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% Section: QFT
% Title: Seiberg-Witten theory
% Last modified: 06.12.2002
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\begin{center}
\large\textbf{SEIBERG\,-WITTEN THEORY}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Nonperturbative solution}
\end{center}

\vspace{.1cm}

The underlying (microscopic) Lagrangian of Seiberg-Witten theory is that of
$N=2$ SUSY Yang-Mills written in terms of $N=1$ superfields,
\begin{equation}
\label{micr1}
{\mathcal L} = \text{Im\ tr\,}\frac{\tau}{8\pi}\left[\int\!d^2\theta\,
W^\alpha W_\alpha + 2\!\int\! d^2\theta d^2\bar\theta\, \Phi^\dag
e^{-2V}\Phi\right]\,,
\end{equation}
with the gauge group $SU(2)$ and both vector
\begin{equation}
\label{VW}
V=-\theta \sigma_\mu\bar{\theta}A_\mu+i\theta^{2}
\bar{\theta}\bar{\lambda} - i \bar{\theta}^{2} \theta \lambda +
\frac{1}{2} \theta^{2} \bar{\theta}^{2} D\,, \ \ \ \ \ \ \
W_{\alpha} = \frac{1}{8} \bar{D}^{2} (e^{2V} D_{\alpha} e^{-2V})
\end{equation}
and chiral
\begin{equation}
\label{Phi}
\Phi = \phi + \sqrt{2}\,\theta\,\psi + \theta\theta F
\end{equation}
supermultiplets belonging to its adjoint representation. Here
\begin{equation}
\label{tau}
\tau = \frac{\vartheta}{2\pi} + \frac{4\pi i}{g^2}
\end{equation}
is a `complexified' coupling constant, and $\vartheta$ the
\emph{vacuum angle}\,. Integrating over Grassmann variables and eliminating
auxiliary fields $D$ and $F$\,, we come to
\begin{multline}
\label{micr2}
\mathcal{L} = \frac{1}{g^2}\text{ tr}\Big(-\frac{1}{4}F_{\mu\nu}F_{\mu\nu}
 + g^2\frac{\vartheta}{32\pi^2}F_{\mu\nu}\widetilde F_{\mu\nu}
 + (D_\mu \phi)^\dag D_\mu \phi -\frac{1}{2} [\phi^\dag\,,\,\phi]^2 \\
 - i\lambda\,\sigma_\mu D_\mu\bar\lambda -i\,\bar{\psi}
 \bar{\sigma}_\mu D_\mu\psi-i\sqrt{2}\,[\lambda,\psi]\,\phi^\dag
 -i \sqrt{2}\,[\bar\lambda,\bar\psi]\,\phi \Big)\,.
\end{multline}
All the component fields in this Lagrangian (vector field $A_\mu$\,,
two Weyl spinors $\lambda$ and $\psi$\,, and a complex scalar $\phi$)
are massless and form $SU(2)$ triplets.

This $SU(2)$ gauge symmetry can be spontaneously broken by the Higgs
mechanism. However, a specific structure,
\,$\sim\text{tr}[\phi^\dag\,,\,\phi]^2$,
of the classical scalar potential in (\ref{micr2}) introduces some
peculiarities. Arranging, by a gauge transformation, a classical vacuum
configuration $\phi_c$ of the scalar field $\phi$ along the third $SU(2)$
generator,
\begin{equation}
\label{a&u}
(\phi_c\!)_{\,b} = \frac{a}{2}\,\delta_{b3}\,, \ \ \ \ \ \ \ \ \ \
u = \text{tr}\,\phi_c^2 = \frac{a^2}{2}\,,
\end{equation}
we discover that the vacuum expectation value $a$ itself is not fixed:
classical potential equals zero for any complex $a$\,. Thus, we obtain
a \emph{classical moduli space} of gauge-nonequivalent vacua,
parametrized by $a$ or, better, by a gauge-invariant parameter $u$. For
non-vanishing $u$, the $SU(2)$ gauge symmetry is broken to $U(1)$\,.
All the component fields of two supermultiplets from the $SU(2)$ triplet
(non-Abelian gauge and Higgs bosons, and accompanying fermions)
acquire mass $M=\sqrt{2}\,|a|$\,, while the rest of the fields comprise a
massless Abelian $N=2$ vector multiplet involving a `photon' with
superpartners.

Besides this (perturbative) part of the spectrum, the full quantum theory
based on the microscopic Lagrangian (\ref{micr2}) is also assumed to develop
a nonperturbative part, consisting of magnetic monopoles and dyons.
These soliton-like nonlocal objects are not seen in the perturbative
expansion, as is evident from the universal mass formula
\begin{equation}
\label{mass}
M = \sqrt{2}\,|a(n + m\tau)|\,,
\end{equation}
where integers $n,m$ count electric and magnetic charges, respectively.
Monopoles and dyons are known to lie in $N=2$ hypermultiplets containing
spins $\leqslant 1/2$\,.

Since it looks extremely difficult to find an exact quantum solution of the
microscopic theory (\ref{micr2})\,, we now proceed to investigate the
corresponding low-energy effective action, where all modes above some
low-energy cutoff are integrated out. Such effective action should involve
massless fields alone and contain only the terms with the minimal number of
derivatives. Fortunately, the structure of these terms is severely restricted
by the $N=2$ supersymmetry. Namely, with Abelian $W_\alpha$ and a single
chiral $\Phi$\,,
\begin{equation}
\label{low}
\mathcal{L}_\text{eff}
= \frac{1}{8\pi}\text{ Im} \left[\int \! d^{2} \theta \,
\mathcal{F}''(\Phi)\,W^{\alpha} W_{\alpha}
 + 2\!\int\! d^2\theta d^2\bar\theta \,\Phi^\dag \mathcal{F}'(\Phi)\right]\,,
\end{equation}
where a \emph{prepotential} $\mathcal{F}$ depends only on $\Phi$ but not on
$\Phi^\dag$ (nor on derivatives of both)\,. In other words, $\mathcal{F}$
should be a holomorphic function. It is an exact (though implicit) expression
for $\mathcal{F}(a)$ described below that is widely known as the
Seiberg-Witten solution.

It is time now to introduce a \emph{dual} notation:
\begin{equation}
\label{duals1}
a_D \doteq \frac{\partial\mathcal{F}(a)}{\partial a}
 \equiv \mathcal{F}'(a)\,, \ \ \ \ \ \ \ \
\tau(a) \doteq \frac{da_D}{da} = \mathcal{F}''(a)\,, \ \ \ \ \ \ \ \
\mathcal{F}_D(a_D) \doteq \mathcal{F}(a) - a\,a_D\,.
\end{equation}
As a simple corollary, we get
\begin{equation}
\label{duals2}
\mathcal{F}'_D(a_D) = -a\,, \ \ \ \ \ \ \ \ \
\tau_D(a_D) \doteq \mathcal{F}''_D(a_D) = -\frac{da}{da_D}
 = -\frac{1}{\tau(a)}\ .
\end{equation}
Here $\tau(a)$ is assumed to be a quantum generalization of the classical
constant $\tau$ from (\ref{tau})\,, and $a_D(a)$ of \,$\tau a$\,.
This is in accordance with the classical (renormalizable) form of
the prepotential, \ $\mathcal{F}_c(a)=\tau a^2/2$\,, and with a quantum
mass formula
\begin{equation}
\label{qmass}
M = \sqrt{2}\,|\,n a + m a_D|\,.
\end{equation}
A transparent electric-magnetic duality displayed by the last relation proves
to be also an intrinsic property of \,$\mathcal{L}_\text{eff}$\ : when
functionally integrated, it remains invariant under the following
duality transformations\,,
\begin{equation}
\label{change}
\Phi \rightarrow \Phi_D\,, \ \ \ \ \ \ \
\mathcal{F} \rightarrow\mathcal{F}_D\,, \ \ \ \ \ \ \
W^\alpha \rightarrow W_D^\alpha\,,
\end{equation}
an explicit form of the first two being evident from (\ref{duals1})\,, whereas
a relation between $W^\alpha$ and its dual $W_D^\alpha$\, should be nonlocal.

It is seen from (\ref{duals2}) that duality swaps the weak and strong
coupling regimes via $\tau\rightarrow -\tau^{-1}$.
Together with $\tau\rightarrow\tau+1$
(which is equivalent to $\vartheta\rightarrow\vartheta+2\pi$
and thus leaves the action invariant)\,, this transformation generates
the full duality group $SL(2,\mathbb{Z})$ of the effective
theory ($\alpha,\beta,\gamma,$ and $\delta$ are integers)\,:
\begin{equation}
\label{SL2Z}
\tau\rightarrow\frac{\alpha\tau + \beta}{\gamma\tau + \delta} \ \ \ \ \
\Longleftrightarrow \ \ \ \ \begin{pmatrix} a_D \\ a \end{pmatrix} \rightarrow
\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}
\begin{pmatrix} a_D \\ a \end{pmatrix}\,, \ \ \ \ \ \ \
\alpha\delta - \beta\gamma =1\,.
\end{equation}
Note that the underlying microscopic theory (\ref{micr1})  cannot be
self-dual, because gauge fields and monopoles belong to different
supermultiplets.

Positive definitness of the component-field kinetic terms requires
\begin{equation}
\label{posit}
\text{Im}\ \tau(a) > 0\,.
\end{equation}
This property, together with holomorphicity of $\mathcal{F}(a)$ and duality,
suffices to solve the problem completely, within a reasonable Ansatz on the
structure of \emph{quantum} moduli space. Like its classical counterpart, it
is parametrized by a complex variable $u=\langle\,\text{tr}\,\phi^2\rangle$\,.
However, an exact quantum functional dependence $a(u)$ is expected to be far
more complicated than (\ref{a&u})\,. Moreover, owing mainly to
(\ref{posit})\,, both $a(u)$ and $a_D(u)$ need to be multivalued functions
of $u$, forming actually an $SL(2,\mathbb{Z})$ bundle. A Seiberg-Witten
Ansatz allows precisely three singular (branch) points for these functions:
$u=\infty\,,\pm\,\Lambda^2$\,, with $\Lambda$ being a dynamical scale.

A singularity at infinity corresponds to the weak coupling (perturbative)
regime where the whole quantum dynamics, including the
formation of the effective action, is dominated by asymptotic freedom
of the original Lagrangian (\ref{micr2})\,. In addition, the $\beta$-function
of this theory, as well as the prepotential, are known to receive
perturbative corrections at the one-loop level only. As a result,
the expansion of $\mathcal{F}(a)$ at large $a$ looks like
\begin{equation}
\label{inf1}
\mathcal {F}(a) = \frac{i}{2\pi}\,a^2\left[\text{ ln}\,\frac{a^2}{\Lambda^2}
 + \sum_{n=1}^{\infty}\,c_n\left(\frac{\Lambda^2}{a^2}\right)^{2n}\,\right]
\end{equation}
(coefficients $c_n$ of nonperturbative instanton contributions are to be
found from the exact solution)\,. This agrees with the perturbative
corrections to $\tau$ due to running of $g^2$ in (\ref{tau})\,. Owing to
(\ref{a&u})\,, one comes to the following asymptotics and monodromies
near $u=\infty$\,:
\begin{gather}
\label{inf2}
a \,\sim\, \sqrt{2u}\,, \ \ \ \ \ \ \ \ \
a_D \,\sim\,\frac{i}{\pi}\sqrt{2u}\text{ ln}\,u \,
\sim\,\frac{2i}{\pi}\,a\,\text{ln}\,a\,, \\
\label{inf3}
\left( \begin{array}{c} a_D \\ a \end{array} \right) \ \rightarrow \
\left( \begin{array}{c}  -a_D + 2a \\ -a \end{array} \right)\doteq M_{\infty}
\left( \begin{array}{c} a_D \\ a \end{array} \right)\,, \ \ \ \ \ \ \
M_{\infty} = \left(\begin{array}{cc} -1 & 2 \\ 0 & -1 \end{array} \right)\,.
\end{gather}

Singularities at finite points $u=\pm\,\Lambda^2$ are interpreted as an
indication on some particles in the quantum spectrum becoming massless.
It is believed that the ``classical" singularity at $u=0$ where $SU(2)$
symmetry gets restored and charged gauge bosons become massless, disappear
in exact quantum picture, and at $u=\pm\,\Lambda^2$ a completely different
physical effect takes place: the mass of a monopole or dyon vanishes.
Furthermore, we expect that in this (strong coupling) region the original
local fields from (\ref{micr1})\,, as well as a variable $a$\,, are no more
good (expansion) parameters, and should be substituted in this role by dual
superfields and $a_D$\,. Specifically, we assume that at
$u=\Lambda^2$\, a magnetic monopole with $m=1$\,, \,$n=0$\, becomes
massless, and $a_D(u)$ vanishes.
To extract the necessary asymptotical information, we use the
$\beta$-function produced by dual (magnetic) photons interacting with
hypermultiplets of (now local) monopoles/dyons. This interaction is not
asymptotically free: rather, it looks like $N=2$ SUSY electrodynamics.
As a result,
\begin{gather}
\label{fin1}
\mathcal {F}_D(a_D)
 = -\frac{i}{4\pi}\left[\,a^2_D\text{ ln}\,\frac{a^2_D}{\Lambda^2}
  + \Lambda^2\sum_{n=1}^{\infty}\,d_n\!\left(\frac{a_D}{\Lambda}\right)^n
   \right]\,, \\
\label{fin2}
a_D \,\sim\, \frac{i}{2}\,(u-\Lambda^2)\,, \ \ \ \ \ \ \ \ \
a - \frac{4}{\pi} \,\sim\ - \frac{(u-\Lambda^2)\,\text{ln}(u-\Lambda^2)}{2\pi}
 \,\sim\, \frac{i}{\pi}\,a_D\,\text{ln}\,a_D\,, \\
\label{fin3}
\left( \begin{array}{c} a_D \\ a \end{array} \right) \ \rightarrow \
\left(\begin{array}{c}a_D \\ a - 2a_D \end{array}\right)\doteq M_{\Lambda^2}
\left( \begin{array}{c} a_D \\ a \end{array} \right)\,, \ \ \ \ \ \ \
M_{\Lambda^2} = \left(\begin{array}{cc} 1 & 0 \\ -2 & 1 \end{array}\right)\,.
\end{gather}
The remaining singular point $u=-\Lambda^2$ corresponds to a massless dyon
with $m=1$\,, \ $n=-1$\,, \ provides $a_D-a$ as a suitable expansion
parameter, and develops a monodromy
\begin{equation}
\label{third}
M_{-\Lambda^2} = M_{\Lambda^2}^{-1} M_{\infty}
 = \left(\begin{array}{cc} -1 & 2 \\ -2 & 3 \end{array}\right)\,.
\end{equation}

We are now in a position to display a celebrated Seiberg-Witten solution.
Actually, we can write down $a(u)$ and $a_D(u)$ in a closed form which is
consistent with all the above. Consider a certain hyperelliptic curve and a
meromorphic differential $dS$ whose $u$-derivative is holomorphic:
\begin{equation}
\label{curve}
y^2 = (x^2 - \Lambda^4)(x-u)\,, \ \ \ \ \ \ \ \ \
dS = \frac{dx\,(x-u)}{\sqrt{2}\,\pi\,y}\,, \ \ \ \ \ \ \ \
\partial_u\,dS \sim \frac{dx}{y}\ .
\end{equation}
The corresponding genus-one Riemann surface has the topology of a torus.
We identify the mutually dual quantities $a(u)$ and $a_D(u)$ as integrals
of $dS$ over two independent cycles (intersecting once)
that constitute a canonical homology basis of a torus:
\begin{gather}
\label{au}
a(u) = \oint dS
 = \frac{\sqrt{2}}{\pi}\int_{-1}^1\,\frac{dx\sqrt{x-u}}{\sqrt{x^2-1}}
 = \sqrt{2(1+u)}\ F\,(-\frac{1}{2}\,,\frac{1}{2}\,,1\,;\frac{2}{1+u})\ , \\
\label{adu}
a_D(u) = \oint_D dS
 = \frac{\sqrt{2}}{\pi}\int_1^u\,\frac{dx\sqrt{x-u}}{\sqrt{x^2-1}}
 = \frac{i}{2}\,(u-1)\,F\,(\frac{1}{2}\,,\frac{1}{2}\,,2\,;\frac{1-u}{2})
\end{gather}
(we have set $\Lambda=1$)\,.
Hence \,$\tau(u)=a'_D(u)/a'(u)$\, is a ratio of \emph{period integrals}
(of the holomorphic differential over basic cycles)\,,
and so $\text{Im}\,\tau>0$\,, as required. As a by-product, one obtains
\begin{equation}
\label{extra}
a\frac{da_D}{du} - \frac{da}{du}\,a_D = \frac{2i}{\pi}\,, \ \ \ \ \ \ \ \ \ \
\mathcal {F}(a) - \frac{1}{2}\,a_D\,a
 = \mathcal {F}_D(a_D) + \frac{1}{2}\,a_D\,a = -\frac{iu}{\pi}\,.
\end{equation}

Now, to compute the coefficients $c_n$ in (\ref{inf1}) to any given order,
one inverts (\ref{au}) expanding $u$ in inverse powers of $a$ (actually, in
$(\Lambda/a)^2$)\,, substitutes it into (\ref{adu}) and then integrates to
obtain $\mathcal{F}(a)$\,. Analogously, to find $d_n$ one needs to re-expand
$u$ in powers of $a_D$ near $u=1$ with the help of (\ref{adu}) and then
substitute into $a(u)$\,. The exact quantum mass spectrum is read off from
(\ref{au})\,, (\ref{adu})\,, and (\ref{qmass})\,. A detailed analysis shows
that in the weak coupling region $(u\sim\infty)$ the only stable massive
states are gauge bosons with superpartners $(n=\pm1,\,m=0)$ and a family of
dyons $(n\in\mathbb{Z},\,m=\pm 1)$\,, whereas at the strong coupling
$(|u|\sim1)$ only a part of them survive: namely, those with
$n=0,\pm1,\ m=\pm1$\,.

To examine more closely the structure of the underlying complex curve
(\ref{curve})\,, we re-express it in terms of different variables
$\{z,\lambda\}$ and consider a more general case of $SU(N)$ gauge group:
\begin{equation}
\label{curveN}
z + \frac{\Lambda^{2N}}{z} = P_N(\lambda)\,, \ \ \ \ \ \ \ \
dS \doteq \lambda\,\frac{dz}{z} = \frac{\lambda P\,{}'(\lambda)\,d\lambda}
      {\sqrt{P^2(\lambda)-4\Lambda^{2N}}}\ ,
\end{equation}
where a polynomial
\begin{equation}
\label{PN}
P_N(\lambda) = \text{det}(\lambda - \phi_c)
 = \lambda^N - \sum_{i=1}^{N-1}u_i \lambda^{N-i-1}
\end{equation}
is determined by a classical vacuum $\phi_c=\text{diag}(a_1,\ldots,a_N)\,, \,
\ \sum a_i=0$\,, with the moduli $u_i$ being gauge invariant combinations
of $a_i$\,. Note also that the masses produced by Higgs effect are in this
case given by $m_{ij}\sim|\,a_i-a_j|$\,. We see that the moduli dependence
of the curve itself \,is entirely specified by the gauge group,
whereas the definition of the differential $dS$ is universal.
Moreover, the case $\Lambda=0$\,, when the hyperelliptic curve
degenerates to a rational one, can be shown to
formally reproduce the perturbative regime. So, the whole wealth of exact
nonperturbative solution is switched on by mere adding a \,$\Lambda^{2N}\!/z$
term to (the perturbative) definition of the complex curve (\ref{curveN})\,!


\end{document}