


\def\unredoffs{} \def\redoffs{\voffset=-.40truein\hoffset=-.40truein}
\def\speclscape{\special{landscape}}

\newbox\leftpage \newdimen\fullhsize \newdimen\hstitle \newdimen\hsbody
\tolerance=1000\hfuzz=2pt\def\fontflag{cm}

\catcode`\@=11
\def\bigans{b }


\def\answ{b }


\ifx\answ\bigans\message{(This will come out unreduced.}
\magnification=1200\unredoffs\baselineskip=16pt plus 2pt minus 1pt
\hsbody=\hsize \hstitle=\hsize




\else\message{(This will be reduced.} \let\l@r=L
\magnification=1000\baselineskip=16pt plus 2pt minus 1pt
\vsize=7truein \redoffs
\hstitle=8truein\hsbody=4.75truein\fullhsize=10truein\hsize=\hsbody
%
\output={\ifnum\pageno=0

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


\newcount\yearltd\yearltd=\year\advance\yearltd by -1900
\def\HUTP#1#2{\Title{HUTP-\number\yearltd/A#1}{#2}}
\def\Title#1#2{\nopagenumbers\abstractfont\hsize=\hstitle\rightline{#1}%
\vskip 1in\centerline{\titlefont #2}\abstractfont\vskip
.5in\pageno=0}
%
\def\Date#1{\vfill\leftline{#1}\tenpoint\supereject\global\hsize=\hsbody%
\footline={\hss\tenrm\folio\hss}}



\def\draft{\draftmode\Date{\draftdate}}
\def\draftmode{\message{ DRAFTMODE }\def\draftdate{{\rm preliminary draft:
\number\month/\number\day/\number\yearltd\ \ \hourmin}}%
\headline={\hfil\draftdate}\writelabels\baselineskip=20pt plus 2pt
minus 2pt
  {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255}
   \multiply\count255 by-60\advance\count255 by\time
   \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}}}


\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string\string##1}}}
\nolabels
%


\global\newcount\secno \global\secno=0 \global\newcount\meqno
\global\meqno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}

\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}
\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1\message{(\secsym\the\subsecno. #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
%
\def\appendix#1#2{\global\meqno=1\global\subsecno=0\xdef\secsym{\hbox{#1.}}
\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}


\def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1\wrlabeL#1}
\def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
\writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
\global\advance\meqno by1\wrlabeL{#1$\{\}$}}
\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}



\newskip\footskip\footskip14pt plus 1pt minus 1pt


\def\footnotefont{\ninepoint}\def\f@t#1{\footnotefont #1\@foot}
\def\f@@t{\baselineskip\footskip\bgroup\footnotefont\aftergroup\@foot\let\next}
\setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt}
%
\global\newcount\ftno \global\ftno=0
\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}


\newwrite\ftfile
\def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
$^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
\immediate\write\ftfile{\noexpand\smallskip%
\noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
\def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
\centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}
\def\footatend{}


\global\newcount\refno \global\refno=1
\newwrite\rfile
%
\def\ref{[\the\refno]\nref}
\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi
\global\advance\refno by1\chardef\wfile=\rfile\immediate
\write\rfile{\noexpand\item{#1\
}\reflabeL{#1\hskip.31in}\pctsign}\findarg}



\def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
\gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
\edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup\else\ifx\next\empty%
\else\immediate\write\wfile{\the\toks0}\fi\let\next=\writ@line\fi\next\relax}}
\def\striprel@x#1{} \def\em@rk{\hbox{}}
%
\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}}


%
\def\listrefs{\footatend\vfill\supereject\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}\nonfrenchspacing}
%
\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel \string#1 is undefined.}%
\nref#1{need to supply reference \string#1.}\fi%
\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
\else#1\count255=1\fi\let\next=\r@fs\fi\next}
%
\def\figures{\centerline{{\bf Figure Captions}}\medskip\parindent=40pt%
\def\fig##1##2{\medskip\item{Fig.~##1.  }##2}}
%



\newwrite\ffile\global\newcount\figno \global\figno=1
%
\def\fig{fig.~\the\figno\nfig}
\def\nfig#1{\xdef#1{fig.~\the\figno}%
\writedef{#1\leftbracket fig.\noexpand~\the\figno}%
\ifnum\figno=1\immediate\openout\ffile=figs.tmp\fi\chardef\wfile=\ffile%
\immediate\write\ffile{\noexpand\medskip\noexpand\item{Fig.\
\the\figno. }
\reflabeL{#1\hskip.55in}\pctsign}\global\advance\figno
by1\findarg}
%
\def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
\baselineskip14pt\centerline{{\bf Figure
Captions}}\nobreak\medskip \escapechar=` \input
figs.tmp\vfill\eject}}
%
\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
\def\figs#1{figs.~\f@gs #1{\hbox{}}}
\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else
\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
%
\newwrite\lfile
{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
\def\writedefs{\immediate\openout\lfile=labeldefs.tmp \def\writedef##1{%
\immediate\write\lfile{\string\def\string##1\rightbracket}}}
%
\def\writestop{\def\writestoppt{\immediate\write\lfile{\string\pageno%
\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
\string\secno\the\secno\string\meqno\the\meqno}\immediate\closeout\lfile}}
%
\def\writestoppt{}\def\writedef#1{}
%
\def\seclab#1{\xdef #1{\the\secno}\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
\def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
%
\newwrite\tfile \def\writetoca#1{}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}


\def\writetoc{\immediate\openout\tfile=SWN.tmp
    \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
    \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}


%       and this lists table of contents on second pass
\def\listtoc{\centerline{\bf CONTENTS}\nobreak
    \medskip{\baselineskip=12pt\parskip=0pt\input SWN.tmp \bigbreak\bigskip}}
%
\catcode`\@=12 % at signs are no longer letters
%
%   Unpleasantness in calling in abstract and title fonts
\edef\tfontsize{\ifx\answ\bigans scaled\magstep3\else
scaled\magstep4\fi} \font\titlerm=cmr10 \tfontsize
\font\titlerms=cmr7 \tfontsize \font\titlermss=cmr5 \tfontsize
\font\titlei=cmmi10 \tfontsize \font\titleis=cmmi7 \tfontsize
\font\titleiss=cmmi5 \tfontsize \font\titlesy=cmsy10 \tfontsize
\font\titlesys=cmsy7 \tfontsize \font\titlesyss=cmsy5 \tfontsize
\font\titleit=cmti10 \tfontsize \skewchar\titlei='177
\skewchar\titleis='177 \skewchar\titleiss='177
\skewchar\titlesy='60 \skewchar\titlesys='60
\skewchar\titlesyss='60
%
\def\titlefont{\def\rm{\fam0\titlerm}% switch to title font
\textfont0=\titlerm \scriptfont0=\titlerms
\scriptscriptfont0=\titlermss \textfont1=\titlei
\scriptfont1=\titleis \scriptscriptfont1=\titleiss
\textfont2=\titlesy \scriptfont2=\titlesys
\scriptscriptfont2=\titlesyss \textfont\itfam=\titleit
\def\it{\fam\itfam\titleit}\rm}
%
\font\authorfont=cmcsc10 \ifx\answ\bigans\else scaled\magstep1\fi
%
\ifx\answ\bigans\def\abstractfont{\tenpoint}\else
\font\abssl=cmsl10 scaled \magstep1 \font\absrm=cmr10
scaled\magstep1 \font\absrms=cmr7 scaled\magstep1
\font\absrmss=cmr5 scaled\magstep1 \font\absi=cmmi10
scaled\magstep1 \font\absis=cmmi7 scaled\magstep1
\font\absiss=cmmi5 scaled\magstep1 \font\abssy=cmsy10
scaled\magstep1 \font\abssys=cmsy7 scaled\magstep1
\font\abssyss=cmsy5 scaled\magstep1 \font\absbf=cmbx10
scaled\magstep1 \skewchar\absi='177 \skewchar\absis='177
\skewchar\absiss='177 \skewchar\abssy='60 \skewchar\abssys='60
\skewchar\abssyss='60
%
\def\abstractfont{\def\rm{\fam0\absrm}% switch to abstract font
\textfont0=\absrm \scriptfont0=\absrms \scriptscriptfont0=\absrmss
\textfont1=\absi \scriptfont1=\absis \scriptscriptfont1=\absiss
\textfont2=\abssy \scriptfont2=\abssys \scriptscriptfont2=\abssyss
\textfont\itfam=\bigit \def\it{\fam\itfam\bigit}\def\footnotefont{\tenpoint}%
\textfont\slfam=\abssl \def\sl{\fam\slfam\abssl}%
\textfont\bffam=\absbf \def\bf{\fam\bffam\absbf}\rm}\fi
%
\def\tenpoint{\def\rm{\fam0\tenrm}% switch back to 10-point type
\textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
\textfont1=\teni  \scriptfont1=\seveni  \scriptscriptfont1=\fivei
\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
\textfont\itfam=\tenit \def\it{\fam\itfam\tenit}\def\footnotefont{\ninepoint}%
\textfont\bffam=\tenbf
\def\bf{\fam\bffam\tenbf}\def\sl{\fam\slfam\tensl}\rm}
%
\font\ninerm=cmr9 \font\sixrm=cmr6 \font\ninei=cmmi9
\font\sixi=cmmi6 \font\ninesy=cmsy9 \font\sixsy=cmsy6
\font\ninebf=cmbx9 \font\nineit=cmti9 \font\ninesl=cmsl9
\skewchar\ninei='177 \skewchar\sixi='177 \skewchar\ninesy='60
\skewchar\sixsy='60
%
\def\ninepoint{\def\rm{\fam0\ninerm}% switch to footnote font
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont\itfam=\ninei \def\it{\fam\itfam\nineit}\def\sl{\fam\slfam\ninesl}%
\textfont\bffam=\ninebf \def\bf{\fam\bffam\ninebf}\rm}
%
%---------------------------------------------------------------------
%
\def\noblackbox{\overfullrule=0pt}
\hyphenation{anom-aly anom-alies coun-ter-term coun-ter-terms}
%
\def\inv{^{\raise.15ex\hbox{${\scriptscriptstyle -}$}\kern-.05em 1}}
\def\dup{^{\vphantom{1}}}
\def\Dsl{\,\raise.15ex\hbox{/}\mkern-13.5mu D} %this one can be subscripted
\def\dsl{\raise.15ex\hbox{/}\kern-.57em\partial}
\def\del{\partial}
\def\Psl{\dsl}
\def\tr{{\rm tr}} \def\Tr{{\rm Tr}}
\font\bigit=cmti10 scaled \magstep1
\def\biglie{\hbox{\bigit\$}} %pound sterling
\def\lspace{\ifx\answ\bigans{}\else\qquad\fi}
\def\lbspace{\ifx\answ\bigans{}\else\hskip-.2in\fi} % $$\lbspace...$$
\def\boxeqn#1{\vcenter{\vbox{\hrule\hbox{\vrule\kern3pt\vbox{\kern3pt
     \hbox{${\displaystyle #1}$}\kern3pt}\kern3pt\vrule}\hrule}}}
\def\mbox#1#2{\vcenter{\hrule \hbox{\vrule height#2in
         \kern#1in \vrule} \hrule}}  %e.g. \mbox{.1}{.1}




\def\tilde{\widetilde}
\def\bar{\overline}
\def\hat{\widehat}




\def\CAG{{\cal A/\cal G}} \def\CO{{\cal O}}
\def\CA{{\cal A}} \def\CC{{\cal C}} \def\CF{{\cal F}} \def\CG{{\cal G}}
\def\CL{{\cal L}} \def\CH{{\cal H}} \def\CI{{\cal I}} \def\CU{{\cal U}}
\def\CB{{\cal B}} \def\CR{{\cal R}} \def\CD{{\cal D}} \def\CT{{\cal T}}
\def\e#1{{\rm e}^{^{\textstyle#1}}}
\def\grad#1{\,\nabla\!_{{#1}}\,}
\def\gradgrad#1#2{\,\nabla\!_{{#1}}\nabla\!_{{#2}}\,}
\def\ph{\varphi}
\def\psibar{\overline\psi}
\def\om#1#2{\omega^{#1}{}_{#2}}
\def\vev#1{\langle #1 \rangle}
\def\lform{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}
\def\darr#1{\raise1.5ex\hbox{$\leftrightarrow$}\mkern-16.5mu #1}
\def\lie{\hbox{\it\$}} %pound sterling
\def\ha{{1\over 2}}
\def\half{{\textstyle{1\over2}}}


\def\roughly#1{\raise.3ex\hbox{$#1$\kern-.75em\lower1ex\hbox{$\sim$}}}

\def\np#1#2#3{Nucl. Phys. {\bf B#1} (#2) #3}
\def\pl#1#2#3{Phys. Lett. {\bf #1B} (#2) #3}
\def\prl#1#2#3{Phys. Rev. Lett. {\bf #1} (#2) #3}
\def\anp#1#2#3{Ann. Phys. {\bf #1} (#2) #3}
\def\pr#1#2#3{Phys. Rev. {\bf #1} (#2) #3}
\def\ap#1#2#3{Ann. Phys. {\bf #1} (#2) #3}
\def\prep#1#2#3{Phys. Rep. {\bf #1} (#2) #3}
\def\rmp#1#2#3{Rev. Mod. Phys. {\bf #1}}
\def\cmp#1#2#3{Comm. Math. Phys. {\bf #1} (#2) #3}
\def\mpl#1#2#3{Mod. Phys. Lett. {\bf #1} (#2) #3}
\def\ptp#1#2#3{Prog. Theor. Phys. {\bf #1} (#2) #3}
\def\jhep#1#2#3{JHEP {\bf#1}(#2) #3}
\def\jmp#1#2#3{J. Math Phys. {\bf #1} (#2) #3}
\def\cqg#1#2#3{Class.~Quantum Grav. {\bf #1} (#2) #3}
\def\ijmp#1#2#3{Int.~J.~Mod.~Phys. {\bf #1} (#2) #3}
\def\atmp#1#2#3{Adv.~Theor.~Math.~Phys.{\bf #1} (#2) #3}
\def\ap#1#2#3{Ann.~Phys. {\bf #1} (#2) #3}
%%%%%%%%%%%%%%%  Rublenye bukvy   %%%%%%%%%%%%%%%%%
\def\IB{\relax\hbox{$\inbar\kern-.3em{\rm B}$}}
\def\IC{\relax\hbox{$\inbar\kern-.3em{\rm C}$}}
\def\ID{\relax\hbox{$\inbar\kern-.3em{\rm D}$}}
\def\IE{\relax\hbox{$\inbar\kern-.3em{\rm E}$}}
\def\IF{\relax\hbox{$\inbar\kern-.3em{\rm F}$}}
\def\IG{\relax\hbox{$\inbar\kern-.3em{\rm G}$}}
\def\IGa{\relax\hbox{${\rm I}\kern-.18em\Gamma$}}
\def\IH{\relax{\rm I\kern-.18em H}}
\def\IK{\relax{\rm I\kern-.18em K}}
\def\IL{\relax{\rm I\kern-.18em L}}
\def\IP{\relax{\rm I\kern-.18em P}}
\def\IR{\relax{\rm I\kern-.18em R}}
\def\IZ{\relax\ifmmode\mathchoice{
\hbox{\cmss Z\kern-.4em Z}}{\hbox{\cmss Z\kern-.4em Z}}
{\lower.9pt\hbox{\cmsss Z\kern-.4em Z}} {\lower1.2pt\hbox{\cmsss
Z\kern-.4em Z}} \else{\cmss Z\kern-.4em Z}\fi}
\def\II{\relax{\rm I\kern-.18em I}}
\def\IX{{\bf X}}
\def\ttb{Type $\II$B string theory}
\def\ndt{{\noindent}}
\def\bx{{\bf G}}
\def\ee#1{{\rm erf}\left(#1\right)}
\def\sssec#1{\ndt$\underline{#1}$}
%%%%%%%% Calligraphic letters  %%%%%%%%%%%%%

\def\CA{{\cal A}}
\def\CB{{\cal B}}
\def\CC{{\cal C}}
\def\CD{{\cal D}}
\def\CE{{\cal E}}
\def\CF{{\cal F}}
\def\CG{{\cal G}}
\def\CH{{\cal H}}
\def\CI{{\cal I}}
\def\CJ{{\cal J}}
\def\CK{{\cal K}}
\def\CL{{\cal L}}
\def\CM{{\cal M}}
\def\CN{{\cal N}}
\def\CO{{\cal O}}
\def\CP{{\cal P}}
\def\CQ{{\cal Q}}
\def\CR{{\cal R}}
\def\CS{{\cal S}}
\def\CT{{\cal T}}
\def\CU{{\cal U}}
\def\CV{{\cal V}}
\def\CW{{\cal W}}
\def\CX{{\cal X}}
\def\CY{{\cal Y}}
\def\CZ{{\cal Z}}
\def\Br{{\bf r}}
%%%%%%%%%%%% Derivatives  %%%%%%%%%%%
\def\p{\partial}
\def\pb{\bar{\partial}}
\def\dir{{\CD}\hskip -6pt \slash \hskip 5pt}
\def\dd{{\rm d}}
%%%%%%%%%%% letters with bar %%%%%%%%
\def\ib{\bar{i}}
\def\jb{\bar{j}}
\def\kb{\bar{k}}
\def\lb{\bar{l}}
\def\mb{\bar{m}}
\def\nb{\bar{n}}
\def\ub{\bar{u}}
\def\wb{\bar{w}}
\def\sb{\bar{s}}
\def\tb{\bar{t}}
\def\zb{\bar{z}}
%%%%%%%%%% Math symbols %%%%%%%%%%%%%
\def\codim{{\mathop{\rm codim}}}
\def\cok{{\rm cok}}
\def\rank{{\rm rank}}
\def\coker{{\mathop {\rm coker}}}
\def\diff{{\rm diff}}
\def\Diff{{\rm Diff}}
\def\Tr{{\rm Tr}}
\def\Id{{\rm Id}}
\def\vol{{\rm vol}}
\def\Vol{{\rm Vol}}
\def\c{\cdot}
\def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule height 4.1pt depth -.3pt
width.25pt\hskip-2pt$\times$}}}
\def\ch{{\rm ch}}
\def\Det{{\rm Det}}
\def\DET{{\rm DET}}
\def\Hom{{\rm Hom}}
\def\dim{{\rm dim}}
\def\imp{$\Rightarrow$}
\def\danger{{NB:}}
\def\HH{{\bf H}}
\def\hn{{\hat n}}
%%%%%%%%%%%%%% Lie algebras %%%%%%%%%%%%%%%%%%%%%%
\def\Lie{{\rm Lie}}
\def\lieg{{\underline{\bf g}}}
\def\liet{{\underline{\bf t}}}
\def\liek{{\underline{\bf k}}}
\def\lies{{\underline{\bf s}}}
\def\lieh{{\underline{\bf h}}}
\def\clieg{{\underline{\bf g}}_{\scriptscriptstyle{\IC}}}
\def\cliet{{\underline{\bf t}}_{\scriptstyle{\IC}}}
\def\cliek{{\underline{\bf k}}_{\scriptscriptstyle{\IC}}}
\def\clies{{\underline{\bf s}}_{\scriptstyle{\IC}}}
\def\CCK{K_{\scriptscriptstyle{\IC}}}
\def\inbar{\,\vrule height1.5ex width.4pt depth0pt}

\font\cmss=cmss10 \font\cmsss=cmss10 at 7pt

\def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule height 4.1pt
depth -.3pt width .25pt\hskip-2pt$\times$}}}
%%%%%%%%%%%% Greek %%%%%%%%%%%%
\def\a{{\alpha}}
\def\ap{{\a}^{\prime}}
\def\b{{\beta}}
\def\d{{\delta}}
\def\G{{\Gamma}}
\def\g{{\gamma}}
\def\e{{\epsilon}}
\def\z{{\zeta}}
\def\ve{{\varepsilon}}
\def\vf{{\varphi}}
\def\m{{\mu}}
\def\n{{\nu}}
\def\u{{\Upsilon}}
\def\l{{\lambda}}
\def\s{{\sigma}}
\def\t{{\theta}}
\def\vt{{\vartheta}}
\def\o{{\omega}}
\def\nc{noncommutative\ }
\def\npt{non-perturbative\ }
\def\hp{\hat\partial}
\def\k{{\kappa}}
%%%%%
\def\bA{{\bf A}}
\def\bF{{\bf F}}
\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern8pt
\vbox{\hbox{\kern8pt}\hbox{\vbox{#1}}\hbox{\kern8pt}}
\kern8pt\vrule}\hrule}}
\def\mathboxit#1{\vbox{\hrule\hbox{\vrule\kern8pt\vbox{\kern8pt
\hbox{$\displaystyle #1$}\kern8pt}\kern8pt\vrule}\hrule}}


%%%%%%%Russian fonts%%%%%%%5

\chardef\tempcat=\the\catcode`\@ \catcode`\@=11
\def\cyracc{\def\u##1{\if \i##1\accent"24 i%
    \else \accent"24 ##1\fi }}
\newfam\cyrfam
\font\tencyr=wncyr10
\def\cyr{\fam\cyrfam\tencyr\cyracc}





%%%%%%%%%% curly letters %%%%%%%%%%

\def\CA{{\cal A}} \def\CC{{\cal C}}
\def\CF{{\cal F}} \def\CG{{\cal G}} \def\CL{{\cal L}}
\def\CH{{\cal H}} \def\CI{{\cal I}} \def\CU{{\cal U}}
\def\CB{{\cal B}} \def\CO{{\cal O}}
\def\CR{{\cal R}} \def\CD{{\cal D}}
\def\CT{{\cal T}}

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

\def\e#1{{\rm e}^{^{\textstyle#1}}}
\def\grad#1{\,\nabla\!_{{#1}}\,}
\def\gradgrad#1#2{\,\nabla\!_{{#1}}\nabla\!_{{#2}}\,}
\def\ph{\varphi} \def\psibar{\overline\psi}
\def\om#1#2{\omega^{#1}{}_{#2}} \def\vev#1{\langle #1 \rangle}
\def\lform{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}
\def\darr#1{\raise1.5ex\hbox{$\leftrightarrow$}\mkern-16.5mu #1}
\def\lie{\hbox{\it\$}}%pound sterling


\def\ha{{1\over 2}}

\def\half{{\textstyle{1\over2}}} %puts a small half in a displayed eqn



\def\roughly#1{\raise.3ex\hbox{$#1$\kern-.75em\lower1ex\hbox{$\sim$}}}


%%%%% REFS %%%%%%%%%%

\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%%%%%%%%% Journals %%%%%%%%%%%%

\def\np#1#2#3{Nucl. Phys. {\bf B#1} (#2) #3}
\def\pl#1#2#3{Phys.Lett. {\bf #1B} (#2) #3}
\def\prl#1#2#3{Phys. Rev. Lett. {\bf #1}(#2) #3}
\def\anp#1#2#3{Ann. Phys. {\bf #1} (#2) #3}
\def\pr#1#2#3{Phys. Rev. {\bf #1} (#2) #3}
\def\ap#1#2#3{Ann. Phys. {\bf #1} (#2) #3}


\def\prep#1#2#3{Phys. Rep. {\bf #1} (#2)
#3} \def\rmp#1#2#3{Rev. Mod. Phys. {\bf #1}}
\def\cmp#1#2#3{Comm.Math. Phys. {\bf #1} (#2) #3}
\def\mpl#1#2#3{Mod. Phys. Lett. {\bf
#1} (#2) #3} \def\ptp#1#2#3{Prog. Theor. Phys. {\bf #1} (#2) #3}
\def\jhep#1#2#3{JHEP {\bf#1}(#2) #3}
\def\jmp#1#2#3{J. Math Phys.
{\bf #1} (#2) #3}
\def\cqg#1#2#3{Class.~Quantum Grav. {\bf #1}
(#2) #3}
\def\ijmp#1#2#3{Int.~J.~Mod.~Phys. {\bf #1} (#2) #3}
\def\atmp#1#2#3{Adv.~Theor.~Math.~Phys.{\bf #1} (#2) #3}
\def\ap#1#2#3{Ann.~Phys. {\bf #1} (#2) #3}



%%%%%%%%%%%%%%% Rublenye bukvy   %%%%%%%%%%%%%%%%%




\def\IB{\relax\hbox{$\inbar\kern-.3em{\rm B}$}}
\def\IC{\relax\hbox{$\inbar\kern-.3em{\rm C}$}}
\def\ID{\relax\hbox{$\inbar\kern-.3em{\rm D}$}}
\def\IE{\relax\hbox{$\inbar\kern-.3em{\rm E}$}}
\def\IF{\relax\hbox{$\inbar\kern-.3em{\rm F}$}}
\def\IG{\relax\hbox{$\inbar\kern-.3em{\rm G}$}}
\def\IGa{\relax\hbox{${\rm I}\kern-.18em\Gamma$}}
\def\IH{\relax{\rm I\kern-.18em H}} \def\IK{\relax{\rm
I\kern-.18em K}} \def\IL{\relax{\rm I\kern-.18em L}}
\def\IP{\relax{\rm I\kern-.18em P}} \def\IR{\relax{\rm
I\kern-.18em R}} \def\IZ{\relax\ifmmode\mathchoice{ \hbox{\cmss
Z\kern-.4em Z}}{\hbox{\cmss Z\kern-.4em Z}}
{\lower.9pt\hbox{\cmsss Z\kern-.4em Z}} {\lower1.2pt\hbox{\cmsss
Z\kern-.4em Z}} \else{\cmss Z\kern-.4em Z}\fi} \def\II{\relax{\rm
I\kern-.18em I}}
\def\ndt{{\noindent}}
\def\tn{{\tilde n}} \def\tk{{\tilde k}}


 %%%%%%%% Calligraphic letters  %%%%%%%%%%%%% \def\CA{{\cal A}}
\def\CB{{\cal B}} \def\CC{{\cal C}} \def\CD{{\cal D}}
\def\CE{{\cal E}} \def\CF{{\cal F}} \def\CG{{\cal G}}
\def\CH{{\cal H}} \def\CI{{\cal I}} \def\CJ{{\cal J}}
\def\CK{{\cal K}} \def\CL{{\cal L}} \def\CM{{\cal M}}
\def\CN{{\cal N}} \def\CO{{\cal O}} \def\CP{{\cal P}}
\def\CQ{{\cal Q}} \def\CR{{\cal R}} \def\CS{{\cal S}}
\def\CT{{\cal T}} \def\CU{{\cal U}} \def\CV{{\cal V}}
\def\CW{{\cal W}} \def\CX{{\cal X}} \def\CY{{\cal Y}}
\def\CZ{{\cal Z}} \def\Br{{\bf r}}

%%%%%%%%%%%% Derivatives %%%%%%%%%%%




\def\p{\partial} \def\pb{\bar{\partial}}
\def\dir{{\CD}\hskip -6pt \slash \hskip 5pt} \def\dd{{\rm d}}




%%%%%%%%%%% letters with bar %%%%%%%%



\def\ib{\bar{i}}
\def\jb{\bar{j}} \def\kb{\bar{k}} \def\lb{\bar{l}}
\def\mb{\bar{m}} \def\nb{\bar{n}} \def\ub{\bar{u}}
\def\wb{\bar{w}} \def\sb{\bar{s}} \def\tb{\bar{t}}
\def\zb{\bar{z}}


%%%%%%%%%% Math symbols %%%%%%%%%%%%%




\def\codim{{\mathop{\rm codim}}} \def\cok{{\rm cok}}
\def\rank{{\rm rank}} \def\coker{{\mathop {\rm coker}}}
\def\diff{{\rm diff}} \def\Diff{{\rm Diff}} \def\Tr{{\rm Tr}}
\def\Id{{\rm Id}} \def\vol{{\rm vol}} \def\Vol{{\rm Vol}}
\def\c{\cdot} \def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule height
4.1pt depth -.3pt width.25pt\hskip-2pt$\times$}}} \def\ch{{\rm
ch}} \def\Det{{\rm Det}} \def\DET{{\rm DET}} \def\Hom{{\rm Hom}}
\def\dim{{\rm dim}} \def\imp{$\Rightarrow$} \def\danger{{NB:}}
\def\HH{{\bf H}} \def\hn{{\hat n}}

%%%%%%%%%%%%%% Lie algebras
%%%%%%%%%%%%%%%%%%%%%%

\def\Lie{{\rm Lie}}

\def\inbar{\,\vrule height1.5ex width.4pt depth0pt}
\font\cmss=cmss10 \font\cmsss=cmss10 at 7pt
\def\sdtimes{\mathbin{\hbox{\hskip2pt\vrule height 4.1pt depth
-.3pt width .25pt\hskip-2pt$\times$}}}

%%%%%%%%%%%% Greek %%%%%%%%%%%%

\def\a{{\alpha}} \def\ap{{\a}^{\prime}}
\def\b{{\beta}} \def\d{{\delta}} \def\G{{\Gamma}} \def\g{{\gamma}}
\def\e{{\epsilon}} \def\z{{\zeta}} \def\ve{{\varepsilon}}
\def\vf{{\varphi}} \def\m{{\mu}} \def\n{{\nu}} \def\u{{\Upsilon}}
\def\l{{\lambda}} \def\s{{\sigma}} \def\t{{\theta}}
\def\vt{{\vartheta}} \def\o{{\omega}} \def\nc{noncommutative\ }
\def\npt{non-perturbative\ } \def\hp{\hat\partial}
\def\kk{{\kappa}}


%%%%%%% bold face %%%%%%%%%%%%%%%

\def\ba{{\bf a}} \def\bb{{\bf b}} \def\bg{{\bf g}} \def\bc{{\bf c}}
\def\bG{{\bf G}} \def\bT{{\bf T}} \def\bX{{\bf X}}
\def\bR{{\bf R}} \def\bZ{{\bf Z}} \def\bC{{\bf C}} \def\bP{{\bf P}}
\def\bS{{\bf S}} \def\be{{\bf e}} \def\bz{{\bf z}} \def\vf{{\bf f}}
\def\bm{{\bf m}} \def\bn{{\bf n}} \def\bk{{\bf k}}
\def\bl{{\bf l}} \def\bs{{\bf s}} \def\bt{{\bf t}}
\def\bo{{\bf o}}
\def\bM{{\bf M}}
\def\bI{{\bf I}}
%%%%%

\def\IA{{\bf A}} \def\IF{{\bf F}}
\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern8pt
\vbox{\hbox{\kern8pt}\hbox{\vbox{#1}}\hbox{\kern8pt}}
\kern8pt\vrule}\hrule}}
\def\mathboxit#1{\vbox{\hrule\hbox{\vrule\kern8pt\vbox{\kern8pt
\hbox{$\displaystyle #1$}\kern8pt}\kern8pt\vrule}\hrule}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%                                          %%%%%%%%
%%%%%%            LITERATURE                    %%%%%%%%
%%%%%%                                          %%%%%%%%
%%%%%%                                          %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\lref\dv{R.~Dijkgraaf, C.~Vafa, hep-th/0206255, hep-th/0207106,
hep-th/0208048\semi R.~Dijkgraaf, S.~Gukov, V.~Kazakov, C.~Vafa,
hep-th/0210238\semi R.~Dijkgraaf, M.~Grisaru, C.~Lam, C.~Vafa,
D.~Zanon, hep-th/0211017\semi M.~Aganagic, M.~Marino, A.~Klemm,
C.~Vafa, hep-th/0211098\semi R.~Dijkgraaf, A.~Neitzke, C.~Vafa,
hep-th/0211194} \lref\cdws{F.~Cachazo, M.~Douglas, N.~Seiberg,
E.~Witten, hep-th/0211170}


\lref\estring{J.~A.~Minahan, D.~Nemeschansky, C.~Vafa, N.P.~Warner,
hep-th/9802168\semi
T.~Eguchi, K.~Sakai, hep-th/0203025, hep-th/0211213}


\lref\bulkbndr{V.~Balasubramanian, P.~Kraus, A.~Lawrence, hep-th/9805171}

\lref\frbranes{D.-E.~Diaconescu, M.~Douglas, J.~Gomis, hep-th/9712230}

\lref\gorodentsevleenzon{A.~Gorodentsev, M.~Leenson, \semi
L.~G\"ottche} \lref\witdgt{E.~Witten, hep-th/9204083}
\lref\ikkt{N.~Ishibashi, H.~Kawai, Y.~Kitazawa, and A.~Tsuchiya,
\np{498}{1997}{467}, hep-th/9612115} \lref\cds{A.~Connes,
M.~Douglas, A.~Schwarz, \jhep{9802}{1998}{003}}
\lref\wtnc{E.~Witten, \np{268}{1986}{253}}

\lref\cstw{L.~Baulieu, A.~Losev, N.~Nekrasov, hep-th/9707174}


\lref\klemmzaslow{ A.~Klemm, E.~Zaslow, hep-th/9906046}


\lref\cftorb{A.~Lawrence, N.~Nekrasov, C.~Vafa, hep-th/9803015}
\lref\booksSW{
A.~Marshakov, ``Seiberg-Witten Theory and Integrable Systems,'' {\it
World Scientific, Singapore (1999)}\semi
``Integrability: The Seiberg-Witten and Whitham Equations",
Eds. H.~Braden and I.~Krichever, {\it Gordon and Breach (2000)}.}


\lref\agmav{M.~Aganagic, M.~Mari\~no, C.~Vafa, hep-th/0206164}

\lref\moorewitten{G.~Moore, E.~Witten, hep-th/9709193}

\lref\gopakumarvafa{R.~Gopakumar, C.Vafa, hep-th/9809187,
hep-th/9812127} \lref\mooreunpublished{G.~Moore, unpublished}
\lref\wittenone{E.~Witten, hep-th/9403195} \lref\cg{E.~Corrigan,
P.~Goddard, ``Construction of instanton and monopole solutions and
reciprocity'', \anp {154}{1984}{253}} \lref\opennc{N.~Nekrasov,
hep-th/0203109\semi K.-Y.Kim, B.-H. Lee, H.S. Yang, hep-th/0205010
} \lref\donaldson{S.K.~Donaldson, ``Instantons and Geometric
Invariant Theory", \cmp{93}{1984}{453-460}}
\lref\nakajima{H.~Nakajima, ``Lectures on Hilbert Schemes of
Points on Surfaces''\semi AMS University Lecture Series, 1999,
ISBN 0-8218-1956-9. } \lref\neksch{N.~Nekrasov, A.~S.~Schwarz,
hep-th/9802068, \cmp{198}{1998}{689}} \lref\freck{A.~Losev,
N.~Nekrasov, S.~Shatashvili, hep-th/9908204, hep-th/9911099}
\lref\rkh{N.J.~Hitchin, A.~Karlhede, U.~Lindstrom, and M.~Rocek,
\cmp{108}{1987}{535}} \lref\branek{H.~Braden, N.~Nekrasov,
hep-th/9912019\semi K.~Furuuchi, hep-th/9912047} \lref\wilson{G.~
Wilson, ``Collisions of Calogero-Moser particles and adelic
Grassmannian", Invent. Math. 133 (1998) 1-41.}
\lref\abs{O.~Aharony, M.~Berkooz, N.~Seiberg, hep-th/9712117,
\atmp{2}{1998}{119-153}} \lref\avatars{A.~Losev, G.~Moore,
N.~Nekrasov, S.~Shatashvili, hep-th/9509151}
\lref\abkss{O.~Aharony, M.~Berkooz, S.~Kachru, N.~Seiberg,
E.~Silverstein, hep-th/9707079, \atmp{1}{1998}{148-157}}

\lref\witsei{N.~Seiberg, E.~Witten, hep-th/9908142,
\jhep{9909}{1999}{032}} \lref\kkn{V.~Kazakov, I.~Kostov,
N.~Nekrasov, ``D-particles, Matrix Integrals and KP hierarchy'',
\np{557}{1999}{413-442}, hep-th/9810035}
\lref\DHf{J.~J.~Duistermaat, G.J.~Heckman, Invent. Math. {\bf 69}
(1982) 259\semi M.~Atiyah, R.~Bott,  Topology {\bf 23} No 1 (1984)
1} \lref\tdgt{M.~Atiyah, R.~Bott,  Phil. Trans. Roy. Soc. London
{\bf A 308} (1982), 524-615\semi E.~Witten, hep-th/9204083\semi
S.~Cordes, G.~Moore, S.~Rangoolam, hep-th/9411210}
\lref\atiyahsegal{M.~Atiyah, G.~Segal, Ann. of Math. {\bf 87}
(1968) 531} \lref\bott{R.~Bott, J.~Diff.~Geom. {\bf 4} (1967) 311}
\lref\torusaction{G.~Ellingsrud, S.A.Stromme, Invent. Math. {\bf
87} (1987) 343-352\semi L.~G\"ottche, Math. A.. {\bf 286} (1990)
193-207} \lref\gravilit{M.~Bershadsky, S.~Cecotti, H.~Ooguri,
C.~Vafa, \cmp{165}{1994}{311}, \np{405}{1993}{279}\semi
I.~Antoniadis, E.~Gava, K.S.~Narain, T.~R.~Taylor,
\np{413}{1994}{162}, \np{455}{1995}{109}} \lref\calculus{N.~Dorey,
T.~J.~Hollowood, V.~V.~Khoze, M.~P.~Mattis, hep-th/0206063}
\lref\instmeasures{N.~Dorey, V.V.~Khoze, M.P.~Mattis,
hep-th/9706007, hep-th/9708036} \lref\twoinst{N.~Dorey,
V.V.~Khoze, M.P.~Mattis, hep-th/9607066} \lref\vafaengine{S.~Katz,
A.~Klemm, C.~Vafa, hep-th/9609239} \lref\connes{A.~Connes,
``Noncommutative geometry'', Academic Press (1994)}
\lref\macdonald{I.~Macdonald, ``Symmetric functions and Hall
polynomials'', Clarendon Press, Oxford, 1979}
\lref\nikfive{N.~Nekrasov, hep-th/9609219 \semi A.~Lawrence,
N.~Nekrasov, hep-th/9706025}

\lref\fivedim{A.~Marshakov, A.~Mironov, hep-th/9711156\semi
H.~Braden, A.~Marshakov, A.~Mironov, A.~Morozov,
%``Seiberg-Witten theory for a non-trivial compactification from five to  four dimensions,''
%Phys.\ Lett.\ B {\bf 448}, 195 (1999)
hep-th/9812078,
%\semi
%H.~Braden, A.~Marshakov, A.~Mironov, A.~Morozov,
%``The Ruijsenaars-Schneider model in the context of Seiberg-Witten  theory,''
%Nucl.\ Phys.\ B {\bf 558}, 371 (1999)
hep-th/9902205\semi
T.~Eguchi, H.~Kanno, hep-th/0005008\semi
H.~Braden, A.~Marshakov,
%``Singular phases of Seiberg-Witten integrable systems: Weak and strong  coupling,''
%Nucl.\ Phys.\ B {\bf 595}, 417 (2001)
hep-th/0009060\semi
 H. Braden, A. Gorsky, A. Odesskii, V. Rubtsov, hep-th/01111066\semi
C.Csaki, J.Erlich, V.V.Khoze, E.Poppitz, Y.Shadmi, Y.Shirman, hep-th/0110188\semi
T.~Hollowood, hep-th/0302165}

\lref\seibergfive{N.~Seiberg, hep-th/9608111}
\lref\ganor{O.~Ganor, hep-th/9607092, hep-th/9608108}


\lref\instlit{Literature on
instantons}


\lref\bcov{ M.~Bershadsky, S.~Cecotti, H.~Ooguri, C.~Vafa, hep-th/9309140}


\lref\op{A.~Okounkov, R.~Pandharipande, math.AG/0207233,
math.AG/0204305} \lref\prtoda{T.~Eguchi, K.~Hori, C.-S.~Xiong,
hep-th/9605225\semi T.~Eguchi, S.~Yang, hep-th/9407134\semi
T.~Eguchi, H.~Kanno, hep-th/9404056}


\lref\givental{A.~Givental, alg-geom/9603021}



\lref\maxim{M.~Kontsevich, hep-th/9405035}
\lref\whitham{A.~Gorsky, A.~Marshakov, A.~Mironov, A.~Morozov,
Nucl.Phys. {\bf B527} (1998) 690-716,
hep-th/9802007} \lref\kricheverwhitham{I.~Krichever,
hep-th/9205110, \cmp{143}{1992}{415}}


\lref\sw{N.~Seiberg, E.~Witten, hep-th/9407087, hep-th/9408099}


\lref\swsol{A.~Klemm, W.~Lerche, S.~Theisen, S.~Yankielowicz,
hep-th/9411048 \semi P.~Argyres, A.~Faraggi, hep-th/9411057\semi
A.~Hanany, Y.~Oz, hep-th/9505074} \lref\hollowood{T.~Hollowood,
hep-th/0201075, hep-th/0202197} \lref\nsvz{V.~Novikov, M.~Shifman,
A.~Vainshtein, V.~Zakharov, \pl{217}{1989}{103}}
\lref\seibergone{N.~Seiberg, \pl{206}{1988}{75}}
\lref\dbound{G.~Moore, N.~Nekrasov, S.~Shatashvili,
hep-th/9803265} \lref\ihiggs{G.~Moore, N.~Nekrasov,
S.~Shatashvili, hep-th/9712241 } \lref\potsdam{W.~Krauth,
H.~Nicolai, M.~Staudacher, hep-th/9803117} \lref\kirwan{F.~Kirwan,
``Cohomology of quotients in symplectic and algebraic geometry'',
Mathematical Notes, Princeton Univ. Press, 1985}
\lref\wittfivebrane{E.~Witten, hep-th/9610234}
\lref\issues{A.~Losev, N.~Nekrasov, S.~Shatashvili,
hep-th/9711108, hep-th/9801061}


\lref\adhm{M.~Atiyah, V.~Drinfeld, N.~Hitchin, Yu.~Manin, Phys.
Lett. {\bf 65A} (1978) 185}


\lref\vafaoo{C.~Vafa, hep-th/0008142}

\lref\seiberghol{N.~Seiberg, hep-th/9408013}

 \lref\warner{A.~Klemm, W.~Lerche, P.~Mayr,
C.~Vafa, N.~Warner, hep-th/9604034}


\lref\wittensolution{E.~Witten, hep-th/9703166}

\lref\witbound{E.~Witten, hep-th/9510153}


\lref\twists{E.~Witten, hep-th/9304026 \semi O.~Ganor,
hep-th/9903110 \semi H.~Braden, A.~Marshakov, A.~Mironov,
A.~Morozov, hep-th/9812078}


\lref\nok{N.~Nekrasov, A.~Okounkov, in progress}

\lref\cmn{S.~Cherkis, G.~Moore, N.~Nekrasov, in progress}


\lref\dijkgraaf{R.~Dijkgraaf, hep-th/9609022}


\lref\wittenm{E.~Witten, hep-th/9503124}

\lref\niklos{A.~Losev, N.~Nekrasov, in progress}


\lref\experiment{G.~Chan, E.~D'Hoker, hep-th/9906193 \semi
E.~D'Hoker, I.~Krichever, D.~Phong, hep-th/9609041\semi
J.~Edelstein, M.~Gomez-Reino, J.~Mas, hep-th/9904087 \semi
J.~Edelstein, M.~Mari\~no, J.~Mas hep-th/9805172 }


\lref\flow{I.~Klebanov, N.~Nekrasov, hep-th/9911096\semi
J.~Polchinski, hep-th/0011193}


\lref\todalit{K.~Ueno, K.~Takasaki, Adv. Studies in Pure Math.
{\bf 4} (1984) 1}

\lref\kharchev{For an excellent review see, e.g. S.~Kharchev,
hep-th/9810091}

\lref\gkmmm{
A.Gorsky, I.Krichever, A.Marshakov, A.Mironov and A.Morozov,
Phys. Lett.  {\bf B355} (1995) 466; hep-th/9505035. }

\lref\witdonaldson{E.~Witten, \cmp{117}{1988}{353}}


\lref\swi{N.~Nekrasov, hep-th/0206161} \lref\fucito{U.~Bruzzo,
F.~Fucito, J.F.~Morales, A.~Tanzini, hep-th/0211108\semi
D.Bellisai, F.Fucito, A.Tanzini, G.Travaglini, hep-th/0002110,
hep-th/0003272, hep-th/0008225} \lref\flume{R.~Flume,
R.~Poghossian, hep-th/0208176\semi R.~Flume, R.~Poghossian,
H.~Storch, hep-th/0110240, hep-th/0112211} \lref\khoze{ N.~Dorey,
T.J.~Hollowood, V.~Khoze, M.~Mattis, hep-th/0206063, and
references therein}

\lref\polyakov{A.~Polyakov, hep-th/9711002, hep-th/9809057}
\lref\ads{J.~Maldacena, hep-th/9711200\semi S.~Gubser,
I.~Klebanov, A.~Polyakov, hep-th/9802109\semi E.~Witten,
hep-th/9802150}


\lref\mmm{A.~Losev, G.~Moore, S.~Shatashvili, hep-th/9707250\semi
N.~Seiberg, hep-th/9705221}


\lref\dbranes{J.~Polchinski, hep-th/9510017}
%%%%%%% Russian fonts%%%%%%%


\chardef\tempcat=\the\catcode`\@ \catcode`\@=11
\def\cyracc{\def\u##1{\if \i##1\accent"24 i
\else \accent"24 ##1\fi }}
\newfam\cyrfam \font\tencyr=wncyr10
\def\cyr{\fam\cyrfam\tencyr\cyracc}



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

\Title{\vbox{\baselineskip 10pt \hbox{} \hbox{ITEP-TH-1/03}
\hbox{MPIM-2003-26}       \hbox{FIAN/TD-05/03}
\hbox{IHES-P/03/09}
%\hbox{hep-th/0302191}
}  } {\vbox{\vskip -30 true pt
\smallskip
   \centerline{\bf SMALL INSTANTONS, LITTLE STRINGS}
\smallskip\smallskip\centerline{\bf AND FREE
FERMIONS} \vskip2pt}}   \centerline{Andrei S. Losev$^{1}$, Andrei
Marshakov$^{231}$, Nikita A. Nekrasov\footnote{$^{\dagger}$}{On
leave of absence from: ITEP, Moscow, 117259, Russia}$^{4}$, }
\medskip\centerline{\it $^1$ ITEP, Moscow, 117259,
Russia}\centerline{\it $^2$ Max Planck Institute of Mathematics, Bonn,
D-53072,
Germany}\centerline{\it $^3$ P.N.Lebedev Physics Institute, Moscow, 117924,
Russia}\centerline{\it $^{4}$ IHES, Bures-sur-Yvette, F-91440,
France}
\bigskip \noindent
We present new evidence for the conjecture that BPS correlation
functions in the ${\CN}=2$ supersymmetric gauge theories are
described by an auxiliary two dimensional conformal field theory.
We study deformations of the ${\CN} =2$ supersymmetric
gauge theory by all gauge-invariant chiral operators. We calculate
the partition function of the ${\CN}=2$ theory on ${\bR}^4$ with
appropriately twisted boundary conditions. For the $U(1)$
theory with instantons (either noncommutative, or D-instantons,
depending on the construction)  the partition function, as it
turns out, has a representation in terms of the
theory of free fermions on a sphere, and coincides with the
tau-function of the Toda lattice hierarchy. Using this result we prove
to all orders in string loop expansion that the effective prepotential
(for $U(1)$ with all chiral couplings
included) is given by the free energy of the topological string on
${\bC\bP}^1$. We show that the gravitational descendants play an
important r\^ole in the gauge fields/string correspondence.  We
then identify this string with the little string bound to the
fivebrane wrapped on the two-sphere. We also discuss the theory
with fundamental matter hypermultiplets.

%\draftmode
\bigskip\bigskip\bigskip
\Date{February 2003}






\newsec{INTRODUCTION}

The Holy Graal of the theoretical physics is the nonperturbative
theory which includes quantum gravity, sometimes called as
M-theory \wittenm.
The current wisdom assumes that there is no coupling constant, and
whatever perturbation
theory is used depends on the particular solution one expands about.
The expansion parameter is one of the geometric characteristics of
the background.
It is of some interest to look for simplified string and field
theoretic models, which have string loop expansion, and where the
string coupling constant has a geometric interpretation.


\bigskip
\noindent{\it String expansion in gauge theory}

Large $N$
gauge theories are the most popular, and the most elusive models with
string representation. In the gauge/string duality \polyakov\ads\
one matches the connected
correlation functions of the gauge theory observables with the
partition function of the string theory in the bulk.  The
closed string dual has $1\over N^2$ as a string
coupling constant. Advances in the studies of the type II string
compactifications
on Calabi-Yau manifolds led to another class of models, which in the
low-energy limit
reduce to ${\CN}=2$ supersymmetric gauge theories, with a novel type
of string loop expansion.
Namely, certain couplings ${\CF}_g$ in the low-energy effective action
are given by the genus $g$ partition function of the topologically
twisted string on Calabi-Yau. The gauge group of the ${\CN}=2$ theory
does not have to be $U(N)$ with large $N$. It is determined by the
geometry of Calabi-Yau manifold \vafaengine. The r\^ole of effective
string coupling is played by the vev of the graviphoton field
strength \gravilit, which is usually assumed to be constant \vafaoo.


In this paper we shall explain that there exists a  natural from the
gauge theory point of view way to flesh out these couplings. The idea
is to put the theory in a nontrivial
geometric background, which we presently describe.  Consider any
Lorentz-invariant field
theory in $k$ dimensions. Suppose the theory can be obtained by
Kaluza-Klein reduction from some theory in $k+1$ dimensions. In
addition, suppose the theory in $k+1$ dimensions had a global
symmetry group $H$. Now compactify the $k+1$ dimensional theory on
a circle ${\bf S}^1$ of circumference $r$, with a twist, so that in going
around the circle, the space-time ${\bf R}^k$ experiences a
Lorentz rotation, by an element ${\exp} \left(r {\Omega}\right)$, and in
addition a Wilson line in the group $H$, ${\exp} \left(r {\bA}\right)$
is turned
on. The resulting theory can be now considered in the $r \to 0$
limit, where for finite ${\Omega}, \bA$ we find extra couplings in
the $k$-dimensional Lagrangian. This is the background we shall
extensively use in the paper. More specifically we shall be mostly
interested in the four dimensional ${\CN}=2$ theories. They all
can be viewed as dimensional reductions of ${\CN}=1$ susy gauge
theories from six or five dimensions. The global symmetry group $H$
is $SU(2)$ (R-symmetry).

These considerations lead to powerful results concerning exact
non-perturbative
calculations in the supersymmetric gauge theories.
In particular, one arrives at the  technique of deriving effective
prepotentials of
the ${\CN}=2$ susy gauge theories with the gauge groups $U(N_1)
\times \ldots \times U(N_k)$  \swi\ (based on
\avatars\nikfive\issues\ihiggs\dbound\freck, see also related work
in \khoze\hollowood\fucito\flume). Previously, the effective
low-energy action and the corresponding prepotential
${\CF}^{SW}(a)$ was determined using the constraints of holomorphy
and electro-magnetic duality \sw\seiberghol\swsol.


\bigskip


\noindent{\it Higher Casimirs in gauge theory}

One of the goals of the present paper is to extend the method \swi\ to
get the correlation functions of ${\CN}=2$ {\it chiral} operators.
This is equivalent to solving for the effective prepotential of
the ${\CN}=2$ theory whose microscopic prepotential (see \sw\ for
introduction in ${\CN}=2$ susy) is given by: \eqn\mcrsc{{\CF}^{UV}
= {\tau}_0 {\Tr} {\Phi}^2 + \sum_{{\vec n}} {\tau}_{\vec n}
\prod_{J=1}^{\infty} \left( {1\over J} {\Tr} {\Phi}^J \right)^{n_J}}
where
$\vec n = (n_1, n_2, \ldots)$ label all possible gauge-invariant
polynomials in the adjoint Higgs field ${\Phi}$ (note that
${\tau}_{0,1,0,\ldots}$ shifts ${\tau}_0$).
Let ${\vec\rho} = (1,2,3,\ldots)$,  $\vert{\vec n}\vert = \sum_J n_J$,
and $\vec n \cdot \vec \rho = \sum_J J n_J$.

In order for the theory defined by \mcrsc\
to avoid vanishing of the second derivatives of prepotential
at large (quasiclassical) values of the Higgs field
%
\eqn\vcavrg{\langle {\Phi} \rangle_{a} \sim a \gg {\Lambda} \sim
e^{2\pi i {\tau}_0} }
%
and not to run into strong coupling singularity,
the couplings ${\tau}_{\vec n}$ should be
treated formally. One could also worry about the nonrenormalizabilty of the
perturbation \mcrsc. This is actually not so, provided the conjugate
prepotential ${\bar\CF}$ is kept classical
${\bar\tau}_0 {\Tr} {\bar\Phi}^2$. The action is no longer real,
however, the effective dimensions of the fields ${\Phi}$ and
${\bar\Phi}$ become $0$ and $2$, thereby justifying an infinite
number of terms in \mcrsc.

We should  note that classically there are relations between the
deformations generated by derivatives w.r.t.
${\tau}_{\vec n}$, which originate in the fact
that there are
polynomial relations between the single-trace operators ${\Tr} {\Phi}^J$
for $J > N$ and the multiple-trace operators.
When instantons are included these classical relations are modified.
It seems convenient to keep all ${\tau}_{\vec n}$ as independent
couplings. The classical prepotential then obeys  additional constraints:
the $N$-independent non-linear ones:
\eqn\trvc{
{\p \over{\p} {\tau}_{\vec n}} {\CF}^{UV} = {{\p}^k \over
{\p}{\tau}_{\vec n_1}\ldots{\p}{\tau}_{\vec n_k}} {\CF}^{UV},
\qquad {\vec n} = {\vec n}_1 + \ldots {\vec n}_k}
and the $N$-dependent linear ones:
\eqn\nntrvc{\sum_{{\vec n}: \ {\vec n} \cdot {\vec \rho} = N + k}
(-1)^{\vert {\vec n} \vert}
{\p \over {\p {\tau}_{\vec n}}} {\CF}^{UV} = 0\ , \qquad k > 0}
The quantum effective prepotential obeys instanton
corrected constraints \issues, which we implicitly determine in this paper.

\bigskip
\noindent{\it Contact terms}

The constraints of holomorphy and electro-magnetic duality are
powerful enough to determine the effective low-energy prepotential
${\CF}^{IR}$ (see \issues), up to a diffeomorphism of the couplings
${\tau}_{\vec n}$, i.e. up to contact terms.
In order to fix the precise mapping between the microscopic couplings
(which we also call ``times'', in accordance with the terminology adopted
in integrable systems) and the macroscopic ones, one needs more refined
methods (see \freck\ for the discussion of the contact terms and their
relation to the topology of the compactifications of the moduli spaces).
As we shall explain in this paper, the
direct instanton calculus is powerful enough to solve for
${\CF}^{IR}$:
\eqn\gnfnc{{\CF}^{IR} ( a, {\tau}_{\vec n} ) =
{\CF}^{SW} (a ; {\tau}_0 ) + \sum_{\vec n} {\tau}_{\vec n}
{\CO}_{\vec n} (a) + \sum_{\vec n, \vec m } {\tau}_{\vec n}
{\tau}_{\vec m} {\CO}_{\vec n\vec m} (a) + \ldots}
where
\eqn\vvs{{\CO}_{\vec n}(a) = \left< \prod_{J=1}^{\infty}\left(
{1\over J}{\Tr} {\Phi}^{J} \right)^{n_J} \right>_{a}}
while
${\CO}_{\vec
n\vec m}$ are the expectation values of the contact terms between
${\CO}_{\vec n}$ and ${\CO}_{\vec m}$ \issues\moorewitten\whitham.


\bigskip
\noindent{\it Dual string theories}

We shall argue that the generalized in this way prepotential \gnfnc,
which
is also a generating function of the correlators of chiral
observables, is encoded in a certain stringy partition function.
We shall demonstrate that the generating function of the
expectation values of the chiral observables in the special
${\CN}=2$ supergravity background are given by the exponential of
the all-genus partition function of the topological string. For
the $"U(1)"$ theory the dual string lives on ${\bP}^1$ (A-model).
To prove this we shall use the recent results of A.~Okounkov and
R.~Pandharipande who related the partition function of the
topological string on ${\bC\bP}^1$ with the tau-function of the Toda
lattice hierarchy. The expression of the generating function of
the chiral operators through the tau-functions of an integrable
system is a straightforward generalization of the experimentally
well-known relation between the Seiberg-Witten prepotentials and
quasiclassical tau-functions \gkmmm\ (see also \booksSW\ and
references therein). For the tau-function giving
the generating function for the correlators of chiral operators we
will present a natural representation in terms of free fermionic
or bosonic system. We think this is a substantial step towards the
understanding the physical origin of the results of \gkmmm.

One may think that this result is yet another example of the local
mirror symmetry \vafaengine. We should stress here that it
is by no means obvious.
Indeed, a powerful method to embed ${\CN}=2$ gauge theories into
string theory is by considering
type II string on local Calabi-Yau manifolds. Almost all of the
results obtained in this way can be viewed as a degeneration of
the theory which exists for global, compact Calabi-Yau manifolds.
In other words, one assumes that the gauge theory decouples from
gravity and excited
string modes, when the Calabi-Yau is about to develope some singularity,
and the global structure of
Calabi-Yau is not relevant; but in this way one cannot really discuss
the higher Casimir deformations \mcrsc.
However, the main claim is there: the prepotential of the gauge
theory, as well as the higher couplings ${\CF}_g$, are given by
the topological string amplitudes on the local Calabi-Yau.

If the local Calabi-Yau is can be viewed as a degeneration of the
compact Calabi-Yau then one simply takes the limit of the
corresponding topological
string amplitudes (effectively all irrelevant K\"ahler classes in
the A-model are sent to infinity, and the worldsheet instantons
do not know about them; however, one has to renormalize the
zero instanton sector level term). In this
case one can, in principle,
take the mirror theory, the B-model on a dual Calabi-Yau manifold,
and try to perform the analogous degeneration there \vafaengine,
this way even leads to some equations on ${\CF}_g$'s \klemmzaslow.
However, the situation here is still unsatisfactory. For the global
Calabi-Yau's
the whole sum ${\sum}_g {\CF}_g {\hbar}^{2g-2}$ is identified with
the logarithm of the partition function of the effective "closed
string field theory" -- the Kodaira-Spencer theory \bcov\ on the B-side
Calabi-Yau manifold.
Nothing of this sort is known for the degenerations corresponding
to local Calabi-Yau's on the A-side.

Our contribution to the subject is the identification of the analogue
of the Kodaira-Spencer theory, at least in the specific context we focus
on in this paper. This is, we claim, the free fermionic (or free bosonic)
theory on a Riemann surface (as sphere for $U(1)$ gauge group), with some
specific $W$-background turned on.

Moreover, we have also something new on the A-side. It is of course
not the first time when the Fano
varieties appear in the context of local mirror symmetry. However,
the topological string
amplitudes, corresponding to the local Calabi-Yau do not coincide
with those for Fano, even if the actual worldsheet instantons land on
Fano subvariety in the local Calabi-Yau. For example,
the resolved conifold is the
%${\CO}(-1,-1)$
${\CO}(-1)\oplus{\CO}(-1)$
bundle over ${\bC\bP}^1$,
all worldsheet instantons
land in ${\bC\bP}^1$ (which is Fano), yet the topological string
amplitude is affected by the zero modes of the fermions, corresponding
to the normal directions.
These zero modes make the contributions of all positive ghost number
observables of topological string on Fano vanish when Fano is embedded
into Calabi-Yau.

In our case, however, we get literally strings on ${\bC\bP}^1$. This model
is much richer
then the strings on conifold. In particular, as we show, the gravitational
descendants of the K\"ahler class of ${\bC\bP}^1$ are dual to the higher
Casimirs in the gauge theory.

It goes without saying that embedding our picture in the general story
of local mirror symmetry
will be beneficial for both. In particular, \bcov\ explains how the
topological string amplitudes arise as the physical string amplitudes
with the insertion of $2g$ powers of the sugra Weyl multiplet ${\CW}$,
the vertex operators for ${\CW}$ effectively twisting the worldsheet
theory.
We claim that the topological string with the gravitational descendants
(which are constructed with the help of the fields of topological gravity)
have direct and clear physical meaning on the gauge theory side. We do not
know at the moment how to embed them in the framework of \bcov. However
we shall make a suggestion.



\bigskip
\noindent{\it Organization of the paper}




The paper is organized as follows. The section $2$ discusses instanton
calculus in the ${\CN}=2$ susy gauge theories, both from the physical
point of view, and with the emphasis on its mathematical aspects,
related to the equivariant cohomology of the moduli spaces. The
equivariant methods lead to the evaluation of the integrals one
encounters in the gauge theory.
As a result one arrives at the generating function of the expectation
values of the chiral operators,
which is expressed as a partition function of a certain auxiliary
statistical model on the
Young diagrams. The section $3$ specifies these results for the
gauge group $U(1)$ and explains their interpretation from the
point of view of the little string theory. This section also
introduces the formalism of free fermions which are very efficient
in packaging the sums over partitions.
The section $4$ identifies the partition function with a simple
correlator of free fermions.



\newsec{${\CN}=2$ THEORY}



\subsec{Gauge theory realizations}


We start our exposition with the case of pure ${\CN}=2$
supersymmetric Yang-Mills theory with the gauge group $U(N)$ and its
maximal torus ${\bf T} = U(1)^N$. The field content of the theory
is given by the vector multiplet, whose components are: the complex
scalar ${\phi}$, two gluions ${\l}_{\a}^{i}$, $i=1,2$; ${\a}=1,2$ their
conjugates ${\bar\l}_{\dot \a i}$, and the gauge field $A_{\m}$ -- all
fields in the adjoint representation of $U(N)$.
The action is given by the integral over the superspace:
%
\eqn\ssyact{S \propto \int d^4 x \left( \int d^4 {\t} {\CF} ({\CA})
+ \int d^4 {\bar\theta} {\bar\CF}({\bar\CA}) \right)}
%
where ${\t}_{\a}^i$, ${\a}=1,2$; $i=1,2$ are the chiral Grassmann
coordinates on the superspace,
${\CA} = {\phi} + {\t}{\l} + {\t}{\t} F^{-} + \ldots$
is the ${\CN}=2$  vector superfield, and ${\CF}$ is
the prepotential (locally, a holomorphic gauge invariant function
of ${\phi}$). Classical supersymmetric Yang-Mills theory has
%
\eqn\prpcl{{\CF}({\phi}) = {\tau}_0 {\Tr} {\phi}^2}
%
where ${\tau}_0$ is a complex constant, whose real and imaginary parts
give the theta angle and the inverse square of the gauge coupling
respectively:
\eqn\brcp{{\tau}_0 = {{\vartheta}_0\over{2\pi}} + {4{\pi i}\over g_0^2} , }
the subscript $0$ reminds us that these are bare quantities, defined at
some high energy scale ${\m}_{UV}$.
 It is well-known that ${\CN}=2$ gauge theory has a
moduli space of vacua, characterized by the expectation value of
the complex scalar ${\phi}$ in the adjoint representation. One can
put $\langle \phi \rangle = a \in {\bf t} = Lie ({\bf
T})$, due to the potential term ${\Tr} [{\phi}, {\bar \phi}]^2$ in
the action of the theory.
We are studying the gauge theory on Euclidean space ${\bR}^4$, and
impose the boundary condition ${\phi} (x) \to a$, for $x \to \infty$.
It is also convenient to accompany the fixing of the asymptotics of
the Higgs field  by the fixing the allowed gauge transformations to
approach unity at infinity.


The ${\CN}=2$ gauge theory in four dimensions is a dimensional
reduction of the ${\CN}=1$ five dimensional theory. The latter theory
needs an ultraviolet completion to be well-defined. However, some
features of its low-energy behaviour are robust \seibergfive.

In particular, the effective gauge coupling runs because of the
one-loop vacuum polarization by the BPS particles. These particles
are W-bosons (for nonabelian theory),  four dimensional instantons,
viewed as solitons in five dimensional theory,  and the bound
states thereof.

To calculate the effective couplings we need to know the
multiplicities, the masses, charges, and the spins of the BPS
particles present in the spectrum of the theory \nikfive. This can
be done, in principle, by careful quantization of the moduli space
of collective coordinates of the soliton solutions (which are four
dimensional gauge instantons). Now suppose the theory is
compactified on a circle. Then the one-loop effect of a given particle
consists of a bulk term, present in the five dimensional
theory, and a new finite-size effect, having to do with the loops
wrapping the circle in space-time \nikfive. If in addition we
rotated the noncompact part of the space-time in going around the
circle, then the loops wrapping the circle would have to be
localized near the origin in the space-time. This localization is
at the core of the method we are employing. Its mathematical
implementation is discussed in the next section.


\lref\cecotti{S.~Cecotti, L.~Girardello, \pl{110}{1982}{39}}
\lref\smilga{A.~Smilga, Yad.Fiz. {\bf 43} (1986), 215-218}
\lref\sethi{S.~Sethi, M.~Stern, hep-th/9705046}
The contribution
of the BPS particles to running of the effective couplings
depends, of course, on their masses, spins, and charges \nikfive.
The generating function of all these effective couplings
can be identified with the
supersymmetric index-like quantity
$$
{\Tr}_{\CH} (-)^F e^{-r m}
e^{r \Omega\cdot {\bM}} e^{r \bA \cdot {\bI}}
$$
where $\bM$ is the
generator of the Lorentz rotations,  $\bI$ is the generator of
the R-symmetry rotations, and $r$ is the circumference of the fifth circle.
Under certain conditions on $\Omega$ and
$\bA$ this trace has some supersymmetry which allows to evaluate it.
In the process one gets some integrals over the moduli space of
instanton collective coordinates, as in
\cecotti\smilga\sethi\dbound\ihiggs. It is to these integrals that
one applies the equivariant localization techniques, described
below.

Before we discuss these integrals we should conclude our discussion of
the reduction of the five dimensional theory down to four dimensions.
Actually, we shall be even more general, and discuss the reduction from six dimensions.

One can
 arrive at the setup we shall use by lifting the theory to
${\CN} = (1,0)$ six dimensional theory, and then compactifying on
a two-torus with the twisted boundary conditions (along both $A$ and
$B$ cycles), such that as we
go around a non-contractable loop ${\ell} \sim n A + m B$, the
space-time and the fields of the gauge theory charged under the
R-symmetry group $SU(2)_I$ are rotated by the element $( e^{i ( n
a_1  + m b_1) } , e^{ i (n a_2 + m b_2 ) } , e^{i (n a_2 + m b_2)}
) \in SU(2)_L \times SU(2)_R \times SU(2)_{I}= Spin(4) \times
SU(2)_{I}$. In other words, we compactify the six dimensional
${\CN}=1$ susy gauge theory on the manifold with the topology
${\bT}^2 \times {\bR}^4$ with the metric and the R-symmetry gauge
field Wilson line:
\eqn\twm{\eqalign{& ds^2 = r^2 dz d{\zb} +
(dx^{\m} + {\Omega}^{\m}_{\n}x^{\n} dz)( dx^{\m} +
{\bar\Omega}^{\m}_{\n}x^{\n} d{\zb}), \cr & \qquad
{\bA}^{a} = ({\Omega}^{\m\n} dz + {\bar\Omega}^{\m\n}
d{\zb}) {\s}_{\m\n}^{a},
\ {\m}=1,2,3,4, \ a=1,2,3 \cr}}
it is convenient to combine $a_{1,2}$ and $b_{1,2}$ into two complex
parameters ${\e}_{1,2}$:
\eqn\eps{{\e}_1 - {\e}_2 = 2(a_1 + i b_1),
\qquad {\e}_1 + {\e}_2 = 2(a_2 + i b_2)}
The antisymmetric matrices ${\Omega}, {\bar\Omega}$ are given by:
\eqn\omgs{{\Omega}^{\m\n} = \pmatrix{ 0 & {\e}_1 & 0 & 0 \cr -{\e}_1
& 0 & 0 & 0 \cr 0 & 0 & 0 & {\e}_2 \cr 0 & 0 & - {\e}_2 & 0 \cr},
\qquad {\bar\Omega}^{\m\n} =
\pmatrix{ 0 & {\bar\e}_1 & 0 & 0 \cr -{\bar\e}_1
& 0 & 0 & 0 \cr 0 & 0 & 0 & {\bar\e}_2 \cr 0 & 0 & - {\bar\e}_2 & 0 \cr}}
In the limit $r \to 0$ we get four dimensional gauge theory. We could also
take the limit to the five dimensional theory, by considering the
degenerate torus ${\bf T}^2$. We note in passing that the complex
structure of the torus ${\bf T}^2$ could be kept finite.
The resulting four dimensional theory (for gauge group $SU(2)$) is
related to the theory of the so-called E-strings \estring\ganor.
The instanton contributions to the correlation functions of the
chiral operators in this theory are related to the elliptic genera
of the instanton moduli space \cstw\ and could be summed up, giving
rise to the Seiberg-Witten curves for these theories.
However, in this paper we shall neither discuss elliptic, nor
trigonometric limits, even though they lead to interesting
integrable systems \fivedim.


The action of the four dimensional theory in the limit $r \to 0$
is not that of the pure supersymmetric
Yang-Mills theory on ${\bR}^4$, but rather is a deformation of it
by the ${\Omega}$, ${\bar\Omega}$-dependent terms.
We shall write down here only the terms with bosonic fields at
$\vartheta_0=0$:
%
\eqn\dfrms{S ({\Omega})^{bos} =
-{1\over{2g_0^2}} {\Tr} \left(  {\half} F_{\m\n}^2 +
( D_{\m}{\phi} - {\Omega}^{\n}_{\l} x^{\l} F_{\m\n} )
(D_{\m}{\bar\phi} - {\bar\Omega}^{\n}_{\l} x^{\l}F_{\m\n} ) +
[{\phi},{\bar\phi}]^2 \right)}
%
It is amusing that this deformation can be described as a
superspace-dependent bare coupling ${\tau}_0$:
\eqn\sspdpc{{\tau}_0 (x, {\theta}; {\m}_{UV}) = {\tau}_0 ({\m}_{UV})
+ {\bar\Omega}^{-} {\t}{\t} + {\Omega}_{\m\n} {\bar\Omega}_{\m\l} x^{\n}
x^{\l}}
We are going to study the correlation functions of chiral observables.
These observables are gauge invariant  holomorphic functions of
the superfield ${\CA}$.
Viewed as a function on the superspace, every such observable ${\CO}$ can
be decomposed:
\eqn\dcmps{{\CO}[{\CA}(x,{\t})] = {\CO}^{(0)} + {\CO}^{(1)}{\t}+\ldots +
{\CO}^{(4)} {\t}^4}
The component ${\CO}^{(4)}$ can be used to deform the action of the
theory, this deformation is equivalent to the addition of ${\CO}$ to
the bare prepotential.

The nice property of the chiral observables is the independence of
their correlation functions of the
anti-chiral deformations of the theory, in particular of
${\bar\tau}_0$\foot{However, one should be aware of the possibility of
holomorphic anomaly.}. We can, therefore, consider the limit
${\bar\tau}_0 \to \infty$. In this limit the term:
$$
{\bar\tau}_0 \Vert F^{+} \Vert^2
$$
in the action localizes the path integral onto the instanton
configurations. The resulting instanton measure can be calculated
quite explicitly and the final integral calculated. We shall discuss
this shortly.


Now we want to pause to discuss other physical realizations of our
${\CN}=2$ theories.

\subsec{String theory realizations}

The ${\CN}=2$ theory can arise as a low energy  limit of the theory on
a stack of D-branes in type II gauge theory. A stack of $N$ parallel
D3 branes in IIB theory in flat ${\bR}^{1,9}$ carries ${\CN}=4$
supersymmetric Yang-Mills theory \witbound.
A stack of parallel D4 branes in IIA theory in flat ${\bR}^{1,9}$
carries ${\CN}=2$ supersymmetric Yang-Mills theory in five dimensions.
Upon compactification on a circle the latter theory reduces to the former
in the limit of zero radius.

Now consider the stack of $N$ D4 branes in the geometry
${\bS}^1 \times {\bR}^{1,8}$
with the metric:
\eqn\stck{ds^2 = dx^{\m} dx^{\m} + r^2 d{\varphi}^2 + d v^2   +
\vert dZ_1 + m Z_1 d{\varphi} \vert^2 +
 \vert d Z_2 - m Z_2 d{\varphi} \vert^2}
Here $x^{\m}$ denote the coordinates on the Minkowski space
${\bR}^{1,3}$, ${\varphi}$ is the periodic coordinate on the circle of
circumference $r$,
%${\varphi} \sim {\varphi} + 1$,
$v$ is a real
transverse direction,
$ Z_1$ and $Z_2$  are the holomorphic coordinates on the
remaining ${\bC}^2$.
The worldvolume of the branes is ${\bS}^1 \times {\bR}^{1,3}$, which is
located at $Z_1 = Z_2 = 0$,
and $v=v_l$, $l=1, \ldots, N$. Together with the Wilson loop eigenvalues
$e^{i{\s}_1}, \ldots, e^{i{\s}_N}$ around ${\bS}^1$ $v_l$'s  form $N$
complex moduli $w_1, \ldots, w_N$, parameterizing the moduli space of
vacua. In the limit $r \to 0$ the $N$ complex moduli loose periodicity.



It is easy to check that the worldvolume theory has ${\CN}=2$ susy,
with the massive hypermultiplet in the adjoint representation. This
realization is T-dual to the standard
realization with the NS5 branes \wittensolution\foot{NN thanks M.~Douglas
for the illuminating discussion on this point.}. Note that the background
\twm\ is similar to \stck\ but the D-branes are
located differently, which leads to very interesting geometries upon
T-dualities
and lifts to M-theory \cmn, which provide another useful insight.

However, in our story we want to analyze the pure ${\CN}=2$
supersymmetric Yang-Mills theory.
This can be achieved by taking $m \to \infty$ limit, at the same time
taking the weak string coupling limit. The resulting brane configuration
can be described using two parallel NS5 branes and $N$ D4 brane suspended
between them, as in \wittensolution, or, alternatively,
as a stack of $N$ D3 (fractional) branes stuck at the ${\bC}^2/{\bZ}_2$
singularity, as in \frbranes.
The relation between these two pictures is through the T-duality of the
resolved ${\bC}^2/{\bZ}_2$ singularity. The fractional D3 branes blow up
into D5 branes wrapping a non-contractable two-sphere.
The resolved space $T^*{\bP}^1$ has a $U(1)$ isometry, with two
fixed points
(the North and South poles of the non-contractable two-sphere).
Upon T-duality these turn into
two NS5 branes. The D5 branes dualize to D4 branes suspended between
NS5's.



The instanton effects in this theory are due to the fractional D(-1)
instantons, which bind to the fractional D3 branes, in the IIB description.
The ``worldvolume'' theory on these D(-1) instantons is the supersymmetric
matrix integral, which we describe with the help of ADHM construction below.
In the IIA picture the instanton effects are due to Euclidean D0 branes,
which ``propagate'' between two NS5 branes.

The IIB picture with the fractional branes corresponds to the metric
(before ${\Omega}$ is turned on):
\eqn\stckii{ds^2 = dx^{\m} dx^{\m} + dwd{\wb}   + ds^2_{{\bC}^2/{\bZ}_2}}

The singularity ${\bC}^2/{\bZ}_2$ has five moduli in IIB string theory:
the three parameters
of the geometric resolution of the singularity, and the fluxes of the NSNS
and RR 2-forms through the two-cycle which appears after blowup.
The latters which are responsible for the gauge couplings on the fractional
D3 branes \cftorb:
\eqn\ggcpl{{\tau}_0 = \int_{{\bS}^2} B_{RR} + {\tau}_{IIB}
\int_{{\bS}^2}B_{NSNS}}

Our conjecture is that turning on the higher Casimirs, (and gravitational
descendants on the dual closed string side) corresponds to a ``holomorphic
wave'', where ${\tau}_0$ starts
holomorphically depend on $w$. This is known to be a solution of
IIB sugra \flow.


We shall return to the fractional brane picture later on. Right now let us
mention another stringy effect. By turning on the constant NSNS B-field
along the worldvolume of the D3-branes we deform the super-Yang-Mills on
${\bR}^4$ to the
super-Yang-Mills on the noncommutative ${\bR}^4_{\t}$ \cds\witsei\connes.
On the worldvolume of the D(-1) instantons the noncommutativity acts as a
Fayet-Illiopoulos term, deforming the ADHM equations \abs\abkss\neksch,
and resolving the singularities of the instanton moduli space, as in
\nakajima. We shall use this deformation as a technical tool, so we shall
not describe it in much detail. The necessary references can be found in
\witsei.

At this point we remark that even for $N=1$ the instantons are present
in the D-brane picture.
They become visible in the gauge theory when noncommutativity is turned on.
Remarkably, the actual value of the noncommutativity parameter ${\t}$
does not affect the expectation values of the chiral observables, thus
simplifying our life enormously.

So far we presented the D-brane realization of ${\CN}=2$ theory. There
exists another useful realization, via local Calabi-Yau manifolds
\vafaengine. This realization, as we already explained
in the introduction is useful in relating the prepotential to the
topological string amplitudes.
If the theory is embedded in the IIA string on local Calabi-Yau,
then the interesting
physics comes from the worldsheet instantons, wrapping some 2-cycles
in the Calabi-Yau.
In the mirror IIB description one gets a string without worldsheet
instantons contributing
to the prepotential, and effectively reducing to some field theory.
This field theory is known in the case of global Calabi-Yau. But it
is not known explicitly in the case of local Calabi-Yau.
As we shall show, it can be sometimes identified with the free fermion
theory on auxiliary Riemann surface (cf. \dijkgraaf).

Relation to the geometrical engineering \vafaengine\ is also useful in
making contact between our ${\Omega}$-deformation and the sugra backgrounds
with graviphoton field strength.
Indeed, our construction involved a lift to five or six dimensions. The
first case embeds easily to IIA string theory where
 this corresponds to the lift to M-theory. To see the whole six dimensional
picture \twm\ one should use IIB language and the lift to F-theory (one
has to set ${\bar\Omega}=0$, though).

Let us consider the five dimensional lift.
We have M-theory on the 11-fold with the metric:
%
\eqn\mmetrc{ds^2 = ( dx^{\m} + {\Omega}^{\m}_{\n}x^{\n} d{\varphi})^2 +
r^2 d{\varphi}^2 +  ds_{CY}^2}
%
Here we assume, for simplicity, that ${\e}_1 = - {\e}_2$, so that
${\Omega} = {\Omega}^{-}$ generates an $SU(2)$ rotation, thus preserving
half of susy.
Now let us reduce on the circle ${\bS}^1$ and interpret the background
\mmetrc\ in the type IIA string. Using \wittenm\ we arrive at the
following IIA background:
\eqn\iias{\eqalign{& g_s = \left( r^2 + \Vert \Omega \cdot x \Vert^2
\right)^{3\over 4} \cr
& A^{grav} = {1\over{r^2 + \Vert \Omega \cdot x \Vert^2}} {\Omega}_{\m\n}
x^{\m} dx^{\n} \cr
& ds_{10}^{2} = {1\over{\sqrt{r^2 + \Vert \Omega \cdot x \Vert^2}}}
\left( r^2 dx^2 + {\Omega}^{\m}_{\n}{\Omega}^{\l}_{\k} \left( x^2 dx^2
{\d}^{\n\k}{\d}_{\m\l} - x^{\n}x^{\k} dx^{\m} dx^{\l} \right)\right) +
\cr & \qquad \qquad \qquad\qquad \qquad \qquad + \sqrt{ r^2 + \Vert
\Omega \cdot x \Vert^2} ds_{CY}^2 \cr}}
where the graviphoton $U(1)$ field is turned on. The IIA string coupling
becomes strong at $x \to \infty$. However, the effective coupling in the
calculations of ${\CF}_g$ is
\eqn\effcpl{{\hbar} \sim g_s \sqrt{ \Vert dA^{grav} \Vert^2} \sim
\left( r^2 +\Vert \Omega \cdot x \Vert^2\right)^{-{1\over 4}} \to 0,
\qquad x \to \infty}


\subsec{Our goal}

Our ultimate goal is the calculation of the partition function
\eqn\partnf{Z({\tau}_{\vec n}; a, {\Omega}) = \int_{{\phi}({\infty}) = a}
D{\Phi}DA D{\l} \ldots e^{-S({\Omega})}}
of the ${\CN}=2$ susy gauge theory
with all the higher couplings \mcrsc\ on the background \twm\ with
the fixed asymptotics of the Higgs field at infininity.
We use the fact that the chiral deformations are not sensitive to
the anti-chiral parameters (up to holomorphic anomaly \niklos).
We take the limit ${\bar\tau}_0 \to \infty$, and the partition function
becomes the sum over the instanton charges of the integrals over the
moduli spaces ${\CM}$ of instantons of the measure, obtained by the
developing the path integral perturbation expansion around instanton
solutions.

On the other hand, if we take instead a low-energy limit, this
calculation should reduce to that of low-energy effective action.
In the standard Seiberg-Witten story the low-energy theory is
characterized by the complexified energy scale
${\Lambda} \sim {\m}_{UV} e^{2\pi i {\tau}_0 ({\m}_{UV})}$.
We now recall \sspdpc. In our setup the low-energy scale is
$(x,{\t})$-dependent:
\eqn\irsc{{
\Lambda}(x,{\t} ) = {\m}_{UV} e^{2\pi i {\tau}_0 (x, {\t};
{\m}_{UV})} = {\Lambda} e^{2\pi i {\bar\Omega}^{-} {\t}^2 -
\Vert {\Omega} \cdot x \Vert^2}}
%
Near $x=0$ it is finite, while at $x \to \infty$ the theory becomes
infinitely weakly coupled.
With \sspdpc\ in mind we can easily relate the partition function
to the prepotential \gnfnc (cf. \swi):
\eqn\rltns{\eqalign{& Z = Z^{pert} \ {\exp} \int d^4 x d^4 {\t}
{\CF}^{inst} \left(a  ; {\tau}_{\vec n}; {\Lambda}(x,{\t}) \right)+
{\it higher \ derivatives} = \cr & \qquad = {\exp}\
{1\over{{\e}_1 {\e}_2}} \left(  {\CF}^{inst} (a, {\tau}_{\vec n};
{\Lambda})+ O ({\e}_1, {\e}_2) \right)\cr}}
where ${\CF}^{inst}$ is the sum of all instanton corrections
to the prepotential, and $Z^{pert}$ is the result of the
perturbative calculation on the ${\Omega}$-background.
The corrections in ${\e}_{1,2}$ come from the ignored higher
derivative terms.
The perturbative part is given by the one-loop contribution from
W-bosons, as well as non-zero modes of the abelian photons.
Recall that in the $\Omega$-background one can integrate out
all non-zero modes, as ${\Omega}$ lifts all massless fields.
Because of the reduced supersymmetry
the determinants do not quite cancel. The simplest way to calculate
them is to go to the basis of spherical harmonics:
$$
W_{lm} \sim \sum_{i,j, {\ib}, {\jb} \geq 1}
W^{lm}_{ij{\ib}{\jb}}\
 z_1^{i-1}z_2^{j-1} {\zb}_1^{{\ib}-1}{\zb}_2^{{\jb}-1},
\qquad l,m = 1, \ldots N
$$
Only the top spin projections contribute, giving rise to:
%
\eqn\prt{Z^{pert} =\ {\prod_{l,m; i,j \geq 1}}^{\kern -.12in\prime}\
%^{\circ}
\left( a_l - a_m + {\e}_1 (i-1) + {\e}_2 (j-1) \right) }
%
times the conjugate term, which depends on $\bar a$. We shall
ultimately take $\bar a \to\infty$, so we ignore this term --- at
any rate, it cancels out in the correlation functions of the chiral
observables . The symbol ${\prod}^\prime$ in \prt\ means that the
contribution of the zero modes $l=m$, $i=j=1$ to the product
is omitted. We shall always understand \prt\
in the sense of $\zeta$-regularization. After regularization one can
analytically continue to ${\e}_1 + {\e}_2= 0$.


In fact, for
${\e}_1 = - {\e}_2 = {\hbar}$ one can expand:
\eqn\gns{Z({\tau}_{\vec n}; a, {\Omega}) =
{\exp} \left(- \sum_{g=0}^{\infty} {\hbar}^{2g-2} {\CF}_{g}
(a; {\tau}_{\vec n}; {\Lambda})\right)}
The higher ``prepotentials'' ${\CF}_{g}$ will turn out later to be
related to the higher genus string amplitudes.

\subsec{Instanton measure}


As will be hopefully clear by the end of these coming sections,
the ${\CN}=2$ instanton measure (together with the
observables ${\CO}$) translates to the ${\bf G}$-equivariant
cohomology class ${\Omega}_{\CO}$ of ${\CM}$. The correlation
function in the gauge theory becomes simply the integral
\eqn\corf{\langle {\CO} \rangle = \int_{\CM} {\Omega}_{\CO}} The
group ${\bf G}$ acts in the instanton moduli space by rotating the
gauge orientation of the instantons at infinity (in other words,
${\CM}$ is the quotient of the space of anti-self-dual gauge
fields by the gauge transformations approaching identity at
infinity)\foot{The equivariant cohomology classes are represented
with the help of the equivariant forms. These are functions on
$\bf g$ with the values in the de Rham complex of ${\CM}$. In
addition, these functions are required to be $\bf G$-equivariant,
i.e. the the adjoint action of $\bf G$ on $\bf g$ must commute
with the action of $\bf G$ on the differential forms on $\CM$. The
differential forms on ${\CM}$ have a degree. The equivariant forms
are bi-graded: by the degree $p$ of the form on ${\CM}$ and by the
degree $q$ of the function on ${\bg}$. The equivariant
differential raises by one the combination $p+2q$ of these
degrees. The equivariant cohomology is graded by this combination
only. Thus, when we speak of a representative of the equivariant
cohomology class, it may be given by an inhomogeneous differential
form on ${\CM}$:
%
\eqn\repr{{\bo} \in H^{d}_{\bG}({\CM})
\leftrightarrow {\o} = \sum_{p + 2q = d} {\o}^{(p)}_{i}
f_{i}^{(q)}}
%
where $f_{i}^{(q)}$ is a homogeneous degree $q$
polynomial on ${\bg}$. Thanks to their equivariance such functions
$\o$ are uniquely determined by their values on $\bf t$ (where the
only additional constraint is invariance under the Weyl group
reflections). So let us fix an
element ${\bf a} \in {\bf t}$. The value of ${\Omega}_{\CO}[a]$ is
an ordinary differential form on $\CM$ and its integral can be
conventionally defined.}. Such integrals can be computed using
localization. In plain words it means that there are given by the
sums over the fixed points of the action of the one-parametric
subgroup ${\exp} (t a) $, $t \in {\bR}$, of ${\bG}$. The
contribution of each fixed point $P \in {\CM}$ (assuming it is
isolated and $\CM$ is smooth at this point) is given by the ratio:
\eqn\contr{Z_{P} =
{{{\Omega}_{\CO}[{\ba}]^{(0)}\vert_{P}}
\over{c(T{\CM})[{\ba}]^{(0)}\vert_{P}}}
} where ${\o}^{(0)}$ denotes the scalar component of the
inhomogeneous differential form corresponding to the equivariant
differential form ${\o}$, and $c(T{\CM})$ is the equivariant Chern
polynomial of $T{\CM}$. It is defined as follows. As $T{\CM}$ is
$\bG$-equivariant, with respect to the maximal torus $\bT$ it
splits as a direct sum of the line bundles,
$$
T{\CM} = \bigoplus_{i} L_i
$$
on which $\bt$ acts with some weight $w_i$ (a
linear function on $\bt$). The equivariant Chern polynomial is
defined simply by: $$ c(T{\CM } )[{\ba}] = \prod_{i} \left( c_1
(L_i) + w_i \left( {\ba} \right) \right) $$ Physicists are
familiar with the Duistermaat-Heckmann \DHf\ formulae like \contr\
in the context of two-dimensional Yang-Mills theory \witdgt, and
in (perhaps less known) the context of sigma models \maxim. In
order to proceed we need to calculate the numerator and the
denominator of \contr\ and to sum over the points $P$.

\bigskip
\noindent{\it Bundles over instanton moduli space}

Here we recall some standard constructions. The problem considered
here is absolutely typical in the soliton physics. One finds some
moduli space of solutions (collective coordinates) which should be
quantized. The supersymmetric theories lead to supersymmetric
quantum mechanics on the moduli spaces. If the gauge symmetry is
present the collective coordinates are defined with the help of
some gauge fixing procedure, which leads to the complications
described below.


\sssec{\bf Tangent \ and \ universal \ bundles.} The tangent space
to the instanton moduli space ${\CM}$ at the point $m$ can be
described as follows. Pick a gauge field $A$ which corresponds to
$m\in\CM$, $F^{+}(A)=0$. Any two such choices differ by a gauge
transformation. Now consider deforming $A$: $$A \to A + {\d} A$$
so that the new gauge field also obeys the instanton equation
$F^{+}(A + {\d} A) = 0$. In other words, ${\d}A$ obeys the linear
equations: \eqn\tngntb{\eqalign{& D^{+}_{A} {\d}A = 0 \cr &
D^{*}_{A} {\d} A = 0 \cr}} where the first equation is the
linearized anti-self-duality equation, while the second is the
gauge choice, to project out the trivial deformations ${\d}A \sim
D_{A} {\ve}$. Let us choose some basis in the (finite-dimensional)
vector space of solutions to \tngntb: ${\d} A = a_{\m}^{K} dx^{\m}
{\z}_{K}$, where $a^K$ obey \tngntb, and, say, are orthonormal
w.r.t to the natural metric
$\langle a^L \vert a^K \rangle \equiv \int_{{\bf R}^4} a^L
\wedge \star a^K = {\d}_{LK}$, $L,K = 1, \ldots, {\rm dim}{\CM}$.
Now suppose we
have a family of instanton gauge fields, parameterized by the
points of ${\CM}$: $A_{\m} (x; m)$, where $x \in {\bf R}^4, \ m
\in {\CM}$. Let us differentiate $A_{\m}$ w.r.t the moduli $m$.
Clearly, one can expand:
\eqn\expns{{{{\p} A}\over{{\p} m^L}} = a^{K} {\z}_{LK} + D_{A} {\ve}_{L}}
The compensating gauge
transformations ${\ve}_L$ together with $A_{\m} (m)$ form a connection
${\bf\CA} = A_{\m} (x; m) dx^{\m} + {\ve}_{L} d m^L$ in the rank $N$
vector bundle ${\CE}$ over ${\CM} \times {\bR}^4$. A
mathematically oriented reader would object at this point, as it
{\it is well-known} that universal bundles together with a nice
connections do not exist over the compactified moduli spaces. We
shall not pay attention to these (fully just) remarks, as
eventually there is a way around. We find it more straightforward
to explain things as if such objects existed over the compactified
moduli space of instantons. Let $p$ denote the projection ${\CM}
\times {\bf R}^4 \to {\CM}$. Suppose we know everything about
${\CE}$. How would we reconstruct $T{\CM}$ from there? We know
already that the tangent space to ${\CM}$ at a point $m$ is
spanned by the solutions to \tngntb. It is plain to identify these
solutions with the cohomology of the Atiyah-Singer complex:
\eqn\atsc{0 \longrightarrow {\Omega}^{0}({\bf R}^4) \otimes
{\bg} \longrightarrow {\Omega}^{1}({\bf R}^4) {\otimes} {\bg}
\longrightarrow {\Omega}^{2,+} ({\bf R}^4) \otimes {\bg}
\longrightarrow 0}
where the first non-trivial arrow is the
infinitesimal gauge transformation: ${\ve} \mapsto D_{A} {\ve}$
and the second it ${\d} A \mapsto D^{+}_{A} {\d} A$. Thanks to
$F^{+}_{A} = 0$ this is indeed a complex, i.e.
$D^{+}_{A} D_{A} = 0$.
The spaces ${\Omega}^{k} \otimes {\bg}$ can be viewed as the
bundles over ${\CM} \times {\bf R}^4$, e.g. for ${\bG} = U(N)$
\eqn\forms{{\Omega}^{k} ({\bf R}^4) \otimes {\bg} = {\CE}
\otimes {\CE}^{*} \otimes {\Lambda}^k T^*{\bf R}^4}
Generically
the complex \atsc\ has only $H^1$ cohomology. We are thus led to
identify K-classes: $T{\CM} = H^{1} - H^{0} - H^{2}$.


\sssec{\bf Framing \ and \ Dirac \ bundles.} We shall need two
more natural bundles over ${\CM}$. As ${\CM}$ is defined by the
quotient w.r.t. the group of gauge transformations, trivial at
infinity, we have a bundle $W$ over ${\CM}$ whose fiber is the
fiber of the original $U(N)$ bundle over ${\bf R}^4$ at infinity.
Another important bundle is the bundle $V$ of Dirac zero modes.
Its fiber over the point $m \in {\CM}$ is the space of
normalizable solutions to the Dirac equation in fundamental
representation in the background of the instanton gauge field,
corresponding to $m$. In the language of K-theory,

\eqn\kthrl{\eqalign{& W = \lim_{x \to \infty} {\CE} \vert_{x} \cr
& V = p_{*} {\CE} \cr}}The pushforward $p_{*}$ is defined here in
$L^2$ sense. In what follows we shall need its equivariant
analogue. Finally, let $S_{\pm}$ denote the bundles of positive
and negative chirality spinors over ${\bR}^4$. These bundles are
trivial topologically. However they are nontrivial as bundles,
equivariant with respect to the group of rotations of ${\bR}^4$.

\sssec{\bf Relations \ among \ bundles.} We arrive at the
following relation among the virtual bundles: \eqn\tang{\eqalign{&
{\CE} = W \oplus V \otimes \left( S_{+} - S_{-} \right) \cr &
T{\CM} = - p_{*} \left( {\CE} \otimes{\CE}^{*} \right) \cr}} The
chiral operators ${\CO}_{\vec n}$ we discussed in the introduction now
are in one-to-one correspondence with the characteristic classes
of the $U(N)$ bundles. A convenient basis in the space of such
classes is given by the skew Schur functions, labeled by the
partitions ${\l} = ( {\l}_1 \geq {\l}_2 \geq \ldots {\l}_N \geq 0
)$: \eqn\shcf{{\bf ch}_{\l} = {\Det} \Vert ch_{{\l}_{i} - i + j}
\Vert}  Another basis is labeled by finite sequences $n_1, n_2,
\ldots, n_k$ of non-negative integers:
\eqn\bsbs{{\CO}_{\vec n} =
{\prod}_{J=1}^{\infty} (ch_J)^{n_J}}
It is this basis that we used in \mcrsc.

\bigskip
\noindent{\it Back to physical fields}

The conclusion of our somewhat formal discussion is approaching. The
importance of ${\CE}$ in theoretical physics is justified by the
following identification:
\eqn\chrn{
{\Tr}{\Phi}^J  = ch_J({\CE}) + \{ Q, \ldots \}}
(as a superfield),
where $\{ Q, \ldots \}$ denote irrelevant supersymmetric variations.
This is justified by the
straightforward analysis of the instanton zero modes.
In particular, up to irrelevant $\{ Q, \ldots \}$ terms one
finds in the background of the instanton gauge field $A$ the
fermionic zero modes:
$$
{\l}_{\a}^{i} \sim {\s}_{\m, \a i } a_{\m}^{K} {\psi}_K
$$
and the expression for the Higgs field
$$
{\Phi} = {1\over{{\Delta}_A}} [ a^K, \star a^L ] {\psi}_K {\psi}_L
$$
where ${\Delta}_A\equiv D^{*}_AD_A$,
which identifies it with the curvature two-form on the moduli space
${\CM}$.
In fact, together with ${\psi}_K$ and $F_{A}$ they combine into a
curvature form on
${\CM} \times {\bR}^4$:
$$
{\Phi} + {\l}{\t} + F_{A}{\t}^2 = {\rm Curvature}\ {\bf\CA}$$
thereby jusifying \chrn.




We shall present the explicit formula for the localized $ch_{J}$,
$J=0,1,2,\ldots$ below.


\bigskip
\noindent{\it Rotational symmetries}


The fixed points of the group
${\bG}$ are the point-like instantons. They do not form a nice
isolated set of points. Moreover, the space of point-like
instantons is non-compact, as they can run away to infinity. One
can, in principle, get around this difficulty by carefully
analyzing the contribution of this singular locus in the instanton
moduli space \hollowood, but one can get more information and save
some time by employing yet another symmetry of the problem. The
symmetry in question is the rotations of the space-time ${\bf
R}^4$. Namely, let us work equivariantly with respect to
$Spin(4)$. In the supersymmetric field theory language it means
that we place our theory in a non-trivial supergravity
 background,
with a non-trivial vev of a graviphoton field strength. Practically this
means that we turn on the ${\Omega}$-deformation described in the
beginning of this chapter.
In the context of equivariant cohomology of the moduli space of
instantons, in the case of two dimensional sigma model, the idea
to employ the rotational symmetries of the worldsheet
was pursued in \givental.

\subsec{ADHM construction} To get a handle on these fixed point
sets and to calculate the characteristic numbers of the various
bundles we have defined above, we need to remind a few facts about
the actual construction of $\CM$, the so-called ADHM construction
\adhm\nakajima. In this construction one starts with two Hermitian
vector spaces $W$ and $V$ (it is not accidental that we label them
in the same way as some of the bundles above). One then looks for
four Hermitian operators ${\bX}^{\m} : V \to V$, ${\m}=1,2,3,4$
and two complex operators ${\l}_{\a} : W \to V$, ${\a}=1,2$ (and
${\bar\l}_{\dot \a}
= {\l}_{\a}^{\dagger}: V \to W$), which can be
combined into a  sequence:
\eqn\adhms{0 \to W \otimes S_{-}
\longrightarrow V \oplus W \otimes S_{+} \to 0}
where the
non-trivial map is given by: $$ {\CD}^{+} = {\l} \oplus {\bX}^{\m}
{\s}_{\m} $$ The ADHM equation requires that ${\CD}{\CD}^{+}$
commutes with the Pauli matrices ${\s}_{\m}$ acting in $S_{-}$. In
addition, one requires that ${\CD}{\CD}^{+}$ has a maximal rank.
The moduli space ${\CM}$ is then identified with the space of such
${\bX}, {\l}$ up to the action of the group $U(V)$ of unitary
transformations in $V$. The group $U(W)$ acts on ${\CM}$ by the
natural action, descending from that on  ${\l}$ (${\bX}$ are
neutral). The group $Spin(4)$ acts on ${\CM}$ by rotating ${\bX}$
in the vector representation and ${\l}$ in the appropriate chiral
spinor representation. From this formalism one can extract the
expressions for the Chern character of the universal bundle.


\subsec{D-brane picture, again}

The ADHM construction becomes very natural when the gauge theory is
realized with the help of D-branes. The space $V$ is the Chan-Paton
space for the D(-1) branes ,
while $W$ is the Chan-Paton space for the D3 branes. The matrices
${\bX}$ are the ground states
of the $(-1,-1)$ strings, while ${\l}_{\a}, {\bar\l}_{\dot a}$ are
those of $(-1,3)$, $(3,-1)$.
The ADHM equations are the conditions for unbroken susy. Their
solutions describe the Higgs branch of the D(-1) instanton
theory\foot{To make this statements literally true one should consider
D2-D6 system instead of D(-1)-D3 (to avoid off-shell string amplitudes,
and the  non-existence of moduli spaces of vacua in the field theories
less then in three dimensions).}
The D(-1) instantons also carry a multiplet responsible for the $U(V)$
``gauge'' group. In particular,
quantization of $(-1,-1)$ strings in addition to $\bX$ gives rise to a
matrix $\phi$ in the adjoint of $U(V)$, which represents the motion of
D(-1) instantons in the directions, transverse to D3 branes.




\subsec{Instanton measure, revisited}

We now write an expression for the instanton measure in terms of the
matrices ${\bX}, {\l}$.
We use the fact that ${\CM}$ is the quotient of a variety given by
some equations in a vector space by a symmetry group. Let us describe
a general construction for the integration theory over such spaces.


Abstractly, given a space ${\CX}$ with coordinates $x^{\m}$, we represent
the exterior differentials
$dx^{\m}$ by ${\psi}^{\m}$. Suppose, in addition, that we have a symmetry
group ${\CG}$ acting on
${\CX}$, the action being generated by the vector fields $V^{\m}_a$,
$a = 1, \ldots, {\rm dim}{\CG}$.
Moreover, we want to impose some equations ${\m}^A(x) =0$,
$A=1, \ldots, n$. We assume
these equations transform in some representation of ${\CG}$, i.e.
$$
{\m}^A (g \cdot x) = T(g)^A_B {\m}^B (x)
$$
Our ultimate goal is to define integration theory over:
$$
{\CM} = {\m}^{-1}(0)/{\CG}
$$
We introduce the Koszul multiplet ${\chi}_{A}, H_{A}$, and the fermionic
gauge fixing multiplet:
${\eta}, {\bar\phi}$, together with the bosonic antighost ${\phi}$ --
all three in the adjoint of ${\CG}$.
The equivariant differential -- non-linearly realized susy
acts as follows:
\eqn\qch{\eqalign{& Q x^{\m} = {\psi}^{\m}, \qquad
Q {\psi}^{\m} = V^{\m}_a (x) {\phi}^a
\cr & Q {\chi}_A = H_A, \qquad Q H_A = (T_a)_A^B{\phi}^a {\chi}_B \cr
& Q {\bar\phi} = {\eta}, \qquad\qquad Q {\eta} =
[ {\phi}, {\bar\phi} ] \cr}}
Suppose, in addition, that  on ${\CM}$ we have a ${\CG}$-invariant
metric $g_{\m\n}$.
Then the ``action'' reads:
%
\eqn\actns{\eqalign{& {\CS} =
Q \left( g_{\m\n} V^{\m}_a {\bar\phi}^a {\psi}^{\n} +
{\eta} [ {\phi}, {\bar\phi}] +i {\chi}_A {\m}^A \right) =
\cr
& \qquad\qquad  = g_{\m\n} V^{\m} ({\phi}) V^{\n} ({\bar\phi}) +
g_{\m\n} V^{\m}_a {\eta}^a {\psi}^{\n} +
{\psi}^{\m}{\psi}^{\n}{\nabla}_{\m} V_{\n}({\bar\phi}) +
\cr &\qquad\qquad + [{\phi},{\bar\phi}]^2 + [{\eta}, {\phi}]{\eta}
+ iH_A {\m}^A (x) + i{\chi}_A {\p}_{\n} {\m}^A  {\psi}^{\n} \cr}}
%
The ``instanton'' measure is:
\eqn\insmsr{{1\over{{\rm Vol}\ \CG }}\
{DxD{\psi}D{\chi} DH D{\eta} D{\bar\phi}D{\phi}}
 \ e^{-{\CS}}}
The actual instanton calculus involves two types of integrals like
\insmsr. The first type corresponds to the gauge groups ${\CG}$.
The space ${\CX}$ in this case is the space of all gauge fields on
${\bR}^4$, the group ${\CG}$ is the group of all gauge transformations,
which are equal to unity at infinity. The equations ${\mu}^A=0$ are
the anti-self-duality equations $F^{+}=0$.

The second type of integrals correspond to ${\CX} = {\CM}$ -- the moduli
space of instantons,
%${\m} = 0$,
and ${\CG}$ being  the global symmetry
group. In our case it is  the product  of the group of global gauge
transformations $U(N)$ and the group of space-time rotations $SO(4)$.
In the case of global symmetry group one does not integrate over
${\eta}, {\bar\phi}, {\phi}$. Moreover,
it is easy to see that the integral \insmsr\ with ${\eta} =0$,
${\bar\phi}$ -- arbitrary, but commuting with ${\phi}$ still has
a fermionic symmetry $Q$. The integral actually depends only on
the conjugacy class of ${\phi}$.  For the global group
$U(N) \times SO(4)$ it means that we can assume:
\eqn\phassm{{\phi} = {\rm diag} ( a_1, \ldots, a_N ) \oplus
\pmatrix{ 0 & {\e}_1 & 0 & 0 \cr -{\e}_1
& 0 & 0 & 0 \cr 0 & 0 & 0 & {\e}_2 \cr 0 & 0 & - {\e}_2 & 0 \cr} }
The maximal torus of $SO(4)$ coincides with that of a $U(2)$ subgroup of
$SO(4)$ which preserves some complex structure on ${\bR}^4$. In the
following we shall only preserve $U(2)$.
The fermionic symmetry $Q$ allows to calculate \insmsr\ by taking
${\bar\phi} \to \infty$. In this case the saddle point approximation
would localize the integral onto the vicinity of the points
$P \in {\CX}$, where $V({\phi}) = 0$.  In this way one arrives precisely
at \contr.

Finally, one encounters yet another representation for \insmsr,
where ${\CX}$ is replaced by the space of all matrices ${\bX}, {\l}$,
${\CG}$ is the product of the ``gauge group'' $U(k)$,
and the global group $U(N) \times U(2)$, so that one integrates
over the $\phi,\bar\phi,\eta$ multiplet for $U(k)$. As a result, the
equations ${\mu}^A=0$ are the ADHM equations \adhm, or, more precisely,
their noncommutative deformation \neksch, when we study the resolved
moduli space (it is at this point that $SO(4)$ reduces to $U(2)$).
These equations transform in the (adjoint, triplet, singlet)
of  $U(k) \times U(2) \times U(N)$.

For details the reader is encouraged to consult
\nsvz\seibergone\calculus\instmeasures\twoinst\hollowood,
as well as \maxim\torusaction\givental\atiyahsegal\adhm\DHf\ihiggs\dbound.

In what follows we set ${\e}_1 = - {\e}_2 = {\hbar}$. Note, that
this Planck constant has nothing to do with the coupling constant of the
gauge theory, where it appears as the parameter of the geometric
background \twm. It
corresponds however exactly to the loop counting in the dual string
theory,
while the gauge theory Planck constant in string theory picture arises
as a {\it worldsheet} parameter, according to the relation between the
world-sheet and gauge theory instantons, described below.

\subsec{Correlation functions of the chiral operators} Now  we are
ready to attack the correlation function \gnfnc. First of all,
using the unbroken supercharges one argues that this correlation
function is independent of the coefficient in front of the term
$\vert F^{+} \vert^2 + \ldots$ which is $\{ Q, \ldots \}$.
Therefore, one can go to the weak coupling regime (with the theta
angle appropriately adjusted, so that ${\tau}_0$ is finite, while
${\bar\tau}_0 \to \infty$ ) in which \gnfnc\ is saturated by
instantons (cf. \nsvz) In this limit the descendants of the chiral
operators become the Chern classes of the universal bundle,
``integrated'' (in the equivariant sense),  over ${\bf R}^4$. Here is
the table
of equivariant integrals \DHf\ (cf. \rltns):
\eqn\tabl{ \int_{{\bf R}^4} {\Omega}^{(4)} =
{{\Omega}^{(0)}(0) \over {{\e}_1 {\e}_2}}}We should then integrate
these classes over ${\CM}$. But then again, we use equivariant
localization,
this time on the fixed points in ${\CM}$. These fixed points are
labeled by partitions ${\bk}$. The
calculation of the expectation values of the chiral operators
becomes equivalent to the calculation of the expectation values of
of some operators in the statistical mechanical model, where the
the basic variables are the $N$-tuples of partitions, ${\bk}_1,
\ldots, {\bk}_N$, where ${\bk}_{l} = \left( k_{l1} \geq k_{l2}
\geq k_{l3} \geq \ldots k_{l\ n_l} > k_{l \ n_{l} +1 } = 0 \ldots
\right)$. In this statistical model, the operator
${\CO}_J^{(0)} = {1\over{J}}
{\Tr}{\Phi}^J$ in the gauge theory translates to the operator
($a_l = {\hbar} M_l$):
\eqn\stmop{\eqalign{& {\CO}_J
[{\vec\bk}] = {{\hbar}^{J}\over J} \times \cr & \sum_{l=1}^{N}
\left[ M_l^J
+ \left( \sum_{i=1}^{\infty} ( M_l + k_{li} - i +1)^J - (M_l +
k_{li} - i)^J - (M_l +  1 - i)^J + (M_l -  i )^J \right)\right]\cr
& \qquad =^{\kern -.2in \rm formally} \ {1\over J} \sum_{l, i}
\left[ \left(
(a_l + {\hbar} (k_{li} + 1- i) \right)^J - \left( a_l + {\hbar}
(k_{li} - i)\right)^J \right]\cr }}
This expression is most simply calculated using the D(-1) instanton
matrix model \swi. Using \tang\ one can write:
$$
Ch({\CE}) = Ch(W) + Ch(V) ( e^{{\hbar}/2} - e^{-{{\hbar}/ 2}} )^2
$$
where $Ch(W) = \sum_l e^{a_l}$, and $Ch(V) = {\rm Tr}\ e^{\phi}$ on the fixed
point set. Recall that here $\phi$ is the matrix in the D(-1) matrix
model, while ${\Phi}$ is, as before, the field on  D3 brane.

Given the single-trace operators ${\CO}_J$ we build arbitrary
gauge-invariant operators ${\CO}_{\vec n}$ as in \mcrsc, \vvs. After
that one can integrate their ${\CN}=2$ descendants
${\CO}_{\vec n}^{(4)}$ using the table of equivariant integrals \tabl.

It is useful to recall here the D-brane interpretation of the partitions
${\bk}$.
In this picture, the fractional D3-branes are separated in the $w$
direction, and are located at
$w = a_l$, $l=1, \ldots, N$. To the $l$'th D3 brane $k_l = \sum_i k_{li}$
D(-1) instantons are attached.
In the noncommutative theory with the noncommutativity parameter ${\t}$,
$$
[x^1, x^2] = [x^3, x^4] = i {\t}
$$
these D(-1) instantons are located near the origin $(z_1, z_2) \sim 0$,
where $z_1 = x^1 + ix^2, z_2 = x^3 + i x^4$. Different partitions
correspond to the different 0-dimensional ``submanifolds''
(in the algebraic geometry sense) of ${\bC}^2$. If we denote by ${\CI}_l$
the algebra of holomorphic functions (polynomials) on ${\bC}^2$ which
vanish on the D(-1) instantons, stuck to the $l$'th D3-brane, then it
can be identified with the ideal in the ring of polynomials
${\bC}[z_1, z_2]$
such that the quotient ${\bC}[z_1, z_2]/{\CI}_l$ is spanned by
the monomials
$$
z_1^{i-1} z_2^{j-1}, \qquad \qquad 1 \leq j \leq k_{li}
$$
Gauge theory generating function of the correlators of the chiral
operators becomes the
statistical model partition function with all the integrated operators
$\int_{{\bR}^4} {\CO}_{\vec n}^{(4)}$ added to the Hamiltonian.
In other words, we sum over the partitions
$\{{\bf k}_l\} = \{ k_{li}\}$ the Bolzmann
weights $\exp\left(-{1\over{{\hbar}^2}} \sum_{\vec n}
t_{\vec n}{\CO}_{\vec n} \right)$, and
the measure on the partitions
is given by the square of the regularized discretized Vandermonde
determinant:
\eqn\msr{\eqalign{& {\m}_{\vec\bk} = \prod_{(li) \neq (mj)}
\left( {\l}_{li} - {\l}_{mj} \right) \cr & {\l}_{li} = a_l +
{\hbar} (k_{li} - i), \quad
 \cr}}
%
The product in \msr\ is taken over all pairs $(li)\neq (mj)$ which is
short for $\{(l\neq m); {\rm or} \ (l=m,i\neq j); \}$ and can be
understood with the help of ${\zeta}$-regularization:
\eqn\rglmsr{{\m}_{\vec\bk} = {\exp}
\left(- {d\over ds}
%\vert_{s=0}
{1\over{{\Gamma}(s)}}
\int_{0}^{\infty} dt\ t^{s-1} \sum_{(li) \ \neq
%\leq
(mj)}
\left.e^{-t ({\l}_{li} - {\l}_{mj})}\right|_{s=0}\right)}
The sum in \rglmsr\ is defined by analityc continuation.



\newsec{ABELIAN THEORY}

\subsec{A little string that could}

Now suppose we take $N=1$. In the pure ${\CN}=2$ gauge theory this is not
the most interesting case, since neither perturbative, nor
non-perturbative corrections affect the low-energy prepotential.
Imagine, then, that we embed the $N=1$ ${\CN}=2$ theory in the
theory with instantons. One possibility is the \nc\ gauge theory,
another possibility is the theory on the D-brane, e.g. fractional
D3-brane at the ADE-singularity, or the D5/NS5 brane wrapping a
${\bP}^1$ in {\bf K3}. In this setup the theory has
non-perturbative effects, coming from \nc\ instantons, or
fractional D(-1) branes, or the worldsheet instantons of D1
strings bound to D5, or the elementary string worldsheet
instantons in the background of NS5 brane, or an $SL_2({\bZ})$
transform thereof. In either case, we shall get the instanton
contributions to the effective prepotential. Let us calculate
them.

We shall slightly change the notation for the times ${\tau}_{\vec n}$
as in this case there is no need to disinguish between ${\Tr}{\Phi}^J$
and $({\Tr}{\Phi})^J$. We set:
\eqn\tmss{\sum_{\vec n} {\tau}_{\vec n} \prod_{J=1}^{\infty} {x^{J n_J}
\over (J)^{n_J}} =
\sum_{J=1}^{\infty}  t_J {x^{J+1} \over (J+1)!}
}
and consider the partition function as a function of the times $t_{J}$.

First, let us not turn off the higher
order Casimirs. Then, we are to calculate:
\eqn\uonecalc{e^{-t_1
{{a^2}\over 2{\hbar}^2}} \sum_{\bk} {\m}_{\bk} e^{t_1 \vert {\bk}
\vert}}

\bigskip
\noindent{\it Partitions and representations}


As it is well-known, the partitions ${\bk}= \left( k_{1} \geq k_{2}
\geq k_{3} \geq \ldots k_{n}\right)$ are in
one-to-one correspondence with the irreducible representations
$R_{\bk}$ of the symmetric group ${\CS}_{k}$, $k = \vert {\bk}
\vert$. Moreover, in the case $N=1$, one gets from \msr: $$
{\m}_{\bk} = \prod_{i\neq j}^\infty{\hbar(k_i-k_j+j-i)\over
\hbar(j-i)} $$ and using the relation between partitions
${\bk}$ and Young diagrams $Y_{\bk}$, whose $i$'th row contains
$k_i > 0$ boxes, $1 \leq i \leq n$ corresponding to the
irreducible representation $R_{\bk}$ of the symmetric group
${\CS}_{k}$ (and to the irreducible representation ${\CR}_{\bk}$
of the group $U({\hat N})$, for any ${\hat N} \geq n$), this can
be rewritten as
$$
{\m}_{\bk} = (-1)^{k}\left[
\prod_{i<j}^n{\left({\hbar} \left(k_i-k_j+j-i\right)\right)}
\prod_{i=1}^n{1\over {\hbar}^{k_i + n -i}
(k_i+n-i)!} \right]^2 = (-1)^k \left[ {{\rm dim}R_{\bk}
\over {\hbar}^k \ k!} \right]^2
$$
where we employ the
rule ${l\cdot (l+1)\cdot (l+2)\dots \over 1\cdot 2\cdot 3\dots
l\cdot (l+1)\cdot (l+2)\dots} = {1\over l!}$.
Hence
$$
{\m}_{\bk} =
\left( {{\rm dim}R_{\bk} \over k!} \right)^2 (-{\hbar}^2)^{-k}
$$
The summation over ${\bk}$ is trivial thanks to Burnside's
theorem, and we conclude:
\eqn\uoneansw{Z = \exp\left[  - {1\over
{\hbar}^2} \left( t_1 {a^2 \over 2} + e^{t_1} \right)\right]}
We see that the gauge theory prepotential or the free energy of
our statistical model coincides with the Gromov-Witten
prepotential of the ${\bP}^1$ topological sigma model.

%\vfill\eject

\bigskip\noindent{\it Back to fractional branes}

At this point the fair question is: where this ${\bP}^1$ came from?
After all, in all physical applications of the topological strings
the target space should be a Calabi-Yau manifold, and ${\bP}^1$ is
definitely not the one. One can imagine the topological string on
a local Calabi-Yau, which is a resolved conifold, i.e. a total
space of the ${\CO}(-1) \oplus {\CO}(-1)$ bundle over ${\bP}^1$.
One can then turn the so-called twisted masses ${\m}_1, {\m}_2$,
or, more mathematically speaking, equivariant parameters with
respect to the rotations of the fiber of the vector bundle. In the
limit ${\m}_{1,2} \to 0$ the sigma model is localized onto the
maps into ${\bP}^1$ proper. Is this the way to embed our model in
a full-fledged string compactification? We doubt it is the case.


Rather, we think the proper model should be that of little string
theory \mmm\ compactified on ${\bP}^1$. Indeed, the discussion in
the beginning of this section suggests a realization of the
abelian gauge theory with instantons by means of the D5 brane
wrapping a ${\bP}^1$ inside the Eguchi-Hanson space
$T^{*}{\bP}^1$, which is the resolution of the ${\bC}^2/{\bZ}_2$
singularity. The wrapped D5 brane is a blown-up  fractional D3
brane stuck at the singularity. It supports an ${\CN}=2$ gauge
theory with a single abelian vector multiplet. In addition, it has
instantons, coming from fractional D(-1) branes, or, after
resolution, D1 string worldsheet instantons. These are bound to
the D5 brane worldvolume. After S-duality and appropriate
decoupling limits these turn into the so-called {\it little
strings}, of which very little is known. In particular, much
debate was devoted to the issue of the tunable coupling constant
in these theories. Our results strongly suggest such a
possibility.






\subsec{Free fermions} Now let us turn on the higher order
Casimirs in the gauge theory. To facilitate the calculus it is
convenient to introduce the formalism of free fermions. Consider
the theory of a single free complex fermion on a two-sphere: $\int
{\tilde\psi} {\pb} {\psi}$. We can expand:
\eqn\frem{\eqalign{& {\psi} (z) = \sum_{r \in {\bZ} + {\half}} \
{\psi}_{r} \  z^{-r} \left( dz \over z \right)^{\half}, \cr &
{\widetilde\psi} (z) = \sum_{r \in {\bZ} + {\half}} {\widetilde
\psi}_{r} \ z^r \left( dz \over z \right)^{\half} \cr &
\qquad\qquad \{ {\psi}_{r}, {\widetilde\psi}_{s} \} = {\d}_{rs}
\cr} }The fermionic Fock space is constructed with the help of the
charge $M$ vacuum state\foot{Any $M$ is good for building the
space.}:
\eqn\vcms{\eqalign{& \vert M \rangle = {\psi}_{M + {1\over
2}} {\psi}_{M+{3\over 2}} {\psi}_{M+ {5\over 2}} \ldots \cr &
{\psi}_{r} \vert M \rangle = 0, \qquad r
> M \cr & {\widetilde\psi}_{r} \vert M \rangle = 0, \qquad r < M
\cr}} It is also convenient to use the basis of the so-called
partition states
(see, e.g. \op\kharchev). For each partition ${\bk} = ( k_1  \geq
k_2 \geq \ldots )$ we introduce the state:
\eqn\prtst{\vert M;
{\bk} \rangle = {\psi}_{M+{1\over 2} -k_1} {\psi}_{M+{3\over 2} -
k_2} \ldots }
One defines the $U(1)$ current as: \eqn\crnt{ \eqalign{& J =
: {\widetilde\psi} {\psi} : = \sum_{n \in {\bZ}} J_{n} z^{-n} {dz
\over z} \cr & J_{n} = \sum_{r < n} {\widetilde\psi}_{r}
{\psi}_{n-r}  - \sum_{r
> n} {\psi}_{n-r} {\widetilde\psi}_{r} \cr}}
Recall the bosonization rules: \eqn\bsnz{{\psi} = : e^{i {\phi}}
:\quad , \quad {\tilde\psi} = : e^{-i{\phi}} :\quad , \quad J =
i{\p} {\phi}} and a useful fact from $U({\hat N})$ group theory:
the famous Weyl correspondence states that
\eqn\wlcrsp{({\bC}^{{\hat N}})^{\otimes k} = \bigoplus_{{\bk}, \vert
{\bk} \vert = k} R_{\bk} \otimes {\CR}_{\bk}} as ${\CS}_{k} \times
U({\hat N})$ representation. Now let $U = {\rm diag} \left( u_1,
\ldots, u_{\hat N} \right)$ be a $U({\hat N})$ matrix. Then one
can easily show using Weyl character formula, and the standard
bosonization rules, that: \eqn\chrt{{\Tr}_{{\CR}_{\bk}} U =
\langle {\hat N}; {\bk} \vert : e^{i \sum_{n=1}^{{\hat N}} {\phi}
( u_n) } : \ : e^{- i {\hat N} {\phi} (0)}:\vert 0 \rangle, \qquad }
From this formula one derives:
\eqn\prtnstts{
e^{J_{-1}\over {\hbar}} \ \vert M \rangle = \sum_{\bk}
{{\rm dim}R_{\bk} \over {\hbar}^k \ k! }\ \vert M; {\bk} \rangle}


\newsec{INTEGRABLE SYSTEM AND ${\bC}{\bP}^1$ SIGMA MODEL}


The importance of the
fermions is justified by the following statement.
The generating function with turned on higher Casimirs equals to
the correlation function:
\eqn\frmrp{\eqalign{& Z = {{\langle M \vert e^{J_{1} \over
{\hbar}} {\exp} \left[  \sum_{p = 1}^{\infty} {\hat t}_{p} W_{p+1}
\right] e^{-{J_{-1}\over{\hbar}}} \vert M \rangle}} \cr}}Here:
\eqn\shft{\eqalign{& \sum_{p=1}^{\infty} {\hat t}_{p} \ x^{p} =
\sum_{p=1}^{\infty} {1\over{(p+1)!}} \ t_{p} \ {{(x+{\hbar \over
2})^{p+1} - (x - {\hbar \over 2})^{p+1}}\over \hbar} \cr}}
and
\eqn\wgen{ W_{p+1}
= {1\over \hbar} \oint \ : \ {\widetilde \psi} \left( {\hbar} D
\right)^{p} {\psi} : \ , \qquad D = z{\p}_z \ }  If only $t_1
\neq 0$ the correlator \frmrp\ is trivially computed and gives
\uoneansw\ with $a=\hbar M$. From comparison of \frmrp\ with the
results of \op\ one gets that generating function \frmrp, as a
function of times $\hat t_p$ is a tau-function of the Toda lattice
hierarchy. Note that the fermionic matrix element \frmrp\ is
very much different from the standard
representation for the Toda tau-function \todalit.
In our case the ``times'' are coupled to the W-generators,
while usually they couple to the components of the $U(1)$ current.


The free fermionic
representation \frmrp\ is useful in several respects. One
of them is the remarkable mapping of the gauge theory correlation
function to the amplitudes of a (topological type A) string,
propagating on ${\bC}{\bP}^1$. Indeed, using the results of \op\
(see also \prtoda)  one can show that:
\eqn\gstrcr{\biggl\langle {\exp} \int_{{\bR}^4}
\sum_{J=1}^{\infty} t_{J} \ {\CO}^{(4)}_{J+1}
\biggr\rangle_{a, {\hbar}}^{\rm gauge \ theory} = {\exp}
\sum_{g=0}^{\infty} {\hbar}^{2g-2} \langle\langle {\exp}
\int_{{\Sigma}_{g}} \ a \cdot {\bf 1} + \sum_{p=1}^{\infty} {\hat
t}_{p} {\s}_{p-1}({\o}) \rangle\rangle_{g}^{\rm string}} It is
tempting to speculate that a similar relation holds for nonabelian
gauge theories. The left hand side of \gstrcr\ is known for the gauge
group $U(N)$ (we essentially described it by the formulae \stmop\msr,
see also \swi) but the right hand side is not, although there are strong
indications that the free fermion representation and relation to the
${\bC\bP}^1$ sigma model holds in this case too \nok.
  Another application of \frmrp\ is the
calculation of the expectation values of ${\CO}_J$. This exercise
is interesting in relation to the recent matrix model/gauge theory
correspondence of R.~Dijkgraaf and C.~Vafa \dv, which predicts,
according to \cdws: \eqn\expct{\langle {\Tr} {\phi}^J \rangle =
\oint \ x^{J} {{d z} \over z}, \qquad z + {{\Lambda}^{2N} \over z}
= P_{N} (x) = x^{N} + u_1 x^{N-1} + u_2 x^{N-2} + \ldots + u_N}
quite in
agreement with the formulae from \whitham, obtained in the context
of the Seiberg-Witten theory.

To compute the
expectation values of ${\CO}_{J}$ in our approach (for $N=1$) it
suffices to calculate $-{\hbar}^2 \
%J
{1\over Z} {\p}_{t_{J-1}}
Z$ at $t_2 = t_3 = \ldots = 0$  and then send ${\hbar} \to 0$ (as
\cdws\ did not look at the equivariance with respect to the
space-time rotations):
\eqn\expv{\eqalign{& \langle {\CO}_{J}
\rangle_{a, 0} = \cr & = \lim_{{\hbar} \to 0}  \ {\hbar}^{J}
\ {{\langle M \vert e^{{1\over{\hbar}} \oint : {\tilde\psi} z
{\psi} : } \oint \ : {\tilde\psi} \left( ( D + {1\over 2} )^{J} -
( D - {1\over 2})^{J} \right) {\psi} : \ e^{-{{\Lambda}^2
\over\hbar} \oint : {\tilde\psi} z^{-1} {\psi} : } \vert M
\rangle}\over{\langle M \vert e^{{1\over{\hbar}} \oint :
{\tilde\psi} z {\psi} :} \ e^{-{{\Lambda}^2 \over\hbar} \oint :
{\tilde\psi} z^{-1} {\psi} : } \vert M \rangle}}= \cr
&\qquad\qquad  = \oint \left( a + z + {{\Lambda}^2 \over z}
\right)^{J} {dz \over z} \cr &  \cr & \qquad\qquad\qquad
{\Lambda}^2 = e^{t_1}, \qquad a = {\hbar} M \cr} } the last
relation proved by bosonization. This reproduces
\expct\ for $N=1$.

The topological string on ${\bC\bP}^1$ actually has even more observables
then the ones presented in \gstrcr. Indeed, we are missing all the
gravitational descendants of the puncture operator ${\s}_k({\bf 1}), k> 0$.
We conjecture, that their gauge theory dual, by analogy with AdS/CFT
correspondence \bulkbndr, is the shift of vevs of the operators
${\Tr} {\phi}^J$, for ${\s}_{J-1}({\bf 1})$. For $J=1$
we are talking about shifting $a$, the vev of ${\phi}$. This is
indeed the case.
When all these couplings are taken into account we would expect
to see the full two-dimensional Toda lattice hierarchy \todalit.

\newsec{THEORY WITH MATTER} In this section we
shall discuss theory with matter in the fundamental
representation. We shall again consider only $U(1)$ case, but as
above we shall be, in general, interested in turning on higher
Casimirs.

\subsec{4d and 2d field theory}

The famous condition of asymptotic freedom, $N_f \leq 2N_c$, if
extrapolated to the case $N_c =1$ suggests that we could add up to
two fundamental hypermultiplets. It is a straightforward exercise
to extend the fixed point calculus to incorporate the effect of
the charged matter. Let us briefly remind the important steps.
Susy equations in the presence of matter hypermultiplet $M =
({\tilde Q}, Q)$ change from $F^{+} = 0$ to $F^{+} + {\bar M}
{\Gamma} M = 0$, ${\dir} M = 0$. The moduli space of solutions to
these equations looks near $M=0$ locus as  a vector bundle over
${\CM}$ -- the instanton moduli, whose fiber is the the bundle of
Dirac zero modes.


It can be shown that the instanton measure gets an extra factor
which the equivariant Euler class of this bundle. The localization
formulae still work, but now each partition ${\bk}$ has an extra
weight \swi. The contribution of the fixed point to the path
integral in the presence of the matter fields is \msr\ multiplied
by the extra factor:

\eqn\extrw{{\tilde\m}_{\bk} (a,m) = Z^{pert} (a, m) \times
\prod_{f=1}^2 \prod_{i=1}^{\infty} \left( a + m_f + {\hbar}( 1 - i
) \right) \ldots\left( a + m_f + {\hbar} (k_i - i) \right)} where
\eqn\zper{\eqalign{& Z^{pert} (a, m) = \prod_{f}
\prod_{i=1}^{\infty} {\Gamma}\left( {a + m_f \over {\hbar}}  + 1-
i \right) \sim {\exp} \int_{0}^{\infty} {{\rm d}t \over t} \sum_f
{e^{- t ( a + m_f )} \over {\rm sinh}^2 \left( {{\hbar} t \over 2
} \right)} = \cr & = \sum_{f} \left[ {( a + m_f)^2
\over 2{\hbar}^2}  {\rm log} (a+m_f) + {1\over 12} {\rm log} ( a +
m_f) + \sum_{g>1} {B_{2g}\over 2g(2g-2)} \left(
{{\hbar}\over{a+m_f}} \right)^{2g-2} \right]\cr}}
The bosonization rule \chrt\ leads
to the following formula:


\eqn\instcntr{Z^{inst} = \biggl\langle e^{i {a + m_2 \over {\hbar}
} {\phi} ({\infty}) } e^{-i {m_2 \over {\hbar}} {\phi} (1) }
e^{\sum_p t_p {\CW}_{p+1} } e^{i {m_1 \over {\hbar}} {\phi}(1)}
e^{ - i {{a+m_1}\over{\hbar}} {\phi}(0)} \biggr\rangle}


It can be shown that the full partition function $Z^{pert}
Z^{inst}$ also has a CFT interpretation, and also obeys Toda
lattice equations. We shall discuss this in a future publication.



\subsec{Relation to geometric engineering}

Now let us turn off the higher Casimirs. Then \instcntr\extrw\
lead to \eqn\prptns{\eqalign{& {\CF}_{0} =
{\half}t_1 a^2 - m_1 m_2
{\rm log} ( 1 - e^{t_1} )+ \sum_f {\half}
\left(a+m_f\right)^2 {\rm log} \left( a + m_f \right)
\cr & {\CF}_1 = {1\over
12} {\rm log} ( a+m_1)( a+m_2) \cr & {\CF}_{g} = {B_{2g} \over 2g
(2g-2) } \sum_f {{\hbar}^{2g-2} \over{( a+m_f)^{2g-2}}} \cr}}


We remark that \prptns\ is a limit of the all-genus topological
string prepotential in the geometry described in \agmav\ (Fig.12,
Eq. (7.34)). The specific limit is to take first $t_1, t_2, g_s$
in their notation to zero, as $t_f = {\b} ( a + m_f), g_s = {\b}
{\hbar}$, ${\b} \to 0$, while $-r^{\prime}$ (their notation) $=
t_1$ (our notation) is finite. The prepotential \agmav\ actually
describes the five dimensional susy gauge theory compactified on a
circle of circumference ${\b}$. The limit ${\b} \to 0$ actually
takes us to the four dimensional theory, which is what we were
studying in this paper. It is clear, from \agmav\ (Fig.12c) that
the geometry corresponds to the $U(1)$ gauge theory with two
fundamental hypermultiplets (two D-branes pulling on the sides).


Our results are, however, stronger. Indeed, we were able to
calculate the prepotential and ${\CF}_g$'s with arbitrary higher
Casimirs turned on. In the limit \eqn\lmts{m_1, m_2 \to \infty,
\quad e^{t_1} \to 0 \quad {\rm (our \ notation)}, \quad{\rm  so \
that} \quad {\Lambda}^2 = m_1 m_2 e^{t_1}= e^{T_1} \quad {\rm  is
\  finite}} we get back the pure $U(1)$ theory, which we
identified with the topological string on ${\bP}^1$ \gstrcr. Note
that this was not ${\bP}^1$ embedded into Calabi-Yau, as in the
latter case no gravitational descendants ever showed up. We are
led, therefore, to the conclusion, that the topological string on
the geometry of Fig.12 of \agmav\ has a deformation, allowing
gravitational descendants, and flowing, in the limit \lmts\ to the
pure ${\bP}^1$ model. This fascinating prediction certainly
deserves further study.



\bigskip
\noindent {\it
Acknowledgements.}

NN acknowledges useful discussions with N.~Berkovitz, S.~Cherkis,
A.~Givental, D.~Gross, M.~Kontsevich, G.~Moore,  A.~Polyakov,
N.~Seiberg, S.~Shatashvili, E.~Witten, C.~Vafa, and especially
A.~Okounkov.
NN is grateful to Rutgers University, Institute
for Advanced Study, Kavli Institute for Theoretical Physics, and
Clay Mathematical Institute for support and hospitality during the
preparation of the manuscript. ASL and AM are grateful to IHES for
hospitality, AM acknowledges also the support of the Ecole Normale
Superieure, CNRS and the Max Planck Institute for Mathematics
where this work was completed. Research
was partially supported by {\cyr RFFI} grants 01-01-00548
(ASL), 01-01-00539 (AM) and 01-01-00549 (NN) and by the INTAS grant
99-590 (ASL and AM).


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