%  Corresction to : Representation of mapping class groups via the universal perturbative invariant  
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% Topmatter
\begin{document}
\baselineskip20pt
% Title
\begin{center}
%\begin{large}
\begin{bf}
{CORRECTION TO: \\
REPRESENTATION OF MAPPING CLASS GROUPS VIA \\ THE UNIVERSAL PERTURBATIVE INVARIANT}
\end{bf}
\\
{Jun Murakami} %\end{large}
\\
Department of Mathematics,
Osaka University\\
Toyonaka, Osaka 560, Japan
\\
{\it e-mail address\/}: jun@@math.sci.osaka-u.ac.jp
\end{center}
\subsection*{p. 578, Figure 10}
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$\begin{matrix}
\begin{picture}(53, 35)(-18, 0)
\put(0, 24){\xdelta}
\put(10, 4){\xright}
\put(0, 0){\xdouble}
\put(1.2, 4){\tline(0, 1){20}}
\put(3.2, 9){\line(0, 1){15}}
\put(2, 4){\framebox(3, 4.8){$\nu$}}
\put(-3, 2){$\varepsilon_1^{}$}
\put(29, 0){$\varepsilon_4^{}$}
\put(27, 24){$\varepsilon_3^{}$}
\end{picture}
\\
x_1^{}
\end{matrix} \quad
\begin{matrix}
\begin{picture}(54, 35)(-15, 0)
\put(0, 24){\xdelta}
\put(10, 4){\xright}
\put(-11, 4){\xleft}
\put(0, 0){\xdouble}
\put(-18, 23){$\varepsilon_{3i-6}^{}$}
\put(-18, 2){$\varepsilon_{3i-5}^{}$}
\put(-6, 0){$\varepsilon_{3i-3}^{}$}
\put(26, 23){$\varepsilon_{3i+1}^{}$}
\put(26, 2){$\varepsilon_{3i}^{}$}
\end{picture}
\\
x_i^{}
\end{matrix} \quad
\begin{matrix}
\begin{picture}(38, 35)(-15, 0)
\put(0, 24){\xdelta}
\put(-11, 4){\xleft}
\put(0, 0){\xdouble}
\put(13.2, 4){\line(0, 1){5}}
\put(6, 9.1){\framebox(8.7, 4.5){$e^{-\zeta/2}$}}
\put(15.2, 4){\tline(0, 1){20}}
\put(13.2, 13.8){\line(0, 1){10.4}}
\put(-18, 23){$\varepsilon_{3g-6}^{}$}
\put(-18, 2){$\varepsilon_{3g-5}^{}$}
\put(16, 18){$\varepsilon_{3g-3}^{}$}
\end{picture}
\\
x_g^{}
\end{matrix}$  
\end{center}
\subsection*{p. 582, ll. 4-8}
\begin{align*}
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_1
=
C_1 +
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_2 + 
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_3 +
\left(
-1 + \dfrac{\theta}{48}
\right)  \, C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_2
=
\left(
1 - \dfrac{\theta}{24}
\right) \, C_2 +
\left(\dfrac{1}{2} -\dfrac{\theta}{96}
\right) \, C_4 + 
 \dfrac{\theta}{12}\, 
C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_3
=
\left(
1 - \dfrac{\theta}{24}
\right) \, C_3 +
\left(\dfrac{1}{2} -\dfrac{\theta}{96}
\right) \, C_4 + 
 \dfrac{\theta}{12}
C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_4
=
\left(
1 - \dfrac{\theta}{12}
\right) \, C_4, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_5
=
\left(
-\dfrac{1}{2}-\dfrac{\theta}{96}\, \right) \, C_4 + 
\left(
1 + \dfrac{\theta}{24}
\right) \, C_5.  
\end{align*}
\subsection*{p. 582, formula (4.7)}
\[
(4.7) 
\qquad
\widetilde\rho_2^{(1)}(\alpha_2^{})
=
-\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16}) \,
\left(
\begin{matrix}
1 & 0 & 0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96}&
1 - \dfrac{\theta}{24}&
0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96}  & 0 & 1 - \dfrac{\theta}{24}\,  & 0 & 0
\\\\
0 & 
\dfrac{1}{2} - \dfrac{\theta}{96} & 
\dfrac{1}{2} - \dfrac{\theta}{96}  &
1 - \dfrac{\theta}{12} & 
\dfrac{1}{2} - \dfrac{\theta}{96}
\\\\
-1 + \dfrac{\theta}{48}  &  \dfrac{\theta}{12}  & \dfrac{\theta}{12}  &  0 & 
1 + \dfrac{\theta}{24}
\end{matrix}
\right). 
\]
\subsection*{p. 583, l. 5}
\begin{align*}
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_2
=
\left(
-2 - \dfrac{\theta}{12} 
\right) \,  C_1 +
\left(
1 + \dfrac{\theta}{24}
\right) \, C_2
- 
\dfrac{\theta}{24} \, C_5, 
\end{align*} 
\subsection*{p. 583, formula (4.8)}
 \[
(4.8) \qquad
\widetilde\rho_2^{(1)}(\beta_1^{})
=
-\sqrt{-1} \,
 \left(
\begin{matrix}
1 + \dfrac{\theta}{24}  & 
-2  - \dfrac{\theta}{12}  &
 0 & 0 & 0
\\\\
0 &
1 + \dfrac{\theta}{24}  &
0 & 0 & 0
\\\\
0 & 0 & 
1 + \dfrac{\theta}{24} & 
-2  - \dfrac{\theta}{12}  & 0
\\\\
0 & 0 & 0 &
1 + \dfrac{\theta}{24}  & 0
\\\\
0 & - \dfrac{\theta}{24} & 0 &  0 & 
1 + \dfrac{\theta}{48}
\end{matrix}
\right). 
\]
\subsection*{p. 584, formula (4.9)}
 \[
(4.9) \qquad
\widetilde\rho_2^{(1)}(\beta_2^{})
=
-\sqrt{-1} \, 
 \left(
\begin{matrix}
1 + \dfrac{\theta}{24}  & 0 &  
-2  - \dfrac{\theta}{12}  &
  0 & 0
\\\\
0 &
1 + \dfrac{\theta}{24}  &
0 & 
-2  - \dfrac{\theta}{12} & 0
\\\\
0 & 0 & 
1 + \dfrac{\theta}{24}  & 
0 & 0
\\\\
0 & 0 & 0 &
1 + \dfrac{\theta}{24} & 0
\\\\
0 & 0 & - \dfrac{\theta}{24} &  0 & 
1 + \dfrac{\theta}{48}
\end{matrix}
\right). 
\]
\end{document}