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\begin{document}
\begin{titlepage}
\hfill
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           hep-th/9609157 \cr
           IPM-96-165   \cr
           Sep 1996   \cr
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\begin{center}
{\LARGE On the Picard-Fuchs equations of the SW models \\}
\vspace*{20mm}
{\ Mohsen Alishahiha \footnote{e-mail:alishah@physics.ipm.ac.ir}}\\
\vspace*{1mm}
{\it Institute for Studies in Theoretical Physics and Mathematics, \\
 P.O.Box 19395-5531, Tehran, Iran } \\
{\it Department of Physics, Sharif University of Technology, \\
\it  P.O.Box 11365-9161, Tehran, Iran }\\
\vspace*{25mm}
%\maketitle
\end{center}
\begin{abstract}
We obtain the closed form of the Picard-Fuchs equations for $N=2$ 
supersymmetric Yang-Mills theories with classical Lie gauge groups. 
For a gauge group of rank $r$, there are $r-1$ regular and an exceptional 
differential equations. We describe the series solutions 
of the Picard-Fuchs equations in the semi-classical regime.
\end{abstract}
\end{titlepage}
\newpage
From duality and holomorphy, Seiberg and Witten\cite{SW} have obtained the
exact prepotential of $N=2$ SYM theory with gauge group $SU(2)$ by studying the
singularities  of its moduli space at strong coupling. 
For a gauge group of rank $r$, the Seiberg and Witten's 
data is a hyperelliptic curve with $r$ complex dimensional moduli space
with certain singulareties and a meromorphic one form, ($E_{u_i},\lambda_{SW}$).
More precisely, the prepotential of $N=2$ SYM in the Coulomb phase can be 
described
with the aid of a family of complex curves with the identification of the
v.e.v.'s $a_i$ and their duals $a^{D}_{i}$ with the periods of the curve
\be
a_{i} = \oint _{\alpha _{i}} \lambda_{SW} \;\;\;\; \mbox{and} \;\;\;\;
a^{D}_{i} = \oint _{\beta_{i}} \lambda_{SW} , \label{integrals}
\ee
where $\alpha_{i}$ and $\beta_{i}$ are the homology cycles of the
corresponding Riemann surface.

To find the periods one should calculate the above integrals, or
one may use the fact that the periods $\Pi=(a_i,a^{D}_i)$
satisfy the Picard-Fuchs equations. So it is
important to find the Picard-Fuchs equations which also help
in the instanton calculus.

In this letter we obtain closed forms for the Picard-Fuchs equations for 
classical Lie gauge groups. Recently, some of these equations have been 
obtained in {\cite{IS} and \cite{KL}}.

The Seiberg-Witten's data ($E_{u_i},\lambda_{SW}$) for classical gauge groups
are known \cite{AA} and \cite{DH}, 
and take the following form
\bea  \l{CU}
y^2&=&W^2(x)-\Lambda ^{2\hat h}x^{2k}\cr
\lambda_{SW}& =&(k W-x{dW \o dx}){{dx} \o y}
\eea
where $\hat h$ is the dual Coxeter number of the Lie gauge group and 
\be \l{PW}
W(x)=x^m -\sum_{i=2}^{m}{u_i x^{m-i}}
\ee
with $m=r+1, i=2,3,...,r+1$ for $A_r$ series and $m=2r, i=2,4,...,2r$ for
$B_r, C_r,D_r$ series, and $u_i$'s, the Casimirs of the gauge groups. 
Also $k=m-\hat h$. Note that the $D_r$ series has
an exceptional Casimir, $t$, of degree $r$, but in our notation we set 
$u_{2r}=t^2$.

From explicit form of $\lambda_{SW}$ and the fact that the $\lambda_{SW}$
is lineary dependent on Casimirs, setting ${\partial \o {\partial{u_i}}}=
\partial_i$ we have
\bea \l{PARTIAL}
\partial_i\lambda_{SW}&=&-{x^{m-i} \o y}dx+d(*), \cr
\partial_i\partial_j\lambda_{SW}&=&-{x^{2m-i-j} \o y^3}W(x)dx+d(*).
\eea

By direct calculation one can see that
\be \l{DI}
{d \o dx}({x^n \o y})=(n-k){x^{n-1} \o y}+({{kx^{n-1} W-x^n{dW \o dx}} \o y^3})
W.
\ee
By inserting equation (\ref{PW}) in (\ref{DI}) we have
\be \l{DIF}
{d \o dx}({x^n \o y})=(n-k){x^{n-1} \o y}-{\hat h}{x^{m+n-1} \o y^3}W+ 
\sum_{i=2}^{m}{(m-k-i)u_i{x^{m+n-1-i} \o y^3}W}
\ee
Now from equations (\ref{PARTIAL}) we can find the second 
order differential equation for the periods $\Pi$ as follows  
\be  \l{PF}
{\cal L}_{n}=(k-n)\partial_{m-n+1}+{\hat h}\partial_2\partial_{m-n-1}-
\sum_{i=2}^{m}{(m-k-i)u_i \partial_i\partial_{m-n+1}}.
\ee
where $n=s-1$ for $A_r$ series and $n=2s-1$ for $B_r,C_r$
and $D_r$ series and $s=1,...,r-1$. Note that in the $A_r$ series, for $s=1$, the
above expression is not valid. In fact we should be careful in the final step 
in the derivation of the equation (\ref{PF}), that ${\cal L}^{A_r}_0$ is
\be  \l{APF}
{\cal L}^{A_r}_0=(r+1)\partial_2\partial_r-\sum_{i=2}^{r}{(r+1-i)u_i 
\partial_{i+1}\partial_{r+1}}
\ee
Moreover for $s>r-1$ equation (\ref{DIF}) does not give
the second order differential equation with respect to $u_i$. So, by this 
 method we can only 
find $r-1$ equations which we call {\it regular} equations. 
Also from the equation (\ref{PARTIAL}) we have the following identity 
\be\ba {ll} \l{ID}
{\cal L}_{i,j;p,q}=\partial_i\partial_j-\partial_p\partial_q, & i+j=p+q
\ea\ee

The $r$th equation, the {\it exceptional} equation, can be obtain from 
the following linear combination
\be
D=(k-m)d({x^{m+1} \o y})+\sum_{i=2}^{m}{(m-k+i)u_i
d({x^{m+1-i} \o y})}
\ee
or
\be
D=\lambda_{SW}-(\sum_{i=2}^{m}{i(i-2)u_i {x^{m-i} \o y}}
+\sum_{j,i=2}^{m}{ij u_iu_j {x^{2m-i-j} \o y^3}W} -{\hat h}^2 
\Lambda^{2{\hat h}} {x^{2k} \o y^3}W)dx.
\ee

So from the equation (\ref{PARTIAL}), the {\it exceptional} differential
equation for the periods $\Pi$ are 
\be
{\cal L}_r=1+\sum_{i=2}^{m}{i(i-2)u_i\partial_i}+\sum_{j,i=2}^{m}{ij u_iu_j
\partial_i\partial_j}-{\hat h}^2 \Lambda^{2{\hat h}}\partial_{{\hat h}}^{2}.
\ee
for $A_r,D_r$ and $C_r$ with odd $r$. For $B_r$ and $C_r$ with even $r$
the last term should be changed 
to $-{\hat h}^2 \Lambda^{2{\hat h}}\partial_{{\hat h}-1}\partial_{{\hat h}+1}$.

This equation together with the equations (\ref{PF}) and (\ref{ID})
give a complete set of the Picard-Fuchs equations for the periods 
$(a_i,a^{D}_i)$.

To study the series solutions of the Picard-Fuchs equations in the 
semi-classical regime, let us rewrite the Picard-Fuchs equations in 
terms of Euler derivative $\vartheta_i=u_i\partial_i$. 
The regular equations for $s\neq r-1$
become
\bea
{\cal L}_n=[(k+1-n)\vartheta_{m-n+1}-1]{\vartheta_{m-n+1} \o u_{m-n+1}}-
\sum_{{i\neq m-n+1},i=2}^{m}{{(m-k-i)\o u_{m-n+1}}\vartheta_i\vartheta_{m-n+1}}\cr
+{{\hat h}\o u_2 u_{m-n-1}}\vartheta_2\vartheta_{m-n-1} 
\eea
If $s=r-1$ the last term should be replaced by ${{\hat h}\o u_2^2}
\vartheta_2(\vartheta_2-1)$. Again, we should be careful for $s=1$ in the
$A_r$ series. From equation (\ref{APF}) we have 
\be
{\cal L}_0^{A_r}=
{(r+1)\o u_2 u_r} \vartheta_2\vartheta_r-{u_r \o u_{r+1}^2}
\vartheta_{r+1}(\vartheta_{r+1}-1) -\sum_{i=2}^{r-1}{{(r+1-i)u_i \o u_{i+1}
u_{r+1}}\vartheta_{i+1}\vartheta_{r+1}}
\ee
The exceptoinal equation changes to
\be
{\cal L}_r=
(1-\sum_{i=2}^m{i\vartheta_i})^2-{{\hat h}^2\Lambda^{2{\hat h}}\o u_{{\hat h}^2}} 
\vartheta_{{\hat h}}(\vartheta_{{\hat h}}-1)
\ee
for $A_r,D_r$ and $C_r$ with odd $r$, and 
\be
{\cal L}_r=
(1-\sum_{i=2}^m{i\vartheta_i})^2-{{\hat h}^2\Lambda^{2{\hat h}}\o u_{{\hat h}-1}
u_{{\hat h}+1}} 
\vartheta_{{\hat h}-1}\vartheta_{{\hat h}+1}
\ee
for $B_r$ and $C_r$ with even $r$. 

Let us define the variables $x_s$  
\be\ba {ll}
x_s={u_{m-n+1} \o {u_2 u_{m-n-1}}} & s=1,...,r-1 \\
& \\
x_r={\Lambda^{2{\hat h}} \o u_{{\hat h}}^2}& (or \,\,\,
x_r={\Lambda^{2{\hat h}} \o {u_{{\hat h}-1}u_{{\hat h}+1}}})
\ea\ee
for $B_r, C_r$ and $D_r$ corresponding to above, and the variables
\be\ba {ll}
x_s={u_{m-n+1} \o {u_2 u_{m-n-1}}} & s=2,...,r-1 \\
&\\
x_1={\Lambda^{2(r+1)} \o u_2u_{r}^2}&x_r={\Lambda^{2(r+1)} \o u_{(r+1)}^2}.
\ea\ee
for the $A_r$ series.  
We can construct the power series solution of the Picard-Fuchs equations
around $(x_i)=(0)$ \cite{IS}
\be \l{SUM}
\omega(a_1,...,a_r;x_1,...,x_r)=\sum_{l_1,...,l_r=0}{C_{l_1,...,l_r}
x_1^{l_1+a_1}...x_r^{l_r+a_r}}
\ee

Let us take $\alpha_i(l_j)=\alpha_i(a_1,...,a_r;l_1,...,l_r)$ be 
the power of $u_i$ when equation (\ref{SUM}) is reexpressed in term  
of $u_i$'s. By inserting $\omega$ in the Picard-Fuchs equations, 
one can obtain the indicial and recursion relations. 
For example the indicial relation for $B_r,C_r$ and $D_r$ are
\bea
[(k+1-n)\alpha_{m-n+1}(0)-1]\alpha_{m-n+1}(0)&-&
\sum_{{i\neq m-n+1},i=2}^{m}{(m-k-i)\alpha_i(0)\alpha_{m-n+1}(0)}=0 \cr
(1-\sum_{i=2}^{m}{i\alpha_i(0))^2}&=&0 
\eea

Also the recursion relations are
\be\ba {ll} \l{RE}
C_{l_1,...,l_r}={{-{\hat h}\alpha_2(l_s-1)\alpha_{m-n-1}(l_s-1)}\o
\Delta_s}C_{l_1,...,l_s-1,...,l_r} & i=1,...,r-2\\
& \\
C_{l_1,...,l_r}={{-{\hat h}\alpha_2(l_{r-1}-1)(\alpha_2(l_{r-1}-1)-1)}\o
\Delta_s} C_{l_1,...,l_{r-1}-1,l_r} & \\
  &   \\
C_{l_1,...,l_r}={{{\hat h}^2\alpha_{{\hat h}}(l_r-1)(\alpha_{{\hat h}}(l_r-1)-1)
} \o (1-\sum_i{i\alpha_i(l_j))^2}}C_{l_1,...,l_r-1}&
\ea\ee
where $\alpha_i(0)$ means that all $l_j=0$ and $\alpha_i(l_s-1)$ means that
the $s$th $l$ should be set equal to $l_s-1$ and other $l$'s are fixed, also
\be
\Delta_s= 
[(k+1-n)\alpha_{m-n+1}(l_j)-1]\alpha_{m-n+1}(l_j)-
\sum_{{i\neq m-n+1},i=2}^{m}{(m-k-i)\alpha_i(l_j)\alpha_{m-n+1}(l_j)}
\ee

Note that in the last relation of the equation (\ref{RE}) the suitable 
change as noted above should be made. By the same 
method one can obtain the indicial and recursion relations for the 
$A_r$ series. This method can be applied for the $N=2$ theories with
massless hypermultiplets which their curves are in the form of (\ref{CU}).

After completion of this work, i recived the paper \cite{MU} which is paid
to the same problem.
\vspace*{5mm}

I would like to thank M. Khorrami for helpful discussions.

\newpage
\begin{thebibliography}{99}
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N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19.

N. Seiberg and E. Witten, Nucl. Phys. B431 (1994) 484.

\bibitem{IS}
K. Ito, N. Sasakura hep-th/9608054

\bibitem{KL}

A. Klemm, W. Lerche, and S. Theisen,
Int. J. Mod. Phys. A11 (1996) 1929.

K. Ito, S. K. Yang hep-th/9603073.

Y. Ohta, hep-th/9604051, hep-th/9604059.

M. Matone, Phys. lett. B357 (1995) 342.

H. Ewen, K. F\"orger, S. Theisen, hep-th/9609062.

\bibitem{AA}

M. R. Abolhasani, M. Alishahiha, A. M. Ghezelbash, hep-th/9606043.

A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen,
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P. Argyres, A. Faraggi, Phys. Rev. Lett. 73 (1995) 3931.

U. H. Danielsson, B. Sundborg, Phys. Lett. B358 (1995) 273.

A. Brandhuber, K. Landsteiner, Phys. Lett. B358 (1995) 73.

P. C. Argyres, A. D. Shapere, Nucl. Phys. B461 (1996) 437.

\bibitem{DH}

E. D'Hoker, I. M. Keichever, D. H. Phong, hep-th/9609145

\bibitem{MU}

J. M. Isidro, A. Mukherjee, J. P. Nunes, H. J. Schnitzer, hep-th/9609116.
\end{thebibliography}
\end{document}

