%Paper: hep-th/9505190
%From: argyres@guinness.ias.edu (Philip Argyres)
%Date: Wed, 31 May 95 17:11:32 EDT


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%								%
%  Curves of Marginal Stability in N=2 Super-QCD		%
%								%
%  By P.C. Argyres, A.E. Faraggi, and A.D. Shapere		%
%								%
%  Plain TeX, with one uuencoded figure included using epsf	%
%								%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\vbox{\hfill hep-th/9505190, IASSNS-HEP-94/103, UK-HEP/95-07}
\vglue0.4cm
\centerline{\tenbf CURVES OF MARGINAL STABILITY IN N=2
SUPER--QCD$^*$}
\vglue0.2cm
\baselineskip=13pt
\centerline{\eightrm Philip C. Argyres and Alon E. Faraggi}
\baselineskip=12pt
\centerline{\eightit Institute for Advanced Study}
\baselineskip=10pt
\centerline{\eightit Princeton, NJ \ 08540, USA}
\centerline{\eightrm E-mail: argyres\@guinness.ias.edu, faraggi\@sns.ias.edu}
\vglue0.2cm
\centerline{\eightrm and}
\vglue0.2cm
\centerline{\eightrm Alfred D. Shapere}
\baselineskip=12pt
\centerline{\eightit Department of Physics and Astronomy,
University of Kentucky}
\baselineskip=10pt
\centerline{\eightit Lexington, KY \ 40506, USA}
\centerline{\eightrm E-mail: shapere\@amoeba.pa.uky.edu}
\vglue0.4cm

\tenrm\baselineskip=13pt
Seiberg and Witten's solution$^1$ to $N{=}2$ $SU(2)$ Yang-Mills with
$N_f{=}0$ flavors has a one-complex-dimensional Coulomb branch of
degenerate vacua labeled by a coordinate $u$.  The  effective $U(1)$
theory is described in terms of two functions $a(u)$ and $a_D(u)$.  The
gauge coupling, $\tau\equiv (\theta/2\pi)+i(4\pi^2/g^2)$ is given by
$\tau=d a_D/d a$; it must satisfy Im$(\tau)>0$.  The theory is governed
by a dynamically generated strong-coupling scale which we set to 1.

The mass of a dyon hypermultiplet with electric and magnetic charges
$n_e$ and $n_m$ is given by $M = \sqrt{2} |n_m a_D(u) + n_e a(u)|$.
Whenever Im$(a_D/a)=0$, any dyon becomes  marginally unstable to decay
into two or more other dyons (conserving electric and magnetic
charges).  This note presents a simple argument that determines the
shape of the curve of  marginal stability Im$(a_D/a)=0$.

The effective theory has a duality group that acts on ${a_D\choose a}$
as a vector under $SL(2,Z)$, and on $\tau$ in the usual way.  Note that
$f(u)\equiv {a_D/ a}$ transforms like $\tau$.

The $U(1)$ effective theory breaks down at $u=\pm 1$ and $\infty$,
where a dyon hypermultiplet becomes massless.  The $SL(2,Z)$
monodromies around these points are $ST^2S^{-1}$, $(TS)T^2(TS)^{-1}$,
and $-T^2$.  These matrices generate the group $\Gamma(2)\subset SL(2,Z)$.
$u(\tau)$ is a one--to--one map of a single fundamental domain of
$\Gamma(2)$ onto the complex plane, which has cusp points at $\tau=0$,
$1$, and $i\infty$.  These cusp points correspond to the three
singularities in the $u$--plane, and are fixed points of the
corresponding $SL(2,Z)$ monodromies---see Fig.\ 1.

The range of the function $f(u)$ is a subset of the complex plane with
similar properties to the fundamental domain of $\tau$.  $\Gamma(2)$ acts
identically on both the $f$--plane and the $\tau$--plane, and its
generators fix the same 3 points.  However, since Im$\,f$ is not
necessarily positive the range of $f$ may extend below the real axis,
unlike $\tau$.  Indeed, since we know (from expanding the explicit
expressions$^1$ for $a$ and $a_D$ around $u=\pm 1$) that there are
whole lines where $f$ is real, it follows that $f^{-1}$ must map an
infinite number of $\Gamma(2)$ domains, both above and below the real
axis, onto the $u$-plane.

There is only one possibility for the shape of the range of $f(u)$, due
to the fact that the generators $ST^2S^{-1}$ and $(T S)T^2(T S)^{-1}$
are of infinite order, which implies that the opening angles of the
corresponding cusps must also be of infinite order, {\it i.e.}, 0 or
$2\pi$.  An opening angle of 0 would correspond to a single fundamental
domain of $\Gamma(2)$, which we have ruled out. Opening angles of
$2\pi$ correspond to the domain shown in Fig.\ 1, a full strip in the
$f$-plane with one $\Gamma(2)$ domain removed. It is easy to see that
the monodromies for this region are correct.  As a check, it is easily
verified using the explicit expressions$^1$ that $f(0)=-(i\pm1)/2$.

The curve of marginal stability
is the image under $f^{-1}$ of the interval [-1,1], which is a simple
closed curve in the $u$--plane (with $f(-1)=\pm 1$ and $f(+1)=0$) as
conjectured in Ref.\ 1.  Outside of this curve are the images of the
infinite number of $ \Gamma(2)$ domains between Re$(f)=+1$ and $-1$ and
with Im$(f)\ge 0$.  Inside the curve are the images of all but one of
the $\Gamma(2)$ domains with Im$(f)<0$.

The curve Im$\,f=0$ has been shown by independent methods to be simple
and closed.$^2$  Also, we have numerically computed it to be the curve
shown in Fig.\ 1.

The methods presented here are easily extended to the massless $N_f=$
1, 2, and 3 cases.$^3$ For nonzero masses, as well as for $N_c>2$,
the curves of marginal stability become dense in moduli space.

\vglue 6pt
\epsfxsize=14 truecm \centerline{\epsfbox{coms.eps}}
{\noindent \baselineskip 13pt Fig.\ 1: The shaded regions are the
images of the $u$--plane in the $\tau$ and $f$--planes.  The dashed
lines are the images of Im$\,f=0$.}
\vglue 9pt

It is a pleasure to thank M. Douglas, M. Matone, R. Plesser, K.
Ranganathan, N. Seiberg, and E. Witten for useful discussions.  P.C.A.
is supported by NSF grant PHY92-45317 and by the Ambrose Monell
Foundation, A.E.F. by DOE grant DE-FG02-90ER40542, and A.D.S. by DOE
grant DE-FC02-91ER75661 and by an A.P. Sloan Fellowship.

\smallskip \eightrm \baselineskip 10pt
\itemitem{$*$.} Talk given by P.C.A. at Strings '95, U.S.C, March 1995.

\itemitem{1.} N. Seiberg and E. Witten, {\eightit Nucl. Phys.}
{\eightbf B426} (1994) 19.

\itemitem{2.} A. Fayyazuddin, hep-th/9504120.

\itemitem{3.} N. Seiberg and E. Witten, {\eightit Nucl. Phys.}
{\eightbf B431} (1994) 484.

\vfill\eject

\bye



