The modular transformation of coordinate on torus In 2D conformal field theory, the modular transformation of torus is defined by the fractional linear transformation
\begin{equation}
\tau'=\frac{a\tau+b}{c\tau+d},~~ad-bc=1,~a,b,c,d\in Z.
\end{equation}
This transformation is performed on the modular parameter $\tau$. However, the corresponding 
transformation for coordinate $z$ on torus is 
\begin{equation}
z'=\frac{z}{c\tau+d}.
\end{equation}
My question is that how the last transformation law come about, which seems just a rescaling of coordinates on torus?
 A: Consider a two-dimensional torus $T^2$ with modular parameter $\tau\in\mathbb{H}$, where $\mathbb{H}$ denotes the upper half-plane.
As you mentioned, the modular transformations of the torus are given by
\begin{equation}\label{eq:tauTransformation}
 \tau\mapsto \tau'=\frac{a\tau+b}{c\tau+d}
\end{equation}
where $a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$. More precisely, these are called Möbius transformations of the modular parameter of the torus.
These transformations generate the group $SL(2,\mathbb{Z})$.
These are symmetries of $T^2$ in the sense that they preserve the lattice
\begin{equation}
 \Lambda_{\tau}=\left\{n+m\tau:n,m\in\mathbb{Z}\right\},
\end{equation}
together with its orientation.
Indeed, recall that $T^2=\mathbb{C}/{\Lambda_{\tau}}$, so that the above statement should be clear.
Let me call this a $\textit{non-geometric}$ definition of $SL(2,\mathbb{Z})$.
The modular transformations of the torus can be equivalently defined in an alternative way, which I will call $\textit{geometric}$.
Let me define a $\tau$-adapted complex coordinate on $T^2$:
\begin{equation}
 z=x+\tau y
\end{equation}
where $x,y\in\mathbb{R}^2/{\mathbb{Z}\oplus\mathbb{Z}}$, namely $x$ and $y$ are real periodic coordinates with period 1.
Given the modular parameter of the torus, one can equivalently work with the pair of real periodic coordinates $(x,y)\sim x+\tau y$.
Consider a modular transformation $\Gamma\in SL(2,\mathbb{Z})$ given by
\begin{equation}
 \Gamma\simeq
 \begin{pmatrix}
  a&b\\
  c&d\\
 \end{pmatrix}
\end{equation}
namely such that the action of $\Gamma$ on $\tau$ is given by $\tau\mapsto \tau'=\frac{a\tau+b}{c\tau+d}$, namely $\Gamma(\tau)=\tau'$.
Let me define the following maps
\begin{equation}
 \begin{aligned}
  \gamma:\,&\mathbb{R}^2/{\mathbb{Z}\oplus\mathbb{Z}}\to\mathbb{R}^2/{\mathbb{Z}\oplus\mathbb{Z}}\\
  &\begin{pmatrix}
   x\\
   y
  \end{pmatrix}
 \mapsto
 \begin{pmatrix}
  d&b\\
  c&a
 \end{pmatrix}
\begin{pmatrix}
 x\\
 y
\end{pmatrix}
 \end{aligned}
\end{equation}
and
\begin{equation}
 \begin{aligned}
  \lambda_\tau:\,&\mathbb{C}\to\mathbb{C}\\
  &z\mapsto\frac{z}{c\tau+d}
 \end{aligned}
\end{equation}
Then the transformed modular parameter $\tau'=\Gamma(\tau)$ satisfies the identity
\begin{equation}
 x+\tau' y=\lambda_{\tau}(\gamma(x,y)),
\end{equation}
where it is understood that $\gamma(x,y)\sim x'+\tau y'$ and $(x',y')=\gamma(x,y)$.
Thus, one can equivalently interpret a modular transformation of $\tau$ as the composition of a $SL(2,\mathbb{Z})$ transformation of the $\tau$-adapted real periodic coordinates on the torus $(x,y)$ with a complex rescaling $z\mapsto\frac{z}{c\tau+d}$ of the complex coordinate on the torus. Notice that the latter is exactly the rescaling you mentioned in your question.
I prefer to refer to this interpretation as $\textit{geometric}$ since it recovers the transformation of the complex structure of the torus starting from a coordinate transformation.
This is the best interpretation I can give to equation (3.4) of the paper you quoted (http://arxiv.org/abs/hep-th/9609022v1).
