$\widetilde{\psi}_{j}:=\psi_{j}\bigg{|}_{L_{j}\rightarrow-
L_{j}}=\left(\\!\\!\begin{array}[]{cc}e^{-L_{j}}&e^{\overline{L}_{j}}\\\
-e^{L_{j}}&e^{-\overline{L}_{j}}\end{array}\\!\\!\right)$ which satisfies

$\displaystyle\widetilde{\psi}_{j}\Lambda_{j}^{k}=E(\psi_{j}\Lambda_{j}^{k})F,~{}~{}E:=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),~{}~{}F:=\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right).$

Then

$\displaystyle\sigma\bigg{|}_{L_{j}\rightarrow-L_{j}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\left|\begin{array}[]{cccccccc}\widetilde{\psi}_{1}&\widetilde{\psi}_{2}&\cdots&\widetilde{\psi}_{n}&1\\\
\widetilde{\psi}_{1}\Lambda_{1}&\widetilde{\psi}_{2}\Lambda_{2}&\cdots&\widetilde{\psi}_{n}\Lambda_{n}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\widetilde{\psi}_{1}\Lambda_{1}^{n}&\widetilde{\psi}_{2}\Lambda_{2}^{n}&\cdots&\widetilde{\psi}_{n}\Lambda_{n}^{n}&$\framebox{0}$\end{array}\right|$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\left|\begin{array}[]{cccccccc}E(\psi_{1})F&E(\psi_{2})F&\cdots&E(\psi_{n})F&E1E\\\
E(\psi_{1}\Lambda_{1})F&E(\psi_{2}\Lambda_{2})F&\cdots&E(\psi_{n}\Lambda_{n})F&E0E\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
E(\psi_{1}\Lambda_{1}^{n})F&E(\psi_{2}\Lambda_{2}^{n})F&\cdots&E(\psi_{n}\Lambda_{n}^{n})F&$\framebox{$E0E$}$\end{array}\right|$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
E\left|\begin{array}[]{cccccccc}\psi_{1}&\psi_{2}&\cdots&\psi_{n}&1\\\
\psi_{1}\Lambda_{1}&\psi_{2}\Lambda_{2}&\cdots&\psi_{n}\Lambda_{n}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\psi_{1}\Lambda_{1}^{n}&\psi_{2}\Lambda_{2}^{n}&\cdots&\psi_{n}\Lambda_{n}^{n}&$\framebox{0}$\end{array}\right|E=E\sigma
E$

by the multiplicative rule of quasideterminant. (Common multiplier of rows is
E, common multipliers columns are F and E.) Now we have

$\displaystyle\sigma\bigg{|}_{L_{j}\rightarrow-
L_{j}}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\frac{-1}{\Delta}\left(\begin{array}[]{cc}\Delta_{11}&\Delta_{12}\\\
\Delta_{21}&\Delta_{22}\end{array}\right)\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)=\frac{-1}{\Delta}\left(\begin{array}[]{cc}\Delta_{22}&\Delta_{21}\\\
\Delta_{12}&\Delta_{11}\end{array}\right).$

Therefore,
$\left(\Delta,\Delta_{11},\Delta_{12},\Delta_{21},\Delta_{22}\right)\bigg{|}_{L_{j}\rightarrow-
L_{j}}=\left(\Delta,\Delta_{22},\Delta_{21},\Delta_{12},\Delta_{11}\right)$.

By the fact that
$\dfrac{df(-x)}{dx}=-\dfrac{df(x)}{dx}\bigg{|}_{x\rightarrow-x},$ we have

$\displaystyle\left(\partial_{\mu}\Delta,\partial_{\mu}\Delta_{11},\partial_{\mu}\Delta_{12},\partial_{\mu}\Delta_{21},\partial_{\mu}\Delta_{22}\right)\bigg{|}_{L_{j}\rightarrow-
L_{j}}$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle-\partial_{\mu}\left(\Delta\big{|}_{L_{j}\rightarrow-
L_{j}},\Delta_{11}\big{|}_{L_{j}\rightarrow-
L_{j}},\Delta_{12}\big{|}_{L_{j}\rightarrow-
L_{j}},\Delta_{21}\big{|}_{L_{j}\rightarrow-
L_{j}},\Delta_{22}\big{|}_{L_{j}\rightarrow-L_{j}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle-\partial_{\mu}\left(\Delta,\Delta_{22},\Delta_{21},\Delta_{12},\Delta_{11}\right)~{}~{}\mbox{by
$\eqref{Symmetry of Delta, Delta_ij}$}$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle-\left(\partial_{\mu}\Delta,\partial_{\mu}\Delta_{22},\partial_{\mu}\Delta_{21},\partial_{\mu}\Delta_{12},\partial_{\mu}\Delta_{11}\right).$

Corollary B.2
Proposition B.1 implies:

(1)
$\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}\right]$
is an even function with respect to $L_{j}$.

(2)
$\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}(\partial_{\rho}\sigma)\sigma^{-1}\right]$
is an odd function with respect to $L_{j}$.
(Proof)
(1)

$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}\right]\bigg{|}_{L_{j}\rightarrow-
L_{j}}$ $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle\frac{-1}{|\sigma|\Delta^{2}}\left\\{\left|\begin{array}[]{cc}\partial_{\mu}\Delta_{11}&\partial_{\mu}\Delta_{12}\\\
\partial_{\nu}\Delta_{21}&\partial_{\nu}\Delta_{22}\end{array}\right|+\left|\begin{array}[]{cc}\partial_{\nu}\Delta_{11}&\partial_{\nu}\Delta_{12}\\\
\partial_{\mu}\Delta_{21}&\partial_{\mu}\Delta_{22}\end{array}\right|-2|\sigma|(\partial_{\mu}\Delta)(\partial_{\nu}\Delta)\right\\}\bigg{|}_{L_{j}\rightarrow-
L_{j}}$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{-1}{|\sigma|\Delta^{2}}\left\\{\left|\begin{array}[]{cc}-\partial_{\mu}\Delta_{22}&-\partial_{\mu}\Delta_{21}\\\
-\partial_{\nu}\Delta_{12}&-\partial_{\nu}\Delta_{11}\end{array}\right|+\left|\begin{array}[]{cc}-\partial_{\nu}\Delta_{22}&-\partial_{\nu}\Delta_{21}\\\
-\partial_{\mu}\Delta_{12}&-\partial_{\mu}\Delta_{11}\end{array}\right|-2|\sigma|(-\partial_{\mu}\Delta)(-\partial_{\nu}\Delta)\right\\}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle\frac{-1}{|\sigma|\Delta^{2}}\left\\{\left|\begin{array}[]{cc}\partial_{\nu}\Delta_{11}&\partial_{\nu}\Delta_{12}\\\
\partial_{\mu}\Delta_{21}&\partial_{\mu}\Delta_{22}\end{array}\right|+\left|\begin{array}[]{cc}\partial_{\mu}\Delta_{11}&\partial_{\mu}\Delta_{12}\\\
\partial_{\nu}\Delta_{21}&\partial_{\nu}\Delta_{22}\end{array}\right|-2|\sigma|(\partial_{\mu}\Delta)(\partial_{\nu}\Delta)\right\\}$
$\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}\right].$

(2)

$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}(\partial_{\rho}\sigma)\sigma^{-1}\right]\bigg{|}_{L_{j}\rightarrow-
L_{j}}$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{1}{2|\sigma|^{2}\Delta^{4}}\left|\begin{array}[]{cccc}\Delta_{11}&\Delta_{12}&\Delta_{21}&\Delta_{22}\\\
\partial_{\mu}\Delta_{11}&\partial_{\mu}\Delta_{12}&\partial_{\mu}\Delta_{21}&\partial_{\mu}\Delta_{22}\\\
\partial_{\nu}\Delta_{11}&\partial_{\nu}\Delta_{12}&\partial_{\nu}\Delta_{21}&\partial_{\nu}\Delta_{22}\\\
\partial_{\rho}\Delta_{11}&\partial_{\rho}\Delta_{12}&\partial_{\rho}\Delta_{21}&\partial_{\rho}\Delta_{22}\end{array}\right|_{L_{j}\rightarrow-
L_{j}}$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{1}{2|\sigma|^{2}\Delta^{4}}\left|\begin{array}[]{cccc}\Delta_{22}&\Delta_{21}&\Delta_{12}&\Delta_{11}\\\
-\partial_{\mu}\Delta_{22}&-\partial_{\mu}\Delta_{21}&-\partial_{\mu}\Delta_{12}&-\partial_{\mu}\Delta_{11}\\\
-\partial_{\nu}\Delta_{22}&-\partial_{\nu}\Delta_{21}&-\partial_{\nu}\Delta_{12}&-\partial_{\nu}\Delta_{11}\\\
-\partial_{\rho}\Delta_{22}&-\partial_{\rho}\Delta_{21}&-\partial_{\rho}\Delta_{12}&-\partial_{\rho}\Delta_{11}\end{array}\right|$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{(-1)^{3}}{2|\sigma|^{2}\Delta^{4}}\left|\begin{array}[]{cccc}\Delta_{22}&\Delta_{21}&\Delta_{12}&\Delta_{11}\\\
\partial_{\mu}\Delta_{22}&\partial_{\mu}\Delta_{21}&\partial_{\mu}\Delta_{12}&\partial_{\mu}\Delta_{11}\\\
\partial_{\nu}\Delta_{22}&\partial_{\nu}\Delta_{21}&\partial_{\nu}\Delta_{12}&\partial_{\nu}\Delta_{11}\\\
\partial_{\rho}\Delta_{22}&\partial_{\rho}\Delta_{21}&\partial_{\rho}\Delta_{12}&\partial_{\rho}\Delta_{11}\end{array}\right|=\frac{-1}{2|\sigma|^{2}\Delta^{4}}\left|\begin{array}[]{cccc}\Delta_{11}&\Delta_{12}&\Delta_{21}&\Delta_{22}\\\
\partial_{\mu}\Delta_{11}&\partial_{\mu}\Delta_{12}&\partial_{\mu}\Delta_{21}&\partial_{\mu}\Delta_{22}\\\
\partial_{\nu}\Delta_{11}&\partial_{\nu}\Delta_{12}&\partial_{\nu}\Delta_{21}&\partial_{\nu}\Delta_{22}\\\
\partial_{\rho}\Delta_{11}&\partial_{\rho}\Delta_{12}&\partial_{\rho}\Delta_{21}&\partial_{\rho}\Delta_{22}\end{array}\right|$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle-\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}(\partial_{\rho}\sigma)\sigma^{-1}\right].$

###  D.2 Data of $n$-Soliton Solutions

In this subsection we present the data of $n$-soliton solutions in terms of
$X_{j},\Theta_{j}$. We introduce the convention $\varepsilon_{j}$ which takes
values in $\left\\{\pm 1,\pm i\right\\}$ and define an informal symbol
$\mathcal{P}\\{(\varepsilon_{1},\cdots,\varepsilon_{n})\\}$ to denote the set
of all permutations of $(\varepsilon_{1},\cdots,\varepsilon_{n})$. The data of
$n$-soliton solutions can be expressed formally as the following $\Delta$,
$\Delta_{ij}$ with some undetermined coefficients
$a(\boldsymbol{\varepsilon}):=a(\varepsilon_{1},\cdots,\varepsilon_{n})$ and
$A(\boldsymbol{\varepsilon}):=A(\varepsilon_{1},\cdots,\varepsilon_{n})$:

$\displaystyle\Delta$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq j\leq
n,\\\ \varepsilon=\pm
1\end{array}}}\\!\\!\\!\\!\\!a(\boldsymbol{\varepsilon})\exp{\left(\sum_{j=1}^{n}\varepsilon_{j}X_{j}\right)}}$
$\displaystyle\\!\\!+\\!\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j}=\pm 1,j\neq k,\ell\\\
(\varepsilon_{k},\varepsilon_{\ell})\in\mathcal{P}\\{(i,-i)\\}\end{array}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!a(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}+\varepsilon_{\ell}\Theta_{\ell}})}+~{}\mbox{terms
involving more $i\Theta_{j}$'s}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
j\leq n,\\\ \varepsilon_{j}=\pm
1\end{array}}}\left[\begin{array}[]{l}\displaystyle{a(\boldsymbol{\varepsilon})\exp{(\sum_{j=1}^{n}\varepsilon_{j}X_{j})}}+\displaystyle{a(-\boldsymbol{\varepsilon})\exp{(-\sum_{j=1}^{n}\varepsilon_{j}X_{j})}}\end{array}\right]}$
$\displaystyle\\!\\!+\underbrace{\frac{1}{2}\\!\\!\\!\\!\\!\\!\\!\\!\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j}=\pm 1,j\neq k,\ell\\\
(\varepsilon_{k},\varepsilon_{\ell})\in\mathcal{P}\\{(i,-i)\\}\end{array}}}\\!\\!\\!\\!\left[\begin{array}[]{l}~{}~{}\displaystyle{a(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}+\varepsilon_{\ell}\Theta_{\ell}})}\\\
+\displaystyle{a(-\boldsymbol{\varepsilon})\exp(-{\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}-\varepsilon_{k}\Theta_{k}-\varepsilon_{\ell}\Theta_{\ell}})}\end{array}\right]+\cdots}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{\footnotesize{\begin{array}[]{l}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j,j\neq k,\ell}=\pm
1\end{array}}}\\!\\!\\!\\!\\!\\!\mathcal{O}\left(\cosh(\sum_{j=1,j\neq
k,\ell}^{n}\varepsilon_{j}X_{j})\right)$

By symmetry (3.77),
$\Delta\Big{|}_{(X_{j},\Theta_{j})\rightarrow(-X_{j},\Theta_{j})}=\Delta$
implies $a(\boldsymbol{\varepsilon})=a(-\boldsymbol{\varepsilon})$, and hence

$\displaystyle\Delta=\\!\\!\\!\\!\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
j\leq n,\\\ \varepsilon_{j}=\pm
1\end{array}}}}\\!\\!\\!\\!a(\boldsymbol{\varepsilon})\cosh{\left(\sum_{j=1}^{n}\varepsilon_{j}X_{j}\right)}+\\!\\!\\!\\!\sum_{\footnotesize{\begin{array}[]{l}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j,j\neq k,\ell}=\pm
1\end{array}}}\\!\\!\\!\\!\\!\\!\mathcal{O}\left(\cosh(\sum_{j=1,j\neq\ell,m}^{n}\\!\\!\\!\\!\varepsilon_{j}X_{j})\right).~{}~{}$
(D.23)

Similarly, we have

$\displaystyle\Delta_{11}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq j\leq
n,\\\ \varepsilon_{j}=\pm
1\end{array}}}\\!\\!\\!\\!\\!\\!\\!\\!A(\boldsymbol{\varepsilon})\exp{\left(\sum_{j=1}^{n}\varepsilon_{j}X_{j}\right)}}$
(D.30)
$\displaystyle\\!\\!\\!\\!+\\!\\!\\!\\!\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j}=\pm 1,j\neq k,\ell\\\
(\varepsilon_{k},\varepsilon_{\ell})\in\mathcal{P}\\{(i,-i)\\}\end{array}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}+\varepsilon_{\ell}\Theta_{\ell}})}+\mbox{terms
involving more $i\Theta_{j}$'s}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
j\leq n,\\\ \varepsilon_{j}=\pm
1\end{array}}}\\!\\!\left[\begin{array}[]{l}\displaystyle{A(\boldsymbol{\varepsilon})\exp{(\sum_{j=1}^{n}\varepsilon_{j}X_{j})}}+\displaystyle{A(-\boldsymbol{\varepsilon})\exp{(-\sum_{j=1}^{n}\varepsilon_{j}X_{j})}}\end{array}\right]}$
(D.43)
$\displaystyle\\!\\!\\!\\!+\underbrace{\frac{1}{2}\\!\\!\\!\\!\\!\\!\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j}=\pm 1,j\neq k,\ell\\\
(\varepsilon_{k},\varepsilon_{\ell})\in\mathcal{P}\\{(i,-i)\\}\end{array}}}\\!\\!\\!\\!\\!\left[\begin{array}[]{l}~{}~{}\displaystyle{A(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}+\varepsilon_{\ell}\Theta_{\ell}})}\\\
+\displaystyle{A(-\boldsymbol{\varepsilon})\exp(-{\sum_{j\neq
k,\ell}^{n}\varepsilon_{j}X_{j}-\varepsilon_{k}\Theta_{k}-\varepsilon_{\ell}\Theta_{\ell}})}\end{array}\\!\\!\right]+\cdots}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{\footnotesize{\begin{array}[]{l}1\leq
k\textless\ell\leq n,\\\ \varepsilon_{j,j\neq k,\ell}=\pm
1\end{array}}}\\!\\!\\!\\!\\!\\!\mathcal{O}\left(\cosh(\sum_{j=1,j\neq
k,\ell}^{n}\varepsilon_{j}X_{j})\right)$

By symmetry (3.77),
$\Delta_{22}=\Delta_{11}\Big{|}_{(X_{j},\Theta_{j})\rightarrow(-X_{j},\Theta_{j})}$
implies that
$\Delta_{22}=\Delta_{11}\Big{|}_{A(\boldsymbol{\varepsilon})\rightarrow
A(-\boldsymbol{\varepsilon})}$.

$\displaystyle\Delta_{12}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\displaystyle{\sum_{\footnotesize{\begin{array}[]{c}1\leq k\leq
n,\\\ \varepsilon_{j}=\left\\{\begin{array}[]{l}\pm 1,j\neq k\\\
+i,j=k\end{array}\right.\end{array}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}})}$ (D.57)
$\displaystyle\\!\\!\\!\\!+\left\\{\underbrace{\begin{array}[]{l}\displaystyle{\\!\\!\\!\\!\sum_{\footnotesize{\begin{array}[]{c}1\leq
k\textless\ell\textless m\leq n,\\\ \varepsilon_{j}=\pm 1,j\neq k,\ell,m\\\
(\varepsilon_{k},\varepsilon_{\ell},\varepsilon_{m})\in\mathcal{P}\\{(i,i,-i)\\}\end{array}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A(\boldsymbol{\varepsilon})\exp({\sum_{j\neq
k,\ell,m}^{n}\varepsilon_{j}X_{j}+\varepsilon_{k}\Theta_{k}+\varepsilon_{\ell}\Theta_{\ell}+\varepsilon_{m}\Theta_{m}})}\\\
+~{}\mbox{terms involving more $i\Theta_{j}$'s}\end{array}}\right\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{\footnotesize{\begin{array}[]{l}1\leq
k\textless\ell\textless m\leq n,\\\ \varepsilon_{j,j\neq k,\ell,m}=\pm
1\end{array}}}\\!\\!\\!\\!\\!\\!\mathcal{O}\left(\cosh(\sum_{j=1,j\neq
k,\ell,m}^{n}\varepsilon_{j}X_{j})\right)$

By symmetry (3.77),
$\Delta_{21}=\Delta_{12}\Big{|}_{(X_{j},\Theta_{j})\rightarrow(-X_{j},\Theta_{j})}$.
Concrete examples are mentioned as follows:
$\bullet$ The data of one-soliton solutions:

$\displaystyle\begin{array}[]{l}\Delta=a(1)\left(e^{X_{1}}+e^{-X_{1}}\right),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Delta_{11}=A(1)e^{X_{1}}+A(-1)e^{-X_{1}},~{}~{}~{}\Delta_{22}=A(-1)e^{X_{1}}+A(1)e^{-X_{1}},~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Delta_{12}=A(i)e^{i\Theta_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{21}=A(i)e^{-i\Theta_{1}},\end{array}$
(D.61)

where
$a(1)=1,~{}~{}A(1)=\lambda_{1},~{}~{}A(-1)=\mu_{1},~{}~{}A(i)=-(\lambda_{1}-\mu_{1}).$
$\bullet$ The data of two-soliton solutions
($\Theta_{jk}:=\Theta_{j}-\Theta_{k},~{}\\!j,k=1,2$) :

$\displaystyle\begin{array}[]{l}\Delta=\left\\{\begin{array}[]{l}~{}~{}a(1,1)\left(e^{X_{1}+X_{2}}+e^{-(X_{1}+X_{2}}\right)+a(1,-1)\left(e^{X_{1}-X_{2}}+e^{-(X_{1}-X_{2}}\right)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+a(i,-i)\left(e^{i\Theta_{12}}+e^{-i\Theta_{12}}\right)\end{array}\right\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\Delta_{11}=\left\\{\begin{array}[]{l}~{}~{}\left[A(1,1)e^{X_{1}+X_{2}}+A(-1,-1)e^{-(X_{1}+X_{2})}\right]\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[A(1,-1)e^{X_{1}-X_{2}}+A(-1,1)e^{-(X_{1}-X_{2})}\right]\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
+\left[A(i,-i)e^{i\Theta_{12}}+A(-i,i)e^{-i\Theta_{12}}\right]\end{array}\right\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\Delta_{22}=\Delta_{11}\Big{|}_{(X_{j},\Theta_{j})\rightarrow(-X_{j},-\Theta_{j})}=\Delta_{11}\Big{|}_{A(\varepsilon_{1},\varepsilon_{2})\rightarrow
A(-\varepsilon_{1},-\varepsilon_{2})},\vskip 12.0pt plus 4.0pt minus 4.0pt\\\
\Delta_{12}=\left\\{\begin{array}[]{l}~{}~{}A(1,i)e^{X_{1}+i\Theta_{2}}\vskip
3.0pt plus 1.0pt minus 1.0pt+A(-1,i)e^{-X_{1}+i\Theta_{2}}\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
+A(i,1)e^{X_{2}+i\Theta_{1}}+A(i,-1)e^{-X_{2}+i\Theta_{1}}\end{array}\right\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\Delta_{21}=\Delta_{12}\Big{|}_{(X_{j},\Theta_{j})\rightarrow(-X_{j},-\Theta_{j})}.\end{array}~{}~{}$
(D.74)

where

$\displaystyle\begin{array}[]{l}\begin{array}[]{l}a(1,1)=(\lambda_{1}-\lambda_{2})(\mu_{1}-\mu_{2}),~{}~{}a(1,-1)=(\lambda_{1}-\mu_{2})(\mu_{1}-\lambda_{2}),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
a(i,-i)=(\lambda_{1}-\mu_{1})(\lambda_{2}-\mu_{2})\end{array}\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\begin{array}[]{l}A(1,1)=-(\lambda_{1}-\lambda_{2})(\mu_{1}-\mu_{2})\lambda_{1}\lambda_{2},~{}~{}A(-1,-1)=-(\lambda_{1}-\lambda_{2})(\mu_{1}-\mu_{2})\mu_{1}\mu_{2},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(1,-1)=-(\lambda_{1}-\mu_{2})(\mu_{1}-\lambda_{2})\lambda_{1}\mu_{2},~{}~{}A(-1,1)=-(\lambda_{1}-\mu_{2})(\mu_{1}-\lambda_{2})\mu_{1}\lambda_{2},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(i,-i)=-(\lambda_{1}-\mu_{1})(\lambda_{2}-\mu_{2})\lambda_{1}\mu_{1},~{}~{}A(-i,i)=-(\lambda_{1}-\mu_{1})(\lambda_{2}-\mu_{2})\lambda_{2}\mu_{2},\end{array}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\begin{array}[]{l}A(1,i)=(\lambda_{1}-\lambda_{2})(\lambda_{1}-\mu_{2})(\lambda_{2}-\mu_{2})\mu_{1},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(-1,i)=(\mu_{1}-\mu_{2})(\mu_{1}-\lambda_{2})(\lambda_{2}-\mu_{2})\lambda_{1},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(i,1)=(\lambda_{1}-\lambda_{2})(\lambda_{1}-\mu_{1})(\mu_{1}-\lambda_{2})\mu_{2},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(i,-1)=(\mu_{1}-\mu_{2})(\lambda_{1}-\mu_{1})(\lambda_{1}-\mu_{2})\lambda_{2}.\end{array}\end{array}$
(D.87)

###  D.3 Exact Calculation of NL$\sigma$M Term (Two-Soliton)

For preparation, we introduce some symmetries between the coefficients of the
soliton data before our calculation. Our observation is as the following
Remark 1 $\sim$ 3 which can be checked simply from (D.74) and (D.87).

Remark 1

$\displaystyle\begin{array}[]{l}A(1,1)=-\lambda_{1}\lambda_{2}~{}\\!a(1,1),~{}~{}A(-1,-1)=-\mu_{1}\mu_{2}~{}\\!a(1,1),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(1,-1)=-\lambda_{1}\mu_{2}~{}\\!a(1,-1),~{}~{}A(-1,1)=-\mu_{1}\lambda_{2}~{}\\!a(1,-1).\end{array}$
(D.90)

Remark 2

$\displaystyle\begin{array}[]{l}A(1,i)A(-1,i)=\lambda_{1}\mu_{1}(\lambda_{2}-\mu_{2})^{2}~{}\\!a(1,1)a(1,-1),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(i,1)A(i,-1)=\lambda_{2}\mu_{2}(\lambda_{1}-\mu_{1})^{2}~{}\\!a(1,1)a(1,-1),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(-1,i)A(i,1)=\lambda_{1}\mu_{2}(\mu_{1}-\lambda_{2})^{2}~{}\\!a(1,1)a(i,-i),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(1,i)A(i,-1)=\mu_{1}\lambda_{2}(\lambda_{1}-\mu_{2})^{2}~{}\\!a(1,1)a(i,-i),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(-1,i)A(i,-1)=\lambda_{1}\lambda_{2}(\mu_{1}-\mu_{2})^{2}~{}\\!a(1,-1)a(i,-i),~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(1,i)A(i,1)=\mu_{1}\mu_{2}(\lambda_{1}-\lambda_{2})^{2}~{}\\!a(1,-1)a(i,-i).\end{array}$
(D.97)

Remark 3

$\displaystyle\begin{array}[]{l}A(1,i)-A(-1,i)=A(i,-1)-A(i,1)=(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})~{}\\!a(i,-i),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(i,1)-A(-1,i)=A(i,-1)-A(1,i)=(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})~{}\\!a(1,-1),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
A(1,i)+A(i,1)=A(-1,i)+A(i,-1)=(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})~{}\\!a(1,1).\end{array}$
(D.101)

which implies the relation

$\displaystyle A(1,i)-A(-1,i)+A(i,1)-A(i,-1)=0$ (D.102)

directly by taking some simple addition and subtraction over (D.101). On the
other hand, by taking sum of squares over $L.H.S.$ and $R.H.S.$ of (D.101) and
using the relation (D.102), we get the following nontrivial identity.
Remark 4

$\displaystyle\\!\\!\\!\\!(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})^{2}a^{2}(1,1)+(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})^{2}a^{2}(1,-1)+(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})^{2}a^{2}(i,-i)~{}~{}~{}~{}$
(D.103) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
A^{2}(1,i)+A^{2}(-1,i)+A^{2}(i,1)+A^{2}(i,-1).$

Now let us start our main calculation of the NL sigma model term for two-
soliton. By using the soliton data (D.74) and after a little bit tedious
calculation, we can conclude that

$\displaystyle\\!\\!\\!\\!\partial_{\mu}\Delta_{11}\partial_{\mu}\Delta_{22}-\partial_{\mu}\Delta_{12}\partial_{\mu}\Delta_{21}-|\sigma|(\partial_{\mu}\Delta)^{2}$
(D.151) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\begin{array}[]{l}~{}(r_{\mu}^{(1)}+r_{\mu}^{(2)})^{2}\left[A(1,1)A(-1,-1)-|\sigma|a^{2}(1,1)\right]\left[e^{2(X_{1}+X_{2})}+e^{-2(X_{1}+X_{2})}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(r_{\mu}^{(1)}-r_{\mu}^{(2)})^{2}\left[A(1,-1)A(-1,1)-|\sigma|a^{2}(1,-1)\right]\left[e^{2(X_{1}-X_{2})}+e^{-2(X_{1}-X_{2})}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(s_{\mu}^{(1)}-s_{\mu}^{(2)})^{2}\left[A(i,-i)A(-i,i)-|\sigma|a^{2}(i,-i)\right]\left(e^{2\Theta_{12}}+e^{-2\Theta_{12}}\right)\end{array}$
$\displaystyle+\\!\left\\{\begin{array}[]{l}~{}(r_{\mu}^{(1)}\\!+r_{\mu}^{(2)})(r_{\mu}^{(1)}\\!-r_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,1)A(-1,1)+A(1,-1)A(-1,-1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,1)a(1,-1)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!-(r_{\mu}^{(1)}\\!+s_{\mu}^{(2)})(r_{\mu}^{(1)}\\!-s_{\mu}^{(2)})A(1,i)A(-1,i)\end{array}\\!\\!\\!\right\\}\\!\\!\left[e^{2X_{1}}+e^{-2X_{1}}\right]$
$\displaystyle-\\!\left\\{\begin{array}[]{l}~{}(r_{\mu}^{(1)}\\!+r_{\mu}^{(2)})(r_{\mu}^{(1)}\\!-r_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,1)A(1,-1)+A(-1,1)A(-1,-1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,1)a(1,-1)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!-(s_{\mu}^{(1)}\\!+r_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-r_{\mu}^{(2)})A(i,1)A(i,-1)\end{array}\\!\\!\\!\right\\}\\!\\!\left[e^{2X_{2}}+e^{-2X_{2}}\right]$
$\displaystyle+\\!\left\\{\begin{array}[]{l}(r_{\mu}^{(1)}\\!+\\!r_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,1)A(-i,i)+A(-1,-1)A(i,-i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,1)a(i,-i)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!\\!\\!-(r_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})(s_{\mu}^{(1)}\\!+\\!r_{\mu}^{(2)})A(-1,i)A(i,1)\end{array}\\!\\!\\!\right\\}\\!\\!\left[\begin{array}[]{l}~{}\\!e^{X_{1}-X_{2}+i\Theta_{12}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!\\!+e^{-(X_{1}-X_{2}+i\Theta_{12})}\end{array}\\!\\!\right]$
$\displaystyle-\\!\left\\{\begin{array}[]{l}(r_{\mu}^{(1)}\\!+\\!r_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,1)A(i,-i)+A(-1,-1)A(-i,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,1)a(i,-i)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!\\!\\!-(r_{\mu}^{(1)}\\!+\\!s_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!r_{\mu}^{(2)})A(1,i)A(i,-1)\end{array}\\!\\!\\!\right\\}\\!\\!\left[\begin{array}[]{l}~{}\\!e^{X_{1}+X_{2}-i\Theta_{12}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!\\!+e^{-(X_{1}+X_{2}-i\Theta_{12})}\end{array}\\!\\!\right]$
$\displaystyle+\\!\left\\{\begin{array}[]{l}(r_{\mu}^{(1)}\\!-\\!r_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,-1)A(-i,i)+A(-1,1)A(i,-i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,-1)a(i,-i)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!\\!\\!-(r_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!r_{\mu}^{(2)})A(-1,i)A(i,-1)\end{array}\\!\\!\\!\right\\}\\!\\!\left[\begin{array}[]{l}~{}\\!e^{X_{1}-X_{2}+i\Theta_{12}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!\\!+e^{-(X_{1}-X_{2}+i\Theta_{12})}\end{array}\\!\\!\right]$
$\displaystyle-\\!\left\\{\begin{array}[]{l}(r_{\mu}^{(1)}\\!-\\!r_{\mu}^{(2)})(s_{\mu}^{(1)}\\!-\\!s_{\mu}^{(2)})\\!\left[\begin{array}[]{l}A(1,-1)A(i,-i)+A(-1,1)A(-i,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!-2|\sigma|a(1,-1)a(i,-i)\end{array}\\!\\!\right]\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\\!\\!\\!\\!-(r_{\mu}^{(1)}\\!+\\!s_{\mu}^{(2)})(s_{\mu}^{(1)}\\!+\\!r_{\mu}^{(2)})A(1,i)A(i,1)\end{array}\\!\\!\\!\right\\}\\!\\!\left[\begin{array}[]{l}~{}\\!e^{X_{1}-X_{2}-i\Theta_{12}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!\\!+e^{-(X_{1}-X_{2}-i\Theta_{12})}\end{array}\\!\\!\right]$
$\displaystyle-\left\\{\begin{array}[]{l}~{}(r_{\mu}^{(1)}+r_{\mu}^{(2)})^{2}\left[A^{2}(1,1)+A^{2}(-1,-1)-2|\sigma|a^{2}(1,1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(r_{\mu}^{(1)}-r_{\mu}^{(2)})^{2}\left[A^{2}(1,-1)+A^{2}(-1,1)-2|\sigma|a^{2}(1,-1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(s_{\mu}^{(1)}-s_{\mu}^{(2)})^{2}\left[A^{2}(i,-i)+A^{2}(-i,i)-2|\sigma|a^{2}(i,-i)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-(r_{\mu}^{(1)}+s_{\mu}^{(2)})^{2}A^{2}(1,i)-(r_{\mu}^{(1)}-s_{\mu}^{(2)})^{2}A^{2}(-1,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-(s_{\mu}^{(1)}+r_{\mu}^{(2)})^{2}A^{2}(i,1)-(s_{\mu}^{(1)}-r_{\mu}^{(2)})^{2}A^{2}(i,-1)\end{array}\right\\}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

By (D.90) and (D.97), we find that the coefficients of the leading terms
$\exp(\pm 2(X_{1}\pm X_{2}))$ are identical to zero and the remaining terms
can be rewritten as

$\displaystyle\\!\\!\\!\\!\partial_{\mu}\Delta_{11}\partial_{\mu}\Delta_{22}-\partial_{\mu}\Delta_{12}\partial_{\mu}\Delta_{21}-|\sigma|(\partial_{\mu}\Delta)^{2}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\begin{array}[]{l}\left\\{\begin{array}[]{l}-(r_{\mu}^{(2)}+s_{\mu}^{(2)})(r_{\mu}^{(2)}-s_{\mu}^{(2)})A(1,i)A(-1,i)\left(e^{2X_{1}}+e^{-2X_{1}}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-(r_{\mu}^{(1)}+s_{\mu}^{(1)})(r_{\mu}^{(1)}-s_{\mu}^{(1)})A(i,1)A(i,-1)\left(e^{2X_{2}}+e^{-2X_{2}}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-(r_{\mu}^{(1)}-s_{\mu}^{(1)})(r_{\mu}^{(2)}+s_{\mu}^{(2)})A(-1,i)A(i,1)\left(e^{X_{1}+X_{2}+i\Theta_{12}}+e^{-(X_{1}+X_{2}+i\Theta_{12})}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-(r_{\mu}^{(1)}+s_{\mu}^{(1)})(r_{\mu}^{(2)}-s_{\mu}^{(2)})A(1,i)A(i,-1)\left(e^{X_{1}+X_{2}-i\Theta_{12}}+e^{-(X_{1}+X_{2}-i\Theta_{12})}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
+(r_{\mu}^{(1)}-s_{\mu}^{(1)})(r_{\mu}^{(2)}-s_{\mu}^{(2)})A(-1,i)A(i,-1)\left(e^{X_{1}-X_{2}+i\Theta_{12}}+e^{-(X_{1}-X_{2}+i\Theta_{12})}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
+(r_{\mu}^{(1)}+s_{\mu}^{(1)})(r_{\mu}^{(2)}+s_{\mu}^{(2)})A(1,i)A(i,1)\left(e^{X_{1}-X_{2}-i\Theta_{12}}+e^{-(X_{1}-X_{2}-i\Theta_{12})}\right)\end{array}\\!\\!\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\begin{array}[]{l}-\left\\{\underbrace{\begin{array}[]{l}~{}(r_{\mu}^{(1)}+r_{\mu}^{(2)})^{2}(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})^{2}~{}\\!a^{2}(1,1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(r_{\mu}^{(1)}-r_{\mu}^{(2)})^{2}(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})^{2}~{}\\!a^{2}(1,-1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+(s_{\mu}^{(1)}-s_{\mu}^{(2)})^{2}(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})^{2}~{}\\!a^{2}(i,-i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-(r_{\mu}^{(1)}+s_{\mu}^{(2)})^{2}A^{2}(1,i)-(r_{\mu}^{(1)}-s_{\mu}^{(2)})^{2}A^{2}(-1,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-(s_{\mu}^{(1)}+r_{\mu}^{(2)})^{2}A^{2}(i,1)-(s_{\mu}^{(1)}-r_{\mu}^{(2)})^{2}A^{2}(i,-1)\end{array}}\right\\}\end{array}\end{array}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=:\Xi$

Next, we want to show that the constant term $\Xi$ above can be absorbed
completely into the non-constant terms. By the definition of
$r_{\mu}^{(j)},s_{\mu}^{(j)}$ above (3.75), we can replace
$r_{\mu}^{(j)},s_{\mu}^{(j)}$ by $\ell_{\mu}^{(j)}$ and obtain

$\displaystyle\Xi$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\left\\{\begin{array}[]{l}~{}\left(\ell_{\mu}^{(1)}+\overline{\ell}_{\mu}^{(1)}+\ell_{\mu}^{(2)}+\overline{\ell}_{\mu}^{(2)}\right)^{2}(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})^{2}~{}\\!a^{2}(1,1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\left(\ell_{\mu}^{(1)}+\overline{\ell}_{\mu}^{(1)}-\ell_{\mu}^{(2)}-\overline{\ell}_{\mu}^{(2)}\right)^{2}(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})^{2}~{}\\!a^{2}(1,-1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\left(\ell_{\mu}^{(1)}-\overline{\ell}_{\mu}^{(1)}-\ell_{\mu}^{(2)}+\overline{\ell}_{\mu}^{(2)}\right)^{2}(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})^{2}~{}\\!a^{2}(i,-i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\left(\ell_{\mu}^{(1)}+\overline{\ell}_{\mu}^{(1)}+\ell_{\mu}^{(2)}-\overline{\ell}_{\mu}^{(2)}\right)^{2}\\!\\!A^{2}(1,i)-\left(\ell_{\mu}^{(1)}+\overline{\ell}_{\mu}^{(1)}-\ell_{\mu}^{(2)}+\overline{\ell}_{\mu}^{(2)}\right)^{2}\\!\\!A^{2}(-1,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\left(\ell_{\mu}^{(1)}-\overline{\ell}_{\mu}^{(1)}+\ell_{\mu}^{(2)}+\overline{\ell}_{\mu}^{(2)}\right)^{2}\\!\\!A^{2}(i,1)-\left(\ell_{\mu}^{(1)}-\overline{\ell}_{\mu}^{(1)}-\ell_{\mu}^{(2)}-\overline{\ell}_{\mu}^{(2)}\right)^{2}\\!\\!A^{2}(i,-1)\end{array}\right\\}$
(D.172)

By (D.103) and (D.101),

$\displaystyle\Xi$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle-4\left\\{\begin{array}[]{l}~{}\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(1)}\left[-(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})^{2}a^{2}(i,-i)+A^{2}(i,1)+A^{2}(i,-1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(2)}\overline{\ell}_{\mu}^{(2)}\left[-(\lambda_{1}\mu_{1}-\lambda_{2}\mu_{2})^{2}a^{2}(i,-i)+A^{2}(1,i)+A^{2}(-1,i)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}\left[-(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})^{2}a^{2}(1,-1)+A^{2}(1,i)+A^{2}(i,-1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\overline{\ell}_{\mu}^{(1)}\ell_{\mu}^{(2)}\left[-(\lambda_{1}\mu_{2}-\mu_{1}\lambda_{2})^{2}a^{2}(1,-1)+A^{2}(-1,i)+A^{2}(i,1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\ell_{\mu}^{(1)}\ell_{\mu}^{(2)}\left[-(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})^{2}a^{2}(1,1)+A^{2}(1,i)+A^{2}(i,1)\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\overline{\ell}_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}\left[-(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})^{2}a^{2}(1,1)+A^{2}(-1,i)+A^{2}(i,-1)\right]\end{array}\right\\}$
(D.179) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle-8\left\\{\begin{array}[]{l}~{}\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(1)}A(i,1)A(i,-1)+\ell_{\mu}^{(2)}\overline{\ell}_{\mu}^{(2)}A(1,i)A(-1,i)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}A(1,i)A(i,-1)+\overline{\ell}_{\mu}^{(1)}\ell_{\mu}^{(2)}A(-1,i)A(i,1)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(1)}\ell_{\mu}^{(2)}A(1,i)A(i,1)+\overline{\ell}_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}A(-1,i)A(i,-1)\end{array}\right\\}.$
(D.183)

Comparing with (D.3), we have

$\displaystyle\\!\\!\\!\\!\partial_{\mu}\Delta_{11}\partial_{\mu}\Delta_{22}-\partial_{\mu}\Delta_{12}\partial_{\mu}\Delta_{21}-|\sigma|(\partial_{\mu}\Delta)^{2}$
(D.191) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\begin{array}[]{l}-4\left\\{\begin{array}[]{l}~{}\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(1)}A(i,1)A(i,-1)\left(e^{X_{2}}+e^{-X_{2}}\right)^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(2)}\overline{\ell}_{\mu}^{(2)}A(1,i)A(-1,i)\left(e^{X_{1}}+e^{-X_{1}}\right)^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}A(1,i)A(i,-1)\left(e^{\frac{X_{1}+X_{2}-i\Theta_{12}}{2}}+e^{-\frac{(X_{1}+X_{2}-i\Theta_{12})}{2}}\right)^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\overline{\ell}_{\mu}^{(1)}\ell_{\mu}^{(2)}A(-1,i)A(i,1)\left(e^{\frac{X_{1}+X_{2}+i\Theta_{12}}{2}}+e^{-\frac{(X_{1}+X_{2}+i\Theta_{12})}{2}}\right)^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\ell_{\mu}^{(1)}\ell_{\mu}^{(2)}A(1,i)A(i,1)\left(e^{\frac{X_{1}-X_{2}-i\Theta_{12}}{2}}-e^{-\frac{(X_{1}-X_{2}-i\Theta_{12})}{2}}\right)^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-\overline{\ell}_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}A(-1,i)A(i,-1)\left(e^{\frac{X_{1}-X_{2}+i\Theta_{12}}{2}}-e^{-\frac{(X_{1}-X_{2}+i\Theta_{12})}{2}}\right)^{2}\end{array}\right\\}\end{array}$

By (2.60) and (D.97), we can conclude that

$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}\right]^{2}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-8\left\\{\begin{array}[]{l}~{}a(1,1)a(1,-1)\left[\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(1)}\Lambda_{11}\cosh^{2}X_{2}+\ell_{\mu}^{(2)}\overline{\ell}_{\mu}^{(2)}\Lambda_{22}\cosh^{2}X_{1}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+a(1,1)a(i,-i)\left[\begin{array}[]{l}~{}\ell_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}\Lambda_{12}\displaystyle{\cosh^{2}\left(\frac{X_{1}+X_{2}-i\Theta_{12}}{2}\right)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\overline{\ell}_{\mu}^{(1)}\ell_{\mu}^{(2)}\Lambda_{21}~{}\\!\displaystyle{\cosh^{2}\left(\frac{X_{1}+X_{2}+i\Theta_{12}}{2}\right)}\end{array}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!-a(1,-1)a(i,-i)\left[\begin{array}[]{l}~{}\ell_{\mu}^{(1)}\ell_{\mu}^{(2)}\Omega_{12}~{}\\!\displaystyle{\sinh^{2}\left(\frac{X_{1}-X_{2}-i\Theta_{12}}{2}\right)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\\!\\!+\overline{\ell}_{\mu}^{(1)}\overline{\ell}_{\mu}^{(2)}\widetilde{\Omega}_{12}~{}\\!\displaystyle{\sinh^{2}\left(\frac{X_{1}-X_{2}+i\Theta_{12}}{2}\right)}\end{array}\right]\end{array}\right\\}}{\displaystyle{\left[\begin{array}[]{l}a(1,1)\cosh(X_{1}+X_{2})+a(1,-1)\cosh(X_{1}-X_{2})+a(i,-i)\cos\Theta_{12}\end{array}\right]^{2}}},$
(D.200)

where

$\displaystyle\displaystyle{\Lambda_{jk}:=\frac{(\lambda_{j}-\mu_{k})^{2}}{\lambda_{j}\mu_{k}},~{}~{}\Omega_{jk}:=\frac{(\lambda_{j}-\lambda_{k})^{2}}{\lambda_{j}\lambda_{k}},~{}~{}\widetilde{\Omega}_{jk}:=\frac{(\mu_{j}-\mu_{k})^{2}}{\mu_{j}\mu_{k}}.}$

###  D.4 Exact Calculation of Wess-Zumino Action Density (Two-Soliton)

By substituting (3.22), (D.74) and (3.75) into (2.75) for
$(m,n,p)=(\mu,\nu,\rho)$, we have

$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}(\partial_{\rho}\sigma)\sigma^{-1}\right]\vskip
3.0pt plus 1.0pt minus
1.0pt=\displaystyle{\frac{1}{2}}(B_{\mu\nu\rho}+B_{\nu\rho\mu}+B_{\rho\mu\nu}),$
(D.201) $\displaystyle
B_{\mu\nu\rho}:=\displaystyle{\frac{1}{|\sigma|^{2}\Delta^{4}}}\left(\left|\\!\\!\begin{array}[]{cc}\Delta_{11}\\!\\!&\Delta_{22}\\!\\!\\\
\partial_{\mu}\Delta_{11}\\!\\!&\partial_{\mu}\Delta_{22}\\!\\!\end{array}\right|\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{12}\\!\\!&\partial_{\nu}\Delta_{21}\\!\\!\\\
\partial_{\rho}\Delta_{12}\\!\\!&\partial_{\rho}\Delta_{21}\\!\\!\end{array}\right|+\left|\\!\\!\begin{array}[]{cc}\Delta_{12}\\!\\!&\Delta_{21}\\!\\!\\\
\partial_{\mu}\Delta_{12}\\!\\!&\partial_{\mu}\Delta_{21}\\!\\!\end{array}\right|\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{11}\\!\\!&\partial_{\nu}\Delta_{22}\\!\\!\\\
\partial_{\rho}\Delta_{11}\\!\\!&\partial_{\rho}\Delta_{22}\\!\\!\end{array}\right|\right).$
(D.210)

where

$\displaystyle\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\Delta_{11}\\!\\!&\Delta_{22}\\!\\!\\\
\partial_{\mu}\Delta_{11}\\!\\!&\partial_{\mu}\Delta_{22}\\!\\!\end{array}\right|$
(D.213) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-\left\\{\begin{array}[]{l}~{}~{}~{}2r_{\mu}^{(1)}ab\mathcal{D}_{11}~{}\\!\mathrm{cosh}(2X_{2})+2r_{\mu}^{(2)}ab\mathcal{D}_{22}~{}\\!\mathrm{cosh}(2X_{1})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[(r_{\mu}^{(1)}+r_{\mu}^{(2)})+(s_{\mu}^{(1)}-s_{\mu}^{(2)})\right]ac\mathcal{D}_{12}~{}\\!\mathrm{cosh}(X_{1}+X_{2}-i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[(r_{\mu}^{(1)}+r_{\mu}^{(2)})-(s_{\mu}^{(1)}-s_{\mu}^{(2)})\right]ac\mathcal{D}_{21}~{}\\!\mathrm{cosh}(X_{1}+X_{2}+i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[(r_{\mu}^{(1)}-r_{\mu}^{(2)})+(s_{\mu}^{(1)}-s_{\mu}^{(2)})\right]bc\mathcal{E}_{12}~{}\\!\mathrm{cosh}(X_{1}-X_{2}-i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-\left[(r_{\mu}^{(1)}-r_{\mu}^{(2)})-(s_{\mu}^{(1)}-s_{\mu}^{(2)})\right]bc\widetilde{\mathcal{E}}_{12}~{}\\!\mathrm{cosh}(X_{1}-X_{2}+i\Theta_{12})\end{array}\right\\}-F}{2\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}-X_{2})+c~{}\\!\mbox{cos}\Theta_{12}\end{array}\right]^{2}}~{},~{}~{}~{}~{}~{}~{}~{}$
(D.220)
$\displaystyle\\!\\!\\!\\!\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\Delta_{12}\\!\\!&\Delta_{21}\\!\\!\\\
\partial_{\mu}\Delta_{12}\\!\\!&\partial_{\mu}\Delta_{21}\\!\\!\end{array}\right|$
(D.223) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-\left\\{\begin{array}[]{l}~{}~{}~{}2s_{\mu}^{(1)}\\!ab\mathcal{d}_{11}~{}\\!\mathrm{cosh}(2X_{2})+2s_{\mu}^{(2)}\\!ab\mathcal{d}_{22}~{}\\!\mathrm{cosh}(2X_{1})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[(r_{\mu}^{(1)}-r_{\mu}^{(2)})+(s_{\mu}^{(1)}+s_{\mu}^{(2)})\right]\\!ac\mathcal{d}_{12}~{}\\!\mathrm{cosh}(X_{1}+X_{2}-i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-\left[(r_{\mu}^{(1)}-r_{\mu}^{(2)})-(s_{\mu}^{(1)}+s_{\mu}^{(2)})\right]\\!ac\mathcal{d}_{21}~{}\\!\mathrm{cosh}(X_{1}+X_{2}+i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+\left[(r_{\mu}^{(1)}+r_{\mu}^{(2)})+(s_{\mu}^{(1)}+s_{\mu}^{(2)})\right]\\!bc\mathcal{e}_{12}~{}\\!\mathrm{cosh}(X_{1}-X_{2}-i\Theta_{12})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-\left[(r_{\mu}^{(1)}+r_{\mu}^{(2)})-(s_{\mu}^{(1)}+s_{\mu}^{(2)})\right]\\!bc\widetilde{\mathcal{e}}_{12}~{}\\!\mathrm{cosh}(X_{1}-X_{2}+i\Theta_{12})\end{array}\right\\}+f}{2\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}-X_{2})+c~{}\\!\mbox{cos}\Theta_{12}\end{array}\right]^{2}}~{},~{}~{}~{}~{}~{}~{}$
(D.230)
$\displaystyle\\!\\!\\!\\!\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{11}\\!\\!&\partial_{\nu}\Delta_{22}\\!\\!\\\
\partial_{\rho}\Delta_{11}\\!\\!&\partial_{\rho}\Delta_{22}\\!\\!\end{array}\right|$
(D.233) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{\left\\{\begin{array}[]{l}~{}2\left[r_{\nu}^{(1)}r_{\rho}^{(2)}-r_{\rho}^{(1)}r_{\nu}^{(2)}\right]ab\left[\mathcal{D}_{11}\sinh{(2X_{2})}-\mathcal{D}_{22}\sinh{(2X_{1})}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[(r_{\nu}^{(1)}+r_{\nu}^{(2)})(s_{\rho}^{(1)}-s_{\rho}^{(2)})-(r_{\rho}^{(1)}+r_{\rho}^{(2)})(s_{\nu}^{(1)}-s_{\nu}^{(2)})\right]\cdot\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
~{}~{}~{}ac\left[\mathcal{D}_{12}~{}\\!\mathrm{sinh}(X_{1}+X_{2}-i\Theta_{12})-\mathcal{D}_{21}~{}\\!\mathrm{sinh}(X_{1}+X_{2}+i\Theta_{12})\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[(r_{\nu}^{(1)}-r_{\nu}^{(2)})(s_{\rho}^{(1)}-s_{\rho}^{(2)})-(r_{\rho}^{(1)}-r_{\rho}^{(2)})(s_{\nu}^{(1)}-s_{\nu}^{(2)})\right]\cdot\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
~{}~{}~{}bc\left[\mathcal{E}_{12}~{}\\!\mathrm{sinh}(X_{1}-X_{2}-i\Theta_{12})+\widetilde{\mathcal{E}}_{12}~{}\\!\mathrm{sinh}(X_{1}-X_{2}+i\Theta_{12})\right]\end{array}\right\\}}{2\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}-X_{2})+c~{}\\!\mbox{cos}\Theta_{12}\end{array}\right]^{2}},~{}~{}~{}~{}~{}~{}$
(D.240)
$\displaystyle\\!\\!\\!\\!\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{12}\\!\\!&\partial_{\nu}\Delta_{21}\\!\\!\\\
\partial_{\rho}\Delta_{12}\\!\\!&\partial_{\rho}\Delta_{21}\\!\\!\end{array}\right|$
(D.243) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{\left\\{\begin{array}[]{l}~{}2\left[\begin{array}[]{l}~{}~{}(s_{\nu}^{(1)}r_{\rho}^{(2)}-s_{\rho}^{(1)}r_{\nu}^{(2)})ab\mathcal{d}_{11}\sinh{(2X_{2})}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-(r_{\nu}^{(1)}s_{\rho}^{(2)}-r_{\rho}^{(1)}s_{\nu}^{(2)})ab\mathcal{d}_{22}\sinh{(2X_{1})}\end{array}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[\begin{array}[]{l}~{}~{}(r_{\nu}^{(1)}+s_{\nu}^{(2)})(s_{\rho}^{(1)}-r_{\rho}^{(2)})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-(r_{\rho}^{(1)}+s_{\rho}^{(2)})(s_{\nu}^{(1)}-r_{\nu}^{(2)})\end{array}\right]\\!ac\mathcal{d}_{12}~{}\\!\mathrm{sinh}(X_{1}+X_{2}-i\Theta_{12})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[\begin{array}[]{l}~{}~{}(r_{\nu}^{(1)}-s_{\nu}^{(2)})(s_{\rho}^{(1)}+r_{\rho}^{(2)})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-(r_{\rho}^{(1)}-s_{\rho}^{(2)})(s_{\nu}^{(1)}+r_{\nu}^{(2)})\end{array}\right]\\!ac\mathcal{d}_{21}~{}\\!\mathrm{sinh}(X_{1}+X_{2}+i\Theta_{12})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[\begin{array}[]{l}~{}~{}(r_{\nu}^{(1)}+s_{\nu}^{(2)})(s_{\rho}^{(1)}+r_{\rho}^{(2)})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-(r_{\rho}^{(1)}+s_{\rho}^{(2)})(s_{\nu}^{(1)}+r_{\nu}^{(2)})\end{array}\right]\\!bc\mathcal{e}_{12}~{}\\!\mathrm{sinh}(X_{1}-X_{2}-i\Theta_{12})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[\begin{array}[]{l}~{}~{}(r_{\nu}^{(1)}-s_{\nu}^{(2)})(s_{\rho}^{(1)}-r_{\rho}^{(2)})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-(r_{\rho}^{(1)}-s_{\rho}^{(2)})(s_{\nu}^{(1)}-r_{\nu}^{(2)})\end{array}\right]\\!bc\widetilde{\mathcal{e}}_{12}~{}\\!\mathrm{sinh}(X_{1}-X_{2}+i\Theta_{12})\end{array}\right\\}}{2\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}+X_{2})+c~{}\\!\mbox{cos}\Theta_{12}\end{array}\right]^{2}}~{}.~{}~{}~{}~{}$
(D.260)

$a,b,c$ are defined in table 1 and
$\mathcal{D}_{jk},\mathcal{d}_{jk},\mathcal{E}_{jk},\mathcal{\widetilde{E}}_{jk},\mathcal{e}_{jk}$
are defined as the following table 2 for each real space respectively. $F$ and
$f$ are some constants. Note that

$\displaystyle(\mathcal{\overline{D}}_{jk},~{}\\!\mathcal{\overline{E}}_{jk})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\left\\{\begin{array}[]{l}(-\mathcal{D}_{kj},~{}\\!\mathcal{\widetilde{E}}_{jk})~{}~{}~{}~{}\mbox{on
$\mathbb{U}_{1}$}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
(\mathcal{D}_{kj},~{}\\!-\mathcal{\widetilde{E}}_{jk})~{}~{}~{}\mbox{on
$\mathbb{E}$}\end{array},\right.$ (D.263)
$\displaystyle(\mathcal{\overline{d}}_{jk},~{}\\!\mathcal{\overline{e}}_{jk})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle(\mathcal{d}_{kj},~{}\\!\mathcal{\widetilde{e}}_{jk})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on
$\mathbb{U}_{1},\mathbb{E}$},$ (D.264)

which implies (D.213) $\sim$ (D.243) are all pure imaginary functions on
$\mathbb{U}_{1}$ and hence (D.201) is real-valued on $\mathbb{U}_{1}$. By
(2.81), the Wess-Zumino term is real-valued on $\mathbb{U}_{1}$. On the other
hand, we find that (D.213), (D.233) are real functions and (D.223), (D.243)
are pure imaginary functions on
$\mathbb{E}-\left\\{\mbox{singularities}\right\\}$. Therefore, (D.201) is a
pure imaginary function on $\mathbb{E}-\left\\{\mbox{singularities}\right\\}$.
This implies the Wess-Zumino term is real-valued on
$\mathbb{E}-\\{\mbox{singularities}\\}$ because of (Cf: (2.81))

$\displaystyle{\cal{L}}_{\scriptsize{\mbox{WZ}}}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\scriptsize{\mathbb{E}}}}{{=}}\\!\\!\\!\\!$
$\displaystyle-\frac{i}{8\pi}\left(\mathrm{Tr}\left(\theta_{1}\theta_{3}\theta_{4}\right)x^{1}+\mathrm{Tr}\left(\theta_{2}\theta_{3}\theta_{4}\right)x^{2}-\mathrm{Tr}\left(\theta_{3}\theta_{1}\theta_{2}\right)x^{3}-\mathrm{Tr}\left(\theta_{4}\theta_{1}\theta_{2}\right)x^{4}\right)~{}~{}~{}~{}$
(D.265)

Table 2: Summary of Coefficients

Space | $\mathbb{U}_{1}$ | $\mathbb{E}$
---|---|---
(Metric) | $(+,+,-,-)$ | $(+,+,+,+)$
$\mathcal{D}_{jk}$ | $\underline{(\lambda_{j}-\overline{\lambda}_{k})(\lambda_{j}+\overline{\lambda}_{k})}$ | $\underline{-(\lambda_{j}\overline{\lambda}_{k}-1)(\lambda_{j}\overline{\lambda}_{k}+1)}$
| $\lambda_{j}\overline{\lambda}_{k}$ | $\lambda_{j}\overline{\lambda}_{k}$
$\mathcal{E}_{jk}$ | $\underline{(\lambda_{j}-\lambda_{k})(\lambda_{j}+\lambda_{k})}$ | $\underline{(\lambda_{j}-\lambda_{k})(\lambda_{j}+\lambda_{k})}$
| $\lambda_{j}\lambda_{k}$ | $\lambda_{j}\lambda_{k}$
$\mathcal{\widetilde{E}}_{jk}$ | $\underline{(\overline{\lambda}_{j}-\overline{\lambda}_{k})(\overline{\lambda}_{j}+\overline{\lambda}_{k})}$ | $\underline{-(\overline{\lambda}_{j}-\overline{\lambda}_{k})(\overline{\lambda}_{j}+\overline{\lambda}_{k})}$
| $\overline{\lambda}_{j}\overline{\lambda}_{k}$ | $\overline{\lambda}_{j}\overline{\lambda}_{k}$
$\mathcal{d}_{jk}$ | $\underline{(\lambda_{j}-\overline{\lambda}_{k})^{2}}$ | $\underline{-(\lambda_{j}\overline{\lambda}_{k}+1)^{2}}$
| $\lambda_{j}\overline{\lambda}_{k}$ | $\lambda_{j}\overline{\lambda}_{k}$
$\mathcal{e}_{jk}$ | $\underline{(\lambda_{j}-\lambda_{k})^{2}}$ | $\underline{(\lambda_{j}-\lambda_{k})^{2}}$
| $\lambda_{j}\lambda_{k}$ | $\lambda_{j}\lambda_{k}$
$\mathcal{\widetilde{e}}_{jk}$ | $\underline{(\overline{\lambda}_{j}-\overline{\lambda}_{k})^{2}}$ | $\underline{(\overline{\lambda}_{j}-\overline{\lambda}_{k})^{2}}$
| $\overline{\lambda}_{j}\overline{\lambda}_{k}$ | $\overline{\lambda}_{j}\overline{\lambda}_{k}$

For convenience to discuss the asymptotic behavior of (D.201), we consider the
asymptotic limit

$\displaystyle\left\\{\begin{array}[]{l}X_{1}~{}\mbox{is a finite real
number}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
|X_{2}|\gg|X_{1}|\end{array}.\right.$ (D.268)

By the same calculation as mentioned in section 4.2, we have

$\displaystyle\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\Delta_{11}\\!\\!&\Delta_{22}\\!\\!\\\
\partial_{\mu}\Delta_{11}\\!\\!&\partial_{\mu}\Delta_{22}\\!\\!\end{array}\right|_{|X_{2}|\gg|X_{1}|}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-2r_{\mu}^{(1)}ab\mathcal{D}_{11}\cosh^{2}{X_{2}}+\mathcal{O}(\cosh{X_{2}})}{\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}+X_{2})+\mathcal{O}(1)\end{array}\right]^{2}}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-2r_{\mu}^{(1)}ab\mathcal{D}_{11}+\mathcal{O}(\mathrm{sech}X_{2})}{\left[(a+b)~{}\\!\mbox{cosh}X_{1}+(a-b)~{}\\!\mbox{sinh}X_{1}\mbox{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})\right]^{2}},$
$\displaystyle\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\Delta_{12}\\!\\!&\Delta_{21}\\!\\!\\\
\partial_{\mu}\Delta_{12}\\!\\!&\partial_{\mu}\Delta_{21}\\!\\!\end{array}\right|_{|X_{2}|\gg|X_{1}|}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-2s_{\mu}^{(1)}ab\mathcal{d}_{11}\cosh^{2}{X_{2}}+\mathcal{O}(\cosh{X_{2}})}{\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}+X_{2})+\mathcal{O}(1)\end{array}\right]^{2}}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{-2s_{\mu}^{(1)}ab\mathcal{d}_{11}+\mathcal{O}(\mathrm{sech}X_{2})}{\left[(a+b)~{}\\!\mbox{cosh}X_{1}+(a-b)~{}\\!\mbox{sinh}X_{1}\mbox{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})\right]^{2}},$
$\displaystyle\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{11}\\!\\!&\partial_{\nu}\Delta_{22}\\!\\!\\\
\partial_{\rho}\Delta_{11}\\!\\!&\partial_{\rho}\Delta_{22}\\!\\!\end{array}\right|_{|X_{2}|\gg|X_{1}|}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{2\left(r_{\nu}^{(1)}r_{\rho}^{(2)}-r_{\rho}^{(1)}r_{\nu}^{(2)}\right)ab\mathcal{D}_{11}\cosh{X_{2}}\sinh{X_{2}}+\mathcal{O}(\cosh{X_{2}})}{\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}+X_{2})+\mathcal{O}(1)\end{array}\right]^{2}}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{2\left(r_{\nu}^{(1)}r_{\rho}^{(2)}-r_{\rho}^{(1)}r_{\nu}^{(2)}\right)ab\mathcal{D}_{11}~{}\\!\mathrm{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})}{\left[(a+b)~{}\\!\mbox{cosh}X_{1}+(a-b)~{}\\!\mbox{sinh}X_{1}\mbox{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})\right]^{2}},$
$\displaystyle\displaystyle{\frac{1}{|\sigma|\Delta^{2}}}\left|\\!\\!\begin{array}[]{cc}\partial_{\nu}\Delta_{12}\\!\\!&\partial_{\nu}\Delta_{21}\\!\\!\\\
\partial_{\rho}\Delta_{12}\\!\\!&\partial_{\rho}\Delta_{21}\\!\\!\end{array}\right|_{|X_{2}|\gg|X_{1}|}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{2\left(s_{\nu}^{(1)}r_{\rho}^{(2)}-s_{\rho}^{(1)}r_{\nu}^{(2)}\right)ab\mathcal{d}_{11}\cosh{X_{2}}\sinh{X_{2}}+\mathcal{O}(\cosh{X_{2}})}{\left[\begin{array}[]{l}a~{}\\!\mbox{cosh}(X_{1}+X_{2})+b~{}\\!\mbox{cosh}(X_{1}+X_{2})+\mathcal{O}(1)\end{array}\right]^{2}}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{2\left(s_{\nu}^{(1)}r_{\rho}^{(2)}-s_{\rho}^{(1)}r_{\nu}^{(2)}\right)ab\mathcal{d}_{11}~{}\\!\mathrm{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})}{\left[(a+b)~{}\\!\mbox{cosh}X_{1}+(a-b)~{}\\!\mbox{sinh}X_{1}\mbox{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})\right]^{2}}.$

Therefore, for fixed $X_{1}$ and $|X_{2}|\gg|X_{1}|$ we can conclude that

$\displaystyle B_{\mu\nu\rho}$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle|X_{2}|\gg|X_{1}|}}{{\simeq}}\\!\\!\\!\\!$
$\displaystyle\displaystyle{\frac{-4a^{2}b^{2}C_{\mu\nu\rho}\mathcal{D}_{11}\mathcal{d}_{11}\mathrm{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})}{\left[(a+b)~{}\\!\mbox{cosh}X_{1}+(a-b)~{}\\!\mbox{sinh}X_{1}\mbox{tanh}X_{2}+\mathcal{O}(\mathrm{sech}X_{2})\right]^{4}}}$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle
X_{2}\rightarrow\pm\infty}}{{\longrightarrow}}\\!\\!\\!\\!$
$\displaystyle~{}~{}\displaystyle{\frac{\mp
32a^{2}b^{2}C_{\mu\nu\rho}\mathcal{D}_{11}\mathcal{d}_{11}}{(ae^{X_{1}}+be^{X_{1}})^{4}}}=\mp
2C_{\mu\nu\rho}D_{11}d_{11}\mathrm{sech}^{4}(X_{1}+\delta_{1})\mbox{ ~{}~{}for
fixed $X_{1}$},$

where the phase shift factor is
$\displaystyle{\delta_{1}:=(1/2)~{}\\!\mathrm{log}(a/b)}$. and
$C_{\mu\nu\rho}:=\left(r_{\mu}^{(1)}s_{\nu}^{(1)}+s_{\mu}^{(1)}r_{\nu}^{(1)}\right)r_{\rho}^{(2)}-\left(r_{\mu}^{(1)}s_{\rho}^{(1)}+s_{\mu}^{(1)}r_{\rho}^{(1)}\right)r_{\nu}^{(2)}$.

###  D.5 Asymptotic Form of WZW4 Action Density ($n$-Soliton)

Without loss of generality, we consider one of the asymptotic regions of type
$\mathscr{R}_{K}$ which is labeled by (the other cases are equivalent to this
one.)

$\displaystyle\varepsilon_{j}=\left\\{\begin{array}[]{l}+1,~{}j=1,\cdots,K-1\\\
-1,~{}j=K+1,\cdots,n\end{array}\right.$ (D.275)

and define

$\displaystyle\boldsymbol{X}_{\\!\widehat{K}}:=\sum_{j=1,j\neq
K}^{n}\varepsilon_{j}X_{j}\stackrel{{\scriptstyle\eqref{specialized
epsilon_j}}}{{=}}X_{1}+\cdots+X_{K-1}-X_{K+1}+\cdots+X_{n}.$

By (D.23), (D.43), and (D.57), we can conclude that

$\displaystyle\begin{array}[]{l}\Delta\stackrel{{\scriptstyle\mathscr{R}_{K}}}{{\simeq}}2\left\\{\begin{array}[]{l}~{}a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})\cosh(\boldsymbol{X}_{\\!\widehat{K}}+X_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\\!\\!+a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\cosh(\boldsymbol{X}_{\\!\widehat{K}}-X_{K})\end{array}\\!\\!\right\\}+\\!\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}}\\!\\!\mathcal{O}\left(\cosh(\boldsymbol{X}_{\\!\widehat{K}}-\varepsilon_{j}X_{j})\right),~{}~{}~{}~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Delta_{11}\stackrel{{\scriptstyle\mathscr{R}_{K}}}{{\simeq}}\\!\\!\left\\{\begin{array}[]{l}~{}A(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})\exp(\boldsymbol{X}_{\\!\widehat{K}}+X_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\\!\\!+A(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\exp(\boldsymbol{X}_{\\!\widehat{K}}-X_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\\!\\!+A(\boldsymbol{-1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{1})\exp(-\boldsymbol{X}_{\\!\widehat{K}}+X_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\\!\\!+A(\boldsymbol{-1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{1})\exp(-\boldsymbol{X}_{\\!\widehat{K}}-X_{K})\end{array}\\!\\!\right\\}\\!+\\!\\!\\!\displaystyle{\sum_{j=1,j\neq
J}^{n}}\\!\\!\mathcal{O}\left(\cosh(\boldsymbol{X}_{\\!\widehat{K}}-\varepsilon_{j}X_{j})\right),~{}~{}~{}~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Delta_{22}\left(\mathscr{R}_{K}\right)=\Delta_{11}\left(\mathscr{R}_{K}\right)\Big{|}_{A(\varepsilon_{1},\cdots,\varepsilon_{n})~{}\rightarrow~{}A(-\varepsilon_{1},\cdots,-\varepsilon_{n})},\end{array}$
(D.285)
$\displaystyle\begin{array}[]{l}\Delta_{12}\stackrel{{\scriptstyle\mathscr{R}_{K}}}{{\simeq}}\\!\\!\left\\{\begin{array}[]{l}~{}A(\boldsymbol{1},~{}\\!\varepsilon_{K}=+i,~{}\\!\boldsymbol{-1})\exp(\boldsymbol{X}_{\\!\widehat{K}}+i\Theta_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\\!\\!+A(\boldsymbol{-1},~{}\\!\varepsilon_{K}=+i,~{}\\!\boldsymbol{1})\exp(-\boldsymbol{X}_{\\!\widehat{K}}+i\Theta_{K})\end{array}\\!\\!\right\\}\\!+\\!\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}}\\!\\!\mathcal{O}\left(\cosh(\boldsymbol{X}_{\\!\widehat{K}}-\varepsilon_{j}X_{j})\right),~{}~{}~{}~{}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Delta_{21}\left(\mathscr{R}_{K}\right)=\Delta_{12}\left(\mathscr{R}_{K}\right)\Big{|}_{(X_{j},\Theta_{K})~{}\rightarrow~{}(-X_{j},-\Theta_{K})}.\end{array}$
(D.290)

####  D.5.1 Asymptotic Form of NL$\sigma$M Action Density

By (D.285), (3.75), and direct calculation, we have

$\displaystyle\\!\\!\\!\\!\\!\\!\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial^{\mu}\sigma)\sigma^{-1}\right]$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle\mathscr{R}_{K}}}{{\simeq}}\\!\\!\\!\\!$
$\displaystyle\frac{\left[\begin{array}[]{l}~{}4d_{KK}~{}\\!a(\boldsymbol{1},\varepsilon_{K}=+1,\boldsymbol{-1})a(\boldsymbol{1},\varepsilon_{K}=-1,\boldsymbol{-1})\cosh(2\boldsymbol{X}_{\\!\widehat{K}})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ +\displaystyle{\sum_{j=1,j\neq
K}^{n}\mathcal{O}\left(\cosh(2\boldsymbol{X}_{\\!\widehat{K}}-\varepsilon_{j}X_{j})\right)}\end{array}\right]}{\left\\{\left[\begin{array}[]{l}~{}~{}a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})\cosh(\boldsymbol{X}_{\\!\widehat{K}}+X_{K})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
+a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\cosh(\boldsymbol{X}_{\\!\widehat{K}}-X_{K})\end{array}\right]\\!+\\!\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}\\!\\!\mathcal{O}\left(\cosh(\boldsymbol{X}_{\\!\widehat{K}}-\varepsilon_{j}X_{j})\right)}\right\\}^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{8d_{KK}~{}\\!a(\boldsymbol{1},\varepsilon_{K}=+1,\boldsymbol{-1})a(\boldsymbol{1},\varepsilon_{K}=-1,\boldsymbol{-1})+\displaystyle{\sum_{j=1,j\neq
K}^{n}\mathcal{O}\left(\mbox{sech}X_{j}\right)}}{\left\\{\begin{array}[]{l}~{}~{}\left[a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})+a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\right]\cosh
X_{K}\par\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\left[a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})-a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\right]\tanh\boldsymbol{X}_{\\!\widehat{K}}~{}\\!\sinh
X_{K}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}\\!\\!\mathcal{O}\left(\mbox{sech}X_{j}\right)}\end{array}\right\\}^{2}}$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle\boldsymbol{X}_{\\!\widehat{K}}\rightarrow\pm\infty}}{{\longrightarrow}}\\!\\!\\!\\!$
$\displaystyle 2d_{KK}\mathrm{sech}^{2}(X_{K}\pm\delta_{K})~{}~{}\mbox{for
fixed real number $X_{K}$},$

where the phase shift factor is
$\displaystyle{\delta_{K}:=\frac{1}{2}~{}\\!\mathrm{log}\left[\frac{a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})}{a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})}\right]}$.
and $d_{KK}$ is defined in table 1.

####  D.5.2 Asymptotic Form of Wess-Zumino Action Density

By (D.285), (3.75), and direct calculation, we have

$\displaystyle
B_{\mu\nu\rho}\stackrel{{\scriptstyle\mathscr{R}_{K}}}{{\simeq}}\displaystyle{\frac{-4\mathcal{A\widetilde{A}}C_{\mu\nu\rho}^{(K)}\mathrm{tanh}\boldsymbol{X}_{\\!\widehat{K}}+\displaystyle{\sum_{j=1,j\neq
K}^{n}\mathcal{O}\left(\mbox{sech}X_{j}\right)}}{|\sigma|^{2}\left\\{\begin{array}[]{l}~{}~{}\left[a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})+a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\right]\cosh
X_{K}\par\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\left[a(\boldsymbol{1},~{}\\!\varepsilon_{K}=+1,~{}\\!\boldsymbol{-1})-a(\boldsymbol{1},~{}\\!\varepsilon_{K}=-1,~{}\\!\boldsymbol{-1})\right]\tanh\boldsymbol{X}_{\\!\widehat{K}}~{}\\!\sinh
X_{K}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}\\!\\!\mathcal{O}\left(\mbox{sech}X_{j}\right)}\end{array}\right\\}^{4}}},~{}~{}~{}~{}~{}~{}~{}~{}$

where

$\displaystyle
C_{\mu\nu\rho}^{(K)}:=\left(r_{\mu}^{(K)}s_{\nu}^{(K)}+s_{\mu}^{(K)}r_{\nu}^{(K)}\right)\boldsymbol{r}_{\rho}^{(\widehat{K})}-\left(r_{\mu}^{(K)}s_{\rho}^{(K)}+s_{\mu}^{(K)}r_{\rho}^{(K)}\right)\boldsymbol{r}_{\nu}^{(\widehat{K})}),~{}r_{\rho}^{(K)}:=\\!\\!\\!\\!\displaystyle{\sum_{j=1,j\neq
K}^{n}}\\!\varepsilon_{j}r_{\rho}^{(j)},$

and

$\displaystyle\mathcal{A}$ $\displaystyle\\!\\!\\!\\!:=\\!\\!\\!\\!$
$\displaystyle\left[\begin{array}[]{l}~{}~{}\\!A({\bf
1},~{}\varepsilon_{K}=+1,~{}{\bf-1})A({\bf-1},~{}\varepsilon_{K}=+1,~{}{\bf
1})\\\ \\!\\!-A({\bf
1},~{}\varepsilon_{K}=-1,~{}{\bf-1})A({\bf-1},~{}\varepsilon_{K}=-1,~{}{\bf
1})\end{array}\right],$ $\displaystyle\mathcal{\widetilde{A}}$
$\displaystyle\\!\\!\\!\\!:=\\!\\!\\!\\!$ $\displaystyle A({\bf
1},~{}\varepsilon_{K}=+i,~{}{\bf-1})A({\bf-1},~{}\varepsilon_{K}=+i,~{}{\bf
1}).$

By (4.47) and the fact that

$\displaystyle
C_{\mu\nu\rho}^{(K)}+C_{\nu\rho\mu}^{(K)}+C_{\rho\mu\nu}^{(K)}=0,$

we have

$\displaystyle\mbox{Tr}\left[(\partial_{\mu}\sigma)\sigma^{-1}(\partial_{\nu}\sigma)\sigma^{-1}(\partial_{\rho}\sigma)\sigma^{-1}\right]\longrightarrow
0$

in the asymptotic region. Therefore, the Wess-Zumino term is asymptotic to
zero for $n$-soliton case.

## References

  * [1]
  * [2] R. Bittleston and D. Skinner, “Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory,” [arXiv:2011.04638].
  * [3] L. L. Chau, M. L. Ge, A. Sinha and Y. S. Wu, Phys. Lett. B 121, 391-396 (1983).
  * [4] K. Costello and M. Yamazaki, “Gauge Theory And Integrability, III,” [arXiv:1908.02289].
  * [5] B. Dai, C.-L. Terng and K. Uhlenbeck “On the space-time Monopole equation,” Surveys in Differential Geometry X (2008) 1 [arXiv:math/0602607].
  * [6] H. J. de Vega, Commun. Math. Phys. 116, 659 (1988).
  * [7] L. Dolan, Phys. Lett. B 113, 387-390 (1982).
  * [8] S. K. Donaldson, Proc. Lond. Math. Soc. 3 (1985) 1.
  * [9] I. Gelfand, S. Gelfand, V. Retakh and R. Wilson, Adv. Math. 193 (2005) 56 [math.QA/0208146].
  * [10] I. Gelfand and V. Retakh, Funct. Anal. Appl. 25 (1991) 91; Funct. Anal. Appl. 2 (1992) 1.
  * [11] C. R. Gilson, M. Hamanaka, S. C. Huang and J. J. C. Nimmo, J. Phys. A 53, no.40, 404002 (2020) [arXiv:2004.01718].
  * [12] C. R. Gilson and J. J. C. Nimmo, J. Phys. A 40, 3839 (2007) [nlin.si/0701027].
  * [13] D. Gluck, Y. Oz and T. Sakai, JHEP 08, 055 (2003) [arXiv:hep-th/0306112].
  * [14] M. Green, J. Schwarz and E. Witten, Superstring theory: volume 1, (1987, Cambridge UP).
  * [15] M. Hamanaka, Nucl. Phys. B 741 (2006) 368 [hep-th/0601209].
  * [16] M. Hamanaka and S.C. Huang, JHEP 10, 101 (2020) [arXiv:2004.09248].
  * [17] M. Hamanaka and S. C. Huang, JHEP 01, 039 (2022) [arXiv:2106.01353].
  * [18] M. Hamanaka and K. Toda, Phys. Lett. A 316, 77 (2003) [hep-th/0211148].
  * [19] J. A. Harvey, P. Kraus and F. Larsen, JHEP 12, 024 (2000) [arXiv:hep-th/0010060].
  * [20] S. C. Huang, “On Soliton Solutions of the Anti-Self-Dual Yang-Mills Equations from the Perspective of Integrable Systems,” A Dissertation in candidacy for the degree of Doctor of Philosophy (2021, Nagoya University) [arXiv:2112.10702].
  * [21] S. C. Huang, “Multi-Soliton scattering of the Anti-Self-Dual Yang-Mills Equations in 4-dimensional split signature,” in Proceedings of the East Asia Joint Symposium on Fields and Strings 2021, pp. 33-42 (2022), [arXiv:2201.13318 [hep-th]].
  * [22] T. Inami, H. Kanno and T. Ueno, Mod. Phys. Lett. A 12, 2757-2764 (1997) [arXiv:hep-th/9704010].
  * [23] T. Inami, H. Kanno, T. Ueno and C. S. Xiong, Phys. Lett. B 399, 97-104 (1997) [arXiv:hep-th/9610187].
  * [24] T. A. Ivanova and O. Lechtenfeld, Int. J. Mod. Phys. A 16, 303-329 (2001) [arXiv:hep-th/0007049].
  * [25] Y. Kodama, KP Solitons and the Grassmannians, (Springer, 2017).
  * [26] Y. Kodama, Solitons in Two-Dimensional Shallow Water, (SIAM, 2018).
  * [27] Y. Kodama and L. Williams, Invent. math. 198, 637 (2014) [arXiv:1106.0023].
  * [28] O. Lechtenfeld, A. D. Popov and and B. Spendig, JHEP 06 (2001) 011 [hep-th/010396].
  * [29] A. Losev, G. W. Moore, N. Nekrasov and S. Shatashvili, Nucl. Phys. B Proc. Suppl. 46, 130-145 (1996) [arXiv:hep-th/9509151].
  * [30] N. Marcus, Nucl. Phys. B 387 (1992) 263 [hep-th/9207024]; “A tour through N=2 strings,” hep-th/9211059.
  * [31] L. J. Mason and N. M. Woodhouse, Integrability, Self-Duality, and Twistor Theory (Oxford UP, 1996).
  * [32] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, (Springer-Verlag, 1991).
  * [33] V. P. Nair, “Kahler-Chern-Simons theory,” [arXiv:hep-th/9110042].
  * [34] V. P. Nair and J. Schiff, Phys. Lett. B 246, 423-429 (1990); Nucl. Phys. B 371, 329-352 (1992).
  * [35] A. Neitzke and C. Vafa, “N=2 strings and the twistorial Calabi-Yau,” [arXiv:hep-th/0402128].
  * [36] N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068].
  * [37] J. J. C. Nimmo, C. R. Gilson and Y. Ohta, Theor. Math. Phys. 122, 239 (2000) [Teor. Mat. Fiz. 122, 284 (2000)].
  * [38] K. Ohkuma and M. Wadati, J. Phys. Soc. Jap. 52, 749 (1983).
  * [39] K. Ohmori, “A Review on tachyon condensation in open string field theories,” [arXiv:hep-th/0102085].
  * [40] H. Ooguri and C. Vafa, Nucl. Phys. B 361 (1991) 469; Nucl. Phys. B 367 (1991) 83.
  * [41] A. Parkes, Nucl. Phys. B 376, 279-296 (1992) [arXiv:hep-th/9110075].
  * [42] M. Sato, RIMS Kokyuroku 439 (1981) 30; M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifold,” in Nonlinear Partial Differential Equations in Applied Sciences (North-Holland, 1983) 259.
  * [43] N. Seiberg and E. Witten, JHEP 09, 032 (1999) [arXiv:hep-th/9908142].
  * [44] R. S. Ward, Phil. Trans. Roy. Soc. Lond. A 315 (1985) 451.
  * [45] E. Witten, Commun. Math. Phys. 252, 189-258 (2004) [arXiv:hep-th/0312171].