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%% 4 pages !! last edited: 09.12.98
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% ========================== def.tex =============================
% my last edition: September 29, 1995 (Hannover)
% 
% Accents and foreign (in text):

\let\under=\unt                 % bar-under (but see \un above)
\let\ced=\ce                    % cedilla
\let\du=\du                     % dot-under
\let\um=\Hu                     % Hungarian umlaut
\let\sll=\lp                    % slashed (suppressed) l (Polish)
\let\Sll=\Lp                    % " L
\let\slo=\os                    % slashed o (Scandinavian)
\let\Slo=\Os                    % " O
\let\tie=\ta                    % tie-after (semicircle connecting two letters)
\let\br=\ub                     % breve
                % Also: \`        grave
                %       \'        acute
                %       \v        hacek (check)
                %       \^        circumflex (hat)
                %       \~        tilde (squiggle)
                %       \=        macron (bar-over)
                %       \.        dot (over)
                %       \"        umlaut (dieresis)
                %       \aa \AA   A-with-circle (Scandinavian)
                %       \ae \AE   ligature (Latin & Scandinavian)
                %       \oe \OE   " (French)
                %       \ss       es-zet (German sharp s)
                %       \$  \#  \&  \%  \pounds  {\it\&}  \dots

% Abbreviations for Greek letters

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

% Varletters

\def\ve{\varepsilon}
\def\vf{\varphi}
\def\vr{\varrho}
\def\vs{\varsigma}
\def\vq{\vartheta}

% 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\car{{\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}}

% Fonts

\def\Sc#1{{\hbox{\sc #1}}}      % script for single characters in equations
\def\Sf#1{{\hbox{\sf #1}}}      % sans serif for single characters in equations

                        % Also:  \rm      Roman (default for text)
                        %        \bf      boldface
                        %        \it      italic
                        %        \mit     math italic (default for equations)
                        %        \sl      slanted
                        %        \em      emphatic
                        %        \tt      typewriter
                        % and sizes:    \tiny
                        %               \scriptsize
                        %               \footnotesize
                        %               \small
                        %               \normalsize
                        %               \large
                        %               \Large
                        %               \LARGE
                        %               \huge
                        %               \Huge

% Math symbols

\def\slpa{\slash{\pa}}                            % slashed partial derivative
\def\slin{\SLLash{\in}}                                   % slashed in-sign
\def\bo{{\raise-.3ex\hbox{\large$\Box$}}}               % D'Alembertian
\def\cbo{\Sc [}                                         % curly "
\def\pa{\partial}                                       % curly d
\def\de{\nabla}                                         % del
\def\dell{\bigtriangledown}                             % hi ho the dairy-o
\def\su{\sum}                                           % summation
\def\pr{\prod}                                          % product
\def\iff{\leftrightarrow}                               % <-->
\def\conj{{\hbox{\large *}}}                            % complex conjugate
\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}   % < or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}   % > or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu \llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu \llap {$\ddot
        \smile$}}}                                      % happy face
\def\dg{\sp\dagger}                                     % hermitian conjugate
\def\ddg{\sp\ddagger}                                   % double dagger
                        % Also:  \int  \oint              integral, contour
                        %        \hbar                    h bar
                        %        \infty                   infinity
                        %        \sqrt                    square root
                        %        \pm  \mp                 plus or minus
                        %        \cdot  \cdots            centered dot(s)
                        %        \oplus  \otimes          group theory
                        %        \equiv                   equivalence
                        %        \sim                     ~
                        %        \approx                  approximately =
                        %        \propto                  funny alpha
                        %        \ne                      not =
                        %        \le \ge                  < or = , > or =
                        %        \{  \}                   braces
                        %        \to  \gets               -> , <-
                        % and spaces:  \,  \:  \;  \quad  \qquad
                        %              \!                 (negative)


\font\tenex=cmex10 scaled 1200

% Math stuff with one argument

\def\sp#1{{}^{#1}}                              % superscript (unaligned)
\def\sb#1{{}_{#1}}                              % sub"
\def\oldsl#1{\rlap/#1}                          % poor slash
\def\slash#1{\rlap{\hbox{$\mskip 1 mu /$}}#1}      % good slash for lower case
\def\Slash#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}      % " upper
\def\SLash#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}    % " fat stuff (e.g., M)
\def\SLLash#1{\rlap{\hbox{$\mskip 6 mu /$}}#1}      % slash for no-in sign
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}   %
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}       %
\def\Tilde#1{\widetilde{#1}}                    % big tilde
\def\Hat#1{\widehat{#1}}                        % big hat
\def\Bar#1{\overline{#1}}                       % big bar
\def\bra#1{\left\langle #1\right|}              % < |
\def\ket#1{\left| #1\right\rangle}              % | >
\def\VEV#1{\left\langle #1\right\rangle}        % < >
\def\abs#1{\left| #1\right|}                    % | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow \mkern-6mu
        \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
        \mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
        \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
        $\hfil\displaystyle{#1}\hfil$\crcr}}}           % <--> accent
\def\dt#1{{\buildrel {\hbox{\LARGE .}} \over {#1}}}     % dot-over for sp/sb
\def\dtt#1{{\buildrel \bullet \over {#1}}}              % alternate "
\def\der#1{{\pa \over \pa {#1}}}                % partial derivative
\def\fder#1{{\d \over \d {#1}}}                 % functional derivative
                % Also math accents:    \bar
                %                       \check
                %                       \hat
                %                       \tilde
                %                       \acute
                %                       \grave
                %                       \breve
                %                       \dot    (over)
                %                       \ddot   (umlaut)
                %                       \vec    (vector)

% Math stuff with more than one argument

\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
        \hbox{$\scriptstyle{#2}$}}}}}                   % fraction
\def\half{\frac12}                                        % 1/2
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
        \vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}} % alternate fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
        \vphantom1\smash{\raise.3ex\hbox{$#2$}}}}       % "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}}    % "
\def\partder#1#2{{\partial #1\over\partial #2}}   % partial derivative of
\def\parvar#1#2{{\d #1\over \d #2}}               % variation of
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}  % second "
\def\on#1#2{\mathop{\null#2}\limits^{#1}}               % arbitrary accent
\def\bvec#1{\on\leftarrow{#1}}                  % backward vector accent
\def\oover#1{\on\circ{#1}}                              % circle accent

\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calD{{\cal D}}
\def\calM{{\cal M}}

\def\szet{{${\scriptstyle \b}$}}
\def\ulA{{\un A}}
\def\ulM{{\underline M}}
\def\cdm{{\Sc D}_{--}}
\def\cdp{{\Sc D}_{++}}
\def\vTheta{\check\Theta}
\def\gg{{\hbox{\sc g}}}
\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\ha{{\fracmm12}}
\def\tr{{\rm tr}}
\def\Tr{{\rm Tr}}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\ula{{\underline a}} \def\ulb{{\underline b}} \def\ulc{{\underline c}}
\def\uld{{\underline d}} \def\ule{{\underline e}} \def\ulf{{\underline f}}
\def\ulg{{\underline g}}
\def\items#1{\\ \item{[#1]}}
\def\ul{\underline}
\def\un{\underline}
\def\fracmm#1#2{{{#1}\over{#2}}}
\def\footnotew#1{\footnote{\hsize=6.5in {#1}}}
\def\low#1{{\raise -3pt\hbox{${\hskip 0.75pt}\!_{#1}$}}}

\def\Dot#1{\buildrel{_{_{\hskip 0.01in}\bullet}}\over{#1}}
\def\dt#1{\Dot{#1}}
\def\DDot#1{\buildrel{_{_{\hskip 0.01in}\bullet\bullet}}\over{#1}}
\def\ddt#1{\DDot{#1}}

\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}
\def\Hat#1{\widehat{#1}}

% Aligned equations

\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
        \ialign{\strut \hfil$\displaystyle{##}$&$
        \displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
        \everycr{\noalign{\ifdtup \global\dtupfalse
        \vskip-\lineskiplimit \vskip\normallineskiplimit
        \else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous                         % eqalignno
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}



% Journal abbreviations (preprints)

\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quantum Grav.~{\bf {#1}} (19{#2}) #3}
\def\cmp#1#2#3{Commun.~Math.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\jmp#1#2#3{J.~Math.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3}
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\ptp#1#2#3{Progr.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~J.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}

%%%%%%%%%%%%%%%%%%%%%% END of my definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%     

\begin {document}
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AdS/CFT correspondence and coincident D-6-branes
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Sergei V. Ketov
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% to numerate address 1 , 2 , etc.           %
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Institut f\"ur Theoretische Physik, Universit\"at Hannover\\
Appelstra\ss{}e 2, 30167 Hannover, Germany
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}
\def\abstracttext{
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A relation between confinement and Maldacena conjecture is briefly
discussed. The gauge symmetry enhancement for two coincident D-6-branes
is analyzed from the viewpoint of the hypermultiplet low-energy 
effective action given by the N=2 supersymmetric non-linear sigma-model
with the Eguchi-Hanson (ALE) target space.
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}
\large
\makefront
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% 4 resp. 6 pages!
% \section{Chapter 1}

The most attractive mechanism of color confinement in QCD is known to be
the dual (type II) superconductivity, i.e. a creation of color-electric
fluxes (or strings) having quarks at their ends \cite{hooft,mandel}.
The usual type-II superconductivity (i.e. the confinement of magnetic charges)
is known to be a solution to the standard Landau-Ginzburg theory, 
whereas the QCD confinement is supposed to be a non-perturbative solution 
to a (1+3)-dimensional quantum $SU(N_c)$ gauge field theory with $N_c=3$. 

The formal proof of the confinement in QCD amounts to a derivation of the area
law for a Wilson loop $W[C]$. It may be based on the following `string' Ansatz 
\cite{polbook}:
$$ W[C] \sim \int_{{\rm surfaces~}B,\atop \pa B=C}\; 
\exp\left(-S_{\rm string}\right)~~.\eqno(1)$$
Eq.~(1) clearly shows that the effective degrees of freedom (or collective
coordinates) in QCD at strong coupling (in the infra-red) are strings whose 
worldsheets are given by surfaces $B$ and whose dynamics is governed by (still 
unknown) action $S_{\rm string}$. The fundamental (Schwinger-Dyson) equations 
of QCD can be reformulated into the equivalent infinite chain of equations for 
the Wilson loops \cite{mig}. The chain of loop equations drastically simplifies
 at large number of colors $N_c$ (i.e. when only planar Feynman 
graphs are taken into account) to a {\it single} closed equation known as the 
{\it Makeenko-Migdal} loop equation \cite{mm}. Eq.~(1) is, therefore, just the 
Ansatz for a solution to the MM loop equation in terms of some string action 
$S_{\rm string}$ to be determined out of it. The main (yet unsolved) problems 
in a realization of this program in the past were (i) taking into account 
quantum renormalization in the MM equation, and (ii) determining the 
corresponding string action \cite{mpriv}. The first problem does not arise if
one replaces QCD by the N=4 supersymmetric Yang-Mills (SYM) theory and 
considers the N=4 supersymmetric MM-type loop equation instead of the original
(N=0) one, just because of the well-known fact that, being a scale invariant 
quantum field theory in 1+3 spacetime dimensions, the N=4 SYM does not 
renormalize at all. The recent Maldacena conjecture \cite{mal}, claiming that
the N=4 SYM theory is dual to the IIB superstring theory in the 
$AdS_5\times S^5$ background, can then be interpreted as the particular Ansatz
$S_{\rm string}=S_{IIB/AdS_5\times S^5}$ for a solution to the N=4 super-MM 
loop equation in the form of eq.~(1). Within the Maldacena conjecture, the 
(1+3)-dimensional spacetime is identified with the boundary of the 
anti-de-Sitter space $AdS_5$, where
$$ AdS_5=\fracmm{SO(4,2)}{SO(4,1)} \quad {\rm and} \quad 
S^5=\fracmm{SO(6)}{SO(5)}~,\eqno(2)$$
while the coupling constants are related to the $AdS_5$ radius as follows
\cite{mal}:
$$ (\a')^{-2}R^4_{\rm AdS}\sim g^2_{\rm YM}N_c~,
\quad g_{\rm string}\sim g^2_{\rm YM}~.\eqno(3)$$
The proposed duality is a strong-weak coupling duality: 
\begin{itemize}
\item for small $\l=g^2_{\rm YM}N_c$ a perturbative SYM description applies,
\item for large $\l$ a perturbative IIB string/AdS supergravity description 
applies.
\end{itemize}
It is in agreement with the holographic proposal \cite{hsu} since physics in 
the $AdS_5$ bulk is supposed to be encoded in terms of the field theory defined
on the $AdS_5$ boundary. The quantum N=4 SYM theory is conformally invariant, 
while its rigid symmetry is given by the supergroup $SU(2,2|4)$ 
that contains 32
supercharges. The isometries of $AdS_5\times S^5$ form the group $SO(4,2)\times
SO(6)\cong SU(2,2)\times SU(4)$ whose extension in the AdS supergravity is also
given by  $SU(2,2|4)$. In addition, both the N=4 SYM and type-IIB superstrings
are believed to be self-dual under the S-duality group $SL(2,{\bf Z})$. In more
practical terms, this CFT/AdS correspondence is just a one-to-one 
correspondence \cite{gkp} between the N=4 SYM correlators and the correlators 
of the certain string theory whose action $S_{\rm string}$ is known and whose
correlators can be computed, in principle, by the methods of two-dimensional
conformal field theory \cite{mybook}. Quantum corrections in powers of 
$(\a'\times{\rm curvature})$ on the string theory side
correspond to corrections in powers of $(g^2_{\rm YM}N_c)^{-1/2}$ on the gauge
field theory side, while the string loop corrections are suppressed by
powers of $N_c^{-2}$.

The close connection between IIB strings and N=4 SYM is also known to exist 
within the modern brane technology. The type-IIB supergravity (= the low-energy
effective field theory of IIB strings) admits extended solitonic BPS-like 
classical solutions known as D-3-branes and D-strings (or `mesons') \cite{host}.
These solutions can spontaneously break the conformal invariance in the 
non-perturbative N=4 SYM theory, and thus may be useful for a simulation of
confinement. When $N_c$ parallel and similarly oriented
D-3-branes coincide, the low-energy effective field theory action in their
common worldvolume appears to be the N=4 SYM with the gauge group $U(N_c)$
\cite{witten}. The brane picture thus provides a classical resolution to the
non-perturbative N=4 SYM, in the very similar way as the M-theory 5-brane
classical dynamics yields exact Seiberg-Witten-type solutions to N=2 
supersymmetric quantum gauge field theories (see e.g., ref.~\cite{myrev} for a
review). The 11-dimensional M-theory compactified on a 2-dimensional torus is 
known to be dual to the 10-dimensional type-IIB strings compactified on a 
circle \cite{schw}, so that the relevant phenomenon is just the gauge symmetry
enhancement for coincident D-branes alone. We would like to understand it from
the field-theoretical point of view, and distinguish between those elements of
this phenomenon that are of perturbative origin and those elements that are 
truly non-perturbative.

As an example, consider the case of two nearly coincident KK monopoles in M 
theory. The metric is essentially given by the 2-centre Taub-NUT metric (or,
equivalently, the mixed Taub-NUT-Eguchi-Hanson metric) characterized by the
harmonic potential \cite{town} 
$$ H(\vec{y})=\l +\fracmm{1}{2}\left\{ \fracmm{1}{\abs{\vec{y}-\x\vec{e}}}+
\fracmm{1}{\abs{\vec{y}+\x\vec{e}}} \right\}~.\eqno(4)$$
In the limit $\x\to 0$ the homology 2-sphere connecting two KK
monopoles contracts to a point. Since the energy of an M-2-brane wrapped about
this 2-sphere is proportional to its area (of order $\x$), a massless vector
particle (= the zero-mode of the M-2-brane) appears. In the type-IIA picture, 
the ground states of the 6--6 strings connecting two D-6-branes become 
massless if the branes coincide. The net effect is called the non-abelian 
gauge symmetry enhancement: $U(1)\times U(1)\to U(2)$, and it is truly 
non-perturbative \cite{ov}. The gauge fields (and their supersymmetric 
partners) related to non-diagonal gauge symmetry generators are the ground 
states of strings connecting {\it different} D-6-branes, with the masses being
proportional to the distance between the D-branes, whereas those related to 
the Cartan subalgebra generators appear as the massless ground states of the 
strings ending on {\it the same} D-6-brane. The existence of the massless 
ground states associated with Cartan subalgebra can be understood  
perturbatively, as a result of dynamical generation of massless vector 
supermultiplets in the (one-loop) quantum perturbation theory \cite{ket}.

The effective field theory in the D-6-brane worldvolume dimensionally reduced
to four dimensions includes the hyper-K\"ahler non-linear sigma-model (NLSM)
for a self-interacting hypermultiplet, whose NLSM metric is dictated by the
potential (4). This NLSM can be written down in harmonic superspace, in terms 
of two hypermultiplets $q^{A+}$, $A=1,2$, and the auxiliary N=2 vector 
superfield $V^{++}$ as a Lagrange multiplier, 
$$ S_{\rm mixed}[q^A,V^{++}] =\int_{\rm analytic}\left\{
\bar{q}^{A+}D^{++}_Zq^+_A +V^{++}\left( \frac{1}{2}\ve^{AB}\bar{q}^+_Aq^+_B 
+\x^{++} \right) +\frac{1}{4}\l(\bar{q}^{A+}q_A^+)^2\right\}~,\eqno(5)$$
where the Fayet-Iliopoulos term $(\sim \x V)$ has been introduced, while both
hypermultiplets are supposed to have {\it different} masses, $m_1$ and $m_2$,
given by the N=2 central charge $(Z)$ eigenvalues. Eq.~(5) is the gauged NLSM 
over the non-compact coset space $SU(1,1)/U(1)$ parametrized by two 
hypermultiplets. Near the core of D-6-branes the parameter $\l$ becomes 
irrelevant, so that the NLSM target space looks like an ALE space with the 
Eguchi-Hanson metric. Setting $\l=0$ in eq.~(5) results in a formally 
renormalizable four-dimensional `linear' NLSM, that allows us to integrate 
over the hypermultiplets in eq.~(5). All one needs is the N=2 superfield 
hypermultiplet propagator in harmonic superspace \cite{ikz},
$$ i\VEV{q^+(1)\bar{q}^+(2)}=-\fracmm{1}{\bo_1^Z}
(D_1^+)^4(D_2^+)^4\left\{ \d^{12}(Z_1-Z_2)\fracmm{e^{v_Z(2)-v_Z(1)}}{
(u^+_1u^+_2)^3}\right\}~,\eqno(6)$$
where $\bo^Z$ is the Klein-Gordon operator, and $v_Z$ is the co-called `bridge'
(see ref.~\cite{ikz} for details). It is now straightforward to calculate the
one-loop gauge effective action $i\Tr\log(\cd_{Z,V}^{++})$ in the low-energy
approximation \cite{ket}. We find that the N=2 vector supermultiplet, 
introduced in the classical action (5) as the Lagrange multiplier (without 
a kinetic term), becomes dynamical in quantum theory due to a dynamical
generation of its kinetic term
$$ S_{\rm induced}[V^{++}]=-\fracmm{1}{2e^2_{\rm ind}(p)}\int_{\rm chiral}\,
W^2 +{\rm h.c.}~,\eqno(7)$$
where we have introduced the standard N=2 superfield strength 
$W=\int du(\bar{D}^-)^2V^{++}$. The induced (momentum-dependent) gauge coupling
constant appears to be \cite{ket}
$$ \fracmm{1}{e^2_{\rm ind}(p)}=\fracmm{1}{16\p^2}\int^1_0dx\,
\ln \fracmm{m_2^2+p^2x(1-x)}{m_1^2+p^2x(1-x)}= \fracmm{1}{e^2_0}+O(p^2/m^2)~.
\eqno(8)$$
The N=2 central charge providing masses to the hypermultiplets can, therefore,
be understood as the origin of the induced gauge coupling (8).

A generalization to higher unitary groups is straightforward \cite{ket}. The
Hooft limit of large $\l$ appears to be equivalent to $\abs{\x/Z^2}\to 0$ in
our approach. The orthogonal gauge groups may also be considered by introducing
orientifolds and the Atiyah-Hitchin metric in M-theory.    

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\end{thebibliography}

\end{document}


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