{\m\n\r\s\t} 
     \big( C^{i j} \Hat C\ud i K L^{j K} 
     + 2 \xi\du{\un\a} i \Hat \xi\ud{\un\a} J L^i L^J  \big) ~~, 
&(5.10) \cr } $$
up to quadratic terms.  After simple algebra, it is easy to show that
(5.9) is desirably proportional to $~\eta^\t F_{\t\m\n\r\s}$~ which is
equivalent to the combination of the usual translation
$~\d_P(\eta^\t) A_{\m\n\r\s}$~ accompanied by a gauge transformation.   
This conclude the linear-order closure of supersymmetry on
$~A_{\m\n\r\s}$,  which provides a non-trivial consistency check of our
system with
$~S_{\rm bulk}$, in particular with non-Abelian gauged groups.  
   
We next consider a possible brane action $~S_{\rm brane}$~ to be added.  
To this end, and for the reason to be clarified later, we truncate the 
hypermultiplets $~(\phi^{\un\a}, \psi^{\un a})$, and we restrict the 
gauged group to be $~SO(2)$~ out of the $~Sp(1)$~ isotropy group in the
coset $~Sp(n',1) /Sp(n') \times Sp(1)$.  We do not have to restrict other
gauged groups in $~G$, but it is only $~SO(2)$~ out of the $~Sp(1)$~
group to be  gauged.  

We next assume that the branes are located at $~y\equiv x^5 = 0$~ and
$~y = b > 0$~ in the 5-th dimension, requiring all the fields to obey the 
usual periodic boundary condition $~f(-b) = f(0) = f(b)$.  Subsequently,
we assign the parities under $~y\leftrightarrow -y$~ on the branes on all
the fields in our system, following \abn\bkvp:  
$$ \li{ & \Pi\big( \Tilde e_\m{}^m \big) 
	= \Pi \big( e_5{}^{(5)} \big) 
	= \Pi \big( A\du 5 I \big) 
	= \Pi \big( \s \big) 
	= \Pi \big( \varphi^\a \big)  
	= \Pi \big( A_{\m\n\r\s} \big) 
	= \Pi \big( \psi_\m \big) 
	= \Pi \big( \e \big) 
	= + 1 {~, ~~~~~ ~~~~~ ~~} 
&(5.11\rma) \cr 
& \Pi \big( e\du 5 m \big) 
	= \Pi \big( e\du\m {(5)}  \big) 
	= \Pi \big( A\du\m I \big) 
	= \Pi \big( G \big) 
	= \Pi \big( \psi_5 \big) 
	= \Pi \big( \l^a \big) 
	= \Pi \big( \chi \big) 
	= - 1 ~, 
&(5.11\rmb) \cr } $$ 
where $~\Tilde e\du\m m$~ denotes the 4D part of the f\"unfbein.  
The parity for an arbitrary bosonic ($\Phi$) or  
fermionic ($\Psi$) field is defined by \abn\bkvp\ 
$$ \li{ & \Phi(-y) = \Pi(\Phi) \, \Phi(y)~~, ~~~~ 
        \Psi (-y) = i \a \Pi(\Psi) \, \g_5 T^2 \Psi(y) ~~.  
&(5.12) \cr } $$
Here the real constant $~\a= \pm 1$~ reflects the signature ambiguity, 
but its sign should be common to all the fermions
\abn\bkvp.

We now consider the brane action 
$$ \li{ & S_{\rm brane} \equiv 
	-2 h \int d^5 x \, \big[ \, \d(y) - \d(y-b) \, \big]
	\Big[ \, \frac 1{\sqrt2} \,  a \Tilde e 
	e^\s V_I L^I 
	+ \frac 1{24} \e^{\m\n\r\s 5} A_{\m\n\r\s}   \, \Big] 
	\equiv \int d^5 x \, \Lag_{\rm brane} {~ , ~~~~~ ~~~~~ ~~}
&(5.13) \cr } $$
where $~h,~a$~ are real constants with $~|a|=1$, and $~\Tilde e$~ is the
4D part of the determinant of the f\"unfbein, while  $~\e^{\m\n\r\s
5}$~  is the $~{\scst\t~=~5}$~ component  of $~\e^{\m\n\r\s\t}$.  The
exponential factor $~e^\s$~ is needed for invariance of $~S_{\rm
brane}$~ under supersymmetry, as will be seen.\footnotew{This  factor
was not considered in \nr.}  In order for the action $~S_{\rm brane}$~
to be invariant under  the scaling transformation (5.2), the constant
$~h$~ should also be  rescaled as 
$$ \li{ & h\rightarrow h^{-c}~~. 
&(5.14) \cr } $$ 
when other fields and constants are transforming like (2.5) and (4.17), {\it 
except for} $~g$~ replaced by $~G$.  

The action $~S_{\rm brane}$~ modifies the 
$~A_{\m\n\r\s}\-$field equation from the original one 
$~\partial_\m G = 0$~ into 
$$ \li{ & \partial_5 G(x) = 2 h \big[ \, \d(y) - \d(y - b) 
           \, \big] ~~, ~~~~\partial_\m G = 0 ~~~(\hbox{for}~~
          {\scst \m ~\neq ~5})~~.  
&(5.15) \cr } $$ 
The solution for this field equation is \abn\bkvp 
$$ \li{ & G(x) = G(y) =  h \, \e(y) = \cases{ + h 
	~~~ \hbox{(for $~0 <  y  < +b$)}~~, \cr   
  - h ~~~ \hbox{(for $~- b < y  < 0$)}~~. \cr} 
&(5.16) \cr } $$
This gives the desirable kinks for the $~SO(2)$~ coupling 
`constant' $~G(x)$~ \abn\bkvp.   

We now take the variation of $~\d_Q S_{\rm brane}$~ under supersymmetry (4.15) and (5.4) for $~\d_Q A_{\m\n\r\s}$:
$$ \li{ \d_Q \Lag_{\rm brane} 
	= & - 2h  \big[\, \d(y) - \d(y-b) \,  \big] \, 
	\frac1{\sqrt 2} \, a \, \Tilde e e^\s  
	 \, \big[ \, \big( \Bar\e \g^\m \psi_\m \big) 
	   - i a^{-1} \big( \Bar\e \g^\m \g_5 T^2 \psi_\m \big) \, \big]  \, 
	   V_I L^I ~~\cr 
& - 2h \big[\, \d(y) - \d(y-b) \,  \big]
	 \, \frac i{\sqrt2} \, a \, \Tilde e \e^\s 
	 \, \big[ \, i \big( \Bar\e\l^a \big) 
	-  a^{-1} \big( \Bar\e \g_5 T^2 \l^a \big) \, \big]  \, 
	 V_I L\du a I  \cr 
& - 2h \big[\, \d(y) - \d(y-b) \,  \big]
	 \, \frac i{\sqrt6} \, a \, \Tilde e \e^\s 
	 \, \big[ \, i \big( \Bar\e \chi \big) 
	-  a^{-1} \big( \Bar\e\g_5 T^2 \chi \big) \, \big]  \, V_I L^I ~~,
&(5.17) \cr } $$ 
Comparing these three lines with (5.12), we see that if 
$$ \li{ & a = \a = \pm 1~~, 
&(5.18) \cr } $$
then (5.17) is simplified to be 
$$ \li{\d_Q \Lag_{\rm brane}  
= & - 2h \big[\, \d(y) - \d(y-b) \,  \big] \, 
	\frac 1{\sqrt2} \, a \Tilde e(y) \, e^{\s(y)} \, \Bar\e(y) \g^m\, 
	\big[ \, \psi_m(y) - \psi_m (-y) \, \big] V_I L^I (y)  \cr  
& - 2h \big[\, \d(y) - \d(y-b) \,  \big]\, 
	\frac i {\sqrt 2} \, a \Tilde e(y) \, e^{\s(y)} \, \Bar\e(y) \, 
	\big[\, \Bar\l{}^a (y) - \l^a (-y) \, \big] V_I L\du a I (y)  \cr 
& - 2h \big[\, \d(y) - \d(y-b) \,  \big]\, 
	\frac i{\sqrt 6} \, a \Tilde e(y) \, e^{\s(y)} \, \Bar\e(y)\,  
	\big[ \, \chi(y) - \chi(-y) \, \big]  V_I L^I (y) 
	{~~ , ~~~~~ ~~~} 
&(5.19) \cr } $$
where each line vanishes after the $~\int d y\-$integration, 
under the periodic boundary condition 
$~f(-b) = f(0) = f(b)$~ for an arbitrary field $~f(y)$~ in (5.19).   
This concludes the proof of the invariance 
$$ \li{ & \d_Q S_{\rm brane} = 0 ~~, 
&(5.20) \cr } $$ 
and therefore that of the total action $~S_{\rm total} \equiv 
S_2\Big|_{g\rightarrow G} + S_{A G} + S_{\rm brane}$~ 
for the $~SO(2)\-$gauging in the absence of hypermultiplets.  

Let us briefly comment on the difficulty of the brane mechanism 
with the gauging $~Sp(1)$~ or with hypermultiplets.   
The difficulty with the $~Sp(1)\-$gauging is that we do not have a good 
analog of $~V_I L^I$~ we can use as the first term in $~S_{\rm brane}$~
as an invariant quantity.  This is because $~\d_Q A_{\m\n\r\s}$~ in (5.4) 
with general $~T^i$~ matrices with general index $~{\scst i}$~ 
can not cancel the variation 
$~\d_Q \Tilde e = \Tilde e \big(\Bar\e \g^\m \psi_\m\big) + \cdots$~ 
considering the parity (5.12).  
As for the inclusion of the hypermultiplets, 
the $~\psi^{\un a}\-$dependent term in $~\d_Q A_{\m\n\r\s}$~ yields 
$$ \li{ & \d_Q \big(\frac1{24} \e^{\m\n\r\s\t}
      A_{\m\n\r\s} \big) \Big|_{\psi^{\un a}}
	   = \frac 1 {\sqrt2} e^\s \big( \Bar\e \g_5 \psi^{\un a}
	\big) V_{\un\a \un a A}\xi^{\un\a 2} L^2  ~~, 
&(5.21) \cr } $$
{\it via} $~S_{\rm brane}$.  
The trouble here is that there seems to be no 
invariant lagrangian that will cancel (5.21).  
For example, a quantity like $~\phi^{\un\a} \xi\du{\un\a} 2 L^2$~ 
is not appropriate, 
because of its lack of gauge invariance under (4.3).  From these
viewpoints, we seem to have no generalization of the brane action
$~S_{\rm brane}$~ (5.13), when $~Sp(1)$~ is gauged and/or
the hypermultiplets are included.  

The brane action $~S_{\rm brane}$~ we gave here is supposed to be the
simplest one based on \bkvp, among other potentially possible lagrangians
invariant under local supersymmetry in singular 5D space-time \abn. 
However, we stress that our alternative 
$~N=2$~ supergravity in 5D is equally applicable to these formulations, 
as well.  


\bigskip\bigskip\bigskip

% \vfill\eject 


\centerline{\bf 6.~~Concluding Remarks}

In this paper, we have completed the non-Abelian gauging of our
alternative $~N=2$~ supergravity to $~n$~ copies of vector multiplets in
5D, and $~n'$~ copies of hypermultiplets, with a simpler coupling structure
compared with the conventional supergravity \gst\gz\cda\gzrecent, 
up to quartic fermion terms in the action.  Our
result is  the combination of considerable works on supergravity
couplings in the past, such as vector multiplet couplings in 9D case with
the scalars forming the coset $~SO(n,1) / SO(n)$~ \gns\agst, together
with the scalars in the  hypermultiplets forming the quaternionic
K\"ahler manifold in $~N=2$~ supergravity in 4D \bw\ and in
5D \gst\gz\cda\gzrecent\ as well as in  6D \ns.   

As in 9D \gns, the scalars in the vector multiplets form the coordinates
of the  $~\s\-$model for the non-Jordan family scalar coset $~H^n \equiv 
SO(n,1)/ SO(n)$, and the vector fields with the total number $~n+1$~ form
the $~({\bf n+1})\-$representation of $~SO(n,1)$, while the gaugini 
$~\l^a$~ form the $~{\bf n}\-$representation of $~SO(n)$.  The scalars in
the hypermultiplets form the $~\s\-$model on the quaternionic K\"ahler
manifold $~Sp(n',1) / Sp(n') \times Sp(1)$.  

Our result is valid for any arbitrary gauge groups of the type
$~G\equiv SO(2) \times Sp(n') \times Sp(1) 
\times \Bar H \times [\,U(1)\,]^{n-p+1}$~  for
any arbitrary group $~\Bar H$~ with $~\dim{\Bar H} = p - n'(2n'+1) -4$,
and $~Sp(n')\times Sp(1)$~ are the isotropy groups of the coset $~Sp(n',1)
/ Sp(n') \times Sp(1)$, with a peculiar potential term in the lagrangian. 
Accordingly, we have obtained a crucial constraint (4.14) required by
consistency between the two different cosets under supersymmetry.  This 
constraint relates the $~Sp(1)$~ curvature $~F\du{\un\a\un\b} i$~ and 
the $~\Hat C^{i J}\-$functions, whenever a non-Abelian group in the
isotropy  groups $~Sp(n')\times Sp(1)$~ is gauged.   
Moreover, the isotropy groups $~Sp(n') \times Sp(1)$~ can be also
reduced into their subgroups, {\it e.g.,} $~Sp(1) \times SO(2)$, where 
$~Sp(1) \subset Sp(n'),~SO(2)\subset Sp(1)$.  This a 
generalization of our previous paper \nr, in which we gauged only the
$~SO(2)$~ subgroup of $~Sp(1) \subset Sp(n')\times Sp(1)$.  Therefore, 
by adjusting the parameters $~n,~n'$~ and $~p$~ for
dimensions appropriately, our results in this paper are considerably
general, and cover a wide range of combinations of gauged groups.      

Since there are two non-trivial coset structures $~SO(n,1) / SO(n)$~ and 
$~Sp(n',1) / Sp(n')\times Sp(1)$~ present in our system, 
our vector fields $~A\du\m I$~ for gaugings are {\it both} 
in the $~({\bf n+1})\-$representation of $~SO(n,1)$~ {\it and} in 
the adjoint representations 
of the gauged groups in $~G$~ at the same time.  The mutual consistency 
of these two structures under supersymmetry requires the constraint
(4.14) which corresponds to analogous equations in \cda\gzrecent.

Even though we did not perform explicitly in this paper, we can 
also combine any Abelian factor groups $~[\,U(1)\,]^{n-p+1}$, by
introducing constant vectors $~V_I$, as has been done in
\gst\gz\cda\gzrecent\nr.  This provides another freedom for practical
applications for phenomenological model building.  

We have also generalized this result to the case of singular 5D space-time
for the case of $~SO(2)\-$gauging as a supersymmetric Randall-Sundrum
brane-world scenario \rs\ similar to the conventional 5D supergravity
\abn\bkvp.  We have applied the prescription in ref.~\bkvp\ in order to
confirm the supersymmetry of our brane action with the singularity of the
type $~S^1/\ZZ_2$.  We have also seen some difficulty, when the gauged
group is $~Sp(1)$~ larger than its subgroup $~SO(2)$, or when the
hypermultiplets are present.  As far as the formulation for singular
space-time is concerned, there seems to be no fundamental difference
between our alternative supergravity and the conventional one
\gst\gz\cda\gzrecent.  

For some readers who wonder why our `larger' supergravity multiplet \nr\ 
has never been studied as a special case in the conventional and
`exhaustively' studied formulation \gst\gz\cda\gzrecent, we repeat the 
following points already given in \nr:  The original result in \gst\ was
presented before the discovery of the importance of superstring in 1984
\gsw, so that there was no strong motivation to include the dilaton or
antisymmetric tensor fields, which are important NS
fields in terms of superstring language.  To put it differently, it is
only superstring \gsw\ or M-theory \mtheory\ that motivates the
peculiar couplings of dilaton and  antisymmetric tensor to supergravity,
as we have done in the present paper.   

Even though we have stressed the difference of our formulation from 
other general matter couplings in \gst\gz\cda\gzrecent, it is fair to 
point out some similarities.  For example, we expect it possible to
generalize the number of the additional tensor fields $~B_{\m\n}$~ in
addition to the one in the supergravity multiplet with a $~\s\-$model
structure  similar to that presented in \gz\cda\gzrecent.  However,  
we also emphasize that our antisymmetric tensor $~B_{\m\n}$~ is still 
to be distinguished from these additional ones which always appear in pairs 
\gz\cda\gzrecent.  This situation is analogous to our dilaton $~\s$~ 
as another NS field separated from other scalar fields.  

We believe that our result in this paper will be of great help
for the study of the Randall-Sundrum brane-world scenario \rs\ associated 
with superstring theory and M-theory.    

\bigskip\bigskip



We are grateful to J.~Bagger, A.~Ceresole, G.~Dall'Agata, S.J.~Gates, Jr.,
M.~G\" unaydin, R.~Kallosh, E.~Sezgin, P.K.~Townsend and M.~Zucker 
for helpful discussions.   


\newpage


\bigskip\bigskip


\centerline{\bf Appendix A: ~Notations and Conventions}

In this Appendix, we deal with important notations and conventions in
our paper.  Typically, our indices $~{\scst
\m,~\n,~\cdots~=~0,~1,~\cdots,~4}$~ are for the curved world indices,
and $~{\scst m,~n,~\cdots~=~(0),~(1),~\cdots,~(4)}$~ are local Lorentz 
indices with the metric $~\big(\eta\low{m n}\big)  =
\hbox{diag.}~(-,+,+,+,+)$, and $~\e^{012345} = + 1$.  Most frequently used
relevant relations are such as 
$$ \li {& \e_{m_1\cdots m_{5-n} \,r_1\cdots r_n} \, 
      \e^{r_1\cdots r_n \, n_1\cdots n_{5-n}} 
     = - (n!)\, [(5-n)! ]\, \d\du{\[ m_1}{\[ n_1} \cdots 
      \d\du{m_{5-n} \]} {n_{5-n} \]} ~~, 
&({\rm A}.1\rma) \cr 
& \e^{m_1\cdots m_{5-n}\, n_1\cdots n_n} \, \g_{n_1\cdots n_n} 
     = i \,(-1)^{n(n-1)/2} \, (n!) \, \g^{m_1\cdots m_{5-n}} ~~.    
&({\rm A}.1\rmb) \cr } $$ 

As the most important notational preparation, we first deal with the 
fermions in our 5D with the signature $~(-,+,+,+,+)$.  In this paper, 
we refer the reader to the general description of arbitrary fermions in 
diverse space-time dimensions \ss.  
Using the notation in \ss, we start with the number of space and 
time dimensions together with the parameters $~\e$~ and $~\eta$~ 
that control the properties of fermions \ss:
$$ \li{& s = 4 ~~, ~~~~t = 1~~, ~~~~\e = - 1~~, ~~~~ \eta = -1~~, 
&({\rm A}.2) \cr } $$
We next define the complex conjugations by \ss
$$ \li{ & \psi^{\dagger A} = \e^{A B} B \psi_B~~, ~~~~
    \psi_B = - B^{-1} \e\low{B A} \psi^{\dagger A} ~~, 
&({\rm A}.3) \cr } $$
with the $~Sp(1)$~ metric $~\e^{A B} = - \e^{B A}$~ 
%%%
\ref\kt{T.~Kugo and P.K.~Townsend, \np{211}{83}{157}.}%      
%%%
\ss. 
Accordingly \ss, 
$$ \li{ & \g_m^\dagger = - A\g_m A^{-1} ~~, ~~~~A\equiv \g\low{(0)}~~,
     ~~~~ \big( \g\low{(0)} \big)^2 = - I ~~, ~~~~
     A^\dagger = - A A A^{-1} = - A~~,\cr 
& \g_m^\dagger = + \g\low{(0)} \g_m \g\low{(0)} 
      = + A  \g_m A  ~~,~~~~
     \Bar\psi = \psi^\dagger A = \psi^\dagger \g\low{(0)} ~~, ~~~~
     B^\dagger B = I ~~, \cr 
& B^T = - B ~~, ~~~~
     \g_m^T = C\g_m C^{-1} ~~, ~~~C^T = - C ~~, ~~~~C = B A ~~.  
&({\rm A}.4) \cr } $$
Relevantly, we have the fermionic quadratic combinations \ss:
$$ \li{ \big( \Bar\chi{}^A \l_A \big)^\dagger & = \l_A^\dagger
      \big(\Bar\chi{}^A \big)^\dagger = \l_A^\dagger 
     \big( \chi^{\dagger A} A\big)^\dagger 
     = \l_A^\dagger A^\dagger \chi^A = - \l_A^\dagger  A \chi^A \cr 
& = - \big( \Bar\l_A \chi^A \big) 
	= - \e^{A B} \big(\Bar\l_A\chi_B \big)
       = - \e^{A B} \big( \Bar\chi_B \l_A\big) 
     = - \big( \Bar\chi^A \l_A\big) {~~.~~~~~ ~~~~~} 
&({\rm A}.5)  \cr } $$
Note that we have taken into account a sign 
error\footnotew{We acknowledge E.~Sezgin for informing about  this error.}
in eq.~(6) in \ss\ that affects the first hand side.  
In a more general case for $~0\le n \le 5$, the hermitian conjugation
works like 
\ss 
$$ \li{ \big( \Bar\chi{}^A \g_{m_1\cdots m_n} \l_A\big)^\dagger 
& = \l_A^\dagger \big(\g_{m_1\cdots m_n}\big)^\dagger
     \big(\Bar\chi{}^A\big)^\dagger \cr 
& = \l_A^\dagger \big( - A \g_{\[m_n| } A^{-1} \big)
      \big( - A \g_{|m_{n-1} | } A^{-1} \big) 
     \cdots \big( - A \g_{|m_1\] } A^{-1} \big) 
    \big( \chi_A^\dagger A\big)^\dagger \cr 
& = (-1)^{n +n(n-1)/2} \big( \l_A^\dagger 
     A \g_{m_1\cdots m_n} A^{-1} A^\dagger \chi^A\big) 
     = (-1)^{n+n(n-1)/2} \big(\Bar\l{}^A 
     \g_{m_1\cdots m_n} \chi_A\big) \cr 
& = (-1)^{n(n+1)/2} \e^{A B} \big( \Bar\l_B \g_{m_1\cdots m_n} 
     \chi_A \big) 
	= (-1)^{n^2/2 + n/2 + n^2/2 - n/2} \e^{A B} 
     \big(\Bar\chi_A\g_{m_1\cdots m_n} \l_B\big) \cr 
& = (-1)^{n^2 +1} \big( \Bar\chi{}^A \g_{m_1\cdots m_n} \l_A\big) ~~,  
&({\rm A}.6) \cr } $$
Hence, omitting the explicit $~{\scst A}\-$indices for contractions as in
section 2, we get  
$$\li{ & \big( \Bar\chi\g_{m_1\cdots m_n} \l\big)^\dagger
     = (-1)^{n+1} \big( \Bar\chi\g_{m_1\cdots m_n} \l\big) ~~. 
&({\rm A}.7) \cr } $$
Typical examples are 
$$\li{ & \big( \Bar\chi\l\big)^\dagger 
     = - \big( \Bar\chi\l\big) ~~, ~~~~
     \big(\Bar\chi\g_m \chi\big)^\dagger 
     = + \big(\Bar\chi\g_m \chi\big)~~, ~~~~
     \big( \Bar\chi\g_{m n} \l\big)^\dagger 
    =  - \big( \Bar\chi\g_{m n} \l\big)~~, ~~~\hbox{\it etc.} {~~~~~ ~~}
&({\rm A}.8) \cr } $$ 
In other words, any combination $~\big(\Bar\chi \g_{m_1\cdots m_n}
\l\big)$~ with an even number of gammas need a pure imaginary unit `i' in
front to be an hermitian expression, while a combination 
with an odd number of gammas is already hermitian.   

In (A.6), we have used the flipping property for $~0\le n\le 5$~ 
$$ \li{ \big(\Bar\l{}^A \g^{m_1\cdots m_n} \chi_B \big)
     & = (-1)^{n+1} (-1)^{(1-n)(2-n)/2} 
     \big( \Bar\chi_B \g^{m_1\cdots m_n} \l^A\big) \cr 
& = (-1)^{n(n-1)/2} 
     \big( \Bar\chi_B \g^{m_1\cdots m_n} \l^A\big) ~~.
&({\rm A}.9) \cr } $$
Therefore we have 
$$\li{ & \big( \Bar\e_1 \g^{m_1\cdots m_n} \e_2\big) \equiv 
	\big(\Bar\e_1^A\g^{m_1\cdots m_n} \e_{2A} \big) 
     = \cases{ ~- \big(\Bar\e_2^A \g^{m_1\cdots m_n}\e_{1A} \big) ~~
      ~~~\hbox{(for $~n = 0,~1 ,~ 4,~5$)} {~~, ~~~~~ ~~~~~} \cr 
     ~ + \big(\Bar\e_2^A \g^{m_1\cdots m_n}\e_{1A} \big) ~~
      ~~~ \hbox{(for $~n = 2,~3$)}  {~~. ~~~~~ ~~~~~} \cr} 
&({\rm A}.10) \cr } $$ 



\bigskip\bigskip\bigskip\bigskip 

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\end{document} 



