%Paper: hep-th/9209002
%From: "Jose M. Figueroa-O'Farrill" <figueroa@pib1.physik.uni-bonn.de>
%Date: Tue, 01 Sep 1992 10:28:29 MEZ
%Date (revised): Mon, 07 Sep 1992 21:58:43 MEZ

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   This is the Plain TeX file for
%
%
%   A geometrical interpretation of classical W-transformations
%
%    by
%
%       J.M. Figueroa-O'Farrill, E. Ramos, and S. Stanciu
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  This is a stripped down version of the macros for submission
%  to the hep-th bulletin board. Last revised: June 10th, 1992.
%
%
\nonstopmode
\catcode`\@=11 % this allows for tricky names
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  First some font definitions
%
\font\seventeenrm=cmr17
\font\fourteenrm=cmr12 at 14pt
\font\twelverm=cmr12
\font\ninerm=cmr9
\font\sixrm=cmr6

\font\seventeenbf=cmbx12 at 17pt
\font\fourteenbf=cmbx12 at 14pt
\font\twelvebf=cmbx12
\font\ninebf=cmbx9
\font\sixbf=cmbx6

\font\seventeeni=cmmi12 at 17pt             \skewchar\seventeeni='177
\font\fourteeni=cmmi12 at 14pt              \skewchar\fourteeni='177
\font\twelvei=cmmi12                        \skewchar\twelvei='177
\font\ninei=cmmi9                           \skewchar\ninei='177
\font\sixi=cmmi6                            \skewchar\sixi='177

\font\seventeensy=cmsy10 scaled\magstep3    \skewchar\seventeensy='60
\font\fourteensy=cmsy10 scaled\magstep2     \skewchar\fourteensy='60
\font\twelvesy=cmsy10 at 12pt               \skewchar\twelvesy='60
\font\ninesy=cmsy9                          \skewchar\ninesy='60
\font\sixsy=cmsy6                           \skewchar\sixsy='60

\font\seventeenex=cmex10 scaled\magstep3
\font\fourteenex=cmex10 scaled\magstep2
\font\twelveex=cmex10 at 12pt
\font\elevenex=cmex10 scaled\magstephalf
%\font\ninex=cmex9
\font\ninex=cmex10 at 9pt
%\font\sevenex=cmex7
\font\sevenex=cmex10 at 9pt
%\font\sixex=cmex7 at 6pt
\font\sixex=cmex10 at 6pt
%\font\fivex=cmex7 at 5pt
\font\fivex=cmex10 at 5pt

\font\seventeensl=cmsl10 scaled\magstep3
\font\fourteensl=cmsl10 scaled\magstep2
\font\twelvesl=cmsl10 scaled\magstep1
\font\ninesl=cmsl10 at 9pt
\font\sevensl=cmsl10 at 7pt
\font\sixsl=cmsl10 at 6pt
\font\fivesl=cmsl10 at 5pt

\font\seventeenit=cmti12 scaled\magstep2
\font\fourteenit=cmti12 scaled\magstep1
\font\twelveit=cmti12
\font\nineit=cmti9
\font\sevenit=cmti7

\font\seventeentt=cmtt12 scaled\magstep2
\font\fourteentt=cmtt12 scaled\magstep1
\font\twelvett=cmtt12

\font\seventeencp=cmcsc10 scaled\magstep3
\font\fourteencp=cmcsc10 scaled\magstep2
\font\twelvecp=cmcsc10 scaled\magstep1
\font\tencp=cmcsc10
%\font\eightcp=cmcsc8
\font\eightcp=cmcsc10 at 8pt
\newfam\cpfam

\font\hugess=cmssbx10 scaled\magstep5
\font\seventeenss=cmss17
\font\fourteenss=cmss12 at 14pt
\font\twelvess=cmss12
\font\tenss=cmss10
\font\niness=cmss9
\font\eightss=cmss8
\font\sevenss=cmss8 at 7pt
\font\sixss=cmss8 at 6pt
\font\fivess=cmss8 at 5pt
\newfam\ssfam
%
\newdimen\b@gheight             \b@gheight=12pt
\newcount\f@ntkey               \f@ntkey=0
\def\f@m{\afterassignment\samef@nt\f@ntkey=}
\def\samef@nt{\fam=\f@ntkey \the\textfont\f@ntkey\relax}
%
\def\rm{\f@m0 }
\def\mit{\f@m1 }         \let\oldstyle=\mit
\def\cal{\f@m2 }
\def\it{\f@m\itfam}
\def\sl{\f@m\slfam}
\def\bf{\f@m\bffam}
\def\tt{\f@m\ttfam}
\def\ssf{\f@m\ssfam}
\def\caps{\f@m\cpfam}
%
\def\seventeenpoint{\relax
    \textfont0=\seventeenrm          \scriptfont0=\twelverm
      \scriptscriptfont0=\ninerm
    \textfont1=\seventeeni           \scriptfont1=\twelvei
      \scriptscriptfont1=\ninei
    \textfont2=\seventeensy          \scriptfont2=\twelvesy
      \scriptscriptfont2=\ninesy
    \textfont3=\seventeenex          \scriptfont3=\twelveex
      \scriptscriptfont3=\ninex
    \textfont\itfam=\seventeenit    %\scriptfont\itfam=\twelveit
    \textfont\slfam=\seventeensl    %\scriptfont\slfam=\twelvesl
      \scriptscriptfont\slfam=\ninesl
    \textfont\bffam=\seventeenbf     \scriptfont\bffam=\twelvebf
      \scriptscriptfont\bffam=\ninebf
    \textfont\ttfam=\seventeentt
    \textfont\cpfam=\seventeencp
    \textfont\ssfam=\seventeenss     \scriptfont\ssfam=\twelvess
      \scriptscriptfont\ssfam=\niness
    \samef@nt
    \b@gheight=17pt
    \setbox\strutbox=\hbox{\vrule height 0.85\b@gheight
                                depth 0.35\b@gheight width\z@ }}
%
\def\fourteenpoint{\relax
    \textfont0=\fourteencp          \scriptfont0=\tenrm
      \scriptscriptfont0=\sevenrm
    \textfont1=\fourteeni           \scriptfont1=\teni
      \scriptscriptfont1=\seveni
    \textfont2=\fourteensy          \scriptfont2=\tensy
      \scriptscriptfont2=\sevensy
    \textfont3=\fourteenex          \scriptfont3=\twelveex
      \scriptscriptfont3=\tenex
    \textfont\itfam=\fourteenit     \scriptfont\itfam=\tenit
    \textfont\slfam=\fourteensl     \scriptfont\slfam=\tensl
      \scriptscriptfont\slfam=\sevensl
    \textfont\bffam=\fourteenbf     \scriptfont\bffam=\tenbf
      \scriptscriptfont\bffam=\sevenbf
    \textfont\ttfam=\fourteentt
    \textfont\cpfam=\fourteencp
    \textfont\ssfam=\fourteenss     \scriptfont\ssfam=\tenss
      \scriptscriptfont\ssfam=\sevenss
    \samef@nt
    \b@gheight=14pt
    \setbox\strutbox=\hbox{\vrule height 0.85\b@gheight
                                depth 0.35\b@gheight width\z@ }}
%
\def\twelvepoint{\relax
    \textfont0=\twelverm          \scriptfont0=\ninerm
      \scriptscriptfont0=\sixrm
    \textfont1=\twelvei           \scriptfont1=\ninei
      \scriptscriptfont1=\sixi
    \textfont2=\twelvesy           \scriptfont2=\ninesy
      \scriptscriptfont2=\sixsy
    \textfont3=\twelveex          \scriptfont3=\ninex
      \scriptscriptfont3=\sixex
    \textfont\itfam=\twelveit    %\scriptfont\itfam=\nineit
    \textfont\slfam=\twelvesl    %\scriptfont\slfam=\ninesl
      \scriptscriptfont\slfam=\sixsl
    \textfont\bffam=\twelvebf     \scriptfont\bffam=\ninebf
      \scriptscriptfont\bffam=\sixbf
    \textfont\ttfam=\twelvett
    \textfont\cpfam=\twelvecp
    \textfont\ssfam=\twelvess     \scriptfont\ssfam=\niness
      \scriptscriptfont\ssfam=\sixss
    \samef@nt
    \b@gheight=12pt
    \setbox\strutbox=\hbox{\vrule height 0.85\b@gheight
                                depth 0.35\b@gheight width\z@ }}
%
\def\tenpoint{\relax
    \textfont0=\tenrm          \scriptfont0=\sevenrm
      \scriptscriptfont0=\fiverm
    \textfont1=\teni           \scriptfont1=\seveni
      \scriptscriptfont1=\fivei
    \textfont2=\tensy          \scriptfont2=\sevensy
      \scriptscriptfont2=\fivesy
    \textfont3=\tenex          \scriptfont3=\sevenex
      \scriptscriptfont3=\fivex
    \textfont\itfam=\tenit     \scriptfont\itfam=\seveni
    \textfont\slfam=\tensl     \scriptfont\slfam=\sevensl
      \scriptscriptfont\slfam=\fivesl
    \textfont\bffam=\tenbf     \scriptfont\bffam=\sevenbf
      \scriptscriptfont\bffam=\fivebf
    \textfont\ttfam=\tentt
    \textfont\cpfam=\tencp
    \textfont\ssfam=\tenss     \scriptfont\ssfam=\sevenss
      \scriptscriptfont\ssfam=\fivess
    \samef@nt
    \b@gheight=10pt
    \setbox\strutbox=\hbox{\vrule height 0.85\b@gheight
                                depth 0.35\b@gheight width\z@ }}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   Next, I define basic spacing parameters.
%
\normalbaselineskip = 15pt plus 0.2pt minus 0.1pt %was 20pt ...
\normallineskip = 1.5pt plus 0.1pt minus 0.1pt
\normallineskiplimit = 1.5pt
\newskip\normaldisplayskip
\normaldisplayskip = 15pt plus 5pt minus 10pt %was 20pt ...
\newskip\normaldispshortskip
\normaldispshortskip = 6pt plus 5pt
\newskip\normalparskip
\normalparskip = 6pt plus 2pt minus 1pt
\newskip\skipregister
\skipregister = 5pt plus 2pt minus 1.5pt
%
\newif\ifsingl@    \newif\ifdoubl@
\newif\iftwelv@    \twelv@true
%
\def\singlespace{\singl@true\doubl@false\spaces@t}
\def\doublespace{\singl@false\doubl@true\spaces@t}
\def\normalspace{\singl@false\doubl@false\spaces@t}
\def\Tenpoint{\tenpoint\twelv@false\spaces@t}
\def\Twelvepoint{\twelvepoint\twelv@true\spaces@t}
%
\def\spaces@t{\relax
      \iftwelv@ \ifsingl@\subspaces@t3:4;\else\subspaces@t1:1;\fi
       \else \ifsingl@\subspaces@t3:5;\else\subspaces@t4:5;\fi \fi
      \ifdoubl@ \multiply\baselineskip by 5
         \divide\baselineskip by 4 \fi }
%
\def\subspaces@t#1:#2;{
      \baselineskip = \normalbaselineskip
      \multiply\baselineskip by #1 \divide\baselineskip by #2
      \lineskip = \normallineskip
      \multiply\lineskip by #1 \divide\lineskip by #2
      \lineskiplimit = \normallineskiplimit
      \multiply\lineskiplimit by #1 \divide\lineskiplimit by #2
      \parskip = \normalparskip
      \multiply\parskip by #1 \divide\parskip by #2
      \abovedisplayskip = \normaldisplayskip
      \multiply\abovedisplayskip by #1 \divide\abovedisplayskip by #2
      \belowdisplayskip = \abovedisplayskip
      \abovedisplayshortskip = \normaldispshortskip
      \multiply\abovedisplayshortskip by #1
        \divide\abovedisplayshortskip by #2
      \belowdisplayshortskip = \abovedisplayshortskip
      \advance\belowdisplayshortskip by \belowdisplayskip
      \divide\belowdisplayshortskip by 2
      \smallskipamount = \skipregister
      \multiply\smallskipamount by #1 \divide\smallskipamount by #2
      \medskipamount = \smallskipamount \multiply\medskipamount by 2
      \bigskipamount = \smallskipamount \multiply\bigskipamount by 4 }
%
\def\normalbaselines{ \baselineskip=\normalbaselineskip
   \lineskip=\normallineskip \lineskiplimit=\normallineskip
   \iftwelv@\else \multiply\baselineskip by 4 \divide\baselineskip by 5
     \multiply\lineskiplimit by 4 \divide\lineskiplimit by 5
     \multiply\lineskip by 4 \divide\lineskip by 5 \fi }
%
\Twelvepoint  % That's the default
%%%%%%%%%%%%%%%%%%%%%%%%%%%%\Tenpoint   % Not in Leuven it wasn't!
%
\interlinepenalty=50
\interfootnotelinepenalty=5000
\predisplaypenalty=9000
\postdisplaypenalty=500
\hfuzz=1pt
\vfuzz=0.2pt
\dimen\footins=24 truecm % 8 truein in SB
\hoffset=10.5truemm % 0 in SB, 6.5mm in Leuven
\voffset=-8.5 truemm % 0in in SB, 5 truemm in Leuven
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Now some output macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% footnote numbering macros
%
%
\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf
      \insert\footins\bgroup\singl@true\doubl@false\Tenpoint
      \interlinepenalty=\interfootnotelinepenalty \let\par=\endgraf
        \leftskip=\z@skip \rightskip=\z@skip
        \splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000
        \smallskip\item{#1}\bgroup\strut\aftergroup\@foot\let\next}
\skip\footins=\bigskipamount % space added when footnote is present
\dimen\footins=24truecm % maximum footnotes per page (8 truein in USA)
%
\newcount\fnotenumber
\def\clearfnotenumber{\fnotenumber=0}
\def\fnote{\advance\fnotenumber by1 \footnote{$^{\the\fnotenumber}$}}
\clearfnotenumber
%
% section and appendix macros
%
\newcount\secnumber
\newcount\appnumber
\newif\ifs@c % this is true if within a section as opposed to an appendix
\newif\ifs@cd % this is true if the article is being section'd
\s@cdtrue % this is the default
\def\unsectioned{\s@cdfalse\let\section=\subsection}
\def\clearappnumber{\appnumber=64}
\def\clearsecnumber{\secnumber=0}
\newskip\sectionskip         \sectionskip=\medskipamount
\newskip\headskip            \headskip=8pt plus 3pt minus 3pt
\newdimen\sectionminspace    \sectionminspace=10pc
\newdimen\referenceminspace  \referenceminspace=25pc
%
\def\Titlestyle#1{\par\begingroup \interlinepenalty=9999
     \leftskip=0.02\hsize plus 0.23\hsize minus 0.02\hsize
     \rightskip=\leftskip \parfillskip=0pt
     \advance\baselineskip by 0.5\baselineskip%this is a test...
     \hyphenpenalty=9000 \exhyphenpenalty=9000
     \tolerance=9999 \pretolerance=9000
     \spaceskip=0.333em \xspaceskip=0.5em
     \seventeenpoint
  \noindent #1\par\endgroup }
%
\def\titlestyle#1{\par\begingroup \interlinepenalty=9999
     \leftskip=0.02\hsize plus 0.23\hsize minus 0.02\hsize
     \rightskip=\leftskip \parfillskip=0pt
     \hyphenpenalty=9000 \exhyphenpenalty=9000
     \tolerance=9999 \pretolerance=9000
     \spaceskip=0.333em \xspaceskip=0.5em
     \fourteenpoint
   \noindent #1\par\endgroup }%  the \Npoint only takes care of spacing.
%                                a font is always specified when calling this
%                                macro.  In computers with little room for
%                                character-size data it is convenient to % out
%                                all the font definitions from the \Npoint
%                                macros.
%
\def\spacecheck#1{\dimen@=\pagegoal\advance\dimen@ by -\pagetotal
   \ifdim\dimen@<#1 \ifdim\dimen@>0pt \vfil\break \fi\fi}
%
\def\section#1{\cleareqnumber \s@ctrue \global\advance\secnumber by1
   \par \ifnum\the\lastpenalty=30000\else
   \penalty-200\vskip\sectionskip \spacecheck\sectionminspace\fi
   \noindent {\caps\enspace\S\the\secnumber\quad #1}\par
   \nobreak\vskip\headskip \penalty 30000 }
%
\def\subsection#1{\par
   \ifnum\the\lastpenalty=30000\else \penalty-100\smallskip
   \spacecheck\sectionminspace\fi
   \noindent\undertext{#1}\enspace \vadjust{\penalty5000}}
\let\subsec=\subsection
%
\def\undertext#1{\vtop{\hbox{#1}\kern 1pt \hrule}}
%
\def\subsubsection#1{\par
   \ifnum\the\lastpenalty=30000\else \penalty-100\smallskip \fi
   \noindent\hbox{#1}\enspace \vadjust{\penalty5000}}
\let\ssubsec=\subsubsection
%
\def\appendix#1{\cleareqnumber \s@cfalse \global\advance\appnumber by1
   \par \ifnum\the\lastpenalty=30000\else
   \penalty-200\vskip\sectionskip \spacecheck\sectionminspace\fi
   \noindent {\caps\enspace Appendix \char\the\appnumber\quad #1}\par
   \nobreak\vskip\headskip \penalty 30000 }
%
\def\endsec{\relax}
%
\clearsecnumber
\clearappnumber
%
% macros for references, acknowledgements, and note added
%
\def\ack{\par\penalty-100\medskip \spacecheck\sectionminspace
   \line{\iftwelv@\fourteencp\else\twelvecp\fi\hfil ACKNOWLEDGEMENTS\hfil}%
\nobreak\vskip\headskip }
\def\refs{\begingroup \par\penalty-100\medskip \spacecheck\sectionminspace
   \line{\iftwelv@\fourteencp\else\twelvecp\fi\hfil REFERENCES\hfil}%
\nobreak\vskip\headskip \frenchspacing }
\def\endrefs{\par\endgroup}
\def\NoteAdded{\noindent{\caps Note added:}\enspace}%--- Note added
%
\newcount\refnumber
\def\clearrefnumber{\refnumber=0}  \clearrefnumber
\newwrite\R@fs                              %This opens a file .refs with
\immediate\openout\R@fs=\jobname.references %the references in order of
                                            %appearance.
\def\closerefs{\immediate\closeout\R@fs} %close file so that TeX can read it
%
\def\refsout{\closerefs\refs
\catcode`\@=11                          % we must do this since the
\input\jobname.references               % references expand to
\catcode`\@=12			        % primitives containing @'s
\endrefs}
%
\def\refitem#1{\item{{\bf #1}}}%just bolds it so that \bf does not expand
%
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
%
%  new reference macros.  Now just say \[_label], and in the
%  ref_defs.tex file type \refdef[_label]{reference}
%
\def\[#1]{\ifundefined{#1R@FNO}%
\global\advance\refnumber by1%
\expandafter\xdef\csname#1R@FNO\endcsname{[\the\refnumber]}%
\immediate\write\R@fs{\noexpand\refitem{\csname#1R@FNO\endcsname}%
\noexpand\csname#1R@F\endcsname}\fi{\bf \csname#1R@FNO\endcsname}}
%
\def\refdef[#1]#2{\expandafter\gdef\csname#1R@F\endcsname{{#2}}}
%
% equation numbering macros
%
%  better than before.  just do \(_label) both to refer or to
%                         define.
%
%        the generic equation is \()
%
%
%
\newcount\eqnumber
\def\cleareqnumber{\eqnumber=0}
\newif\ifal@gn \al@gnfalse  % this is true if within an \eqalignno
% at some point try the following:
%\def\eqnalign#1{\al@gntrue \vbox{\eqalignno{#1}} \al@gnfalse}
% but meanwhile let`s define a new macro...
\def\veqnalign#1{\al@gntrue \vbox{\eqalignno{#1}} \al@gnfalse}
\def\eqnalign#1{\al@gntrue \eqalignno{#1} \al@gnfalse}
%
\def\(#1){\relax%
\ifundefined{#1@Q}
 \global\advance\eqnumber by1
 \ifs@cd
  \ifs@c
   \expandafter\xdef\csname#1@Q\endcsname{{%
\noexpand\rm(\the\secnumber .\the\eqnumber)}}
  \else
   \expandafter\xdef\csname#1@Q\endcsname{{%
\noexpand\rm(\char\the\appnumber .\the\eqnumber)}}
  \fi
 \else
  \expandafter\xdef\csname#1@Q\endcsname{{\noexpand\rm(\the\eqnumber)}}
 \fi
 \ifal@gn
    & \csname#1@Q\endcsname
 \else
    \eqno \csname#1@Q\endcsname
 \fi
\else%
\csname#1@Q\endcsname\fi\global\let\@Q=\relax}
%
% macros for running heads and page numbering
%
\newif\iffrontpage \frontpagefalse
%
\headline={\hfil}
%
\footline={\iffrontpage\hfil\else \hss\twelverm
-- \folio\ --\hss \fi }
%
\def\flushleft#1{\line{#1\hss}}
\def\flushright#1{\line{\hss#1}}
%
\def\monthname{\relax\ifcase\month 0/\or January\or February\or
   March\or April\or May\or June\or July\or August\or September\or
   October\or November\or December\else\number\month/\fi}
%
\hsize=14 truecm
\vsize=22 truecm
\skip\footins=\bigskipamount
\normalspace
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   Here come macros for title pages.
%
\newskip\frontpageskip
\newif\ifp@bblock \p@bblocktrue
\newif\ifm@nth \m@nthtrue
\newtoks\pubnum
\newtoks\pubtype
\newtoks\m@nthn@me
\newcount\Ye@r
\advance\Ye@r by \year
\advance\Ye@r by -1900
%
\def\Year#1{\Ye@r=#1}%--- set the year by hand
%
\def\Month#1{\m@nthfalse \m@nthn@me={#1}}
\def\m@nthname{\ifm@nth\monthname\else\the\m@nthn@me\fi}
%
\def\titlepage{\global\frontpagetrue\hrule height\z@ \relax
               \ifp@bblock\pubblock\fi\relax }
\def\endtitlepage{\vfil\break
                  \frontpagefalse} %I took a \pageno=1 from here
%
\def\bonntitlepage{\global\frontpagetrue\hrule height\z@ \relax
               \ifp@bblock\pubblock\fi\relax }
%\def\endbonntitlepage{\vfil%
%\vcenter{\hfill\hbox to 5.5cm{%
%\special{psfile=is.ps voffset=-155}\hfill}\vfill\centering}
%\break\frontpagefalse} %I took a \pageno=1 from here
%
\frontpageskip=12pt plus .5fil minus 2pt
\pubtype={\iftwelv@\twelvesl\else\tensl\fi\ (Preliminary Version)}
\pubnum={?}
\def\nopubblock{\p@bblockfalse}
%
\def\pubblock{\line{\hfil\iftwelv@\twelverm\else\tenrm\fi%
BONN--HE--\number\Ye@r--\the\pubnum\the\pubtype}
              \line{\hfil\iftwelv@\twelverm\else\tenrm\fi%
\m@nthname\ \number\year}}
%
\def\title#1{\vskip\frontpageskip\Titlestyle{\caps #1}\vskip3\headskip}
%                                 ^---notice capital Titlestyle...
\def\author#1{\vskip.5\frontpageskip\titlestyle{\caps #1}\nobreak}
\def\andauthor{\vskip.5\frontpageskip\centerline{and}\author}
\def\authors{\vskip\frontpageskip\noindent}
%
\def\address#1{\par\kern 5pt\titlestyle{%\iftwelv@\twelvepoint\else\tenpoint\fi
\it #1}}
\def\andaddress{\par\kern 5pt \centerline{\sl and} \address}
\def\addresses{\vskip\frontpageskip\noindent\interlinepenalty=9999}
%
\def\KUL{\address{Instituut voor Theoretische Fysica, Universiteit Leuven\break
Celestijnenlaan 200 D, B--3001 Heverlee, BELGIUM}}
%
\def\Bonn{\address{Physikalisches Institut der Universit\"at Bonn\break
Nu{\ss}allee 12, W--5300 Bonn 1, GERMANY}}
%
\def\Santiago{\address{Departamento de F{\'\i}sica de Part{\'\i}culas
Elementales\break
Universidad de Santiago, Santiago de Compostela 15706, SPAIN}}
%
\def\abstract#1{\par\dimen@=\prevdepth \hrule height\z@ \prevdepth=\dimen@
   \vskip\frontpageskip\spacecheck\sectionminspace
   \centerline{\iftwelv@\fourteencp\else\twelvecp\fi ABSTRACT}\vskip\headskip
   {\noindent #1}}
%
\def\submit#1{\par\nobreak\vfil\nobreak\medskip
   \centerline{Submitted to \sl #1}}
\def\PACS#1{\par\nobreak\vfil\nobreak\medskip\line{PACS numbers: #1\hfil}}
\def\toappear#1{\par\nobreak\vfil\nobreak\medskip
   \centerline{To appear in \sl #1}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% macros for leaders, boxes, underline, ...
%
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}%--- leading ...
\def\under#1{$\underline{\smash{\hbox{#1}}}$} %--- underline
\def\boxit#1{\vcenter{\hrule\hbox{\vrule\kern8pt
      \vbox{\kern8pt#1\kern8pt}\kern8pt\vrule}\hrule}}%--- box
\def\Boxed#1{\boxit{\hbox{$\displaystyle{#1}$}}} %--- box in $$...$$
\def\Item{\par\hang}
\def\unnatural{\message{Unnatural Page Break}\vfil\eject}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Now come basic non-math macros
%
\def\ref#1{{\bf [#1]}}%--- [ref]
\def\etal{{\it et al.\/}}%--- et al.
\def\ie{{\it i.e.\/}}%--- i.e.
\def\eg{{\it e.g.\/}}%--- e.g.
\def\Cf{{\it Cf.\ }}%--- Cf.
\def\cf{{\it cf.\ }}%--- cf.
\def\dlq{\lq \lq} %--- double left quote
\def\th{{\rm th}}%--- th as in fifth
\def\nl{\hfil\break}%--- new line
\def\ni{\noindent}%--- just an abbrev.
\def\half{{1\over 2}}%--- 1/2
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Now some math macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% First macros for theorems, definitions, ...
%
\newif\ifm@thstyle \m@thstylefalse
\def\mathstyle{\m@thstyletrue}
\def\proclaim#1#2\par{\smallbreak\begingroup%        small --> med???
\advance\baselineskip by -0.25\baselineskip%
\advance\belowdisplayskip by -0.35\belowdisplayskip%
\advance\abovedisplayskip by -0.35\abovedisplayskip%
    \noindent{\caps#1.\enspace}{#2}\par\endgroup%
\smallbreak}%--- defs, thms, ...                     small --> med???
\def\m@kem@th<#1>#2#3{%
\ifm@thstyle \global\advance\eqnumber by1
 \ifs@cd
  \ifs@c
   \expandafter\xdef\csname#1\endcsname{{%
\noexpand #2\ \the\secnumber .\the\eqnumber}}
  \else
   \expandafter\xdef\csname#1\endcsname{{%
\noexpand #2\ \char\the\appnumber .\the\eqnumber}}
  \fi
 \else
  \expandafter\xdef\csname#1\endcsname{{\noexpand #2\ \the\eqnumber}}
 \fi
 \proclaim{\csname#1\endcsname}{#3}
\else
 \proclaim{#2}{#3}
\fi}
%
%
%  To use the new Math macros...
%
%         \Thm<_label>{Statemenet of the Thm} etc...
%
%     where _label is the label of the Thm.  The generic
%     Thm has an empty label.
%
%     To refer to it, just say \<_label>
%
\def\Thm<#1>#2{\m@kem@th<#1M@TH>{Theorem}{\sl#2}}%--- Theorem
\def\Prop<#1>#2{\m@kem@th<#1M@TH>{Proposition}{\sl#2}}%--- Proposition
\def\Def<#1>#2{\m@kem@th<#1M@TH>{Definition}{\rm#2}}%--- Definition
\def\Lem<#1>#2{\m@kem@th<#1M@TH>{Lemma}{\sl#2}}%--- Lemma
\def\Cor<#1>#2{\m@kem@th<#1M@TH>{Corollary}{\sl#2}}%--- Corollary
\def\Conj<#1>#2{\m@kem@th<#1M@TH>{Conjecture}{\sl#2}}%--- Conjecture
\def\Rmk<#1>#2{\m@kem@th<#1M@TH>{Remark}{\rm#2}}%--- Remark
\def\Exm<#1>#2{\m@kem@th<#1M@TH>{Example}{\rm#2}}%--- Example
\def\Qry<#1>#2{\m@kem@th<#1M@TH>{Query}{\it#2}}%--- Query
%
\def\Proof{\noindent{\caps Proof:}\enspace}%--- Proof
%
\let\Pf=\Proof
\let\Example=\Exm
\def\<#1>{\csname#1M@TH\endcsname}
%
% We then continue with basic mathematics
%
\def\defn{\buildrel \rm def \over =}%--- def over =
\def\qed{\vrule width 0.7em height 0.6em depth 0.2em}%--- Halmos Q.E.D.
\def\QED{\enspace\qed}
\def\implies{\Rightarrow}%--- implies
\def\implied{\Leftarrow}%--- is implied by
\def\iff{\Leftrightarrow}%--- if and only if
\def\lapprox{\hbox{\lower3pt\hbox{$\buildrel<\over\sim$}}}% approx lt
\def\gapprox{\hbox{\lower3pt\hbox{$\buildrel<\over\sim$}}}% approx gt
\def\quotient#1#2{#1/\lower0pt\hbox{${#2}$}}%--- factor objects
\def\lsemidir{\mathbin{\hbox{\hskip2pt\vrule height 5.7pt depth -.3pt
width .25pt\hskip-2pt$\times$}}}
\def\rsemidir{\mathbin{\hbox{$\times$\hskip-2pt\vrule height 5.7pt
depth -.3pt width .25pt\hskip2pt}}}
%
% Arrow stuff
%
\def\into{\hookrightarrow}%--- injective map
\def\onto{\rightarrow\!\!\!\!\!\rightarrow}%--- surjective map
\def\iso{\into\!\!\!\!\!\rightarrow}%--- bijective map
\def\to{\rightarrow}%--- mapping
\def\tto{\longrightarrow}%--- long mapping
\def\isomap{\buildrel \cong \over \tto}%--- isom over -->
\def\lra{\leftrightarrow}%--- just an abbrev.
%
\def\mapright#1{\smash{
    \mathop{\tto}\limits^{#1}}}
\def\mapleft#1{\smash{
    \mathop{\longleftarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow
  \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow
  \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapsw#1{\swarrow
\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\def\commdiag#1{
\def\normalbaselines{\baselineskip20pt \lineskip3pt \lineskiplimit3pt }
\matrix{#1}} %--- commutative diagram macro
\def\cmap#1{\buildrel {\Phi_{#1}} \over \longrightarrow}%-- lin map over arrow
\def\comap#1{\buildrel {#1} \over\longrightarrow} %--- map in complex
%
% Numbers...
%
\def\reals{\mathord{\bf R}} %--- reals
\def\comps{\mathord{\bf C}} %--- complex nos.
\def\quats{\mathord{\bf H}} %--- quaternions
\def\integ{\mathord{\bf Z}} %--- integers
\def\rats{\mathord{\bf Q}} %--- rationals
\def\nats{\mathord{\bf N}} %--- naturals
\def\field{\mathord{\bf F}} %--- ground field
%
% Algebra
%
\def\Hom{\mathop{\rm Hom}}%--- Hom(omorphisms)
\def\tr{\mathop{\rm tr}}%--- tr(ace)
\def\Tr{\mathop{\rm Tr}}%--- Tr(ace)
\def\End{\mathop{\rm End}}%--- End(omorphisms)
\def\Mor{\mathop{\rm Mor}}%--- Mor(phisms)
\def\Aut{\mathop{\rm Aut}}%--- Aut(omorphisms)
\def\aut{\mathop{\rm aut}}%--- aut(omorphisms)
\def\str{\mathop{\rm str}}%--- supertrace
\def\sdet{\mathop{\rm sdet}}%--- superdeterminant
\def\ker{\mathop{\rm ker}}%--- kernel
\def\coker{\mathop{\rm coker}}%--- cokernel
\def\im{\mathop{\rm im}}%--- image
\def\underrightarrow#1{\vtop{\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr
      \noalign{\kern-\p@\nointerlineskip}
      \rightarrowfill\crcr}}} %--- modification of \overrightarrow
\def\underleftarrow#1{\vtop{\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr
      \noalign{\kern-\p@\nointerlineskip}
      \leftarrowfill\crcr}}}  %--- modification of \overleftarrow
\def\liminv{\mathop{\underleftarrow{\rm lim}}}
\def\limdir{\mathop{\underrightarrow{\rm lim}}}
%
% Brackets,...
%
\def\comm#1#2{\left[#1\, ,\,#2\right]}%--- [ , ]
\def\anticomm#1#2{\left\{#1\, ,\,#2\right\}}%--- { , }
\def\gradcomm#1#2{\left[\, ,\,#2\right\}}%--- [ , }
\def\stc#1#2#3{{f_{#2#3}}^{#1}}%--- structure const.
%
% Analysis anyone?
%
\def\lied#1#2{{\cal L}_{#1}{#2}}%--- Lie derivative
\def\lvdr#1#2{{{{\delta^{\bf L}}{#1}}\over{{\delta}{#2}}}}% lft var derivative
\def\rvdr#1#2{{{{\delta^{\bf R}}{#1}}\over{{\delta}{#2}}}}% rgt var derivative
\def\vder#1#2{{{{\delta}{#1}}\over{{\delta}{#2}}}}%--- vartnl derivative
\def\ptdr#1#2#3{{{\delta^2#1}\over{{\delta#2}{\delta#3}}}}% double vartl deriv.
\def\lpdr#1#2{{{\partial^{\bf L}#1}\over{\partial#2}}}%--- rgt prtl derivative
\def\pder#1#2{{{\partial #1}\over{\partial #2}}}%--- partial derivative
\def\dpder#1#2#3{{{\partial^2#1}\over{{\partial#2}{\partial#3}}}}%
%                                                           dble partl deriv.
\def\lder#1#2{{d^{\bf L}{#1}}\over{d{#2}}}%--- full lft derivative
\def\rder#1#2{{d^{\bf R}{#1}}\over{d{#2}}}%--- full rgt derivative
\def\der#1#2{{{d #1}\over {d #2}}}%--- full derivative
\def\laplacian{\bigtriangleup}%--- laplacian
\def\dcov{d^{\nabla}}%--- cov. ext. der.
%
% Dirac slashes
%
\def\Dslash{D\!\!\!\!/}%--- D slash
\def\dslash{\partial{\mkern-10.5mu}/}%--- del slash
\def\Aslash{A{\mkern-10.5mu}/}%--- A slash
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  These are the macros to make Young tableaux
%
\newdimen\unit
\newdimen\redunit
%
%   this puts the ref. point of #1 at coordinates (#2,#3)
%
\def\p@int#1:#2 #3 {\rlap{\kern#2\unit
     \raise#3\unit\hbox{#1}}}
%
% this defines the sides of the tableau
% notice that \rver and \lver coincide
% It would have been natural for \rver to have negative
% width but that does not print in TeX...
%
\def\th@r{\vrule height0\unit depth.1\unit width1\unit}
\def\bh@r{\vrule height.1\unit depth0\unit width1\unit}
\def\lv@r{\vrule height1\unit depth0\unit width.1\unit}
\def\rv@r{\vrule height1\unit depth0\unit width.1\unit}
%
% this is the tableau: the .9 is due to the unnatural definition
% of \rver
%
\def\t@ble@u{\hbox{\p@int\bh@r:0 0
                   \p@int\lv@r:0 0
                   \p@int\rv@r:.9 0
                   \p@int\th@r:0 1
                   }
             }
%
% we now define the tableau at a particular location
%
\def\t@bleau#1#2{\rlap{\kern#1\redunit
     \raise#2\redunit\t@ble@u}}
%
%  Now a macro to make a column of #1 tableaux down at (#2,#3)
%
\newcount\n
\newcount\m
\def\makecol#1#2#3{\n=0 \m=#3
  \loop\ifnum\n<#1{}\advance\m by -1 \t@bleau{#2}{\number\m}\advance\n by 1
\repeat}
%
%   Now a macro to make a row of #1 tableaux at (#2,#3) to the right
%
\def\makerow#1#2#3{\n=0 \m=#3
 \loop\ifnum\n<#1{}\advance\m by 1 \t@bleau{\number\m}{#2}\advance\n by 1
\repeat}
%
% Some useful ready made Young tableaux
%
\def\checkunits{\ifinner \unit=6pt \else \unit=8pt \fi
                \redunit=0.9\unit } %these are the basic sizes
\def\ytsym#1{\checkunits\kern-.5\unit
  \vcenter{\hbox{\makerow{#1}{0}{0}\kern#1\unit}}\kern.5em} % #1 symmetrized
%                                                             tableaux
\def\ytant#1{\checkunits\kern.5em
  \vcenter{\hbox{\makecol{#1}{0}{0}\kern1\unit}}\kern.5em} % #1 antisymmetrized
%                                                            tableaux
\def\ytwo#1#2{\checkunits
  \vcenter{\hbox{\makecol{#1}{0}{0}\makecol{#2}{1}{0}\kern2\unit}}
                  \ } % 2 column #1 #2 (left->right) Young tableau
\def\ythree#1#2#3{\checkunits
  \vcenter{\hbox{\makecol{#1}{0}{0}\makecol{#2}{1}{0}\makecol{#3}{2}{0}%
\kern3\unit}}
                  \ } % 3 column #1 #2 #3 (left->right) Young tableau
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Finally some useful macros for journals, ...
%
\def\PRL#1#2#3{{\sl Phys. Rev. Lett.} {\bf#1} (#2) #3}
\def\NPB#1#2#3{{\sl Nucl. Phys.} {\bf B#1} (#2) #3}
\def\NPBFS#1#2#3#4{{\sl Nucl. Phys.} {\bf B#2} [FS#1] (#3) #4}
\def\CMP#1#2#3{{\sl Comm. Math. Phys.} {\bf #1} (#2) #3}
\def\PRD#1#2#3{{\sl Phys. Rev.} {\bf D#1} (#2) #3}
\def\PLA#1#2#3{{\sl Phys. Lett.} {\bf #1A} (#2) #3}
\def\PLB#1#2#3{{\sl Phys. Lett.} {\bf #1B} (#2) #3}
\def\JMP#1#2#3{{\sl J. Math. Phys.} {\bf #1} (#2) #3}
\def\JMM#1#2#3{{\sl J. Math. Mech.} {\bf #1} (#2) #3}
\def\PTP#1#2#3{{\sl Prog. Theor. Phys.} {\bf #1} (#2) #3}
\def\SPTP#1#2#3{{\sl Suppl. Prog. Theor. Phys.} {\bf #1} (#2) #3}
\def\AoP#1#2#3{{\sl Ann. of Phys.} {\bf #1} (#2) #3}
\def\PNAS#1#2#3{{\sl Proc. Natl. Acad. Sci. USA} {\bf #1} (#2) #3}
\def\RMP#1#2#3{{\sl Rev. Mod. Phys.} {\bf #1} (#2) #3}
\def\PR#1#2#3{{\sl Phys. Reports} {\bf #1} (#2) #3}
\def\AoM#1#2#3{{\sl Ann. of Math.} {\bf #1} (#2) #3}
\def\UMN#1#2#3{{\sl Usp. Mat. Nauk} {\bf #1} (#2) #3}
\def\RMS#1#2#3{{\sl Russian Math Surveys} {\bf #1} (#2) #3}
\def\FAP#1#2#3{{\sl Funkt. Anal. Prilozheniya} {\bf #1} (#2) #3}
\def\FAaIA#1#2#3{{\sl Functional Analysis and Its Application} {\bf #1} (#2)
#3}
\def\BSMF#1#2#3{{\sl Bull. Soc. Mat. France} {\bf #1} (#2) #3}
\def\BAMS#1#2#3{{\sl Bull. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\TAMS#1#2#3{{\sl Trans. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\AIHP#1#2#3{{\sl Ann. Inst. Henri Poincar\'e} {\bf #1} (#2) #3}
\def\AIF#1#2#3#4{{\sl Ann. Inst. Fourier} {\bf #1,#2} (#3) #4}
\def\PAMS#1#2#3{{\sl Proc. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\CMJ#1#2#3{{\sl Czechosl. Math. J.} {\bf #1} (#2) #3}
\def\CompM#1#2#3{{\sl Compositio Math.} {\bf #1} (#2) #3}
\def\Invm#1#2#3{{\sl Invent. math.} {\bf #1} (#2) #3}
\def\LMP#1#2#3{{\sl Letters in Math. Phys.} {\bf #1} (#2) #3}
\def\IJMPA#1#2#3{{\sl Int. J. Mod. Phys.} {\bf A#1} (#2) #3}
\def\AdM#1#2#3{{\sl Advances in Math.} {\bf #1} (#2) #3}
\def\RevMP#1#2#3{{\sl Reviews in Math. Phys.} {\bf #1} (#2) #3}
\def\RMaP#1#2#3{{\sl Reports on Math. Phys.} {\bf #1} (#2) #3}
\def\IJM#1#2#3{{\sl Ill. J. Math.} {\bf #1} (#2) #3}
\def\APP#1#2#3{{\sl Acta Phys. Polon.} {\bf #1} (#2) #3}
\def\TMP#1#2#3{{\sl Theor. Mat. Phys.} {\bf #1} (#2) #3}
\def\JPA#1#2#3{{\sl J. Physics} {\bf A#1} (#2) #3}
\def\JSM#1#2#3{{\sl J. Soviet Math.} {\bf #1} (#2) #3}
\def\MPLA#1#2#3{{\sl Mod. Phys. Lett.} {\bf A#1} (#2) #3}
\def\JETP#1#2#3{{\sl Sov. Phys. JETP} {\bf #1} (#2) #3}
\def\CMH#1#2#3{{\sl Comment. Math. Helv.} {\bf #1} (#2) #3}
\def\PJAS#1#2#3{{\sl Proc. Jpn. Acad. Sci.} {\bf #1} (#2) #3}
\def\JPSJ#1#2#3{{\sl J. Phys. Soc. Jpn.} {\bf #1} (#2) #3}
\def\JETPL#1#2#3{{\sl  Sov. Phys. JETP Lett.} {\bf #1} (#2) #3}
\def\JDG#1#2#3{{\sl J. Diff. Geometry} {\bf #1} (#2) #3}
\def\AdiM#1#2#3{{\sl Annali di Matematica} {\bf #1} (#2) #3}
%
%
\catcode`\@=12 % @ no longer a letter
%
%
%
%   These are the local macros for the KdV notes
%
\def\W{\mathord{\ssf W}}
\def\w{\mathord{\ssf w}}
\def\wB{\mathord{\ssf wB}}
\def\wC{\mathord{\ssf wC}}
\def\gd{\mathord{\ssf gd}}
\def\diffs#1{\mathord{\ssf diff}(S^{#1})}
\def\d{\partial}
\let\pb=\anticomm
\def\pdo{{\hbox{$\Psi$DO}}}
\def\res{{\rm res\,}}
\def\Tr{{\rm Tr\,}}
\def\fr#1/#2{\hbox{${#1}\over{#2}$}}
\def\mod{\mathop{\rm mod}}
%
\refdef[Adler]{M.~Adler, \Invm{50}{1981}{403}.}
\refdef[Dickey]{L.~A.~Dickey, {\sl Integrable equations and Hamiltonian
systems}, World Scientific Publ.~Co.}
\refdef[GD]{I.~M.~Gel'fand and L.~A.~Dickey, {\sl A family of Hamiltonian
structures connected with integrable nonlinear differential equations},
Preprint 136, IPM AN SSSR, Moscow (1978).}
\refdef[DS]{V.~G.~Drinfel'd and V.~V.~Sokolov, \JSM{30}{1984}{1975}.}
\refdef[FL]{V.~A.~Fateev and S.~L.~Lukyanov, \IJMPA{3}{1988}{507}.}
\refdef[ClassLim]{J. M. Figueroa-O'Farrill and E. Ramos,
\PLB{282}{1992}{357} ({\tt hep-th/9202040}.}
\refdef[Zam]{A. B. Zamolodchikov, \TMP{65}{1986}{1205}.}
\refdef[Class]{J.~M.~Figueroa-O'Farrill and E.~Ramos, {\sl
Classical $\W$-algebras from dispersionless Lax hierarchies},
Preprint-KUL-TF-92/6.}
\refdef[Krich]{I. Krichever, \CMP{143}{1992}{415}.}
\refdef[WGeom]{C.M. Hull, \PLB{269}{1991}{257};\nl
G.~Sotkov and M.~Stanishkov, \NPB{356}{1991}{439};\nl
G.~Sotkov, M.~Stanishkov and C.~J.~Zhu, \NPB{356}{1991}{245};\nl
J.~Gervais \& Y.~Matsuo, \PLB{282}{1992}{309} ({\tt hep-th/9110028});
Preprint LPTENS-91/35, NBI-HE-91-50 ({\tt hep-th/9201026});\nl
J.~de~Boer and J.~Goeree, THU--92/14 ({\tt hep-th/9206098}).}
\refdef[Radul]{A.~O.~Radul, \FAaIA{25}{1991}{25}.}
%
\overfullrule=0pt
\unsectioned
%
\def\pubblock{ \line{\hfil\twelverm BONN--HE--92--27}
               \line{\hfil\twelverm KUL--TF--92/34}
               \line{\hfil\twelvett hep-th/9209002}
               \line{\hfil\twelverm August 1992}}
\titlepage
\title{A Geometrical Interpretation of Classical $\W$-Transformations}
\vskip 1.cm
\author{Jos\'e~M.~Figueroa-O'Farrill\footnote{$^\natural$}{\tt
e-mail: figueroa@pib1.physik.uni-bonn.de\hfil},
Sonia~Stanciu\footnote{$^\flat$} {\tt e-mail:
stanciu@pib1.physik.uni-bonn.de\hfil}}
\Bonn
\vskip 0.5cm
\centerline{\it and}
\author{Eduardo Ramos\footnote{$^\sharp$}{{\tt e-mail:
fgbda06@blekul11.bitnet\hfil}\break {\rm (Address after October 1992:
Queen Mary and Westfield College, UK)\hfil}}}
\KUL

\abstract{We give a simple geometrical interpretation of
classical $\W$-transformations as deformations of constant energy
surfaces by canonical transformations on a two-dimensional phase
space.}

\endtitlepage
%

\section{Introduction}

The purpose of this letter is to give a simple and readable account of
the geometric significance of classical $\W$-transformations\fnote{For
different approaches to this problem see \[WGeom].} ($\w$-morphisms).
We should first explain what is meant by a $\w$-morphism from the
strictly algebraic point of view.

$\W$-algebras were first introduced in \[Zam], where it is shown,
using the bootstrap method, that the extension of the Virasoro algebra
by a field of spin 3 ($W$) yielded a non-linear associative algebra,
denoted since then by $\W_3$.  It is well-known that
$$Q_{\epsilon}=\oint dz \,\epsilon (z)T(z)\(diff)$$
is the generator of conformal transformations (or, equivalently, of
diffeomorphisms of the circle). But, what is the geometrical
significance, if any, of the transformation generated by
$$Q_{\eta}=\oint dz\,\eta(z)W(z)~?\(Wtransf)$$

Soon after Zamolodchikov's paper, Fateev and Lukyanov \[FL]
recognized that the second hamiltonian structure of the Boussinesque
hierarchy (the Gel'fand--Dickey algebra associated to the Boussinesque
operator) is a classical realisation of the $\W_3$-algebra.  Then
they were able, using the formalism of Drinfel'd and
Sokolov \[DS], to generalise the results of Zamolodchikov to
construct $\W_n$-algebras; \ie, extended conformal algebras with
fields of integer spins from 3 to $n$. Of course, from the geometrical
point of view this development was not of much help: it only seemed to
complicate matters even further opening the question of what is the
geometrical meaning of all these new $\W_n$-transformations.
Nevertheless, it was a crucial development from the point of view of
the algebraic theory, putting at our disposal all the powerful
machinery of integrable systems of the KdV-type. (For a comprehensive
review see \[Dickey].)

Recently, we became interested in the problem of looking for simpler
algebraic structures that would still retain the essential features of
$\W$-algebras.  In \[ClassLim] it was proven that it is possible to
define a classical limit of the Gel'fand--Dickey algebras and their
reductions; thus providing a natural simplification.  These are
nonlinear extensions of $\diffs1$ by tensors $\{u_j\}$ of weights
$3,4,\cdots,n$.  The name classical $\W$-algebras is justified in the
sense that upon quantisation the full structure of the $\W$-algebra is
recovered.  We would also like to stress that these classical
$\W$-algebras are of great interest in their own right.  For example,
they play a fundamental r\^ole in the context of planar $2{-}D$
gravity \[Class], as well as in $2{-}D$ topological field
theory \[Krich].

The simplifications introduced by the classical limit notwithstanding,
it remains to elucidate the geometrical meaning of the $\w$-morphisms
generated, under Poisson bracket, by
$$Q^j_{\epsilon}=\int dx\,\epsilon (x)u_j(x)~.\(wmorph)$$
We will see in what follows that $\w$-morphisms can be interpreted as
deformations of constant-energy surfaces in a two-dimensional
phase-space induced by infinitesimal canonical transformations.

But before getting into the details we would like to remark that this
result has been inspired by the relation found by Radul \[Radul]
between the algebra of differential operators on the circle and the
Gel'fand--Dickey algebras; although our presentation will not make
this explicit.

\section{Geometric Setup}

Consider a two-dimensional phase space $M$ and a smooth function $H$
on $M$, which to fix ideas we can think of as a hamiltonian.  Let
$\lambda$ be a (regular) value of $H$, so that the constant energy
(one-dimensional) surface $Z=H^{-1}(\lambda)$ is a submanifold of $M$.
Let $L\equiv H-\lambda$.  Then $Z$ is the zero locus of $L$ and $L$
generates the ideal ${\cal I}_Z$ of functions vanishing on $Z$.  This
ideal consists of functions $FL$, where $F$ is any smooth function on
$M$.  It is clear that such functions vanish on $Z$ and it can be
proven that these are all the functions which do.  Any function on $Z$
extends to a function on all of $M$ and the difference of any two such
extensions is a function vanishing on $Z$.  In other words, there is a
one-to-one correspondence between the functions ${\cal F}(Z)$ on $Z$
and the quotient ${\cal F}(M)/{\cal I}_Z$.  We let $\pi: {\cal F}(M)
\to {\cal F}(M)/{\cal I}_Z$ denote the map which sends a function on
$M$ to its equivalence class modulo ${\cal I}_Z$.  In the next
section, and for the class of functions we shall consider, we exhibit
an explicit model for this quotient.

We now investigate the effect of canonical transformations
(symplectomorphisms) on the constant energy surface $Z$.
We can analyze deformations of $Z$ by looking at how the function $L$
behaves on $Z$ under symplectomorphisms.

Infinitesimal symplectomorphisms are locally generated by functions on
$M$.  In fact, given a function $S$ on $M$, it gives rise to a vector
field $\delta_S$ defined such that acting on a function $F$,
$$\delta_S F = \pb{S}{F}~.\()$$
If $S$ vanishes on $Z$, then $\delta_S$ is tangent to $Z$.  In fact,
such an $S$ can be written as $GL$ and hence
$$\delta_S L = \pb{GL}{L} = \pb{G}{L}L~,\()$$
which vanishes on $Z$.  (Physically this is nothing but energy
conservation.)  Therefore infinitesimal symplectomorphisms generated
by functions in ${\cal I}_Z$ do not change $Z$.  In other words,
nontrivial deformations of $Z$ induced from symplectomorphisms are
locally generated by ${\cal F}(M)/{\cal I}_Z$.
Therefore, on $Z$, the function $L$ transforms as
$$\delta_S L \equiv \pi(\pb{\pi(S)}{L})~.\(gendef)$$
For a specific choice of hamiltonian, we will now see that \(gendef)
defines $\w$-morphisms associated to the classical $\W$-algebras:
$\gd_n$ and its reduction $\w_n$.

\section{Classical $\W$-Transformations}

To fix the ideas, we now specialize to $M = S^1\times \reals$ a
cylinder.\fnote{This represents no loss of generality.  The only other
connected two-dimensional phase space (\ie, cotangent bundle) is the
plane, and we can recover this case by simply working locally on the
cylinder.}  In other words, $M$ is the phase space whose configuration
space is a circle.  A coordinate system $q$ for the circle gives rise
to a coordinate system $(q,p)$ for $M$ in such a way that
$\pb{p}{q}=1$.  Moreover any other coordinate system $Q$ on the circle
is related to $q$ by a diffeomorphism, and the associated
coordinate system $(Q,P)$ is related to $(q,p)$ by a canonical
transformation.  Explicitly, if $q \mapsto Q(q)$, then $p\mapsto P =
p/Q'$, where $Q' = \der{Q}{q}$.  This preserves the fundamental
one-form $\theta = pdq = PdQ$ and hence the symplectic form $\omega =
d\theta$ whence the Poisson brackets.

As our function $L$ we choose one of the form $L(q,p) = p^n +
\sum_{i=1}^n u_i(q) p^{n-i}$, where $u_i$ are arbitrary functions.
Under a change of coordinates $(q,p) \to (Q,P)$,
$$L(q,p) = (Q')^n\left(P^n + \sum_{i=1}^n U_i(Q)
P^{n-i}\right)~,\(newL)$$
where $U_i$ and $u_i$ are related by
$$u_i(q) = (Q')^i U_i(Q)~.\(utransf)$$
Since $q\mapsto Q(q)$ is a diffeomorphism, $Q'$ is nowhere vanishing,
hence the submanifold $Z$ which is defined as the zero locus of $L$ in
the coordinates $(q,p)$ is defined, in the coordinates $(Q,P)$, as the
zero locus of the function $P^n + \sum_{i=1}^n U_i(Q) P^{n-i}$, which
has the same form.  Thus these constant-energy surfaces have an
invariant geometric meaning.

In order to have an algebraic handle on the situation, we will work
with functions whose dependence on $p$ is polynomial.  Under a change
of coordinates $(q,p) \to (Q,P)$, polynomials in $p$ go over to
polynomials in $P$.  Let ${\cal E}$ denote the subring of these
functions.  Notice that $L$ belongs to ${\cal E}$.  We let ${\cal
J}_Z$ denote the ideal of ${\cal E}$ generated by $L$.  Since $p^n = L
- \sum_{i=1}^n u_i(q) p^{n-i}$, we notice that modulo ${\cal J}_Z$ we
can always reduce any function in ${\cal E}$ to one with at most $n-1$
powers of $p$.  In other words, ${\cal E}/{\cal J}_Z$ is in one-to-one
correspondence with the functions of the form $\sum_{i=0}^{n-1}f_i(q)
p^i$.  We now give an explicit expression for this representative.
For this we will have to introduce $L^{-1}_{(r)}$---polynomial
functions in $p$ and $p^{-1}$ and which correspond to finite
truncations of the formal inverse of $L$.  Explicitly,
$$L^{-1} = p^{-n}\sum_{k\geq 0}^{\infty}(-1)^k \left(\sum_{j=0}^{n-1}
u_j(q)p^{j-n}\right)^k~,\(linv)$$
and $L_{(r)}^{-1}$ is defined by $L^{-1}= L_{(r)}^{-1} \mod
O(p^{-n-r-1})$ with $r\geq 0$, such that
$$L^{-1}_{(r)}\cdot L= 1 +O(p^{-r-1}).\()$$
Given any polynomial function $F$ in $p$ and $p^{-1}$ we denote by
$F_+$ the part polynomial in $p$ and $F_- = F - F_+$.

\Prop<projection>{Any element $R$ of ${\cal E}$ of order $r$ is
equivalent modulo ${\cal J}_Z$ to a unique polynomial of order at most
$n-1$ given by
$$\pi_L(R) = R-(RL^{-1}_{(r)})_+L
=\left((RL_{(r)}^{-1})_-L\right)_+~.\(projector)$$}

\Pf It is obvious that $\pi_L(R)$ is polynomial in $p$ of order
smaller than $n$ and, moreover, $\pi_L(R) - R \in {\cal J}_Z$.
Uniqueness follows because the order of any function in ${\cal J}_Z$
is equal or bigger than $n$.\QED

This provides us with a concrete model for the equivalence space
${\cal E}/{\cal J}_Z$---namely the space ${\cal E}_{<n}$ of functions
polynomial in $p$ with order strictly less than $n$.  In the sequel,
and in order not to clutter the notation, we will write $L^{-1}$ in
\(projector) to mean the appropriate truncation $L^{-1}_{(r)}$.

We now have at our disposal all the ingredients to establish the link
between the algebraic $\w$-morphisms alluded to in the introduction
and the deformation of constant-energy surfaces.   To this effect, we
compute \(gendef) in this concrete example, where we now make use of
our explicit projector $\pi_L$ instead of $\pi$.  Since
$$\pi_L(S) = ((SL^{-1})_-L)_+~,\()$$
it is natural to reparametrize $\w$-morphisms by
$$X = (SL^{-1})_- \mod p^{-n-1}~,\(reparam)$$
with $\pi_L(S) = (XL)_+$.  We can then write \(gendef) as follows
$$\veqnalign{\delta_X L \equiv \delta_S L &= \pb{(XL)_+}{L} -
(\pb{(XL)_+}{L}L^{-1})_+L\cr
&= \pb{(XL)_+}{L} - \pb{(XL)_+L^{-1}}{L}_+L\cr
&= \pb{(XL)_+}{L} - \pb{X}{L}_+L~,\(cladler)\cr}$$
which as shown in \[ClassLim] is the classical limit of the Adler
map or, equivalently, of the Gel'fand--Dickey brackets---namely
$\gd_n$.  This establishes the equivalence between the algebraic
and geometric approaches to $\w$-morphisms.

In order to obtain now the classical limit $\w_n$ of the $\W_n$
algebras, we need to restrict ourselves to functions $L$ of the form
$$L(q,p) = p^n + \sum_{i=2}^n u_i(q) p^{n-i}~.\(redL)$$
One can always achieve this by a symplectomorphism of the form
$$\eqalign{p &\mapsto p -{1\over n} u_1(q)\cr
q &\mapsto q~,\cr}\()$$
which puts the coefficient of $p^{n-1}$ to zero.  Notice moreover
that, under coordinate changes induced from diffeomorphisms of the
circle, this form of $L$ is preserved.  It then follows that if we
restrict ourselves to infinitesimal symplectomorphisms which preserve
the constraint, \(cladler) define $\w$-morphisms associated with
$\w_n$ \[ClassLim].

Finally, if we restrict to functions $L$ which are odd or even under
the transformation $p\mapsto -p$, and we again only consider
symplectomorphism preserving such property, \(cladler) will induce
$\w$-morphisms associated with the $\wB$ or $\wC$ series,
respectively.

\section{A Simple Example: $\w_3$.}

Consider now, as an example, the function
$$L(q,p) = p^3 + T(q) p + W(q)~.\(wiiiL)$$
The associated classical $\W$-algebra is the $\w_3$-algebra:
$$\veqnalign{\pb{T(x)}{T(y)}^{c\ell} &= - \left[ 2 T(x)\d +
T'(x)\right] \cdot \delta(x-y)~,\cr
\pb{W(x)}{T(y)}^{c\ell} &= - \left[ 3 W(x)\d + W'(x)\right] \cdot
\delta(x-y)~,\()\cr
\noalign{\hbox{and}}
\pb{W(x)}{W(y)}^{c\ell} &= \left[ {2\over3} T(x)\d T(x)\right]
\cdot \delta(x-y)~,\cr}$$
which corrects a typographical error in \[ClassLim].

The algebraic $\w$-morphisms generated by $T$ and $W$ under the above
algebra are given by the usual formulas
$$\veqnalign{\delta_\epsilon^{(T)} F(y) &= \int dx\, \epsilon(x)
\pb{T(x)}{F(y)}^{c\ell}~,\()\cr
\noalign{\hbox{and}}
\delta_\alpha^{(W)} F(y) &= \int dx\, \alpha(x)
\pb{W(x)}{F(y)}^{c\ell}~.\()\cr}$$
With them we can compute the effect of $\w$-morphisms on the
generators themselves.  We obtain
$$\eqalign{\delta_\epsilon^{(T)} T &= 2T\epsilon' + T'\epsilon\,\cr
\delta_\epsilon^{(T)} W &= 3W\epsilon' + W' \epsilon\,\cr
\delta_\alpha^{(W)} T &= 2W'\alpha + 3W\alpha'\,\cr
\delta_\alpha^{(W)} W &= -\fr2/3 (\alpha T)' T~.\cr}\(algwmor)$$

We now compute the deformation of the constant-energy surface $Z$
defined by $L$ using the geometric procedure introduced earlier.  The
most general infinitesimal symplectomorphism which yields a nontrivial
deformation of $Z$ is generated by functions of the form
$$\pi_L(S) = \alpha p^2 + \epsilon p + \beta~.\()$$
Demanding that the symplectomorphism preserve the form \(wiiiL) of L
requires that $\beta = \fr2/3 \alpha T$.  We can now compute \(gendef)
yielding
$$\delta_S L = \left(\delta_\epsilon^{(T)} T + \delta_\alpha^{(W)}
T\right) p + \delta_\epsilon^{(T)} W + \delta_\alpha^{(W)} W~,\()$$
with the variations given by \(algwmor).

\ack

We would like to thank S. Schrans for a careful reading of a previous
draft of the \TeX script. JMF and SS take great pleasure in expressing
their thanks to the Instituut voor Theoretische Fysica of the
Universiteit Leuven for its hospitality and support during the start
of this collaboration.

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