state vectors $|\phi _{1}>$ and $<\phi
_{2}|$ in terms of coherent state, $$\begin{aligned}
|\phi _{1}>&= &\frac{1}{\pi }\int_{-\infty }^{\infty
}dxdy|z><z|\phi _{1}>
\nonumber  \label{eq:28} \\
&\equiv &\frac{1}{\pi }\int_{-\infty }^{\infty }dxdyf_{1}(z)|z>, \\
<\phi _{2}| &=&\frac{1}{\pi }\int_{-\infty }^{\infty }dxdy<\phi
_{2}|z><z|
\nonumber \\
&=&\frac{1}{\pi }\int_{-\infty }^{\infty }dxdyf_{2}(z)<z|.
\label{eq:29}\end{aligned}$$

The condition (\[eq:50\]) is satisfied if and only if $$f_{1}(\omega_{N}^{-1}z)=f_{1}(z)e^{i\alpha _{1}},$$$$f_{2}(\omega_{N}^{-1}z)=f_{2}(z)e^{-i\alpha _{2}}.$$

Here $\omega_{N}=e^{-i\frac{2\pi }{N}}$. We have $$\begin{aligned}
  \label{eq:31}
F_{st}(k,q_{0}) &=&\frac{1}{\pi ^{2}}\sum_{n=0}^{A-1}\int <k,q_{0}+\frac{%
l(n+s)}{A}|z_{1}>f_{1}(z_{1})dx_{1}dy_{1}  \nonumber \\
&&\times \int
<z_{2}|k,q_{0}+\frac{ln}{A}>f_{2}(z_{2})dx_{2}dy_{2}\times e^{2\pi
i(q_{0}/l+n/A)t}  \nonumber \\
&=&\frac{1}{\pi ^{2}}\int
dx_{1}dy_{1}dx_{2}dy_{2}g_{st}(k,q_{0},z_{1},z_{2})f_{1}(z_{1})f_{2}(z_{2}),\end{aligned}$$where $$g_{st}(k,q_{0},z_{1},z_{2})=\sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}%
|z_{1}><z_{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t}.$$We call the kernel $g$ as generating function in coherent state representation. Next, we study the expression of $g$ for $Z_{4}$ case. Through $g$ we can give the integration expression for all the projection operators on the $T^{2}/Z_{4}$ with minimal trace and continuous eigen value function. Consider the equation $$\theta (z,\tau )^{\ast }=\theta (z^{\ast },-\tau ^{\ast }).$$For the $z_{4}$ case, $\tau =i,A=\frac{l^{2}}{2\pi }$, from (\[eq:27\])(\[eq:28\]) and (\[eq:29\]) we get $$\begin{aligned}
&<&k,q+\frac{ls}{A}|z_{1}><z_{2}|k,q+\frac{ls^{\prime }}{A}>  \nonumber \\
&=&\frac{1}{l\sqrt{\pi }}\theta (\frac{q}{l}+\frac{i}{l}k+\frac{s}{A}-\frac{i%
\sqrt{2}z_{1}}{l},\frac{i}{A})\theta (\frac{q}{l}-\frac{i}{l}k+\frac{%
s^{\prime }}{A}+\frac{i\sqrt{2}z_{2}^{\ast }}{l},\frac{i}{A})  \nonumber \\
&&\times e^{-k^{2}+\frac{k}{2}(z_{1}+z_{2}^{\ast })+ik\frac{l(s-s^{\prime })%
}{A}}e^{-\frac{1}{2}(z_{1}^{2}+(z_{2}^{\ast
})^{2}+z_{1}z_{1}^{\ast }+z_{2}z_{2}^{\ast })}\equiv K_{ss^{\prime
}}.\end{aligned}$$Let $u=\frac{lk}{2\pi },v=\frac{q}{l},\mu =-i\frac{\sqrt{2}A}{l}z_{1},\nu =i%
\frac{\sqrt{2}A}{l}z_{2}^{\ast }$, then $$\begin{aligned}
K_{ss^{\prime }} &=&\frac{C_{1}}{l\sqrt{\pi }}\theta (v+\frac{iu}{A}+\frac{%
s+\mu }{A},\frac{i}{A})\theta (v-\frac{iu}{A}+\frac{s^{\prime }+\nu }{A},%
\frac{i}{A})  \nonumber \\
&&\times e^{\pi i\frac{2i}{A}u^{2}+2\pi i(\frac{s+\mu 