
% This file uses harvmac and btxmac.  That way I can have the
% convenience of BibTeX without using LaTeX.
%
% For it to TeX properly, one needs several extra files, which can
% all reside in the same directory: 
%  thisfile.tex, thisfile.bib, labeldefs.tmp, macros.tex,
%  and harvmac.tex, btxmac.tex, psfig.sty, and ssg.bst 
% labeldefs.tmp can initially be empty.  If you omit it you have to 
% answer null when tex asks for it.
%
% To TeX from scratch, use these commands:
%  tex thisfile
%  bibtex thisfile
%  tex thisfile
%  tex thisfile
%
% If you want to send the files to someone (or to xxx) in a form that
% will tex right the first time, include also:
%  thisfile.aux, thisfile.bbl, and thisfile.blg
%
% If you want to get fancy, you can put 
%  harvmac.tex, btxmac.tex, and macros.tex in <dir1>, 
%  thisfile.bib in <dir2>, and 
%  ssg.bst in <dir3>
% and then make sure 
%  TEXINPUTS includes the directories . and <dir1>, 
%  BIBINPUTS includes the directories . and <dir2>, and 
%  BSTINPUTS includes the directories . and <dir3>.
% These commands would be suitable:
%  echo $TEXINPUTS                        #see what you have in TEXINPUTS
%  setenv TEXINPUTS $TEXINPUTS:<dir1>     #Add <dir1>
%  echo $BIBINPUTS                        #Probably not defined
%  setenv BIBINPUTS .:<dir2>              #Assuming it wasn't defined before
%  echo $BSTINPUTS                        #Probably not defined
%  setenv BSTINPUTS .:<dir3>              #Assuming it wasn't defined before
% I put commands similar to these setenv commands in my .cshrc file.
%
\input harvmac
%\input btxmac

%% @texfile{
%%   author = "Karl Berry and Oren Patashnik",
%%   version = "0.99j",
%%   date = "14 Mar 1992",
%%   filename = "btxmac.tex",
%%   address = "Please use electronic mail",
%%   checksum = "834    4503   33061",
%%   email = "opbibtex@cs.stanford.edu",
%%   codetable = "ISO/ASCII",
%%   supported = "yes",
%%   docstring = "Defines macros that make BibTeX work with plain TeX",
%% }
% BibTeX-for-TeX macros, version 0.99j, for BibTeX 0.99c, TeX 3.0 or later.
% Copyright (C) 1990--92 by Karl Berry and Oren Patashnik; all rights reserved.
% You may copy this file provided: that it's accompanied by the
% "BibTeXing" document, whose text is contained in the file `btxdoc.tex';
% that any documentation you write for these macros also gives a
% reference for "BibTeXing"; and that either you make absolutely no
% changes to your copy, or if you do make changes, (1) you name the file
% something other than `btxmac.tex' and you remove all occurrences of
% `btxmac.tex' from the file, (2) you put, somewhere in the first twenty
% lines of the file, your name, along with an electronic address at which
% others who might use the file may reach you, and (3) you remove each
% occurrence of Oren's name and electronic address from this file.  These
% restrictions help ensure that all standard versions of these macros are
% identical, and that Oren doesn't get deluged with inappropriate e-mail.
%
% This file, btxmac.tex, contains TeX macros that allow BibTeX, a
% bibliography program that was originally designed for use with LaTeX,
% to work with plain TeX.  Please report any bugs (outright goofs,
% improvable macros, misfeatures, or unclear documentation) to Oren
% Patashnik (opbibtex@cs.stanford.edu).  These macros will become frozen
% shortly after BibTeX version 1.00 is released.
%
% AMS-TEX WARNING: We tried very hard, for version .99i of these macros,
% to make them compatible with AmS-TeX.  We succeeded to the extent
% that, if you use one of the standard bibliography styles, you probably
% won't notice any problems with version 0.99i of btxmac.tex.  But
% ultimately we failed, in that the inherent incompatibilities between
% plain TeX and AmS-TeX kept making these macros break, for certain
% inputs or certain styles.  Examples:  (1) AmS-TeX treats at-signs as
% special, in ways that plain TeX and LaTeX don't, so that, for example,
% you can't have any `@' characters in an argument to the \cite command,
% the way you can in TeX or LaTeX; (2) AmS-TeX decided that plain TeX's
% and LaTeX's macron-accent control sequence `\=' should be undefined;
% so you'll need to define `\=' to be `\B' to get the xampl.bib example
% suggested below to work with AmS-TeX; (3) AmS-TeX redefines the tie
% character `~' of plain TeX, and AmS-TeX's `amsppt' style redefines
% plain TeX's `\nobreak' macro, so that if you use an author-date style
% like `apalike' and you have a multiple-author reference for which the
% author-date style automatically produces a citation in the text like
% `(Jones et~al., 1992)' you will throw AmS-TeX's `amsppt' style into
% an infinite loop, exceeding its input stack size.  In practice, such
% incompatibilities surface infrequently; but it is now clear to us that
% it's not worth the effort (perhaps it's not even possible) to make the
% btxmac.tex macros robust when used both with plain TeX and Ams-TeX.
% If the BibTeX/AmS-TeX results attainable with the current btxmac.tex
% macros are sufficient, fine.  But if there's a demand for more robust
% BibTeX/AmS-TeX behavior, then someone who's very familiar with the
% AmS-TeX package should probably make an amsbtxmc.tex version of the
% macros (remembering to follow the copyright restrictions above).
% Until then, if you're an AmS-TeX user, or a LaTeX or plain TeX user
% sharing files with an AmS-TeX user, beware.
% END OF AMS-TEX WARNING.
%
% To use these macros you should be familiar with how BibTeX interacts
% with LaTeX, since BibTeX's interaction with TeX is very similar; that
% interaction is explained in the LaTeX manual.  It also helps to
% have read "BibTeXing", the documentation that accompanies BibTeX.
% Then, if you want, you should redefine any of the macros that begin
% with `\bbl' or `\biblabel' or `\print' that you need to get formatting
% different from the default (the default settings are designed to
% accompany a bibliography style like BibTeX's standard style `plain').
% The macros you might want to change are described briefly a few
% paragraphs hence.  [To get started without reading any documentation,
% try running the nine-line .tex file below through TeX and BibTeX.
% Remember the general scheme: Running (La)TeX writes information on
% the .aux (auxiliary) file; then running BibTeX reads information from
% the .aux, .bst (style), and .bib (database) files, and writes
% information (the bibliography) on a .bbl file; then running (La)TeX
% incorporates the bibliography; then running (La)TeX once more fixes
% the remaining forward references into the bibliography.  Thus, to get
% everything incorporated into your output, you'll have to run (La)TeX,
% BibTeX, (La)TeX, (La)TeX.  (Standup, sitdown, fight, fight, fight.)]
%
% These macros can stand alone or they can be \input into a macro
% package, like Eplain, that is sufficiently compatible with plain TeX.
% To use these macros to format the 0.99 version of the xampl.bib file
% that's distributed with BibTeX (that version of the file has no
% self-identification), you'll need to define \mbox, which is a LaTeX
% command, to be \hbox, as in the example below.
%
% Here's a nine-line plain TeX file for trying out btxmac.tex; of course
% you'll have to remove the comment characters at the beginning of each
% line, and, depending on your system, you might have to take steps so
% that BibTeX can "see" the files xampl.bib and plain.bst (BibTeX will
% give you two empty-field warning messages that you should ignore).
%
%     \def\mbox#1{\leavevmode\hbox{#1}}
%     \input btxmac
%     \noindent This cites Aamport's gnominious article~\cite{article-full}.
%     \medskip
%     \leftline{\bf References}
%     \nocite{*}   % put all database entries into the reference list
%     \bibliography{xampl}   % specify the database files; here, just xampl.bib
%     \bibliographystyle{plain}   % specify plain.bst as the style file
%     \bye
%
%
%   HISTORY
%
% Karl Berry wrote the original version of these macros in 1989 and
% 1990, for use in his `Eplain' package.  Oren Patashnik modified them
% slightly in July 1990, as part of the official BibTeX distribution.
%
%    1-Aug-90  Version 0.99a, not released to the general public.
%   14-Aug-90  0.99b, first general release.
%   26-Aug-90  0.99c, made \@undefinedmessage work with other macro packages.
%    6-Sep-90  0.99d, allowed for general formatting of bibliography labels,
%                     for general formatting of (in-text) citations, and for
%                     changing certain catcodes while reading the .aux file.
%   14-Nov-90  0.99e, changed the way \@setletters works, made some \new...'s
%                     non-outer, and changed the way Eplain reads this file.
%   12-Dec-90  0.99f, made \@resetnumerals change the `,' and `.' catcodes; and
%                     added \biblabelextrahang, \@getoptionalarg, and \bblsc.
%   11-Mar-91  0.99g, made a few minor changes required by the way Eplain reads
%                     this file, but no functional changes.
%   24-Apr-91  0.99h, inhibited the reading and writing of the .aux file if it
%                     isn't used or if the \noauxfile macro is defined, and
%                     removed some .aux-file-opening detritus; printed the
%                     cite-key of undefined citations in \tt font; changed the
%                     catcode of `_' inside \cite; and called \@resetnumerals
%                     from inside a group.
%   29-Feb-92  0.99i, made these macros semi-compatible with AmS-TeX; removed
%                     \@resetnumerals, \@setletters, \@tokstostring, and
%                     friends; changed the way \cite handles catcodes; changed
%                     \@getoptionalarg, and had \bibitem and \newcommand use
%                     it; added \@futurenonspacelet and (to facilitate the use
%                     of multiple reference lists) \bblfilebasename; changed
%                     \biblabelprint to use the new macros \biblabelprecontents
%                     and \biblabelpostcontents, and to, by default, right-
%                     justify numeric labels; and renamed \biblabelextrahang to
%                     the more descriptive \biblabelextraspace.
%   14-Mar-92  0.99j, made 0.99i's use of `\\' local to btxmac.tex.
%
%
% The LaTeX-related commands defined in this file include (a) the four
% commands that a user types (\bibliography, \bibliographystyle, \cite,
% and \nocite); (b) the three commands that BibTeX looks for in the .aux
% file (\bibdata, \bibstyle, and \citation---there is a fourth command
% that BibTeX looks for, but that command is related to LaTeX's \include
% facility, so these macros ignore that command); and (c) a LaTeX
% command (\newcommand) that's written by one of the four standard
% bibliography styles (alpha).  The definitions here are not exactly the
% same as the corresponding LaTeX definitions (those eight LaTeX
% definitions depend on a significant fraction of LaTeX itself).  But
% the only substantial differences are with \newcommand, which here,
% without complaining, lets you redefine a preexisting control sequence
% (in LaTeX, \newcommand won't let you redefine a preexisting command),
% and which here doesn't make the control sequences it defines \long (in
% LaTeX, that happens automatically); there may also be other minor
% differences.  To summarize: Unless you know what you're doing, you
% shouldn't define any control sequences with these eight names:
%
% \bibdata
% \bibliography
% \bibliographystyle
% \bibstyle
% \citation
% \cite
% \newcommand
% \nocite
%
% There are three other commands written by one or more of the four
% standard (plain, abbrv, alpha, unsrt) or four semistandard (acm,
% apalike, ieeetr, siam) bibliography styles; they take effect only
% within the bibliography, and are redefinable, as explained later:
%
% \em
% \newblock
% \sc
%
% There's one control sequence you might want to use (but not redefine)
% in redefining \biblabelprint:
%
% \biblabelwidth
%
% There are fifteen other control sequences (explained later in more detail)
% that the macros of this file will use if you define them---you should
% define them after the \input btxmac command but before the \bibliography
% command.  The first six begin with `\bbl' and affect fonts, spacing,
% perhaps other characteristics of the bibliography, and which .bbl files
% get read; the next five begin with `\biblabel' and determine how labels
% are formatted in the bibliography; and the last four begin with `\print'
% and determine how the in-text citations are formatted:
%
% \bblem
% \bblfilebasename
% \bblhook
% \bblnewblock
% \bblrm
% \bblsc
% \biblabelcontents
% \biblabelprecontents
% \biblabelprint
% \biblabelpostcontents
% \biblabelextraspace
% \printbetweencitations
% \printcitefinish
% \printcitenote
% \printcitestart
%
% If it's defined before the \input btxmac command, the control sequence
% below inhibits the reading and writing of the .aux file(s), and the
% issuing of related warning messages.  Any definition will do.  This
% feature might help when you're working on draft stages of a document:
%
% \noauxfile
%
%
% Here's another control sequence (it's described later) that you
% probably won't want to redefine unless you are writing another macro
% package; if you do redefine it, however, do it before the \input btxmac
% command (and notice that it has an `@' in its name):
%
% \@undefinedmessage
%
% Any other control sequence in this file that might conflict with
% something you've defined will have an `@' in its name, so such conflicts
% are unlikely; but if you're worried about a specific control sequence
% name, do a text search of this file to look for it.
%
%
% So to start things off we turn `@' into a letter (category code 11),
% keeping track of the old category code for future restoration.
% (Simply resetting it to 12 when we leave these macros is
% insufficient.)  The use of `\cite' as a temporary control sequence is
% a kludge, but it's a reasonably simple way to accomplish what we need
% without possibly overwriting something (without an `@' in its name)
% that might already be defined.
%
\edef\cite{\the\catcode`@}%
\catcode`@ = 11
\let\@oldatcatcode = \cite
\chardef\@letter = 11
\chardef\@other = 12
%
%
% Next come some things that will be useful later.
%
% Make an outer definition into an inner one (due to Chris Thompson).
% The arguments should be the control sequence to be defined, and the
% new of the \outer control sequence, as characters; the control
% sequence #1 is defined to be just the same as \csname#2\endcsname, but
% not \outer.  For example, \@innerdef\innernewcount{newcount} would
% define \innernewcount to be a non-outer version of \newcount.
%
\def\@innerdef#1#2{\edef#1{\expandafter\noexpand\csname #2\endcsname}}%
%
% We use \@innerdef to make some of our allocations local, because
% Eplain includes our code inside a conditional.  We put @'s in the
% names to minimize the (already small) chance of conflicts.
%
\@innerdef\@innernewcount{newcount}%
\@innerdef\@innernewdimen{newdimen}%
\@innerdef\@innernewif{newif}%
\@innerdef\@innernewwrite{newwrite}%
%
%
% Swallow one parameter.
%
\def\@gobble#1{}%
%
%
% Use TeX 3.0's \inputlineno to get the line number, for better error
% messages, but if we're using an old version of TeX, don't do anything.
%
\ifx\inputlineno\@undefined
   \let\@linenumber = \empty % Pre-3.0.
\else
   \def\@linenumber{\the\inputlineno:\space}%
\fi
%
%
% The following macro \@futurenonspacelet (from the TeXbook) behaves
% essentially like \futurelet except that it discards any implicit or
% explicit space tokens that intervene before a nonspace is scanned:
%
\def\@futurenonspacelet#1{\def\cs{#1}%
   \afterassignment\@stepone\let\@nexttoken=
}%
\begingroup % The grouping here avoids stepping on an outside use of `\\'.
\def\\{\global\let\@stoken= }%
\\ % now \@stoken is a space token (\\ is a control symbol, so that
   % space after it is seen).
\endgroup
\def\@stepone{\expandafter\futurelet\cs\@steptwo}%
\def\@steptwo{\expandafter\ifx\cs\@stoken\let\@@next=\@stepthree
   \else\let\@@next=\@nexttoken\fi \@@next}%
\def\@stepthree{\afterassignment\@stepone\let\@@next= }%
%
%
% \@getoptionalarg\CS gets an optional argument from the input, enclosed
% in brackets, then expands \CS.  We set \@optionalarg to \empty if we
% don't find one, otherwise to the text of the argument.  This assumes
% the brackets don't have a funny category code.
%
\def\@getoptionalarg#1{%
   \let\@optionaltemp = #1%
   \let\@optionalnext = \relax
   \@futurenonspacelet\@optionalnext\@bracketcheck
}%
%
% The \expandafter's in this macro let us avoid the use of \aftergroup,
% which is somewhat more expensive.
%
\def\@bracketcheck{%
   \ifx [\@optionalnext
      \expandafter\@@getoptionalarg
   \else
      \let\@optionalarg = \empty
      % We can't do the \temp after the \fi, because then the \temp gets
      % in the way of reading the optional argument from the input, if
      % we do have one.
      \expandafter\@optionaltemp
   \fi
}%
%
\def\@@getoptionalarg[#1]{%
   \def\@optionalarg{#1}%
   \@optionaltemp
}%
%
%
% From LaTeX.
%
\def\@nnil{\@nil}%
\def\@fornoop#1\@@#2#3{}%
%
\def\@for#1:=#2\do#3{%
   \edef\@fortmp{#2}%
   \ifx\@fortmp\empty \else
      \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}%
   \fi
}%
%
\def\@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@iforloop #3\@@#4{#5}\fi\fi
}%
%
\def\@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@fornoop \else
      #4\relax\let\@nextwhile=\@iforloop\fi\@nextwhile#2\@@#3{#4}%
}%
%
%
% This macro tests if a file \jobname.#1 exists, and sets \if@fileexists
% appropriately.  If an optional argument is given, it is used as the
% root part of the filename instead of \jobname.
%
\@innernewif\if@fileexists
%
\def\@testfileexistence{\@getoptionalarg\@finishtestfileexistence}%
\def\@finishtestfileexistence#1{%
   \begingroup
      \def\extension{#1}%
      \immediate\openin0 =
         \ifx\@optionalarg\empty\jobname\else\@optionalarg\fi
         \ifx\extension\empty \else .#1\fi
         \space
      \ifeof 0
         \global\@fileexistsfalse
      \else
         \global\@fileexiststrue
      \fi
      \immediate\closein0
   \endgroup
}%
%
%
%% [[[start of BibTeX-specific stuff]]]
%
% Now come the four main LaTeX commands and their associated .aux
% commands.  Just as in LaTeX, \bibliographystyle defines the BibTeX
% style name (.bst file, that is), and \bibliography defines the
% database (.bib) file(s).  The corresponding .aux-file commands are
% \bibstyle and \bibdata, which are there only for BibTeX's (but not
% LaTeX's) use.
%
\def\bibliographystyle#1{%
   \@readauxfile
   \@writeaux{\string\bibstyle{#1}}%
}%
\let\bibstyle = \@gobble
%
% As well as writing the \bibdata command to tell BibTeX which .bib
% files to read, we read the .bbl file that BibTeX (or a person,
% conceivably) has produced.  We use \bblfilebasename as the root of the
% filename to read; this defaults to \jobname.
%
\let\bblfilebasename = \jobname
\def\bibliography#1{%
   \@readauxfile
   \@writeaux{\string\bibdata{#1}}%
   \@testfileexistence[\bblfilebasename]{bbl}%
   \if@fileexists
      % We just output a non-discardable item (the `whatsit' with the
      % \bibdata command).  This means that the glue that will be
      % inserted next (\parskip or \baselineskip, most likely) will be a
      % legal breakpoint.  Most likely, this is after some kind of
      % heading, where we don't want to allow a page break.  So:
      \nobreak
      \@readbblfile
   \fi
}%
\let\bibdata = \@gobble
%
% The \nocite{label,label,...} command writes its argument to \@auxfile,
% unless instructed not to, but produces no text in the document.  Both
% the \nocite and \cite commands produce \citation commands in the .aux file.
%
\def\nocite#1{%
   \@readauxfile
   \@writeaux{\string\citation{#1}}%
}%
%
\@innernewif\if@notfirstcitation
%
% \cite[note]{label,label,...} produces the citations for the labels as
% well.  If the optional argument `note' is present, it's added after
% the labels.  Since \cite calls \nocite to do its .aux-file writing,
% \cite doesn't need to call \@readauxfile (\nocite does).
%
\def\cite{\@getoptionalarg\@cite}%
%
% Typeset the citations for the labels in #1, followed by the note, if
% it exists.  To change the citation's format in the text, redefine one
% or more `\print...' macros, whose defaults appear later in this file.
%
\def\@cite#1{%
   % Remember the optional argument, in case one of the macros we call
   % below ends up looking for an optional argument itself.  For
   % example, if a \cite[note] triggers reading the .aux file, then the
   % [note] would be clobbered, since \@testfileexistence looks for an
   % optional arg.
   \let\@citenotetext = \@optionalarg
   % Start printing the text, beginning with a left bracket by default.
   \printcitestart
   % It's complicated, but because \nocite puts a `whatsit' onto the list,
   % \nocite should follow \printcitestart.  It's conceivable, but very
   % unlikely, that this `whatsit' will cause a problem (glue that doesn't
   % disappear when you want it to is the most likely symptom), requiring
   % a change either to \printcitestart or to the label that the .bst file
   % produces.
   \nocite{#1}%
   \@notfirstcitationfalse
   \@for \@citation :=#1\do
   {%
      \expandafter\@onecitation\@citation\@@
   }%
   \ifx\empty\@citenotetext\else
      \printcitenote{\@citenotetext}%
   \fi
   \printcitefinish
}%
%
\def\@onecitation#1\@@{%
   \if@notfirstcitation
      \printbetweencitations
   \fi
   %
   \expandafter \ifx \csname\@citelabel{#1}\endcsname \relax
      \if@citewarning
         \message{\@linenumber Undefined citation `#1'.}%
      \fi
      % Give it a dummy definition:
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
         {\tt
            \escapechar = -1
            \nobreak\hskip0pt
            \expandafter\string\csname#1\endcsname
            \nobreak\hskip0pt
         }%
      }%
   \fi
   % Now produce the text, whether it was undefined or not.
   \@printcitelabel{#1}%
   \@notfirstcitationtrue
}%
%
% This is in case some macro package wants to redefine \@citelabel to be
% something that is not fully expandable.  Suggested Victor Eijkhout for
% Lollipop.
\def\@printcitelabel#1{%
   \csname\@citelabel{#1}\endcsname
}%
%
% Given a label `foo', the macro `\b@foo' is supposed to
% hold the text that should be produced.
%
\def\@citelabel#1{b@#1}%
%
% So, how does a citation label get defined?  When we read the .bbl file
% (below), a \bibitem writes out a \@citedef command.  And when we read
% the \@citedef, we define \@citelabel{#1}, where #1 is the user's
% label.
%
\def\@citedef#1#2{\expandafter\gdef\csname\@citelabel{#1}\endcsname{#2}}%
%
%
% Reading the .bbl file also produces the typeset bibliography.  Please
% notice, however, that we do not produce the title for the references
% (e.g., `References'), as LaTeX does.  The formatting and spacing of
% that title, whether it should go into the headline, and so on, are all
% things determined by your format.  We cannot know those things in
% advance.  If you wish, you can define \bblhook to produce the title.
% Or just do it before the \bibliography command.
%
\def\@readbblfile{%
   % Define a counter to tell us which item number we are on, unless
   % we've already defined it (because the document has more than one
   % bibliography).
   \ifx\@itemnum\@undefined
      \@innernewcount\@itemnum
   \fi
   %
   \begingroup
      % If another package has already defined \begin, don't define our
      % own simplistic \begin and \end; assume they want to take care of
      % it themselves.  (That way, their \begin's and \end's for other
      % things can be used in the bib files.)
      \ifx\begin\undefined
         \def\begin##1##2{%
            % ##1 is just `thebibliography'.
            % ##2 is the widest label.
            % We set (new dimen) \biblabelwidth based on the widest label
            \setbox0 = \hbox{\biblabelcontents{##2}}%
            \biblabelwidth = \wd0
         }%
         \let\end = \@gobble % The arg is `thebibliography' again.
      \fi
      %
      % Here we have two possibilities:
      % \bibitem[typesetlabel]{citationlabel}
      % \bibitem{citationlabel}
      % If we have the second of these, the citations are numbered, starting
      % from one; we use our own count register \@itemnum for this.
      %
      \@itemnum = 0
      \def\bibitem{\@getoptionalarg\@bibitem}%
      \def\@bibitem{%
         \ifx\@optionalarg\empty
            \expandafter\@numberedbibitem
         \else
            \expandafter\@alphabibitem
         \fi
      }%
      \def\@alphabibitem##1{%
         % Need \xdef here for various reasons.
         \expandafter \xdef\csname\@citelabel{##1}\endcsname {\@optionalarg}%
         % Left-justify alpha labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \relax
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \hss
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@numberedbibitem##1{%
         \advance\@itemnum by 1
         \expandafter \xdef\csname\@citelabel{##1}\endcsname{\number\@itemnum}%
         % Right-justify numeric labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \hss
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \relax
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@finishbibitem##1{%
         \biblabelprint{\csname\@citelabel{##1}\endcsname}%
         \@writeaux{\string\@citedef{##1}{\csname\@citelabel{##1}\endcsname}}%
         \ignorespaces
      }%
      %
      % Do the printing (we're producing the bibliography, remember).
      %
      \let\em = \bblem
      \let\newblock = \bblnewblock
      \let\sc = \bblsc
      % Punctuation won't affect spacing;
      \frenchspacing
      % the penalties below are from LaTeX's [article,book,report].sty;
      \clubpenalty = 4000 \widowpenalty = 4000
      % the next two values come from LaTeX's \sloppy command;
      \tolerance = 10000 \hfuzz = .5pt
      \everypar = {\hangindent = \biblabelwidth
                      \advance\hangindent by \biblabelextraspace}%
      \bblrm
      % the \parskip is a guess at what looks good;
      \parskip = 1.5ex plus .5ex minus .5ex
      % and the space between label and text comes from LaTeX's \labelsep.
      \biblabelextraspace = .5em
      \bblhook
      %
      \input \bblfilebasename.bbl
   \endgroup
}%
%
% The widest label's width is useful for redefining \biblabelprint;
% you redefine \biblabelwidth, in effect, by redefining the
% \biblabelcontents macro that appears below.  And \biblabelextraspace,
% which is redefinable inside \bblhook, is added to \biblabelwidth to
% determine the amount of hanging indentation.
%
\@innernewdimen\biblabelwidth
\@innernewdimen\biblabelextraspace
%
% Now come the main macros that are related to the printing of the
% bibliography.  Since you might want to redefine them, they are given
% default definitions outside of \@readbblfile.
%
% The first one controls the printing of a bibliography entry's label.
% If you change it, make sure that it starts with something like
% \noindent or \indent or \leavevmode that puts TeX into horizontal mode
% (even if the label itself is empty); otherwise, the hanging
% indentation will get messed up in certain circumstances.
%
\def\biblabelprint#1{%
   \noindent
   \hbox to \biblabelwidth{%
      \biblabelprecontents
      \biblabelcontents{#1}%
      \biblabelpostcontents
   }%
   \kern\biblabelextraspace
}%
%
% If you are using numeric labels, and you want them left-justified
% (numeric labels by default are right-justified), do something like:
%     \def\biblabelprecontents{\relax}
%     \def\biblabelpostcontents{\hss}
%
% By default the labels are typeset in \bblrm, and enclosed in brackets.
%
\def\biblabelcontents#1{{\bblrm [#1]}}%
%
% The main text, too, is typeset using \bblrm, which is \rm by default.
%
\def\bblrm{\rm}%
%
% Emphasis for producing, e.g., titles, is done with \it by default.
%
\def\bblem{\it}%
%
% Some styles use a caps-and-small-caps font for author names.  LaTeX
% defines an \sc command but plain TeX doesn't, so we need one here.
% The definition below doesn't load the font unless it's needed, but it
% tries to load only the 10pt version, because it might not exist at
% other point sizes.
%
\def\bblsc{\ifx\@scfont\@undefined
              \font\@scfont = cmcsc10
           \fi
           \@scfont
}%
%
% The major parts of an entry are separated with \bblnewblock.  The
% numbers below are taken from LaTeX's `article' style.
%
\def\bblnewblock{\hskip .11em plus .33em minus .07em }%
%
% Here's where you stick any other bibliography-formatting goodies, or
% redefine the values above.
%
\let\bblhook = \empty
%
%
% Here are the four default definitions for formatting the in-text
% citations.  These are what you redefine (after your \input btxmac but
% before your \bibliography) to get parens instead of brackets, or
% superscripts, or footnotes, or whatever.
%
\def\printcitestart{[}%         left bracket
\def\printcitefinish{]}%        right bracket
\def\printbetweencitations{, }% comma, space
\def\printcitenote#1{, #1}%     comma, space, note (if it exists)
%
% That scheme is pretty flexible.  For example you could use
%     \def\printcitestart{\unskip $^\bgroup}
%     \def\printcitefinish{\egroup$}
%     \def\printbetweencitations{,}
%     \def\printcitenote#1{\hbox{\sevenrm\space (#1)}}
%     \font\eighttt = cmtt8
%     \scriptfont\ttfam = \eighttt
% to get superscripted in-text citations.  (The scriptfont stuff
% exists only to print an undefined citation; it's in cmtt8 because
% there is no cmtt7.)  To get something radically different, however,
% you'll have to define your own \cite command.
%
% When we read `\citation' from the .aux file, it means nothing.
%
\let\citation = \@gobble
%
%
% Now comes the stuff for dealing with LaTeX's \newcommand.  As
% mentioned earlier, this \newcommand will redefine a preexisting
% command; that's different from how LaTeX's \newcommand behaves.
%
\@innernewcount\@numparams
%
% \newcommand{\foo}[n]{text} defines the control sequence \foo to have
% n parameters, and replacement text `text'.
%
\def\newcommand#1{%
   \def\@commandname{#1}%
   \@getoptionalarg\@continuenewcommand
}%
%
% Figure out if this definition has parameters.
%
\def\@continuenewcommand{%
   % If no optional argument, we have zero parameters.  Otherwise, we
   % have that many.
   \@numparams = \ifx\@optionalarg\empty 0\else\@optionalarg \fi \relax
   \@newcommand
}%
%
% \@numparams is how many arguments this command has.  The name of the
% command is \@commandname.  The replacement text for the new macro is #1.
%
\def\@newcommand#1{%
   \def\@startdef{\expandafter\edef\@commandname}%
   \ifnum\@numparams=0
      \let\@paramdef = \empty
   \else
      \ifnum\@numparams>9
         \errmessage{\the\@numparams\space is too many parameters}%
      \else
         \ifnum\@numparams<0
            \errmessage{\the\@numparams\space is too few parameters}%
         \else
            \edef\@paramdef{%
               % This is disgusting, but \loop doesn't work inside \edef,
               % because \body isn't defined.
               \ifcase\@numparams
                  \empty  No arguments.
               \or ####1%
               \or ####1####2%
               \or ####1####2####3%
               \or ####1####2####3####4%
               \or ####1####2####3####4####5%
               \or ####1####2####3####4####5####6%
               \or ####1####2####3####4####5####6####7%
               \or ####1####2####3####4####5####6####7####8%
               \or ####1####2####3####4####5####6####7####8####9%
               \fi
            }%
         \fi
      \fi
   \fi
   \expandafter\@startdef\@paramdef{#1}%
}%
%
%% [[[end of BibTeX-specific stuff]]]
%
%
% Names of references (arguments given in the \cite and \nocite
% commands) and file names (arguments given in the \bibliography and
% \bibliographystyle commands) are recorded in \jobname.aux, called the
% \@auxfile in these macros.  Here's how they get read in.
%
\def\@readauxfile{%
   \if@auxfiledone \else % remember: \@auxfiledonetrue if \noauxfile is defined
      \global\@auxfiledonetrue
      \@testfileexistence{aux}%
      \if@fileexists
         \begingroup
            % Because we might be in horizontal mode when \@readauxfile
            % is called (if it's in response to a \cite or \nocite), we
            % want to ignore all the would-be spaces at the ends of
            % lines in the aux file.  Fortunately, it's highly unlikely
            % an end-of-line might actually be desired.
            % And because we don't change the category code of anything
            % but @, primitives like \gdef can't be used to define labels
            % in the aux file.  The solution adopted by btxmac.tex is to
            % write `\@citedef{LABEL}{DEFINITION}' to the aux file, and
            % use \csname on LABEL.
            \endlinechar = -1
            \catcode`@ = 11
            \input \jobname.aux
         \endgroup
      \else
         \message{\@undefinedmessage}%
         \global\@citewarningfalse
      \fi
      \immediate\openout\@auxfile = \jobname.aux
   \fi
}%
%
% The \@readauxfile macro does all that work the first time it's called.
% Since it's called once for every \cite, \nocite, \bibliography, and
% \bibliographystyle command that the user issues, we need to remember
% whether the work's been done.  It's considered done if we're not to do
% it---that is, if \noauxfile is defined.
%
\newif\if@auxfiledone
\ifx\noauxfile\@undefined \else \@auxfiledonetrue\fi
%
% It's conceivable you'd want to change how other characters are read;
% to do that, change their category code before doing \input btxmac.
%
%
% After reading the .aux file, \@readauxfile opens it for writing.
% The \@writeaux macro does the actual writing (as long as
% \noauxfile is undefined).
%
\@innernewwrite\@auxfile
\def\@writeaux#1{\ifx\noauxfile\@undefined \write\@auxfile{#1}\fi}%
%
%
% A macro package that uses btxmac.tex might define
% \@undefinedmessage (before doing an \input btxmac).
%
\ifx\@undefinedmessage\@undefined
   \def\@undefinedmessage{No .aux file; I won't give you warnings about
                          undefined citations.}%
\fi
%
% Even if citations are undefined, we want to complain only if
% \@citewarningtrue.  The default is to set \@citewarningtrue unless
% \noauxfile is defined.  Again, a macro package that uses
% btxmac.tex might want to redefine this.
%
\@innernewif\if@citewarning
\ifx\noauxfile\@undefined \@citewarningtrue\fi
%
%
% Finally, before leaving we restore @'s old category code.
%
\catcode`@ = \@oldatcatcode
%\input psfig.sty
\def\href#1#2{#2}  %For use with utphys.bst
%\input labeldefs.tmp
\def\Introduction{1}
\def\fields{(1)}
\def\ws{(2)}
\def\gaugeop{(3)}
\def\stringop{(4)}
\def\dilvert{(5)}
\def\gravvert{(6)}
\def\confan{(7)}
\def\DThreeGeom{(8)}
\def\DThreeVars{(9)}
\def\NearGeom{(10)}
\def\zphi{(11)}
\def\Ident{(12)}
\def\BndCond{(13)}
\def\SymBC{1}
\def\AnyADS{(14)}
\def\AllowedSize{(15)}
\def\AllowedZeta{(16)}
\def\LieVar{(17)}
\def\CKE{(18)}
\def\TwoPt{2}
\def\MinAction{(19)}
\def\MinEOM{(20)}
\def\CompleteSet{(21)}
\def\WVCoup{(22)}
\def\SimpIdent{(23)}
\def\KBound{(24)}
\def\FluxDef{(25)}
\def\TwoPtO{(26)}
\def\EvalFlux{(27)}
\def\poscorr{(28)}
\def\TwoPtFct{(29)}
\def\XFourDef{(30)}
\def\SInt{(31)}
\def\TwoTT{(32)}
\def\MassiveString{3}
\def\PartAction{(33)}
\def\PartEOM{(34)}
\def\PartSols{(35)}
\def\PhiBehave{(36)}
\def\GotFlux{(37)}
\def\higher{(38)}
\def\Coulone{(39)}
\def\Coultwo{(40)}
\def\anom{(41)}
\def\massivewave{(42)}
\def\genexpand{(43)}
\def\OOFct{(44)}
\def\opdim{(45)}
\def\anomform{(46)}

\writedefs
\sequentialequations
%\input macros

%--------+---------+---------+---------+---------+---------+---------+
%My usual macros

%A macro to call as \rf{EqnLabel} to get \EqnLabel
\def\rf#1{\csname #1\endcsname{}}

%Macros to facilitate use of halign for complicated equations:
\def\TL{\hfil$\displaystyle{##}$}
\def\TR{$\displaystyle{{}##}$\hfil}
\def\TC{\hfil$\displaystyle{##}$\hfil}
\def\TT{\hbox{##}}
%Example: the \noalign command gives an extra bit of vertical space
%  \eqn\One{\vcenter{\openup1\jot
%    \halign{\strut\span\TL & \span\TR & \span\TT & \span\TL & \span\TR\cr
%     x^2 &> 1 & \quad when $x$ satisfies\ \ & x &> 1 \cr\noalign{\vskip1\jot}
%     y^2 &< 1 & \quad when $y$ satisfies\ \ & y &< 1 \cr
%   }}}

%Simple macro for making lists when you're not in Latex:
\def\litem#1#2{{\leftskip=20pt\noindent\hskip-10pt {\bf #1} #2\par}}
%Example: you can put a blank line between items.  Might want some vskip too.
%\litem{1)}{First item in list.}
%\litem{2)}{Second item in list.}

%Blank macros:
\def\comment#1{}
\def\fixit#1{}

%For controlling the size of fractions:
\def\tf#1#2{{\textstyle{#1 \over #2}}}
\def\df#1#2{{\displaystyle{#1 \over #2}}}

%More math operators: (add as needed)
\def\coth{\mathop{\rm coth}\nolimits}
\def\csch{\mathop{\rm csch}\nolimits}
\def\sech{\mathop{\rm sech}\nolimits}
\def\Vol{\mathop{\rm Vol}\nolimits}
\def\vol{\mathop{\rm vol}\nolimits}
\def\diag{\mathop{\rm diag}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
\def\Disc{\mathop{\rm Disc}\nolimits}
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\ad{\mathop{\rm ad}\nolimits}

%For adding more math operators:
\def\mop#1{\mathop{\rm #1}\nolimits}

%Group symbols:
\def\SU{{\rm SU}}
\def\USp{{\rm USp}}            

%Approximately less than operators:
\def\lsim{\mathrel{\mathstrut\smash{\ooalign{\raise2.5pt\hbox{$<$}\cr\lower2.5pt\hbox{$\sim$}}}}}
\def\gsim{\mathrel{\mathstrut\smash{\ooalign{\raise2.5pt\hbox{$>$}\cr\lower2.5pt\hbox{$\sim$}}}}}
%Used to use this:
%\def\lsim{\mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}}}
%\def\gsim{\mathrel{\raise2pt\hbox{$\mathop>\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}}}

%Nicest general slashing macro I can come up with:
\def\slashed#1{\ooalign{\hfil\hfil/\hfil\cr $#1$}}
%Used to use this: \def\slashed#1{\hskip2pt/\hskip-5.9pt#1} 

%To produce a box for a Dalembertian (adapted from p. 320 of TeXbook):
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
         \hbox{\vrule width.#2pt height#1pt \kern#1pt
            \vrule width.#2pt}
         \hrule height.#2pt}}}}
\def\square{\mathop{\mathchoice\sqr56\sqr56\sqr{3.75}4\sqr34}\nolimits}

%Young Tableaux macros:
\def\idget{$\sqr55$\hskip-0.5pt}
\def\endrow{\hskip0.5pt\cr\noalign{\vskip-1.5pt}}
\def\endyoung{\hskip0.5pt\cr}
%Example: in a paragraph or in mathmode, say
%\oalign{\idget\idget\idget\idget\endrow
%        \idget\idget\idget\endyoung}
%See young.tex for more examples.

%List harvmac references as a continuation of the present page:
\def\shortlistrefs{\footatend\bigskip\bigskip\bigskip%
\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}\nonfrenchspacing}

%Now, some miscellany:
%Added for partial waves paper:
\def\eff{{\rm eff}}
\def\abs{{\rm abs}}
\def\hc{{\rm h.c.}}
\def\+{^\dagger}
%Added for dyadic index notes:
\def\omicron{o}
\def\O{{\cal O}}
%Added for exact coefficients notes:
\def\cl{{\rm cl}}
%Added for 5-d black holes letter
\def\str{{\rm str}}
%Added for partial waves into 3-brane notes:
\def\M{\cal M}
\def\D#1#2{{\partial #1 \over \partial #2}}
%Added for absorption by extremal 3-branes paper.  Adapted from the 
%TeXbook, p. 359.
\def\overleftrightarrow#1{\vbox{\ialign{##\crcr
     $\leftrightarrow$\crcr\noalign{\kern-0pt\nointerlineskip}
     $\hfil\displaystyle{#1}\hfil$\crcr}}}
%Added for 5d fermion notes and papers:
\def\eps{\epsilon}

\def\em{\it}   %utphys.bst uses \em for book titles




%--------+---------+---------+---------+---------+---------+---------+
%Title page

\Title{
 \vbox{\baselineskip10pt
  \hbox{PUPT-1767}
  \hbox{hep-th/9802109}
 }
}
{
 \vbox{
  \centerline{Gauge Theory Correlators from Non-Critical String Theory}
 }
}
\vskip -25 true pt

\centerline{
 S.S.~Gubser,\footnote{$^1$}{e-mail:  ssgubser@viper.princeton.edu}
 I.R.~Klebanov\footnote{$^2$}{e-mail:  klebanov@viper.princeton.edu}
and
 A.M.~Polyakov\footnote{$^3$}{e-mail:  polyakov@puhep1.princeton.edu}
}

\centerline{\it Joseph Henry Laboratories, 
Princeton University, Princeton, NJ  08544}

\bigskip\bigskip

\centerline {\bf Abstract}
\smallskip
\baselineskip12pt
\noindent

We suggest a means of obtaining certain Green's functions in
$3+1$-dimensional ${\cal N} = 4$ supersymmetric 
Yang-Mills theory with a large number of colors via non-critical
string theory. The non-critical string theory
is related to critical string theory in anti-deSitter background.
We introduce a boundary of the
anti-deSitter space analogous to a cut-off on
the Liouville coordinate of the two-dimensional
string theory. 
Correlation functions of operators in the gauge
theory are related to the dependence of the
supergravity action on the boundary conditions.
From the quadratic terms in supergravity 
we read off the anomalous dimensions.
For operators that couple to massless string states it has been
established through absorption calculations that the anomalous
dimensions vanish, and we rederive this result. 
The operators that couple to massive string states at level $n$
acquire anomalous dimensions that grow as
$2\left (n g_{YM} \sqrt {2 N} \right )^{1/2}$  for large `t Hooft coupling.
This is a new prediction about the strong coupling behavior
of large $N$ SYM theory.


\Date{February 1998}

\noblackbox
\baselineskip 14pt plus 1pt minus 1pt




%--------+---------+---------+---------+---------+---------+---------+
%Body

\newsec{Introduction}
\seclab\Introduction

Relations between gauge fields and strings present an old, fascinating
and unanswered question. The full answer to this question is of great
importance for theoretical physics. It will provide us with a
theory of quark confinement  by explaining the dynamics of
color-electric fluxes. On the other hand, it will perhaps uncover
the true ``gauge'' degrees of freedom of the fundamental string theories,
and therefore of gravity.

The Wilson loops of gauge theories satisfy the loop equations
which translate the Schwinger-Dyson equations into variational
equations on the loop space \cite{AP,mm}. 
These equations should have a solution
in the form of the sum over random surfaces bounded by the loop.
These are the world surfaces of the color-electric fluxes.
For the $SU(N)$ Yang-Mills theory they are expected to carry the 
`t Hooft factor \cite{GT}, $N^\chi$, where $\chi$ is the Euler
character. Hence, in the large $N$ limit 
where $g_{YM}^2 N$ is kept fixed only the simplest
topologies are relevant.

Until recently, the action for the ``confining string''
had not been known. In \cite{Sasha} is was suggested that it
must have a rather unusual structure. Let us describe it briefly.
First of all, the world surface of the electric flux propagates
in at least 5 dimensions. This is because the non-critical strings
are described by the fields
\eqn\fields{X^\mu (\sigma)\ ,\qquad g_{ij} (\sigma) =
e^{\varphi (\sigma)} \delta_{ij}\ ,
}
where $X^\mu$ belong to 4-dimensional (Euclidean) space and
$g_{ij} (\sigma)$ is the world sheet metric in the conformal gauge.
The general form of the world sheet lagrangian compatible with
the 4-dimensional symmetries is
\eqn\ws{
{\cal L}= {1\over 2} (\partial_i \varphi)^2 + a^2 (\varphi)
(\partial_i X^\mu)^2 + \Phi (\varphi) ^{(2)}R + {\rm Ramond-Ramond
\ backgrounds}\ ,
}
where $^{(2)}R$ is the world sheet curvature, $\Phi(\varphi)$
is the dilaton \cite{ft}, while the field 
$$ \Sigma (\varphi) = a^2 (\varphi)
$$
defines a variable string tension. In order to reproduce the zig-zag
symmetry of the Wilson loop, the gauge fields must be
located at a certain value $\varphi=\varphi_*$ such 
that $a(\varphi_*)=0$.
We will call this point ``the horizon.''

The background fields $\Phi(\varphi)$, $a(\varphi)$ and others must be
chosen to satisfy the conditions of conformal invariance on the
world sheet \cite{cmpf}. 
After this is done, the relation between gauge fields and 
strings can be described as an isomorphism between the general Yang-Mills
operators of the type
\eqn\gaugeop{
\int d^4 x e^{ip\cdot x}\tr \left (\nabla_{\alpha_1}\ldots F_{\mu_1 \nu_1}
\ldots \nabla_{\alpha_n}\ldots F_{\mu_n \nu_n} (x)\right )
}
and the algebra of vertex operators of string theory, which have the form
\eqn\stringop{ V^{\alpha_1 \ldots \alpha_n} (p)= \int d^2 \sigma 
\Psi_p^{i_1\ldots i_n j_1\ldots j_m} \big (\varphi (\sigma) \big )
e^{ip\cdot X(\sigma)} \partial_{i_1} X^{\alpha_1}
\ldots \partial_{i_n} X^{\alpha_n}\partial_{j_1}\varphi\ldots
 \partial_{j_m} \varphi  
\ ,
}
where the wave functions $\Psi_p^{i_1\ldots i_n j_1\ldots j_m} (\varphi)$
are again determined by the conformal invariance on the world sheet.
The isomorphism mentioned above implies the coincidence of the correlation 
functions of these two sets of vertex operators.


Another, seemingly unrelated, development is connected with the
Dirichlet brane \cite{brane} description of black 3-branes
in \cite{gkp,kleb,gukt,gkThree}. 
The essential observation is that, on the one hand, the black
branes
are solitons which curve space \cite{hs} 
and, on the other hand, the world volume of $N$
parallel D-branes is described by supersymmetric $U(N)$ gauge theory with
16 supercharges \cite{Witten}.
A particularly interesting system is 
provided by the limit of
a large number $N$ of coincident D3-branes \cite{gkp,kleb,gukt,gkThree}, 
whose world volume is described by ${\cal N}=4$ supersymmetric $U(N)$ 
gauge theory in $3+1$ dimensions. For large $g_{YM}^2 N$ the
curvature of the classical geometry becomes small compared to the
string scale \cite{kleb}, 
which allows for comparison
of certain correlation functions between the supergravity and the gauge
theory, with perfect agreement \cite{kleb,gukt,gkThree}. 
Corrections in powers of $\alpha'$ times the curvature
on the string theory side correspond to corrections in powers of
$(g_{YM}^2 N)^{-1/2}$ on the gauge theory side.
The string loop corrections are suppressed by powers 
of $1/N^2$.

The vertex operators introduced in \cite{kleb,gukt,gkThree} describe
the coupling of massless closed string fields to the world volume.
For example, the vertex operator for the dilaton is
\eqn\dilvert{ 
{1\over 4 g_{YM}^2}
\int d^4 x e^{ip\cdot x}
\left (\tr F_{\mu\nu} F^{\mu\nu} (x)+ \ldots \right )\ , }
while that for the graviton polarized along the 3-branes is
\eqn\gravvert{ \int d^4 x e^{ip\cdot x} T^{\mu\nu}(x)\ ,
}
where $T^{\mu\nu}$ is the stress tensor.
The low energy absorption cross-sections are related to the
2-point functions of the vertex operators, and turn out to be
in complete agreement with conformal invariance and supersymmetric
non-renormalization theorems \cite{gkThree}. An earlier
calculation of the entropy as a function of temperature 
for $N$ coincident D3-branes \cite{gkp} exhibits a dependence 
expected of a field theory with $O(N^2)$ massless fields, 
and turns out to be $3/4$ of the free field answer. 
This is not a discrepancy since the free field result is valid
for small $g_{YM}^2 N$, while the result of \cite{gkp} is applicable as
$g_{YM}^2 N\rightarrow \infty$. We
now regard this result as a non-trivial prediction of
supergravity concerning the strong coupling behavior of ${\cal N} = 4$
supersymmetric gauge theory at large $N$ and finite temperature.

The non-critical string and the D-brane approaches to 3+1
dimensional gauge theory have been synthesized in \cite{jthroat}
by rescaling the 3-brane metric and taking the limit in which it has 
conformal symmetry, being the direct product $AdS_5\times S^5$.
\foot{The papers that put an early
emphasis on the anti-deSitter nature of the near-horizon region of
certain brane configurations, 
and its relation with string and M-theory, are
\cite{gt,ght}. Other ideas on the relation between branes and AdS
supergravity were recently pursued in
\cite{Hyun,Boonstra,Sfetsos}.}
This is exactly the confining string ansatz \cite{Sasha} with
\eqn\confan{ a(\varphi) = e^{\varphi/R}\ ,
}
corresponding to the case of constant negative curvature of order
$1/R^2$. The horizon is located at $\varphi_*=-\infty$.
The Liouville field is thus related to the radial coordinate
of the space transverse to the 3-brane. The extra $S^5$
part of the metric is associated with the 6 scalars and the $SU(4)$
R-symmetry present in the 
${\cal N}=4$ supersymmetric gauge theory.


In the present paper we
make the next step and show how the 
masses of excited states of the ``confining
string'' are related to the anomalous dimensions of the SYM theory.
Hopefully this analysis will help future explorations of asymptotically
free gauge theories needed for quark confinement. 

We will suggest a potentially very rich and detailed means of
analyzing the throat-brane correspondence: we propose an
identification of the generating function of the Green's functions of
the superconformal world-volume theory and the supergravity action in
the near horizon background geometry.
We will find it necessary to introduce a boundary of the $AdS_5$
space near the place where the throat turns into the asymptotically
flat space. Thus, the anti-deSitter coordinate $\varphi$ is 
defined on a half-line $(-\infty, 0]$,
similarly to the Liouville coordinate of the 2-dimensional string
theory \cite{Polch,DJ}. 
The correlation functions are specified by the dependence of
the action on the boundary conditions, 
again in analogy with the $c=1$ case. 
One new prediction that we will be able to extract this way is for the 
anomalous dimensions of the gauge
theory operators that correspond to massive string
states. For a state at level $n$ we find that, for large 
$g_{YM}^2 N$, the anomalous dimension grows as
$2 \sqrt n (2 g_{YM}^2 N)^{1/4}$.




\newsec{Green's functions from the supergravity action}

The geometry of a large number $N$ of coincident D3-branes is
  \eqn\DThreeGeom{
   ds^2 = \left( 1 + {R^4 \over r^4} \right)^{-1/2}
    \left( -dt^2 + d\vec{x}^2 \right) + 
    \left( 1 + {R^4 \over r^4} \right)^{1/2} 
    \left( dr^2 + r^2 d\Omega_5^2 \right) \ .
  }
 The parameter $R$, where
  \eqn\DThreeVars{
   R^4 = {N \over 2\pi^2 T_3}\ , \qquad\qquad 
   T_3 = {\sqrt{\pi} \over \kappa}
  }
 is the only length scale involved in all of what we will say.  $T_3$
is the tension of a single D3-brane, and $\kappa$ is the
ten-dimensional gravitational coupling.  The near-horizon geometry of
$N$ D3-branes is $AdS_5 \times S^5$, as one can see most easily by
defining the radial coordinate $z = R^2/r$.  Then 
  \eqn\NearGeom{
   ds^2 = {R^2 \over z^2} \left( -dt^2 + d\vec{x}^2 + dz^2 \right) + 
    R^2 d\Omega_5^2 \ .
  }
The relation to the coordinate $\varphi$ used in the previous section
is
\eqn\zphi{ z= R e^{-\varphi/R}\ .
}
 Note that the limit $z \to 0$ is far from the brane.  Of course, for
$z \lsim R$ the $AdS$ form \NearGeom\ gets modified, and for $z \ll R$
one obtains flat ten-dimensional Minkowski space.  We will freely use
phrases like ``far from the brane'' and ``near the brane'' to describe
regimes of small $z$ and large $z$, despite the fact that the geometry
is geodesically complete and nonsingular.

The basic idea is to identify the generating functional of connected
Green's functions in the gauge theory with the minimum of the
supergravity action, subject to some boundary conditions at 
$z=R$ and $z=\infty$:
  \eqn\Ident{
   W[g_{\mu\nu}(x^\lambda)] = K[g_{\mu\nu}(x^\lambda)] = 
     S[g_{\mu\nu}(x^\lambda,z)] \ .
  }
 $W$ generates the connected Green's functions of the gauge theory; $S$ is 
the supergravity action on the $AdS$ space; while $K$ is the minimum of $S$ 
subject to the boundary conditions.
 We have kept only the metric $g_{\mu\nu}(x^\lambda)$ of the
world-volume as an explicit argument of $W$.  The boundary conditions
subject to which the supergravity action $S$ is minimized are
  \eqn\BndCond{
   ds^2 = {R^2 \over z^2} 
    \left( g_{\mu\nu} dx^\mu dx^\nu + dz^2 \right) + O(1) \qquad
    \hbox{as $z \to R$} \ .
  }
All fluctuations have to vanish as $z\rightarrow \infty$.

 A few refinements of the identification \Ident\ are worth commenting
on.  First, it is the generator of connected Green's functions which
appears on the left hand side because the supergravity action on the
right hand side is expected to follow the cluster decomposition
principle.  Second, because classical supergravity is reliable only
for a large number $N$ of coincident branes, \Ident\ can only be
expected to capture the leading large $N$ behavior.  Corrections in
$1/N$ should be obtained as loop effects when one replaces the
classical action $S$ with an effective action $\Gamma$.  This is
sensible since the dimensionless expansion parameter $\kappa^2/R^8
\sim 1/N^2$. We also note that, since $(\alpha')^2/R^4 \sim
(g^2_{YM} N)^{-1}$, 
the string theoretic $\alpha'$ corrections
to the supergravity action translate into gauge theory
corrections proportional to inverse powers of $g^2_{YM} N$. 
Finally, the fact that there is
no covariant action for type IIB supergravity does not especially
concern us: to obtain $n$-point Green's functions one is actually
considering the $n^{\rm th}$ variation of the action, which for $n>0$
can be regarded as the $(n-1)^{\rm th}$ variation of the covariant
equations of motion.

In section \TwoPt\ we will compute two-point functions 
of massless vertex operators from \Ident, compare them
with the absorption calculations in \cite{kleb,gukt,gkThree}
and find exact agreement.
However it is instructive first to examine
boundary conditions.


\subsec{Preliminary: symmetries and boundary conditions}
\subseclab\SymBC

As a preliminary it is useful to examine the appropriate boundary
conditions and how they relate to the conformal symmetry.  In this
discussion we follow the work of Brown and Henneaux \cite{hen}.  In
the consideration of geometries which are asymptotically anti-de
Sitter, one would like to have a realization of the conformal group on
the asymptotic form of the metric.  Restrictive or less restrictive
boundary conditions at small $z$ (far from the brane) corresponds, as
Brown and Henneaux point out in the case of $AdS_3$, to smaller or
larger asymptotic symmetry groups.  On an $AdS_{d+1}$ space,
  \eqn\AnyADS{
   ds^2 = G_{mn} dx^m dx^n  
     = {R^2 \over z^2} \left( -dt^2 + d\vec{x}^2 + dz^2 \right)
  }
 where now $\vec{x}$ is $d-1$ dimensional, the boundary conditions
which give the conformal group as the group of asymptotic symmetries
are 
  \eqn\AllowedSize{
   \delta G_{\mu\nu} = O(1)\ , \qquad
   \delta G_{z\mu} = O(z)\ , \qquad
   \delta G_{zz} = O(1) \ .
  }
 Our convention is to let indices $m,n$ run from $0$ to $d$ (that is,
over the full $AdS_{d+1}$ space) while $\mu,\nu$ run only from $0$ to
$d-1$ (ie excluding $z = x^d$).  Diffeomorphisms which preserve
\AllowedSize\ are specified by a vector $\zeta^m$ which for small $z$
must have the form
  \eqn\AllowedZeta{\eqalign{
   \zeta^\mu &= \xi^\mu - {z^2 \over d} \eta^{\mu\nu} 
     \partial_\nu (\xi^\kappa{}_{,\kappa}) + O(z^4)  \cr
   \zeta^z &= {z \over d} \xi^\kappa{}_{,\kappa} + O(z^3) \ .
  }}
 Here $\xi^\mu$ is allowed to depend on $t$ and $\vec{x}$ but not $r$.
\AllowedZeta\ specifies only the asymptotic form of $\zeta^m$ at large
$r$, in terms of this new vector $\xi^\mu$.

Now the condition that the variation
  \eqn\LieVar{
   \delta G_{mn} = {\cal L}_\zeta G_{mn} = 
     \zeta^k \partial_k G_{mn} + G_{kn} \partial_m \zeta^k + 
      G_{mk} \partial_n \zeta^k
  }
 be of the allowed size specified in \AllowedSize\ is equivalent to 
  \eqn\CKE{
   \xi_{\mu,\nu} + \xi_{\nu,\mu} = {2 \over d} \xi^\kappa_{,\kappa}
  }
 where now we are lowering indices on $\xi^\mu$ with the flat space
Minkowski metric $\eta_{\mu\nu}$.  Since \CKE\ is the conformal
Killing equation in $d$ dimensions ($d=4$ for the 3-brane), we see
that we indeed recover precisely the conformal group from the set of
permissible $\xi^\mu$.

The spirit of \cite{hen} is to determine the central charge of an
$AdS_3$ configuration by considering the commutator of deformations
corresponding to Virasoro generators $L_m$ and $L_{-m}$.  This method
is not applicable to higher dimensional cases because the conformal
group becomes finite, and there is apparently no way to read off a
Schwinger term from commutators of conformal transformations.
Nevertheless, the notion of central charge can be given meaning in
higher dimensional conformal field theories, either via a curved space
conformal anomaly (also called the gravitational anomaly) or as the
normalization of the two-point function of the stress energy tensor
\cite{erd}.  
We shall see in section \TwoPt\ that a calculation
reminiscent of absorption probabilities allows us to read off the two
point function of stress-energy tensors in
${\cal N} = 4$ super-Yang-Mills, and with it the
central charge.  




\subsec{The two point functions}
\subseclab\TwoPt

First we consider the case of a 
minimally coupled massless
scalar propagating in the anti-de Sitter near-horizon geometry 
(one example of such a scalar is the dilaton $\phi$ \cite{kleb}).
As a further simplification
we assume for now that $\phi$ is in the $s$-wave
(that is, there is no variation over $S^5$).  Then the action becomes
  \eqn\MinAction{\eqalign{
   S &= {1 \over 2 \kappa^2} \int d^{10} x \sqrt{G} 
    \left[ \tf{1}{2} G^{MN} \partial_M \phi \partial_N \phi \right]  \cr
     &= {\pi^3 R^8 \over 4 \kappa^2} \int d^4 x \,
      \int_R^\infty {dz \over z^3} \left[ (\partial_z \phi)^2 +  
       \eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi \right] \ .
  }}
 Note that in \MinAction, as well as in all the following equations,
we take $\kappa$ to be the ten-dimensional gravitational constant.
The equations of motion resulting from the variation of $S$ are
  \eqn\MinEOM{
   \left[ z^3 \partial_z {1 \over z^3} \partial_z + 
    \eta^{\mu\nu} \partial_\mu \partial_\nu \right] \phi = 0 \ .
  }
 A complete set of normalizable solutions is
  \eqn\CompleteSet{
   \phi_k(x^\ell) = \lambda_k e^{i k \cdot x} \tilde\phi_k(z) \quad 
    \hbox{where} \quad \tilde\phi_k(z) = 
{z^2 K_2(kz)\over R^2 K_2 (k R)} \ ,
   }
$$ k^2 = \vec k^2 - \omega^2
\ .$$
We have chosen the modified Bessel function $K_2(kz)$
rather than $I_2(kz)$ because the functions $K_\nu (kz)$ fall off
exponentially for large $z$, 
while the functions $I_\nu (kz)$ grow exponentially.
In other words, the requirement of regularity at the horizon (far
down the throat) tells us which solution to keep.
A connection of this choice with the absorption calculations of
\cite{kleb} is provided by the fact that,
for time-like momenta, this is the incoming wave which
corresponds to absorption from the small $z$ region. 
$\lambda_k$ is a coupling constant, and
the normalization factor has been chosen so that $\tilde\phi_k
=1$ for $z=R$.

Let us consider a coupling 
  \eqn\WVCoup{
   S_{\rm int} = \int d^4 x \, \phi(x^\lambda) {\cal O}(x^\lambda)
  }
 in the world-volume theory.  If $\phi$ is the dilaton then according
to \cite{kleb} one would have 
${\cal O} = {1\over 4 g_{YM}^2}\tr F^2+ \ldots$.  Then
the analogue of \Ident\ is the claim that 
  \eqn\SimpIdent{
   W[\phi(x^\lambda)] = K[\phi(x^\lambda)] = S[\phi(x^\lambda,z)]
  }
 where $\phi(x^\lambda,z)$ is the unique solution of the equations of
motion with $\phi(x^\lambda,z) \to \phi(x^\lambda)$ as $z \to R$.
Note that the existence and uniqueness of $\phi$ are guaranteed
because the equation of motion is just the laplace equation on the
curved space.  (One could in fact compactify $x^\lambda$ on 
very large $T^4$ and
impose the boundary condition $\phi(x^\lambda,z) = \phi(x^\lambda)$ at
$z = R$.  Then the determination of $\phi(x^\lambda,z)$ is just the
Dirichlet problem for the laplacian on a compact manifold with
boundary).  

Analogously to the work of \cite{Polch} on the $c=1$ matrix model, we
can obtain the quadratic part of
$K[\phi(x^\lambda)]$ as a pure boundary term through
integration by parts,
  \eqn\KBound{\eqalign{
   K[\phi(x^\lambda)] &= {\pi^3 R^8 \over 4 \kappa^2} 
     \int d^4 x \, \int_R^\infty {dz \over z^3} 
     \left[ -\phi \left( z^3 \partial_z {1 \over z^3} \partial_z + 
      \eta^{\mu\nu} \partial_\mu \partial_\nu \right) \phi  + 
      z^3 \partial_z \left( \phi {1 \over z^3} \partial_z \phi \right)
     \right]  \cr
    &={1\over 2}\int d^4 k d^4 q \lambda_k \lambda_q
(2\pi)^4 \delta^4(k+q) {N^2 \over 16 \pi^2} {\cal F}
\ ,  }}
where we have expanded
$$\phi (x^\lambda) = \int d^4 k \lambda_k e^{i k\cdot x}\ .
$$ 
The ``flux factor'' ${\cal F}$ (so named because of its
resemblance to the particle number flux in a scattering calculation)
is
  \eqn\FluxDef{
   {\cal F} = 
\left[ \tilde \phi_k {1 \over z^3} \partial_z 
\tilde \phi_k \right]_R^\infty \ .
  }
 In \KBound\ we have suppressed the boundary terms in the $x^\lambda$
directions---again, one can consider these compactified on 
very large $T^4$ so
that there is no boundary.  We have also used \DThreeVars\ to simplify
the prefactor.  Finally, we have cut off the integral at $R$ as a
regulator of the small $z$ divergence.  This is in fact appropriate
since the D3-brane geometry is anti-de Sitter only for $z \gg R$.
Since there is exponential falloff in $\tilde\phi_k$ as $z\to\infty$,
only the behavior at $R$ matters.

To calculate the two-point function of ${\cal
O}$ in the world-volume theory, we differentiate $K$ twice
with respect to the coupling constants $\lambda$:\foot{The 
appearance of the logarithm 
here is analogous to the logarithmic scaling violation in the
$c=1$ matrix model.}
  \eqn\TwoPtO{\eqalign{
   \langle {\cal O}(k) {\cal O}(q) \rangle &= 
    \int d^4 x d^4 y \, e^{i k \cdot x + i q \cdot y} 
     \langle {\cal O}(x) {\cal O}(y) \rangle  \cr
    &= {\partial^2 K\over \partial \lambda_k \partial \lambda_q }=
(2\pi)^4 \delta^4(k+q) {N^2 \over 16 \pi^2} {\cal F}  \cr
    &= -(2\pi)^4 \delta^4(k+q) {N^2 \over 64 \pi^2} 
        k^4 \ln (k^2 R^2) + \hbox{(analytic in $k^2$)}
  }}
 where now the flux factor has been evaluated as
  \eqn\EvalFlux{
   {\cal F} = \left[ \tilde\phi_k
{1 \over z^3} \partial_z \tilde\phi_k 
    \right]_{z=R} =
\left[ {1 \over z^3} \partial_z  \ln (\tilde\phi_k) \right]_{z=R}
     = -\tf{1}{4} k^4 \ln (k^2 R^2) + 
      \hbox{(analytic in $k^2$)} \ .
  }
Fourier transforming back to position space, we find 
\eqn\poscorr{
\langle {\cal O} (x) {\cal O} (y) \rangle \sim {N^2\over
\vert x - y \vert ^8 }\ .
}
This is consistent with the free field result for small $g_{YM}^2 N$.
Remarkably, supergravity tells us that this formula continues to
hold as $g_{YM}^2 N\rightarrow \infty$.

Another interesting application of this analysis is to the two-point
function of the stress tensor, which with the normalization
conventions of \cite{gkThree} is
  \eqn\TwoPtFct{
   \langle T_{\alpha\beta}(x) T_{\gamma\delta}(0) \rangle
     = {c \over 48 \pi^4} X_{\alpha\beta\gamma\delta}
       \left( {1 \over x^4} \right) \ ,
  }
 where the central charge (the conformal anomaly) is $c = N^2/4$ and 
  \eqn\XFourDef{
   \vcenter{\openup1\jot
   \halign{\strut\span\TL & \span\TR\cr
     X_{\alpha\beta\gamma\delta} &= 
      2 \square^2 \eta_{\alpha\beta} \eta_{\gamma\delta} - 
       3 \square^2 (\eta_{\alpha\gamma} \eta_{\beta\delta} + 
        \eta_{\alpha\delta} \eta_{\beta\gamma}) -
      4 \partial_\alpha \partial_\beta 
        \partial_\gamma \partial_\delta  \cr
      &\quad - 2 \square 
        (\partial_\alpha \partial_\beta \eta_{\gamma\delta} +
         \partial_\alpha \partial_\gamma \eta_{\beta\delta} +
         \partial_\alpha \partial_\delta \eta_{\beta\gamma} +
         \partial_\beta \partial_\gamma \eta_{\alpha\delta} +
         \partial_\beta \partial_\delta \eta_{\alpha\gamma} +
         \partial_\gamma \partial_\delta \eta_{\alpha\beta}) \ .  \cr
   }}}
 For metric perturbations $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$
around flat space, the coupling of $h_{\mu\nu}$ at linear order is
  \eqn\SInt{
   S_{\rm int} = \int d^4 x \, \tf{1}{2} h^{\mu\nu} T_{\mu\nu} \ .
  }
 Furthermore, at quadratic order the supergravity
action for a graviton polarized along the brane,
$h_{xy}(k)$, is exactly the minimal scalar action, 
%with the normalization indicated in \MinAction.  
provided the momentum $k$ is orthogonal to the $xy$ plane.
We can therefore carry over the result \TwoPtO\ to obtain
  \eqn\TwoTT{
   \langle T_{xy}(k) T_{xy}(q) \rangle 
    = - (2\pi)^4 \delta^4(k+q) {N^2 \over 64 \pi^2} 
        k^4 \ln (k^2 R^2) + 
         \hbox{(analytic in $k^2$)} \ ,
  }
 which upon Fourier transform can be compared with \TwoPtFct\ to
give $c = N^2/4$.  In view of the conformal symmetry of both the
supergravity and the gauge theory, the evaluation of this one
component is a sufficient test.

The conspiracy of overall factors to give the correct normalization of
\TwoTT\ clearly has the same origin as the successful prediction of
the minimal scalar $s$-wave absorption cross-section
\cite{kleb,gukt,gkThree}. The absorption cross-section is, up to
a constant of proportionality, the imaginary part of \TwoTT.
In \cite{kleb} the absorption cross-section was calculated in
supergravity using propagation of a scalar field in the entire
3-brane metric, including the asymptotic region far from the brane. 
Here we have, in effect, replaced communication of the throat region with
the asymptotic region by a boundary condition at one end
of the throat. The physics of this is clear: signals coming
from the asymptotic region excite the part of the throat near
$z=R$. Propagation of these excitations into the throat
can then be treated just in the anti-deSitter approximation.
Thus, to extract physics from anti-deSitter space we 
introduce a boundary at $z=R$ and take careful account of 
the boundary terms that contain the dynamical information.

It now seems clear how to proceed to three-point functions: on
the supergravity side one must expand to third order in the perturbing
fields, including in particular the three point vertices.  
At higher
orders the calculation is still simple in concept (the classical
action is minimized subject to boundary conditions), but the
complications of the ${\cal N}=8$ supergravity theory seem likely to
make the computation of, for instance, the four-point function,
rather tedious. We leave the details of such calculations for
the future. It may be very useful to compute at least the three point
functions in order to have a consistency check on the normalization of
fields.  

One extension of the present work is to consider what fields couple to
the other operators in the $N=4$ supercurrent multiplet.  The
structure of the multiplet (which includes the supercurrents, the
$SU(4)$ $R$-currents, and four spin $1/2$ and one scalar field)
suggests a coupling to the fields of gauged $N=4$ supergravity.  The
question then becomes how these fields are embedded in $N=8$
supergravity.  We leave these technical issues for the future, but
with the expectation that they are ``bound to work'' based on
supersymmetry. 

The main lesson we have extracted so far is that, for
certain operators that couple to the massless string states, the
anomalous dimensions vanish. We expect this to hold for all
vertex operators that couple to the fields of
supergravity. This may be the complete set of operators
that are protected by supersymmetry. As we will
see in the next section, other operators acquire anomalous dimensions
that grow for large `t Hooft coupling.




\newsec{Massive string states and anomalous dimensions}
\seclab\MassiveString

Before we proceed to the massive string states, 
a useful preliminary is to discuss the higher partial waves of a
minimally coupled massless scalar.
The action in five
dimensions (with Lorentzian signature), the equations of motion, and
the solutions are
  \eqn\PartAction{
   S = {\pi^3 R^8 \over 4 \kappa^2} \int d^4 x \int_R^\infty
        {dz \over z^3} \left[ (\partial_z \phi)^2 +
         (\partial_\mu \phi)^2 + {\ell (\ell + 4) \over z^2} \phi^2
        \right]
  }
  \eqn\PartEOM{
   \left[ z^3 \partial_z {1 \over z^3} \partial_z +
    \eta^{\mu\nu} \partial_\mu \partial_\nu -
    {\ell (\ell + 4) \over z^2} \right] \phi = 0
  }
  \eqn\PartSols{
   \phi_k(x^\ell) = e^{i k \cdot x} \tilde\phi_k(z) \quad
    \hbox{where} \quad
   \tilde\phi_k(z) = {z^2 K_{\ell+2}(kz) \over
    R^2 K_{\ell+2}(kR)} \ .
  }
We have chosen the normalization such that
$\tilde\phi_k(z) =1$ at $z=R$.
The flux factor is evaluated by expanding
  \eqn\PhiBehave{
   K_{\ell+2}(kz) = 2^{\ell+1}\Gamma(\ell+2)
(k z)^{-(\ell+2)} \left (1 + \ldots +
     {(-1)^\ell \over 2^{2\ell+3} (\ell+1)! (\ell+2)!}
     (kz)^{2(\ell+ 2)} \ln kz + \ldots \right )\ ,
  }
where in parenthesis we exhibit the leading non-analytic term.
We find
  \eqn\GotFlux{
   {\cal F} =
\left[ {1 \over z^3} \partial_z  \ln (\tilde\phi_k) \right]_{z=R}=
 {(-1)^\ell \over 2^{2 \ell + 2}
    [(\ell+1)!]^2 } k^{4+ 2l} R^{2\ell} \ln kR \ .
  }
 As before, we have neglected terms containing analytic powers of
$k$ and focused on the leading nonanalytic term.
This formula indicates that the operator that couples to $\ell$-th
partial wave has dimension $4+\ell$.
In \cite{kleb} it was shown that such operators with the $SO(6)$
quantum numbers of the $\ell$-th partial wave have the form
\eqn\higher{
\int d^4 x e^{ik\cdot x} \tr \left [ \left (X^{(i_i}\ldots X^{i_\ell )}
+ \ldots \right )
F_{\mu\nu} F^{\mu\nu} (x) \right ]\ , }
where in parenthesis we have a traceless symmetric tensor of $SO(6)$.
Thus, supergravity predicts that
their non-perturbative dimensions equal their bare dimensions.

Now let us consider massive string states.  
Our goal is to use supergravity to calculate the anomalous dimensions of
the gauge theory operators that couple to them.
To simplify
the discussion, let us focus on excited string states which are
spacetime scalars of mass $m$.  The propagation equation for such a
field in the background of the 3-brane geometry is
\eqn\Coulone{
\left [{d^2\over d
r^2} + {5\over r} {d\over dr} - k^2 \left (1 + {R^4\over r^4}\right )
-m^2 \left (1 + {R^4\over r^4}\right )^{1/2} \right ]
\tilde \phi_k =0\ . } 
For the state at excitation level $n$,
$$ m^2 = {4 n\over
\alpha'} \ . $$ 
In the throat region, $z\gg R$, \Coulone\ simplifies to 
\eqn\Coultwo{ \left [{d^2\over d
z^2} - {3\over z} {d\over dz} -k^2 
-{m^2 R^2\over z^2} \right ]\tilde \phi_k =0\ . } 
Note that a massive particle with small energy $\omega \ll m$, which would
be far off shell in the asymptotic region $z\ll R$, can nevertheless
propagate in the throat region (i.e. it is described by an oscillatory
wave function).
 
Equation \Coultwo\
is identical to the equation encountered in the analysis of higher
partial waves, except
the effective angular momentum is not in general an integer:
in the centrifugal barrier term $\ell (\ell+4)$ is replaced by $(m R)^2$.
Analysis of the choice of wave function goes through as before,
with $\ell+2$ replaced by $\nu$, where
\eqn\anom{\nu = \sqrt{ 4 + (m R)^2 } \ .}
In other words, the wave function falling off exponentially for
large $z$ and normalized to $1$ at $z=R$ is\foot{
These solutions are reminiscent of the loop correlators
calculated for $c\leq 1$ matrix models in \cite{mss}.}
\eqn\massivewave{
   \tilde\phi_k(z) = {z^2 K_\nu(kz) \over
    R^2 K_\nu (k R)} \ .}
Now we recall that
$$ K_\nu = {\pi\over 2\sin (\pi\nu)} \left (I_{-\nu} - I_\nu \right )\ ,
$$
$$ I_\nu (z) = \left ({z\over 2}\right )^\nu
\sum_{k=0}^\infty {(z/2)^{2k} \over k! \Gamma (k+\nu+ 1)}\ .
$$
Thus, 
\eqn\genexpand{ K_\nu (kz) = 2^{\nu -1}\Gamma (\nu)
(k z)^{-\nu} \left (1+ \ldots -  
\left ({z k\over 2 }\right )^{2\nu } 
{\Gamma (1-\nu)\over \Gamma (1+\nu )}+\ldots \right )\ ,
} 
where in parenthesis we have exhibited the leading non-analytic term.
Calculating the flux factor as before, 
we find that the leading non-analytic term of the
2-point function is
  \eqn\OOFct{
   \langle {\cal O}(k) {\cal O}(q) \rangle =-(2\pi)^4 \delta^4(k+q)
{N^2 \over 8\pi^2} {\Gamma (1-\nu)\over \Gamma (\nu )}
    \left ({kR\over 2}\right )^{2\nu} R^{-4}\ .
  }
This implies that the dimension of the corresponding SYM
operator is equal to 
\eqn\opdim{
\Delta= 2+\nu= 2+\sqrt{ 4 + (m R)^2 }\ .
}

Now, let us note that
$$ R^4 = 2 N g_{YM}^2 (\alpha')^2\ ,
$$
which implies
$$ (m R)^2 = 4 n g_{YM} \sqrt {2 N} 
\ .
$$
Using \anom\ we find that the spectrum of dimensions
for operators that couple to massive string states is,
for large $g_{YM} \sqrt N$,
\eqn\anomform{ 
h_n \approx 2\left (n g_{YM} \sqrt {2 N} \right )^{1/2}
\ .}
Equation \anomform\ is a new 
non-trivial prediction of the string theoretic
approach to large $N$ gauge theory.\foot{We do not expect this equation
to be valid for arbitrarily large $n$, because application of linearized
local effective actions to arbitrarily excited string states 
is questionable.
However, we should be able to trust our approach for moderately excited
states.}

We conclude that, for large `t Hooft coupling,
the anomalous dimensions of the
vertex operators corresponding to massive string
states grow without bound.
By contrast, the vertex operators that couple to the massless string
states do not acquire any anomalous dimensions. This has been checked
explicitly for gravitons, dilatons and RR scalars \cite{kleb,gukt,gkThree},
and we believe this to be a general statement.\foot{A more general
set of such massless fields is contained in the supermultiplet of 
AdS gauge fields, whose boundary couplings were recently
studied in \cite{ff}.} Thus, there are several $SO(6)$ towers of
operators that do not acquire anomalous dimensions, such as
the dilaton tower \higher. The absense of
the anomalous dimensions
is probably due to the fact that they are protected by SUSY. The rest
of the operators are not protected and can receive arbitrarily large
anomalous dimensions. While we cannot yet write down the
explicit form of these operators in the gauge theory, 
it seems likely that they
are the conventional local operators, such as 
\gaugeop.  Indeed,
the coupling of a highly excited string state to the world volume may
be guessed on physical grounds.  Since a string in a D3-brane is a
path of electric flux, it is natural to assume that a string state
couples to a Wilson loop ${\cal O} = \exp \left( i \oint_\gamma A
\right)$. Expansion of a small loop in powers of $F$ yields
the local polynomial operators.

There is one potential problem with our treatment of massive
states. The minimal linear equation \Coulone\ where higher
derivative terms are absent may be true only for a particular field
definition (otherwise corrections in positive
powers of $\alpha' \nabla^2$ will be present in the equation).
Therefore, it is possible that there are energy dependent leg
factors relating the operators $\cal O$ in 
\OOFct\ and the gauge theory operators of the form \gaugeop.
We hope that these leg factors do not change our conclusion about the
anomalous dimensions. However, to completely settle this issue we
need to either find an exact sigma-model which incorporates all
$\alpha'$ corrections or to calculate the 3-point functions
of massive vertex operators.



\newsec{Conclusions}

There are many unanswered questions that we have left for the future.
So far, we have considered the limit of large `t Hooft coupling,
since we used the one-loop sigma-model calculations for
all operators involved. If this coupling is not large, then we have
to treat the world sheet theory as an exact conformal field theory
(we stress, once again, that the string loop corrections are
$\sim 1/N^2$ and, therefore, vanish in the large $N$ limit).
This conformal field theory is a sigma-model on a hyperboloid.
It is plausible that, in addition to the global $O(2,4)$ symmetry,
this theory possesses the $O(2,4)$ Kac-Moody algebra.
If this is the case, then the sigma model is tractable
with standard methods of conformal field theory.

Throughout this work we detected many formal similarities of our
approach with that used in $c\leq 1$ matrix models. These models
may be viewed as early examples of gauge theory -- non-critical string
correspondence, with the large $N$ matrix models playing the role
of gauge theories. Clearly, a deeper understanding of
the connection between the present work and the $c\leq 1$ matrix models
is desirable.





\bigbreak\bigskip\bigskip\centerline{{\bf Acknowledgements}}\nobreak
\bigskip

We would like to thank C.~Callan and the participants of the
``Dualities in String Theory'' program at ITP, Santa Barbara, for useful
discussions. I.R.K. is grateful to ITP for
hospitality during the completion of this paper.
This research was supported in part by the National Science 
Foundation under Grants No. PHY94-07194 and
PHY96-00258, by the Department
of Energy under Grant No.  
DE-FG02-91ER40671, 
and by the James S.~McDonnell Foundation under Grant No. 91-48.  
S.S.G. also thanks the Hertz Foundation for its support.


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