$\displaystyle=\varphi_{\alpha}^{\mathrm{Ext}}\circ\begin{pmatrix}v(a_{\alpha}^{11}/\lambda)&v(a^{12}_{\alpha}/\lambda)\\\
v(a^{\text{App}}_{\alpha}/\lambda)b^{22}_{\alpha}&v(a^{22}_{\alpha}/\lambda)\end{pmatrix}\circ(\varphi_{\alpha}^{\mathrm{Ext}})^{-1}.$

Here $\\{b^{22}_{\alpha}\\}_{\alpha}$ is the coboundary in (4.6). Since we fix
the determinant bundle $L_{0}$, we may assume that the coboundary
$\\{b^{22}_{\alpha}\\}_{\alpha}$ is independent of the moduli space
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$.

Now we calculate the class

(4.18)
$[(\\{\mathrm{tr}(u_{\alpha\beta}(v)u_{\beta\gamma}(v^{\prime}))\\},-\\{\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)\\})]$

in
${\mathbf{H}}^{2}(\mathcal{O}_{C}\xrightarrow{d}\Omega_{C}^{1})\cong\mathbb{C}$.
First we calculate $u_{\alpha\beta}(v)u_{\beta\gamma}(v^{\prime})$ as follows:

$\displaystyle u_{\alpha\beta}(v)u_{\beta\gamma}(v^{\prime})$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&\frac{v(b^{\text{Bun}}_{\alpha\beta})}{b^{22}_{\alpha}}\\\
0&0\end{pmatrix}\circ(\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}\circ\varphi_{\gamma}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&\frac{v(b^{\text{Bun}}_{\beta\gamma})}{b^{22}_{\alpha}}\\\
0&0\end{pmatrix}\circ(\varphi_{\gamma}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&\frac{v(b^{\text{Bun}}_{\alpha\beta})}{b^{22}_{\alpha}}\\\
0&0\end{pmatrix}B_{\beta\gamma}\begin{pmatrix}0&\frac{v(b^{\text{Bun}}_{\beta\gamma})}{b^{22}_{\alpha}}\\\
0&0\end{pmatrix}\circ(\varphi_{\gamma}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&0\\\
0&0\end{pmatrix}\circ(\varphi_{\gamma}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}=0.$

So we may take a representative of the class (4.18) so that

$[-\\{\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)\\}]$

is in $H^{1}(C,\Omega^{1}_{C})$. By using equalities (4.16) and (4.17), we
have the following equality

(4.19)
$\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)=v(b^{\text{Bun}}_{\alpha\beta})v^{\prime}\left(\frac{a^{\text{App}}_{\beta}}{\lambda}\right)-v\left(\frac{a^{\text{App}}_{\alpha}}{\lambda}\right)v^{\prime}(b^{\text{Bun}}_{\alpha\beta}).$

We take bases

$a^{\text{App}(0)},a^{\text{App}(1)},\ldots,a^{\text{App}(N_{0})}\in
H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D))$

of $H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D))$ and

$[\\{b_{\alpha\beta}^{\text{App}(0)}\\}],[\\{b_{\alpha\beta}^{\text{App}(1)}\\}],\ldots,[\\{b_{\alpha\beta}^{\text{App}(N_{0})}\\}]$

of $H^{1}(C,L_{0}^{-1}(-D))$ so that these bases give the homogeneous
coordinates $(a_{0}:\cdots:a_{N_{0}})$ on
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}$ and $(b_{0}:\cdots:b_{N_{0}})$ on
$\mathbb{P}_{\boldsymbol{b}}^{N_{0}}$. We may assume that these bases are
independent of the moduli space $M_{X}({L_{0}},\nabla_{L_{0}})_{0}$. We set

$a^{\text{App}}_{\alpha}=a_{0}a^{\text{App}(0)}|_{U_{\alpha}}+a_{1}a^{\text{App}(1)}|_{U_{\alpha}}+\cdots+a_{N_{0}}a^{\text{App}(N_{0})}|_{U_{\alpha}}$

and

$b^{\text{App}}_{\alpha\beta}=b_{0}b^{\text{App}(0)}_{\alpha\beta}+b_{1}b^{\text{App}(1)}_{\alpha\beta}+\cdots+b_{N_{0}}b^{\text{App}(N_{0})}_{\alpha\beta}.$

By (4.19), we have that

$\displaystyle\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)$
$\displaystyle=v\left(\sum_{k=0}^{N_{0}}b_{k}b^{\text{App}(k)}_{\alpha\beta}\right)v^{\prime}\left(\frac{\sum_{k=0}^{N_{0}}a_{k}a^{\text{App}(k)}|_{U_{\alpha}}}{\lambda}\right)$
$\displaystyle\qquad-v\left(\frac{\sum_{k=0}^{N_{0}}a_{k}a^{\text{App}(k)}|_{U_{\alpha}}}{\lambda}\right)v^{\prime}\left(\sum_{k=0}^{N_{0}}b_{k}b^{\text{App}(k)}_{\alpha\beta}\right)$
$\displaystyle=\left(\sum_{k=0}^{N_{0}}v\left(b_{k}\right)b^{\text{App}(k)}_{\alpha\beta}\right)\left(\sum_{k=0}^{N_{0}}v^{\prime}\left(\frac{a_{k}}{\lambda}\right)a^{\text{App}(k)}|_{U_{\alpha}}\right)$
$\displaystyle\qquad-\left(\sum_{k=0}^{N_{0}}v\left(\frac{a_{k}}{\lambda}\right)a^{\text{App}(k)}|_{U_{\alpha}}\right)\left(\sum_{k=0}^{N_{0}}v^{\prime}\left(b_{k}\right)b^{\text{App}(k)}_{\alpha\beta}\right).$

Since $(b_{0}:\cdots:b_{N_{0}})$ is dual of $(a_{0}:\cdots:a_{N_{0}})$ with
respect to the natural pairing

$H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D))\times
H^{1}(C,L_{0}^{-1}(-D))\longrightarrow
H^{1}(C,\Omega^{1}_{C})\cong\mathbb{C},$

we have that

$\displaystyle\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)$
$\displaystyle=\sum_{k=0}^{N_{0}}v\left(b_{k}\right)v^{\prime}\left(\frac{a_{k}}{\lambda}\right)-\sum_{k=0}^{N_{0}}v^{\prime}\left(b_{k}\right)v\left(\frac{a_{k}}{\lambda}\right).$

On the other hand, we have that

$\lambda=\frac{\langle[\\{a^{\text{App}}_{\alpha}\\},[\\{b^{\text{Bun}}_{\alpha\beta}\\}]\rangle}{-\sum_{i}\theta_{i,-1}^{-}}=\frac{a_{0}b_{0}+a_{1}b_{1}+\cdots+a_{N_{0}}b_{N_{0}}}{-\sum_{i}\theta_{i,-1}^{-}}.$

Then we have

$\displaystyle H^{1}(C,\Omega^{1}_{C})$ $\displaystyle\xrightarrow{\ \cong\
}\mathbb{C}$
$\displaystyle[-\\{\mathrm{tr}\left(u_{\alpha\beta}(v)v_{\beta}(v^{\prime})\right)-\mathrm{tr}\left(v_{\alpha}(v)u_{\alpha\beta}(v^{\prime})\right)\\}]$
$\displaystyle\longmapsto d\eta(v,v^{\prime}).$

This means the statement. ∎

##  5\. Companion normal forms for an elliptic curve with two poles

In Section 2, we introduced the companion normal form of a rank 2 meromorphic
connection with some assumption. The purpose of the present section is to
detail the case of an elliptic curve with two simple poles, or with an
unramified irregular singularity of order $2$. The latter case arises by
confluence from the first one, up to some modification in the arguments. We
will give explicit description of the companion normal form for an elliptic
curve in these cases. Moreover, we will calculate the canonical coordinates
introduced in Section 3.5. First we start from construction of the companion
normal form $(\mathcal{O}_{C}\oplus(\Omega_{C}^{1}(D))^{-1},\nabla_{0})$. Next
we will construct a rank 2 meromorphic connection $(E,\nabla)$ by transforming
the companion normal form.

Let $C$ be the elliptic curve constructed by gluing affine cubic curves

$U_{0}:=(y_{1}^{2}-x_{1}(x_{1}-1)(x_{1}-\lambda)=0)\quad\text{and}\quad
U_{\infty}:=(y_{2}^{2}-x_{2}(1-x_{2})(1-\lambda x_{2})=0)$

with the relations $x_{1}=x_{2}^{-1}$ and $y_{1}=y_{2}x_{2}^{-2}$. We fix some
$t\in\mathbb{C}$ and set $D=t_{1}+t_{2}$ where $t_{1}=(t,s)$ and
$t_{2}=(t,-s)$, so that $D$ is the positive part of $\mathrm{div}(x-t)$. Let
$q_{1},q_{2},q_{3}$ be points on $C$:

$q_{j}\colon(x_{1},y_{1})=(u_{j},v_{j})$

for each $j=1,2,3$. Now we assume that
$u_{j}\not\in\\{0,1,\lambda,\infty,t\\}$ for any $j$.

We take trivialization of the line bundle $(\Omega^{1}_{C}(D))^{-1}$ over $C$
as follows:

(5.1) $\mathcal{O}_{U_{0}}\xrightarrow{\ \sim\
}(\Omega^{1}_{C}(D))^{-1}|_{U_{0}};\quad
1\longmapsto\left(\frac{\operatorname{d}\\!x_{1}}{(x_{1}-t)y_{1}}\right)^{-1}$

and

(5.2) $\mathcal{O}_{U_{\infty}}\xrightarrow{\ \sim\
}(\Omega^{1}_{C}(D))^{-1}|_{U_{\infty}};\quad
1\longmapsto\left(\frac{\operatorname{d}\\!x_{2}}{(1-tx_{2})y_{2}}\right)^{-1}.$

Then the corresponding transition function $f_{0\infty}$ is as follows:

(5.3) $\displaystyle f_{\infty 0}\colon\mathcal{O}_{U_{0}}|_{U_{0}\cap
U_{\infty}}$ $\displaystyle\xrightarrow{\ \sim\
}\mathcal{O}_{U_{\infty}}|_{U_{0}\cap U_{\infty}}$ $\displaystyle 1$
$\displaystyle\longmapsto-\frac{1}{x_{2}}.$

###  5.1. Definition of a connection $\nabla_{0}$ on
$\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}$

For $\zeta_{1},\zeta_{2},\zeta_{3}\in\mathbb{C}$, we define $1$-forms
$\omega_{12}$, $\omega_{21}$, and $\omega_{22}$ as follows:

(5.4)
$\displaystyle\omega_{12}=\sum_{j=1}^{3}\frac{\zeta_{j}}{2}\cdot\frac{y_{1}+v_{j}}{x_{1}-u_{j}}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}}+\left(\frac{A_{1}+A_{2}y_{1}}{x_{1}-t}+A_{3}+A_{4}x_{1}\right)\frac{\operatorname{d}\\!x_{1}}{y_{1}}$
$\displaystyle\omega_{21}:=\frac{1}{x_{1}-t}\frac{\operatorname{d}\\!x_{1}}{y_{1}}$
$\displaystyle\omega_{22}:=\sum_{j=1}^{3}\frac{1}{2}\cdot\frac{y_{1}+v_{j}}{x_{1}-u_{j}}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}}+\left(\frac{B_{1}+B_{2}y_{1}}{x_{1}-t}+B_{3}\right)\frac{\operatorname{d}\\!x_{1}}{y_{1}}.$

Here $A_{1},\ldots,A_{4}\in\mathbb{C}$ and $B_{1},\ldots,B_{3}\in\mathbb{C}$
are parameters. Notice that $\omega_{12}\otimes\omega_{21}$ is a global
section of $(\Omega_{C}^{1})^{\otimes 2}(2D+B)$ and $\omega_{22}$ is a global
section of $\Omega_{C}^{1}(D+B+\infty)$.

####  5.1.1. Fixing the polar parts in the logarithmic case

We start by analyzing the case where $t\notin\\{0,1,\lambda,\infty\\}$. In
this case, we have $s\neq 0$, so $t_{1}\neq t_{2}$. We fix complex numbers
$\theta_{1}^{\pm},\theta_{2}^{\pm}$ such that
$\sum_{i=1}^{2}(\theta^{+}_{i}+\theta^{-}_{i})=-1$, which is called Fuchs’
relation. Now we assume that the eigenvalues of the matrix

$\mathrm{res}_{t_{1}}\begin{pmatrix}0&\omega_{12}\\\
\omega_{21}&\omega_{22}\end{pmatrix}$

are given by $\theta_{1}^{+},\theta_{1}^{-}$ and the eigenvalues of the matrix

$\mathrm{res}_{t_{2}}\begin{pmatrix}0&\omega_{12}\\\
\omega_{21}&\omega_{22}\end{pmatrix}$

are given by $\theta_{2}^{+},\theta_{2}^{-}$. (To be coherent with Definition
2, we should write $\theta_{1,-1}$ and $\theta_{2,-1}$ for elements of the
Cartan subalgebra, and $\theta_{1,-1}^{\pm}$ and $\theta_{2,-1}^{\pm}$ for
their eigenvalues; however, we drop the subscript $-1$ for ease of notation,
because there are only poles of order $1$, so no confusion is possible.)
Specifically, these conditions read as

(5.5)
$\mathrm{res}_{(t,s)}\omega_{12}\cdot\mathrm{res}_{(t,s)}\omega_{21}=\theta^{+}_{1}\cdot\theta^{-}_{1},\qquad\mathrm{res}_{(t,-s)}\omega_{12}\cdot\mathrm{res}_{(t,-s)}\omega_{21}=\theta^{+}_{2}\cdot\theta^{-}_{2},$

and

(5.6)
$\mathrm{res}_{(t,s)}\omega_{22}=\theta^{+}_{1}+\theta^{-}_{1},\qquad\mathrm{res}_{(t,-s)}\omega_{22}=\theta^{+}_{2}+\theta^{-}_{2}.$

Notice that $\mathrm{res}_{(u_{j},v_{j})}\omega_{22}=1$ for each $j$. By the
residue theorem, $\mathrm{res}_{\infty}\omega_{22}=-2$. By the assumption
(5.5) and (5.6), we may determine the parameters $A_{1},A_{2},B_{1}$, and
$B_{2}$.

######  Lemma 32.

Let complex numbers $\theta_{1}^{\pm},\theta_{2}^{\pm}$ satisfying Fuchs’
relation be given. Then, there exist unique values of the parameters
$A_{1},A_{2},B_{1}$, and $B_{2}$ such that (5.5) and (5.6) are fulfilled.
Moreover, these parameter values are independent of $u_{1},u_{2},u_{3}$,
$\zeta_{1},\zeta_{2}$, and $\zeta_{3}$. So the polar parts of
$\omega_{12},\omega_{21}$, and $\omega_{22}$ at $t_{i}$ are independent of
$u_{1},u_{2},u_{3}$, $\zeta_{1},\zeta_{2}$, and $\zeta_{3}$.

###### Proof.

By the equalities (5.5), we have

$\frac{A_{1}+A_{2}s}{s}\cdot\frac{1}{s}=\theta^{+}_{1}\cdot\theta^{-}_{1}\quad\text{and}\quad\frac{A_{1}-A_{2}s}{-s}\cdot\frac{1}{-s}=\theta^{+}_{2}\cdot\theta^{-}_{2}.$

By the equalities in (5.6), we have

$\frac{B_{1}+B_{2}s}{s}=\theta^{+}_{1}+\theta^{-}_{1}\quad\text{and}\quad\frac{B_{1}-B_{2}s}{-s}=\theta^{+}_{2}+\theta^{-}_{2}.$

By these equalities, $A_{1},A_{2},B_{1}$, and $B_{2}$ are determined, and
$A_{1},A_{2},B_{1}$, and $B_{2}$ are independent of $u_{1},u_{2},u_{3}$,
$\zeta_{1},\zeta_{2}$, and $\zeta_{3}$. It is clear that the polar parts of
$\omega_{12},\omega_{21}$, and $\omega_{22}$ at $t_{i}$ are independent of
$u_{1},u_{2},u_{3}$, $\zeta_{1},\zeta_{2}$, and $\zeta_{3}$. ∎

####  5.1.2. Fixing the polar part in the irregular case

We now study the situation $t\in\\{0,1,\lambda,\infty\\}$. For sake of
concreteness, we let $t=0$, the other cases being similar. Then, $s=0$ and
$t_{1}=t_{2}$, so the divisor $D$ is reduced of length $2$. A local
holomorphic coordinate of the elliptic curve $C$ in a neighbourhood of $t_{1}$
is given by $y_{1}$.

We fix $\theta_{-2}^{\pm},\theta_{-1}^{+}\in\mathbb{C}$ so that
$\theta_{-2}^{+}\neq\theta_{-2}^{-}$ and set
$\theta_{-1}^{-}=-1-\theta_{-1}^{+}$. (To be coherent with Definition 2, we
should write $\theta_{1,-2}$ and $\theta_{1,-1}$ for elements of the Cartan
subalgebra, and $\theta_{1,-2}^{\pm}$ and $\theta_{1,-1}^{\pm}$ for their
eigenvalues; however, we omit the subscript $1$ for ease of notation, because
there is only one singular point, so no confusion is possible.)

######  Lemma 33.

Fix $\theta_{-2}^{\pm},\theta_{-1}^{\pm}$ as above. Then, there exist unique
values $A_{1},A_{2},B_{1},B_{2}\in\mathbb{C}$ such that the eigenvalues of

$\mathrm{res}\begin{pmatrix}0&\omega_{12}\\\
\omega_{21}&\omega_{22}\end{pmatrix}$

admit Laurent expansions of the form

$\left(\theta_{-2}^{\pm}\frac{1}{y_{1}^{2}}+\theta_{-1}^{\pm}\frac{1}{y_{1}}+O(1)\right)\otimes\operatorname{d}\\!y_{1}.$

Moreover, the values of the solutions are independent of $u_{i},\zeta_{i}$.

###### Proof.

By the inverse function theorem, there exists an analytic open subset
$U\subset\mathbb{C}$ and a holomorphic function $h\colon U\to\mathbb{C}$
satisfying $h(0)=0$ such that $C$ is given by the explicit equation
$x_{1}=h(y_{1}^{2})$. It is obvious that this function $h$ is independent of
the choice of $u_{i},\zeta_{i}$, and it is easy to see that
$h^{\prime}(0)=\frac{1}{\lambda}\neq 0$. From the defining equation of $C$ we
get

$\frac{\operatorname{d}\\!x_{1}}{y_{1}}=\frac{2\operatorname{d}\\!y_{1}}{3x_{1}^{2}-2(1+\lambda)x_{1}+\lambda},$

so $\frac{\operatorname{d}\\!x_{1}}{y_{1}}$ is a holomorphic $1$-form around
$t_{1}$. Moreover,

$\frac{\operatorname{d}\\!x_{1}}{x_{1}y_{1}}=\frac{\operatorname{d}\\!y_{1}}{y_{1}^{2}}g(y_{1}^{2})$

for some holomorphic function $g\colon U\to\mathbb{C}$ satisfying $g(0)=2$.
The polar parts of the coefficients can be separated as

$\displaystyle\omega_{12}$
$\displaystyle=(A_{1}+A_{2}y_{1})\frac{\operatorname{d}\\!x_{1}}{x_{1}y_{1}}+O(1)=2(A_{1}+A_{2}y_{1})\frac{\operatorname{d}\\!y_{1}}{y_{1}^{2}}+O(1)$
$\displaystyle\omega_{21}$
$\displaystyle=2\frac{\operatorname{d}\\!y_{1}}{y_{1}^{2}}+O(1)$
$\displaystyle\omega_{22}$
$\displaystyle=(B_{1}+B_{2}y_{1})\frac{\operatorname{d}\\!x_{1}}{x_{1}y_{1}}+O(1)=2(B_{1}+B_{2}y_{1})\frac{\operatorname{d}\\!y_{1}}{y_{1}^{2}}+O(1).$

Now, the sum of the eigenvalues must be

$(\theta_{-2}^{+}+\theta_{-2}^{-})\frac{1}{y_{1}^{2}}+(\theta_{-1}^{+}+\theta_{-1}^{-})\frac{1}{y_{1}}.$

These conditions determine

$B_{1}=\frac{1}{2}(\theta_{-2}^{+}+\theta_{-2}^{-}),\qquad
B_{2}=\frac{1}{2}(\theta_{-1}^{+}+\theta_{-1}^{-})=-\frac{1}{2}.$

Moreover, we have

$-\omega_{12}\omega_{21}=-4(A_{1}+A_{2}y_{1})\frac{\left(\operatorname{d}\\!y_{1}\right)^{\otimes
2}}{y_{1}^{4}}+O\left(\frac{1}{y_{1}^{2}}\right).$

On the other hand, the product of the eigenvalues must have the expansion (up
to a global factor $\left(\operatorname{d}\\!y_{1}\right)^{\otimes 2}$)

$\theta_{-2}^{+}\theta_{-2}^{-}\frac{1}{y_{1}^{4}}+(\theta_{-2}^{+}\theta_{-1}^{-}+\theta_{-2}^{-}\theta_{-1}^{+})\frac{1}{y_{1}^{3}}.$

These condition then determine

$A_{1}=-\frac{1}{4}\theta_{-2}^{+}\theta_{-2}^{-},\qquad
A_{3}=-\frac{1}{4}(\theta_{-2}^{+}\theta_{-1}^{-}+\theta_{-2}^{-}\theta_{-1}^{+}).$

This finishes the proof. ∎

####  5.1.3. Construction of the connection

We define

$\displaystyle\beta\colon(\Omega^{1}_{C}(D))^{-1}\longrightarrow\Omega^{1}_{C}(D+B)$
($\mathcal{O}_{C}$-morphism)
$\displaystyle\delta\colon(\Omega^{1}_{C}(D))^{-1}\longrightarrow(\Omega^{1}_{C}(D))^{-1}\otimes\Omega^{1}_{C}(D+B)$
(connection)
$\displaystyle\gamma\colon\mathcal{O}_{C}\longrightarrow(\Omega^{1}_{C}(D))^{-1}\otimes\Omega^{1}_{C}(D)$
($\mathcal{O}_{C}$-morphism)

by using the trivializations (5.1) and (5.2) of $(\Omega^{1}_{C}(D))^{-1}$ as
follows:

$\beta=\begin{cases}\omega_{12}\colon\mathcal{O}_{U_{0}}\rightarrow\mathcal{O}_{U_{0}}\otimes\Omega^{1}_{C}(D+B)|_{U_{0}}\\\
\mathrm{id}\circ\omega_{12}\circ f^{-1}_{\infty
0}\colon\mathcal{O}_{U_{\infty}}\rightarrow\mathcal{O}_{U_{\infty}}\otimes\Omega^{1}_{C}(D+B)|_{U_{\infty}},\end{cases}$
$\delta=\begin{cases}\operatorname{d}+\omega_{22}\colon\mathcal{O}_{U_{0}}\rightarrow\mathcal{O}_{U_{0}}\otimes\Omega^{1}_{C}(D+B)|_{U_{0}}\\\
\operatorname{d}+f_{\infty 0}\circ\omega_{22}\circ f^{-1}_{\infty 0}+f_{\infty
0}\circ\operatorname{d}\\!f^{-1}_{\infty
0}\colon\mathcal{O}_{U_{\infty}}\rightarrow\mathcal{O}_{U_{\infty}}\otimes\Omega^{1}_{C}(D+B)|_{U_{\infty}},\end{cases}$
$\gamma:=\begin{cases}\omega_{21}\colon\mathcal{O}_{U_{0}}\rightarrow\mathcal{O}_{U_{0}}\otimes\Omega^{1}_{C}(D+B)|_{U_{0}}\\\
f_{\infty
0}\circ\omega_{21}\circ\mathrm{id}\colon\mathcal{O}_{U_{\infty}}\rightarrow\mathcal{O}_{U_{\infty}}\otimes\Omega^{1}_{C}(D+B)|_{U_{\infty}}.\end{cases}$

Here $f_{\infty 0}$ is the transition function of $(\Omega^{1}_{C}(D))^{-1}$
described in (5.3). Notice that

$f_{\infty 0}\circ\omega_{22}\circ f^{-1}_{\infty 0}+f_{\infty
0}\circ\operatorname{d}\\!f^{-1}_{\infty
0}=\omega_{22}+\frac{\operatorname{d}\\!x_{2}}{x_{2}},$

which is holomorphic at $\infty\in C$, since we have
$\mathrm{res}_{\infty}\omega_{22}=-2$. We define a connection as follows:

(5.7) $\nabla_{0}:=\operatorname{d}+\begin{pmatrix}0&\beta\\\
\gamma&\delta\end{pmatrix}\colon\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}\longrightarrow\left(\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}\right)\otimes\Omega^{1}_{C}(D+B),$

which is the companion normal form. Remark that

$\mathrm{res}_{q_{j}}(\nabla_{0})=\begin{pmatrix}0&\zeta_{j}\\\
0&1\end{pmatrix}$

for $j=1,2,3$.

######  Lemma 34.

The fact that $\nabla_{0}$ has apparent singular points at $q_{1},q_{2},q_{3}$
imposes 3 linear conditions on $A_{3},A_{4},B_{3}$ in terms of spectral data,
and $((u_{j},v_{j}),\zeta_{j})$’s; we can uniquely determine
$A_{3},A_{4},B_{3}$ from these conditions if, and only if, we have

(5.8) $\det\begin{pmatrix}1&u_{1}&\zeta_{1}\\\ 1&u_{2}&\zeta_{2}\\\
1&u_{3}&\zeta_{3}\end{pmatrix}\not=0.$

###### Proof.

It is just Lemma 7 specified to the present elliptic case with 2 poles. We set

(5.9)
$C_{j}=\sum_{j^{\prime}\in\\{1,2,3\\}\setminus\\{j\\}}\frac{\zeta_{j^{\prime}}-\zeta_{j}}{2}\cdot\frac{v_{j}+v_{j^{\prime}}}{u_{j}-u_{j^{\prime}}}+\frac{A_{1}+A_{2}v_{j}-\zeta_{j}(B_{1}+B_{2}v_{j})-\zeta_{j}^{2}}{u_{j}-t}.$

We denote by $((a_{j})_{j},(b_{j})_{j},(c_{j})_{j})$ the $3\times 3$-matrix

$((a_{j})_{j},(b_{j})_{j},(c_{j})_{j})=\begin{pmatrix}a_{1}&b_{1}&c_{1}\\\
a_{2}&b_{2}&c_{2}\\\ a_{3}&b_{3}&c_{3}\end{pmatrix}.$

The condition where $q_{1},q_{2},q_{3}$ are apparent singularities means that

(5.10) $((1)_{j},(u_{j})_{j},(-\zeta_{j})_{j})\begin{pmatrix}A_{3}\\\ A_{4}\\\
B_{3}\end{pmatrix}=-\begin{pmatrix}C_{1}\\\ C_{2}\\\ C_{3}\end{pmatrix}.$

By Cramer’s rule, the parameters $A_{3},A_{4},B_{3}$ of the family of
connections $\nabla_{0}$ are uniquely determined

$\displaystyle
A_{3}=-\frac{\det((C_{j})_{j},(u_{j})_{j},(\zeta_{j})_{j})}{\det(((1)_{j},(u_{j})_{j},(\zeta_{j})_{j}))}$
$\displaystyle
A_{4}=-\frac{\det((1)_{j},(C_{j})_{j},(\zeta_{j})_{j})}{\det((1)_{j},(u_{j})_{j},(\zeta_{j})_{j})}$
$\displaystyle
B_{3}=\frac{\det((1)_{j},(u_{j})_{j},(C_{j})_{j})}{\det((1)_{j},(u_{j})_{j},(\zeta_{j})_{j})},$

if and only if (5.8). ∎

######  Lemma 35.

We have:

$\det\begin{pmatrix}1&u_{1}&\zeta_{1}\\\ 1&u_{2}&\zeta_{2}\\\
1&u_{3}&\zeta_{3}\end{pmatrix}=0$

if, and only if, $E$ is not stable.

###### Proof.

The vanishing of the determinant gives that $\zeta_{j}=\sigma(q_{j})$ for a
global section $\sigma\in H^{0}(C,\Omega_{C}^{1}(D))$. In other words, the
quasi-parabolic structure on $E_{0}$ given over each $q_{j}$ by the
eigenvectors corresponding to eigenvalue $1$ lie on a subbundle
$(\Omega^{1}_{C}(D))^{-1}\subset E_{0}$. After elementary transformations at
each $q_{j}$, we get $L\subset E$ with $\deg(L)=1$ (in fact $L=\det(E)$). ∎

###  5.2. Definition of a rank 2 vector bundle $E$

We set

$\tilde{U}_{0}:=U_{0}\setminus\\{q_{1},q_{2},q_{3}\\}\quad\text{and}\quad\tilde{U}_{\infty}:=U_{\infty}\setminus\\{q_{1},q_{2},q_{3}\\}.$

We take an analytic open subsets $\tilde{U}_{q_{j}}$ ($j=1,2,3$) of $C$ such
that $q_{j}\in\tilde{U}_{q_{j}}$ and $\tilde{U}_{q_{j}}$ are small enough. In
particular, $(u_{j},-v_{j})\not\in\tilde{U}_{q_{j}}$. On $\tilde{U}_{q_{j}}$,
the apparent singular point $q_{j}$ is defined by $x_{1}-u_{j}=0$. We have an
open covering $(\tilde{U}_{k})_{k\in\\{0,1,q_{1},q_{2},q_{3}\\}}$ of $C$. We
define transition functions $B_{k_{1}k_{2}}$
($k_{1},k_{2}\in\\{0,1,q_{1},q_{2},q_{3}\\}$) as follows:

$B_{0q_{j}}:=\begin{pmatrix}1&\frac{\zeta_{j}}{x_{1}-u_{j}}\\\
0&\frac{1}{x_{1}-u_{j}}\end{pmatrix}\colon\mathcal{O}^{\oplus
2}_{\tilde{U}_{q_{j}}}|_{\tilde{U}_{0}\cap\tilde{U}_{q_{j}}}\xrightarrow{\
\sim\ }\mathcal{O}^{\oplus
2}_{\tilde{U}_{0}}|_{\tilde{U}_{0}\cap\tilde{U}_{q_{j}}};$
$B_{0\infty}:=\begin{pmatrix}1&0\\\
0&-x_{2}\end{pmatrix}\colon\mathcal{O}^{\oplus
2}_{\tilde{U}_{\infty}}|_{\tilde{U}_{0}\cap\tilde{U}_{\infty}}\xrightarrow{\
\sim\ }\mathcal{O}^{\oplus
2}_{\tilde{U}_{0}}|_{\tilde{U}_{0}\cap\tilde{U}_{\infty}}.$

Then we have a vector bundle

$E=\left((\tilde{U}_{k})_{k\in\\{0,1,q_{1},q_{2},q_{3}\\}},\
(B_{k_{1}k_{2}})_{k_{1},k_{2}\in\\{0,1,q_{1},q_{2},q_{3}\\}}\right),$

where $E$ is trivial on each $\tilde{U}_{k}$ and the transition function from
$\tilde{U}_{k_{2}}$ to $\tilde{U}_{k_{1}}$ is $B_{k_{1}k_{2}}$.

###  5.3. Definition of a connection $\nabla$ on $E$

We define matrices $A_{0},A_{q_{j}},A_{\infty}$ as follows:

$\displaystyle A_{0}:=\begin{pmatrix}0&\omega_{12}\\\
\omega_{21}&\omega_{22}\end{pmatrix},$ $\displaystyle
A_{\infty}:=\begin{pmatrix}0&-x_{2}\omega_{12}\\\
-\frac{\omega_{21}}{x_{2}}&\omega_{22}+\frac{\operatorname{d}\\!x_{2}}{x_{2}}\end{pmatrix},$
$\displaystyle
A_{q_{j}}:=\begin{pmatrix}\omega^{(j)}_{11}&\frac{\omega^{(j)}_{12}}{x_{1}-u_{j}}\\\
(x_{1}-u_{j})\omega_{21}&\omega^{(j)}_{22}\end{pmatrix}.$

The 1-form $\omega_{12}$, $\omega_{21}$, and $\omega_{22}$ are defined in
(5.4). The 1-form $\omega_{12}^{(j)}$, $\omega_{21}^{(j)}$, and
$\omega_{22}^{(j)}$ are defined as follows:

$\displaystyle\omega^{(j)}_{11}$
$\displaystyle=-\frac{\zeta_{j}}{x_{1}-t}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}},$
$\displaystyle\omega^{(j)}_{12}$
$\displaystyle=\sum_{j^{\prime}\in\\{1,2,3\\}\setminus\\{j\\}}\frac{\zeta_{j^{\prime}}-\zeta_{j}}{2}\cdot\frac{y_{1}+v_{j^{\prime}}}{x_{1}-u_{j^{\prime}}}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}}$
$\displaystyle\qquad+\left(\frac{A_{1}+A_{2}y_{1}-\zeta_{j}(B_{1}+B_{2}y_{1})-\zeta_{j}^{2}}{x_{1}-t}+A_{3}+A_{4}x_{1}-\zeta_{j}B_{3}\right)\frac{\operatorname{d}\\!x_{1}}{y_{1}},$
$\displaystyle\omega^{(j)}_{22}$
$\displaystyle=\frac{1}{2}\cdot\frac{-y_{1}+v_{j}}{x_{1}-u_{j}}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}}+\sum_{j^{\prime}\in\\{1,2,3\\}\setminus\\{j\\}}\frac{1}{2}\cdot\frac{y_{1}+v_{j^{\prime}}}{x_{1}-u_{j^{\prime}}}\cdot\frac{\operatorname{d}\\!x_{1}}{y_{1}}$
$\displaystyle\qquad+\left(\frac{B_{1}+B_{2}y_{1}}{x_{1}-t}+B_{3}+\frac{\zeta_{j}}{x_{1}-t}\right)\frac{\operatorname{d}\\!x_{1}}{y_{1}}.$

######  Proposition 36.

  * •

The $(1,2)$-entry of $A_{q_{j}}$ is a section of
$\Omega_{C}^{1}(D)|_{\tilde{U}_{q_{j}}}$ for each $j=1,2,3$.

  * •

We define a local connection on each $\tilde{U}_{k}$
$(k\in\\{0,1,q_{1},q_{2},q_{3}\\})$ by

$\begin{cases}\operatorname{d}+A_{0}\colon\mathcal{O}_{\tilde{U}_{0}}^{\oplus
2}\longrightarrow\mathcal{O}_{\tilde{U}_{0}}^{\oplus
2}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{0}}&\text{on $\tilde{U}_{0}$}\\\
\operatorname{d}+A_{q_{j}}\colon\mathcal{O}_{\tilde{U}_{q_{j}}}^{\oplus
2}\longrightarrow\mathcal{O}_{\tilde{U}_{q_{j}}}^{\oplus
2}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{q_{j}}}&\text{on
$\tilde{U}_{q_{j}}$}\\\
\operatorname{d}+A_{\infty}\colon\mathcal{O}_{\tilde{U}_{\infty}}^{\oplus
2}\longrightarrow\mathcal{O}_{\tilde{U}_{\infty}}^{\oplus
2}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{\infty}}&\text{on
$\tilde{U}_{\infty}$}.\end{cases}$

Then we can glue these local connections. So we have a global connection
$\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D)$ on $E$.

###### Proof.

Since $A_{3},A_{4},B_{3}$ are determined so that these parameters satisfy the
condition (5.10), we have

$\omega_{12}^{(j)}|_{q_{j}}=\left(C_{j}+A_{3}+A_{4}u_{j}-\zeta_{j}B_{3}\right)\frac{\operatorname{d}\\!x_{1}|_{q_{j}}}{v_{j}}=0.$

Here, $C_{j}$ is in (5.9). So $\frac{\omega^{(j)}_{12}}{x_{1}-u_{j}}$ has no
pole at $q_{j}$ for each $j=1,2,3$. Since we have

$B_{k_{1}k_{2}}^{-1}A_{k_{1}}B_{k_{1}k_{2}}+B_{k_{1}k_{2}}^{-1}\operatorname{d}\\!B_{k_{1}k_{2}}=A_{k_{2}}$

for each $k_{1},k_{2}\in\\{0,\infty,q_{1},q_{2},q_{3}\\}$, the connection
$\nabla$ acting on $E$ is defined globally. ∎

######  Remark 37.

By Definition 13 in Section 3.3, we have trivializations of $E$. On
$C\setminus\\{t_{1},t_{2}\\}$, the trivializations in Definition 13 coincide
with the trivializations described in the present section. We have defined the
trivialization in Definition 13 at $t_{i}$ $(i=1,2)$ so that the residue
matrix (respectively, the polar part in the reduced case) is a diagonal
matrix. On the other hand, by the trivializations described in the present
section, the residue matrix at $t_{i}$ $(i=1,2)$ (respectively, the polar
part) is not a diagonal matrix. The reason why the residue matrix at $t_{i}$
$(i=1,2)$ is a diagonal matrix is that the corresponding description of the
variation (3.11) satisfies the compatibility conditions of the quasi-parabolic
structure in $\mathcal{F}^{0}$ and $\mathcal{F}^{1}$ of (3.1). On the other
hand, now we are interested in behavior of the connection $\nabla$ around
$q_{j}$ $(j=1,2,3)$. So now we do not consider the diagonalization of the
residue matrices at $t_{i}$ $(i=1,2)$ (respectively, of the polar part when
$D$ is reduced).

###  5.4. Canonical coordinates

We will calculate the canonical coordinates introduced in Section 3.5. For the
transition functions $B_{k_{1}k_{2}}$
($k_{1},k_{2}\in\\{0,1,q_{1},q_{2},q_{3}\\}$) of $E$, we have transition
functions of $\det(E)$ as follows:

$\det(B_{0q_{j}})=\frac{1}{x_{1}-u_{j}}\colon\mathcal{O}_{\tilde{U}_{q_{j}}}|_{\tilde{U}_{0}\cap\tilde{U}_{q_{j}}}\xrightarrow{\
\sim\ }\mathcal{O}_{\tilde{U}_{0}}|_{\tilde{U}_{0}\cap\tilde{U}_{q_{j}}};$
$\det(B_{0\infty})=-x_{2}\colon\mathcal{O}_{\tilde{U}_{\infty}}|_{\tilde{U}_{0}\cap\tilde{U}_{\infty}}\xrightarrow{\
\sim\ }\mathcal{O}_{\tilde{U}_{0}}|_{\tilde{U}_{0}\cap\tilde{U}_{\infty}}.$

So we have a cocycle
$(\det(B_{k_{1}k_{2}}))_{k_{1},k_{2}\in\\{0,1,q_{1},q_{2},q_{3}\\}}$, which
gives a class of $H^{1}(C,\mathcal{O}_{C}^{*})$. We have

$\operatorname{d}\log(\det(B_{0q_{j}}))=-\frac{\operatorname{d}\\!x_{1}}{x_{1}-u_{j}}\quad\text{and}\quad\operatorname{d}\log(\det(B_{0\infty}))=\frac{\operatorname{d}\\!x_{2}}{x_{2}},$

and these 1-forms give a class of $H^{1}(C,\Omega^{1}_{C})$. We denote by
$c_{1}$ and $\boldsymbol{\Omega}(D,c_{1})$ the class of
$H^{1}(C,\Omega^{1}_{C})$ and the total space of the twisted cotangent bundle
corresponding to $c_{1}$, respectively. We have the following description of
$\mathrm{tr}(\nabla)$:

$\mathrm{tr}(\nabla)=\begin{cases}\operatorname{d}+\omega_{22}\colon\mathcal{O}_{\tilde{U}_{0}}\longrightarrow\mathcal{O}_{\tilde{U}_{0}}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{0}}&\text{on
$\tilde{U}_{0}$}\\\
\operatorname{d}+\omega_{11}^{(j)}+\omega_{22}^{(j)}\colon\mathcal{O}_{\tilde{U}_{q_{j}}}\longrightarrow\mathcal{O}_{\tilde{U}_{q_{j}}}^{\oplus
2}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{q_{j}}}&\text{on
$\tilde{U}_{q_{j}}$}\\\
\operatorname{d}+\omega_{22}+\frac{\operatorname{d}\\!x_{2}}{x_{2}}\colon\mathcal{O}_{\tilde{U}_{\infty}}\longrightarrow\mathcal{O}_{\tilde{U}_{\infty}}^{\oplus
2}\otimes\Omega_{C}^{1}(D)|_{\tilde{U}_{\infty}}&\text{on
$\tilde{U}_{\infty}$}.\end{cases}$

Notice that we have

$\omega_{11}^{(j)}+\omega_{22}^{(j)}=\omega_{22}+\operatorname{d}\log(\det(B_{0q_{j}})),\text{
and}$
$\omega_{22}+\frac{\operatorname{d}\\!x_{2}}{x_{2}}=\omega_{22}+\operatorname{d}\log(\det(B_{0\infty})).$

So these connection matrices of $\mathrm{tr}(\nabla)$ give an explicit global
section of $\boldsymbol{\Omega}(D,c_{1})\rightarrow C$. We consider a section
of
$\boldsymbol{\Omega}(D,c_{1})|_{\tilde{U}_{q_{j}}}\rightarrow\tilde{U}_{q_{j}}$

$\frac{\zeta_{j}\operatorname{d}\\!x_{1}}{(x_{1}-t)y_{1}}+\omega_{11}^{(j)}+\omega_{22}^{(j)}.$

For this section on $\tilde{U}_{q_{j}}$, we define $p_{j}$ ($j=1,2,3$) by

$p_{j}=\mathrm{res}_{q_{j}}\left(\frac{\zeta_{j}}{x_{1}-u_{j}}\cdot\frac{\operatorname{d}\\!x_{1}}{(x_{1}-t)y_{1}}\right)+\mathrm{res}_{q_{j}}\left(\frac{\omega_{11}^{(j)}+\omega_{22}^{(j)}}{x_{1}-u_{j}}\right).$

Then we have a map

$(E,\nabla)\longmapsto(u_{1},u_{2},u_{3},\zeta_{1},\zeta_{2},\zeta_{3})\longmapsto(u_{1},u_{2},u_{3},p_{1},p_{2},p_{3}),$

where

$\displaystyle p_{j}$
$\displaystyle=\frac{\zeta_{j}}{(u_{j}-t)v_{j}}-\frac{K^{\prime}(u_{j})}{4v_{j}^{2}}+\sum_{j^{\prime}\in\\{1,2,3\\}\setminus\\{j\\}}\frac{1}{2}\cdot\frac{v_{j}+v_{j^{\prime}}}{u_{j}-u_{j^{\prime}}}\cdot\frac{1}{v_{j}}$
$\displaystyle\qquad+\left(\frac{B_{1}+B_{2}v_{j}}{u_{j}-t}+B_{3}\right)\frac{1}{v_{j}}$

Here we set $K(x_{1}):=x_{1}(x_{1}-1)(x_{1}-\lambda)$. Notice that $B_{1}$ and
$B_{2}$ are determined by Lemma 32 and $B_{3}$ is determined by Lemma 34.
Notice that $B_{3}$ depends on $\zeta_{1},\zeta_{2}$ and $\zeta_{3}$. The
symplectic structure is
$\sum_{j=1}^{3}\operatorname{d}\\!p_{j}\wedge\operatorname{d}\\!u_{j}$ by
Theorem 20.

##  6\. Canonical coordinates revised and another proof for birationality

In this section, we will give another proof of Proposition 17. For simplicity,
we will consider the cases where $D$ is a reduced effective divisor. Let
$(E,\nabla)\in M_{X}^{0}$ be a connection on a fixed irregular curve
$X=(C,D,\\{z_{i}\\}_{i\in I},\\{\theta_{i}\\}_{i\in I},\theta_{res})$ with
genericity conditions as before.

We set $D=t_{1}+\cdots+t_{n}$ and the connection is given by

$\nabla\colon E\longrightarrow E\otimes\Omega^{1}_{C}(D).$

In this section, we assume that $g=g(C)\geq 1$ and $n\geq 1$ as in the
previous sections. Moreover if $g=g(C)=1$, we assume that $n\geq 2$.

Note that we have the unique extension

(6.1) $0\longrightarrow{\mathcal{O}}_{C}\longrightarrow E\longrightarrow
L_{0}\longrightarrow 0$

with $L_{0}=\det(E)$. Moreover for $(E,\nabla)\in M^{0}_{X}$ we have $\deg
L_{0}=2g-1$ and $\dim_{\mathbb{C}}H^{0}(C,E)=1$. Then we can define apparent
singularities $q_{1},\ldots,q_{N}\in C$ where $N=4g-3+n$. Since $\deg
D=2g-2+n\geq 1$ and $\deg L_{0}=2g-1\geq 1$, we see that
$\dim_{\mathbb{C}}H^{0}(C,\Omega^{1}_{C}(D))=g-1+n\geq 2$. We can choose
$\gamma\in H^{0}(C,\Omega^{1}_{C}(D))$ and $s\in H^{0}(C,L_{0})$ whose zeros
are given by

$\\{\gamma=0\\}=\\{c_{1},\ldots,c_{2g-2+n}\\}\quad\text{and}\quad\\{s=0\\}=\\{u_{1},\ldots,u_{2g-1}\\}.$

We assume the following genericity conditions:

  1. (1)

$u_{i_{1}}\not=u_{i_{2}}$ (for $i_{1}\not=i_{2}$), and
$c_{k_{1}}\not=c_{k_{2}}$ (for $k_{1}\neq k_{2}$);

  2. (2)

$\\{u_{1},\ldots,u_{2g-1}\\}\cap\\{c_{1},\ldots,c_{2g-2+n}\\}=\emptyset$;

  3. (3)

$\\{q_{1},\ldots,q_{N}\\}\cap\\{u_{1},\ldots,u_{2g-1},c_{1},\ldots,c_{2g-2+n}\\}=\emptyset.$

Set

$U_{0}=C\setminus\\{u_{1},\ldots,u_{2g-1},c_{1},\ldots,c_{2g-2+n}\\}.$

Moreover we take small an analytic neighborhood $U_{i}$ of $u_{i}$ for $1\leq
i\leq 2g-1$ and $U_{2g-1+k}$ of $c_{k}$ for $1\leq k\leq 2g-2+n$. For
$i=1,\ldots,4g-3+n$, we can identify $U_{i}$ with a unit disc
$\Delta=\\{z\in\mathbb{C}\mid|z|<1\\}$ with the origin corresponding to
$u_{i}$ ($1\leq i\leq 2g-1$) and $c_{i-2g+1}$ ($2g\leq i\leq 4g-3+n$). We can
assume that $U_{i_{1}}\cap U_{i_{2}}=\emptyset$ for $i_{1}\not=i_{2}$,
$i_{1},i_{2}\geq 1$. Note that since $U_{0}$ is an affine variety and
$U_{0}\cap U_{i}\cong\Delta\setminus\\{0\\}$ for $i=1,\ldots,4g-3+n$, the
covering $C=U_{0}\cup U_{1}\cup\cdots\cup U_{4g-3+n}$ gives a Stein covering
of $C$. For $0\leq i\leq 4g-3+n$, we have nonzero sections
$\boldsymbol{e}_{1}^{(i)}\in\mathcal{O}_{U_{i}},\boldsymbol{e}_{2}^{(i)}\in(L_{0})_{|U_{i}}$
giving trivializations of $E$ on $U_{i}$ respectively:

$E_{|U_{i}}\simeq{\mathcal{O}}_{|U_{i}}\boldsymbol{e}_{1}^{(i)}\oplus{\mathcal{O}}_{|U_{i}}\boldsymbol{e}_{2}^{(i)}.$

Moreover we have a transition matrix $H_{0i}$ on $U_{0}\cap U_{i}$ of the form

(6.2) $H_{0i}=\begin{pmatrix}1&h_{0i}\\\ 0&g_{0i}\end{pmatrix}$

satisfying

(6.3)
$(\boldsymbol{e}_{1}^{(i)},\boldsymbol{e}_{2}^{(i)})=(\boldsymbol{e}_{1}^{(0)},\boldsymbol{e}_{2}^{(0)})H_{0i}=(\boldsymbol{e}_{1}^{(0)},h_{0i}\boldsymbol{e}_{1}^{(0)}+g_{0i}\boldsymbol{e}_{2}^{(0)}).$

Here $\\{h_{0i}\\}_{i}\in\operatorname{Ext}^{1}(L_{0},{\mathcal{O}}_{C})\cong
H^{1}(C,L_{0}^{-1})$ corresponds to the extension class of (6.1) and
$\\{g_{0i}\\}_{i}\in H^{1}(C,{\mathcal{O}}_{C}^{*})$ gives the transition
function of $L_{0}=\det(E)$. With these trivializations we have connection
matrices $A^{(i)}$:

(6.4)
$\nabla(\boldsymbol{e}_{1}^{(i)},\boldsymbol{e}_{2}^{(i)})=(\boldsymbol{e}_{1}^{(i)},\boldsymbol{e}_{2}^{(i)})A^{(i)}$

of the form

(6.5) $A^{(i)}=\begin{pmatrix}a_{11}^{(i)}\gamma_{i}&a_{12}^{(i)}\gamma_{i}\\\
a_{21}^{(i)}\gamma_{i}&a_{22}^{(i)}\gamma_{i}\end{pmatrix}.$

Here $a_{kl}^{(i)}\in\Gamma(U_{i},{\mathcal{O}}_{U_{i}})$ and
$\gamma_{i}\in\Gamma(U_{i},\Omega^{1}_{U_{i}}(D))$. We set
$\gamma_{0}=\gamma_{|U_{0}}$ as above.

From (6.3) and (6.4), we can verify the following

######  Lemma 38.

For $1\leq i\leq 4g-3+n$, on $U_{0}\cap U_{i}$, we gave

(6.6) $A^{(i)}=H_{0i}^{-1}A^{(0)}H_{0i}+H_{0i}^{-1}\operatorname{d}\\!H_{0i}.$

Specifically, we have the following identities:

(6.7)
$a_{21}^{(i)}\gamma_{i}=a_{21}^{(0)}\gamma_{0}g_{0i}^{-1};\quad\text{and}$
(6.8)
$a_{22}^{(i)}\gamma_{i}=a_{22}^{(0)}\gamma_{0}+a_{21}^{(0)}\gamma_{0}h_{0i}g_{0i}^{-1}+\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}.$

The identity (6.7) shows that $a_{21}^{(i)}\gamma_{i}$ defines a section of
$H^{0}(C,\Omega_{1}(D)\otimes L_{0})$ and the zeros of this section are
nothing but the apparent singularities $q_{1},\ldots,q_{N}$. Evaluating the
identity (6.8) at $q_{j}$ ($j=1,\ldots,N$), we then have

(6.9)
$(a_{22}^{(i)}\gamma_{i})_{q_{j}}=(a_{22}^{(0)}\gamma_{0})_{q_{j}}+\left(\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}\right)_{q_{j}}$

Noting that the cohomology class of the cocycle
$\left\\{\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}\right\\}_{i}$ corresponds to
$c_{d}=c_{1}(L_{0})$, from (6.9), we have the following

######  Proposition 39.

For each $0\leq j\leq N$, the data $(E,\nabla)\in M^{0}_{X}$ defines $N$
points $(q_{j},\tilde{p}_{j})$ on the total space of
$\boldsymbol{\Omega}(D,c_{d})$ by the formula

(6.10)
$\tilde{p}_{j}=(a_{22}^{(0)}\gamma_{0})_{q_{j}}\in\Omega^{1}_{C}(D,c_{d})_{|q_{j}}$

The above definition of $\tilde{p}_{j}$ does not depend on the choice of the
sections $s\in H^{0}(C,L_{0})$ and $\gamma\in H^{0}(C,\Omega^{1}_{C}(D))$ and
defines the same map as in Definition 16:

(6.11) $f_{\mathrm{App}}\colon
M^{0}_{X}\longrightarrow\operatorname{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d})).$

Now we consider $q_{j}$ as a local coordinate near $q_{j}$ and we write
$\gamma=c(q_{j})\operatorname{d}\\!q_{j}$ for some local holomorphic function
$c(q_{j})$. Then we have

$\tilde{p}_{j}=p_{j}\operatorname{d}\\!q_{j}$

with

$p_{j}=a_{22}^{(0)}(q_{j})c(q_{j}).$

As we have proved in Theorem 20, the map $f_{\mathrm{App}}$ is symplectic.

###  6.1. From a connection to a Higgs field

Keeping the notation, let us consider the section $s\in H^{0}(C,L_{0})$ as
before, and set $s^{(0)}=s$. Take trivialization of $L_{0|U_{i}}$ over $U_{i}$
we have a holomorphic function $s^{(i)}\in\Gamma(U_{i},{\mathcal{O}}_{U_{i}})$
such that

$s^{(0)}=g_{0i}s^{(i)}.$

Note that $s^{(i)}$ has zeros at $u_{i}\in U_{i}$ for $1\leq i\leq 2g-1$. Set
$D(s)=u_{1}+\cdots+u_{2g-1}$. We can show the following

######  Lemma 40.

There exists a connection

$\nabla_{1}\colon E\longrightarrow E\otimes\Omega_{C}^{1}(D(s))$

such that for each $0\leq i\leq N=4g-3+n$, on $U_{i}$ it has the form

$\nabla_{1}^{(i)}=\operatorname{d}+\,S^{(i)}=\operatorname{d}+\begin{pmatrix}0&-\frac{\beta_{i}}{s^{(i)}}\\\
0&-\frac{\operatorname{d}\\!s^{(i)}}{s^{(i)}}\end{pmatrix}$

with respect to the trivialization
$(\boldsymbol{e}_{1}^{(i)},\boldsymbol{e}_{2}^{(i)})$. Here
$\beta_{i}\in\Gamma(U_{i},\Omega^{1}_{U_{i}})$.

###### Proof.

Since $s^{(0)}=g_{0i}s^{(i)}$, one has

$\frac{\operatorname{d}\\!s^{(0)}}{s^{(0)}}=\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}+\frac{\operatorname{d}\\!s^{(i)}}{s^{(i)}}$

in $U_{0i}=U_{0}\cap U_{i}$. The compatibility condition for connection
matrices $S^{(i)}$ is

(6.12)
$S^{(i)}=H_{0i}^{-1}S^{(0)}H_{0i}+H_{0i}^{-1}\operatorname{d}\\!H_{0i}.$

The right hand side of (6.12) is

(6.13)
$\begin{pmatrix}0&-g_{0i}\frac{\beta_{0}}{s^{(0)}}+h_{0i}\left(\frac{\operatorname{d}\\!s^{(0)}}{s^{(0)}}-\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}\right)+\operatorname{d}\\!h_{0i}\\\
0&-\frac{\operatorname{d}\\!s^{(0)}}{s^{(0)}}+\frac{\operatorname{d}\\!g_{0i}}{g_{0i}}\end{pmatrix}$

Since $\\{h_{0i}\\}_{i}$ is a class in $H^{1}(C,L_{0}^{-1})$ and $s\in
H^{0}(C,L_{0})$, the class $\\{s^{(i)}h_{0i}\\}_{i}$ defines a class in
$H^{1}(C,{\mathcal{O}}_{C})$. Then, by the Hodge theory, the derivative
$\\{\operatorname{d}(s^{(i)}h_{0i})\\}_{i}\in H^{1}(C,\Omega^{1}_{C})$
vanishes, so there exist $\beta_{i}\in\Gamma(U_{i},\Omega^{1}_{U_{i}})$ such
that

$\operatorname{d}(s^{(i)}h_{0i})=\beta_{0}-\beta_{i}.$

Choose such $\beta_{i}$’s for the formula. Then we have

$\operatorname{d}\\!h_{0i}=-h_{0i}\frac{\operatorname{d}\\!s^{(i)}}{s^{(i)}}+g_{0i}\frac{\beta_{0}}{s^{(0)}}-\frac{\beta_{i}}{s^{(i)}}.$

Then the right hand side of (6.13) becomes

$\begin{pmatrix}0&-\frac{\beta_{i}}{s^{(i)}}\\\
0&-\frac{\operatorname{d}\\!s^{(i)}}{s^{(i)}}\end{pmatrix}$

as desired. ∎

For any $(E,\nabla)\in M^{0}_{X}$, the difference

$\nabla-\nabla_{1}\colon E\longrightarrow E\otimes\Omega^{1}_{C}(D+D(s))$

defines an ${\mathcal{O}}_{C}$-homomorphism, that is a rational Higgs fields
on $E$. We reprove Proposition 17.

######  Theorem 41.

For generic $(E,\nabla)\in M_{X}^{0}$, the point
$(q_{j},\tilde{p}_{j})_{j=1,\ldots,N}\in\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$
determines $(E,\nabla)$. So the map $f_{\mathrm{App}}$ is birational.

###### Proof.

Consider the Higgs field

$\Phi=\Phi_{\nabla}=\nabla-\nabla_{1}\colon E\longrightarrow
E\otimes\Omega^{1}_{C}(D+D(s))$

where $D=t_{1}+\cdots+t_{n}$ and $D(s)=u_{1}+\cdots+u_{2g-1}$ as in the
notation above. We assume that the set of apparent singularities
$q_{1},\ldots,q_{N}$ of $(E,\nabla)$ is disjoint from $D$ and $D(s)$. We will
consider the characteristic curve of $\Phi$. On $U_{i}$, we have

$\Phi_{i}=A^{(i)}-S^{(i)}=\begin{pmatrix}\tilde{a}_{11}&\tilde{a}_{12}+\frac{\beta_{i}}{s^{(i)}}\\\
\tilde{a}_{21}&\tilde{a}_{22}+\frac{\operatorname{d}\\!s^{(i)}}{s^{(i)}}\end{pmatrix}.$

The characteristic curve $C_{s}$ can be defined in the total space of
$\boldsymbol{\Omega}(D+D(s))$ of the line bundle $\Omega^{1}_{C}(D+D(s))$ by

$C_{s}:x^{2}-b_{1}x-b_{2}=0$

with $b_{i}\in H^{0}(C,(\Omega^{1}_{C}(D+D(s)))^{\otimes i})$, and $x$ the
canonical section. The dimension of the family of spectral curves is thus
given by

$\displaystyle\dim H^{0}(C,\Omega^{1}_{C}(D+D(s)))+\dim
H^{0}(C,(\Omega^{1}_{C}(D+D(s)))^{\otimes 2})$ $\displaystyle=$ $\displaystyle
N+1-g+2N+1-g$ $\displaystyle=$ $\displaystyle 3N+2-2g=3(4g-3+n)+2-2g$
$\displaystyle=$ $\displaystyle 10g-7+3n.$

Then $\Phi$ is constrained by the following conditions.

  1. (1)

At $t_{i},i=1,\ldots,n$, $\Phi$ has eigenvalues fixed by data $X$. These
impose $2n-1$ conditions because of the Fuchs relation.

  2. (2)

At $u_{k}$, $k=1,\ldots,2g-1$, take a local coordinate $z_{k}$ such that
$z_{k}(u_{k})=0$. Then $\Phi$ has the following form near $z_{k}=0$

$\Phi=\begin{pmatrix}0&\frac{\beta_{i}(0)}{z_{k}}\\\
0&\frac{\operatorname{d}\\!z_{k}}{z_{k}}\end{pmatrix}+\mbox{holomorphic}.$

Then eigenvalues of the residue matrix are $0,1$ and the $\beta_{i}(0)$ gives
a restriction on $C_{s}$. Then totally we have $3\times(2g-1)$ conditions.

  3. (3)

At $q_{j},j=1,\ldots,N$, the points
$\tilde{a}_{22}(q_{j})+\frac{ds^{(i)}}{s^{(i)}}(q_{j})=\tilde{p}_{j}+c_{j}\in\boldsymbol{\Omega}(D+D(s))$
lie on the characteristic curve $C_{s}$. These give $N=4g-3+n$ conditions.

For generic choice of $q_{1},\cdots,q_{N}$ and $s\in H^{0}(C,L_{0})$, we can
see using the method of Lemma 7 and Proposition 17 that these conditions are
independent, so we obtain a total of

$2n-1+3(2g-1)+(4g-3+n)=10g-7+3n$

conditions, so these determine the spectral curve $C_{s}$. Now the divisor
$\mu=\sum_{j=1}^{N}(\tilde{p}_{j}-c_{j})+\sum_{k=1}^{2g-1}(1_{k})$ determines
the rank 1 sheaf ${\mathcal{O}}_{C_{s}}(\mu)$ where $(1_{k})\in C_{s}$ denotes
the point over $u_{k}$ corresponding to the eigenvalue $-1$ of the residue of
$\Phi$ at $u_{k}$. Then $(\pi\colon C_{s}\longrightarrow
C,{\mathcal{O}}_{C_{s}}(\mu))$ determines $(E,\Phi)$ uniquely by [3,
Proposition 3.6]. Hence $E$ and $\nabla=\Phi+\nabla_{1}$ is determined
uniquely. ∎

## References

  * [1] D. Arinkin, S. Lysenko. Isomorphisms between moduli spaces of SL(2)-bundles with connections on $\boldsymbol{P}^{1}\setminus\\{x_{1},\ldots,x_{4}\\}$. Math. Res. Lett., 4 (2-3): 181–190, 1997.
  * [2] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
  * [3] A. Beauville, M.S. Narasimhan, S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.
  * [4] I. Biswas, On the moduli space of holomorphic $G$-connections on a compact Riemann surface, Euro. Jour. Math. 6 (2020), 321–335.
  * [5] I. Biswas, V. Heu, J. Hurtubise, Isomonodromic deformations of logarithmic connections and stability, Math. Ann., 366 (2016), 121–140.
  * [6] I. Biswas, V. Heu, J. Hurtubise, Isomonodromic deformations of irregular connections and stability of bundles Comm. Anal. Geom. 29 (2021), no. 1, 1–18.
  * [7] P. Boalch, Symplectic manifolds and isomonodromic deformations. Adv. Math. 163 (2), (2001), 137–205
  * [8] F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. École Norm. Sup. 28 (1995), 391–433.
  * [9] K. Diarra, F. Loray, Normal forms for rank two linear irregular differential equations and moduli spaces. Period. Math. Hungar. 84 (2022) 303–320.
  * [10] B. Dubrovin, M. Mazzocco, Canonical structure and symmetries of the Schlesinger equations. Comm. Math. Phys. 271 (2007), no.2, 289–373.
  * [11] O. Dumitrescu, M. Mulase, Quantum curves for Hitchin fibrations and the Eynard–Orantin theory. Letters in Mathematical Physics 104 (2014) 635–671.
  * [12] T. Fassarella, F. Loray Flat parabolic vector bundles on elliptic curves. J. Reine Angew. Math. 761 (2020) 81–122.
  * [13] T. Fassarella, F. Loray, A. Muniz, Flat parabolic vector bundles on elliptic curves II. Math. Zeichr. 301 (2022) 4079–4118.
  * [14] R. Fedorov, Algebraic and Hamiltonian approaches to isoStokes deformations. Transform. Groups 11 (2006), no. 2, 137–160.
  * [15] I. Gaiur, M. Mazzocco, V. Rubtsov, Isomonodromic Deformations: Confluence, Reduction and Quantisation Commun. Math. Phys. 400, (2023), 1385–1461.
  * [16] A. Gorsky, N. Nekrasov, V. Rubtsov, Hilbert schemes, separated variables, and D-branes. Comm. Math. Phys. 222 (2001), no. 2, 299–318.
  * [17] J. Harnad, Dual Isomonodromic Deformations and Moment Maps to Loop Algebras. Commun. Math. Phys. 166, (1994), 337–365.
  * [18] K. Hiroe, D. Yamakawa, Moduli spaces of meromorphic connections and quiver varieties. Adv. Math. 266, (2014), 120–151.
  * [19] J. Hurtubise, Integrable systems and algebraic surfaces. Duke. Math. J. 83, 19–50 (1996).
  * [20] J. Hurtubise, On the geometry of isomonodromic deformations. J. Geom. Phys. 58 (2008), no.10, 1394–1406.
  * [21] M.-A. Inaba, Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence. J. Algebraic Geom. 22 (2013), no. 3, 407–480.
  * [22] M.-A. Inaba, K. Iwasaki, M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I. Publ. Res. Inst. Math. Sci. 42 (2006), no. 4, 987–1089.
  * [23] M.-A. Inaba, K. Iwasaki, M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II. Moduli spaces and arithmetic geometry, 387-432, Adv. Stud. Pure Math., 45, Math. Soc. Japan, Tokyo, 2006.
  * [24] M.-A. Inaba, M.-H. Saito, Moduli of unramified irregular singular parabolic connections on a smooth projective curve. Kyoto J. Math. 53 (2013) 433–482.
  * [25] P. Ivanics, A. Stipsicz, Sz. Szabó. Two-dimensional moduli spaces of rank 2 Higgs bundles over $\mathbb{C}P^{1}$ with one irregular singular point, Journal of Geometry and Physics 130 (2018) pp. 184–212.
  * [26] P. Ivanics, A. Stipsicz, Sz. Szabó. Hitchin fibrations on moduli of irregular Higgs bundles and motivic wall-crossing, Journal of Pure and Applied Algebra (9) 223 (2019) 3989–4064.
  * [27] P. Ivanics, A. Stipsicz, Sz. Szabó. Hitchin fibrations on two-dimensional moduli spaces of irregular Higgs bundles with one singular fiber, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) (85) 15 (2019) 28 pages.
  * [28] K. Iwasaki, Moduli and deformation for Fuchsian projective connections on a Riemann surface. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), no.3, 431–531.
  * [29] K. Iwasaki, Fuchsian moduli on a Riemann surface—its Poisson structure and Poincaré-Lefschetz duality. Pacific J. Math. 155 (1992), no.2, 319–340.
  * [30] M. Jimbo, T. Miwa, Y. Mori, M. Sato, Density Matrix of an Impenetrable Bose Gas and the Fifth Painlevé Equation. Phys. D 1(1), (1980), 80–158.
  * [31] S. Kawai, Isomonodromic deformation of Fuchsian projective connections on elliptic curves. Nagoya Math. J. 171 (2003), 127–161.
  * [32] H. Kawakami, A. Nakamura, H. Sakai, Degeneration scheme of $4$-dimensional Painlevé-type equations, MSJ Memoir, 37, (2018), 25–111.
  * [33] A. Komyo, Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections. J. Math. Soc. Japan 74 (2022), no. 2, 473–519.
  * [34] A. Komyo, Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities, to appear in Publications of the Research Institute for Mathematical Sciences (arXiv:2003.08045).
  * [35] A. Komyo, F. Loray, M.-H. Saito, Moduli space of irregular rank two parabolic bundles over the Riemann sphere and its compactification. Adv. Math. 410 (2022), Paper No. 108750, 76 pp.
  * [36] A. Komyo, M.-H. Saito, Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces. Kyoto J. Math. 59 (2019), no.3, 515–552.
  * [37] M. Kontsevich, A. Odesskii, Multiplication kernels. Letters in Mathematical Physics (2021) 111–152.
  * [38] M. Kontsevich, Y. Soibelman, Wall-crossing structures in Donaldson–Thomas invariants, integrable systems and Mirror Symmetry. In: Castano-Bernard, R., Catanese, F., Kontsevich, M., Pantev, T., Soibelman, Y., Zharkov, I. (eds) Homological Mirror Symmetry and Tropical Geometry. Lecture Notes of the Unione Matematica Italiana, vol 15 (2014). Springer
  * [39] I. Krichever, Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Mosc. Math. J. 2 (2002), no.4, 717–752, 806.
  * [40] M. Logares and J. Martens, Moduli of parabolic Higgs bundles and Atiyah algebroids, Jour. Reine Angew. Math. 649 (2010), 89–116.
  * [41] F. Loray, V. Ramírez A map between moduli spaces of connections. SIGMA 16 (2020) 125, 42 pages.
  * [42] F. Loray, M.-H. Saito, Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line. Internat. Math. Res. Notices (2015), no. 4, 995–1043.
  * [43] F. Loray, M.-H. Saito, C. Simpson. Foliations on the moduli space of rank two connections on the projective line minus four points. In Geometric and differential Galois theories, volume 27 of Sémin. Congr., pages 115–168. Soc. Math. France, Paris, 2012.
  * [44] T. Matsumoto Birational geometry of moduli spaces of rank 2 logarithmic connections, arXiv:2105.06892.
  * [45] V. B. Mehta, C. S. Seshadri. Moduli of vector bundles on curves with parabolic structures. Math. Ann., 248 (3): 205–239, 1980.
  * [46] S. Oblezin, Isomonodromic deformations of $sl(2)$ Fuchsian systems on the Riemann sphere. Mosc. Math. J. 5 (2005), no.2, 415–441.
  * [47] K. Okamoto, Isomonodromic deformation and Painlevé equations, and the Garnier system. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), no.3, 575–618.
  * [48] K. Okamoto, The Hamiltonian structure derived from the holonomic deformation of the linear ordinary differential equations on an elliptic curve, Sci. Papers College Arts Sci. Univ. Tokyo, 37 (1987), 1–11.
  * [49] K. Okamoto, On the holonomic deformation of linear ordinary differential equations on an elliptic curve, Kyushu J. Math., 49 (1995), 281–308.
  * [50] M.-H. Saito, S. Szabó, Apparent singularities and canonical coordinates for moduli of parabolic connections and parabolic Higgs bundles, I, in preparation.
  * [51] S. Szabó, The dimension of the space of Garnier equations with fixed locus of apparent singularities. Acta Sci. Math. (Szeged) 79 (2013), no.1-2, 107–128.
  * [52] S. Szabó, The birational geometry of unramified irregular Higgs bundles on curves. International Journal of Mathematics (6) 28 (2017).
  * [53] N. M. J. Woodhouse, Duality for the general isomonodromy problem. J. Geom. Phys. 57 (4), (2007), 1147–1170.
  * [54] K. Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves. Journal of Mathematics of Kyoto University 33, no. 2 (1993): 451–504.


aainstitutetext: Institute of Physics, Jagiellonian University, Łojasiewicza
11, 30-348 Kraków, Polandbbinstitutetext: Instituto de Física Teórica UAM-
CSIC, c/ Nicolás Cabrera 13-15, 28049, Madrid, Spainccinstitutetext: Center
for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, NY 11794-3800ddinstitutetext: Departamento de Física Teórica,
Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

#  Critical and near-critical relaxation of holographic superfluids

Mario Flory c,b,d Sebastian Grieninger b,d Sergio Morales-Tejera
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the relaxation of holographic superfluids after quenches, when
the end state is either tuned to be exactly at the critical point, or very
close to it. By solving the bulk equations of motion numerically, we
demonstrate that in the former case the system exhibits a power law falloff as
well as an emergent discrete scale invariance. The later case is in the regime
dominated by critical slowing down, and we show that there is an intermediate
time-range before the onset of late time exponential falloff, where the system
behaves similarly to the critical point with its power law falloff. We further
postulate a phenomenological Gross–Pitaevskii-like equation that is able to
make quantitative predictions for the behavior of the holographic superfluid
after near-critical quenches. Intriguingly, all parameters of our
phenomenological equation which describes the non-linear time evolution may be
fixed with information from the static equilibrium solutions and linear
response theory.

††preprint: IFT-UAM/CSIC-22-108

##  1 Introduction

So-called holographic superconductors have been a steady source of inspiration
and target of research for the holography community since their inception in
Hartnoll:2008vx ; Hartnoll:2008kx ; Herzog:2008he ; Herzog:2009xv ;
Gubser:2008px . Despite the established nomenclature, it is commonly
understood that these systems might more correctly be understood to be
holographic superfluids, because the gauge-symmetry present in the bulk should
translate into a global symmetry according to common AdS/CFT wisdom.

However, it has been demonstrated in Domenech:2010nf  that the Maxwell
equations on the boundary can be imposed, leading to a genuine holographic
superconductor. Another interesting perspective was offered in Maeda:2010br ,
where it was argued that we should understand large gauge transformations in
the bulk to give rise to transformations that can be interpreted as
”background-gauge” transformations in the boundary theory, affecting the
sources imposed there but not being tied to a dynamical photon. This
perspective will be especially influential for our work.

The goal of this work is to study how holographic superconductors relax to
their equilibrium state after a quench when the final state is close to the
critical point. Generically, for a near-critical end-state this relaxation
will of course be characterized at late times by an exponential falloff, where
the half-life time of said falloff diverges as the end-state is taken towards
the critical point. This is known as critical slowing down RevModPhys.49.435 ,
see Maeda:2008hn ; Maeda:2009wv ; Natsuume:2010vb ; Ewerz:2014tua  for works
in the holographic context.

But what happens when the end-state is tuned to lie exactly at the critical
point? In this case, the exponential falloff behavior is replaced by a power
law falloff PhysRevB.96.054303 ; PhysRevB.81.012303 . This makes sense
intuitively, because a power law falls off slower than any exponential, hence
this is a physically sensible result in the limit where the half-life time
diverges. From the holographic bulk perspective this may be seen as the system
trying to balance the condensate floating above the horizon exactly, hence the
slow decay. The condensate is affected by electrostatic forces driving it into
the bulk and gravitational forces trying to pull it into the horizon
Gubser:2008wv . In the following, we will study such critical and near-
critical quenches both from the bulk and the boundary perspective. Note,
however, that unlike in the discussion about the Kibble-Zurek mechanism
Chesler:2014gya ; Sonner:2014tca ; Bhaseen:2012gg ; delCampo:2021rak ;
Li:2021jqk  we are interested in quenches with initial state in the ordered
phase, i.e. the quenches we are studying do not cross the critical point at
any finite rate.

##  2 Bulk results

We consider the common bulk gravitational model of a superconductor in
AdS4/CFT3 Hartnoll:2008vx ; Hartnoll:2008kx ; Herzog:2008he

$S\,=\,S_{\text{grav}}+\frac{1}{2\kappa^{2}}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{-g}\left[-\frac{1}{4\,q^{2}}F_{\mu\nu}F^{\mu\nu}-|D\varphi|^{2}-m^{2}|\varphi|^{2}\right]+S_{\text{nf}}\,.$
(1)

The field content consists of the metric field $g_{\mu\nu}$, a $U(1)$ gauge
field $A_{\mu}$ ($F=dA$) and a complex scalar field $\varphi$ of mass
$m^{2}=-2$ charged under the gauge field with
$D_{\mu}\equiv\partial_{\mu}-i\,q\,A_{\mu}$ with charge $q=+1$. We will
neglect backreaction in this work, and hence fix the metric

$\displaystyle\mathrm{d}s^{2}=\frac{1}{u^{2}}\left[-f(u)\,\mathrm{d}t^{2}-2\,\mathrm{d}t\mathrm{d}u+\mathrm{d}x^{2}+\mathrm{d}y^{2}\right],$
(2)

with $f(u)=1-u^{3}$ where we already set $u_{h}=1$ and $L=1$. We measure all
physical quantities in terms of the fixed temperature $\bar{T}=4\pi T/3$ where
$T=|f^{\prime}(1)|/(4\pi)$. To keep the notation simple we assume that all
appearing quantities have already been divided by the appropriate power of
$\bar{T}$ and hence are dimensionless.111The dimensionless ratios are
$\mu/{\bar{T}},\,\rho/{\bar{T}}^{2},\,\langle\mathcal{O}\rangle/{\bar{T}}^{2},\,t\,\bar{T}.$
$S_{\text{nf}}$ is interpreted as an external source which we use to perform a
quench in the charge density. The equations of motion (eoms) are solved by a
fully pseudo-spectral code (in space and time) as previously employed in
Ammon:2016fru . See appendix A for more details on this setup (including
$S_{\text{nf}}$) and the numerical methods used to solve it.

The near boundary expansions of the bulk matter fields read

$\displaystyle\varphi$ $\displaystyle\sim\langle\mathcal{O}\rangle
u^{2}+\ldots,$ (3) $\displaystyle A_{t}$
$\displaystyle\sim\mathcal{A}_{t}-\rho u+\ldots,$ (4)

where we set the source of the scalar field to zero in order to engineer
spontaneous symmetry breaking and
$2\kappa^{2}\langle\mathcal{O}\rangle\equiv\Psi(t)=\phi(t)e^{i\psi(t)}$ is the
complex expectation value of the dual operator Arean:2021tks . At static
equilibrium we identify $\mathcal{A}_{t}=\mu$ where $\mu$ is the chemical
potential. The subleading component $\rho(t)$ is the charge density. There is
a second order phase transition to a phase with non-zero condensate at
$\rho=\rho_{c}\approx 4.06371$. We perform quenches of the system by giving
$\rho$ a step-function-like time dependence. See appendix A.2 for more
details.

Some of our numerical results are shown in figure 1. It is clear that we
observe late time power-law falloffs in $\phi(t)$, $|\dot{\psi}(t)|$, and
$|\mathcal{A}_{t}(t)-\rho_{c}|$ that are universal, i.e. independent on the
initial state or other details of the quench. At late times, where $\rho$ is
held constant in time, the bulk equations of motion are not explicitly
dependent on $t$, and so for any collective solution $y(t)$ of the bulk eoms,
$y(t+\delta t)$ will also be a solution for any $\delta t$. We hence make an
ansatz of the form

$\displaystyle\phi(t)$ $\displaystyle=A(t+\delta t)^{\alpha}$ (5)
$\displaystyle\dot{\psi}(t)-(\mathcal{A}_{t}(t)-\rho_{c})$
$\displaystyle=B(t+\delta t)^{\gamma}$ (6)

which we fit to our numerical curves at late times ($t\geq 5\times 10^{4}$).
We only consider the combination on the left hand side of equation (6) because
it is gauge invariant under background-gauge transformations Maeda:2010br .
With a high degree of consistency between the five different quenches plotted
in figure 1, we obtain

(9)

with only the value of $\delta t$ obtained from the fit varying significantly
from quench to quench.

Figure 1:  Numerical results for $|\mathcal{A}_{t}(t)-\rho_{c}|$ (solid
lines), $|\langle\mathcal{O}\rangle|\equiv\phi(t)$ (dashed lines), and
$|\dot{\psi}(t)|$ (dotted lines) for multiple exactly critical quenches
starting from $\rho_{\text{initial}}=\\{4.124,4.302,4.951,6.981,8.184\\}$
(purple, blue, green, orange, red). We clearly observe a universal power-law
like late time behavior with $\phi\propto 1/\sqrt{t}$ and
$|\mathcal{A}_{t}-\rho_{c}|,|\dot{\psi}|\propto 1/t$.

The observed behavior $|\dot{\psi}|\propto 1/t$ indicates
$\psi(t)\propto\log{t}$, i.e. oscillations of the real- and imaginary part of
$|\langle\mathcal{O}\rangle|$ that are periodic on a logarithmic time axis.
This signifies the presence of a discrete scale invariance, in contrast to the
continuous scale invariance inherent to ordinary power laws SORNETTE1998239 .
Among other contexts, such discrete scale invariance has also previously been
observed numerically after critical quenches in a holographic Kondo model in
Erdmenger:2016msd , as well as in the formation of black holes through the
collapse of charged scalar fields PhysRevD.51.4198 .

##  3 Boundary model

From a boundary point of view, it is of course known that the phenomenology of
superconductors is well-described by the (complex) Ginzburg–Landau equation,
while for superfluids with their global symmetry breaking the Gross–Pitaevskii
equation takes a similar role RevModPhys.74.99 ; tsuneto_1998 . Consequently,
there has been some recent activity in PhysRevLett.127.101601 ; Yan:2022jfc
trying to fit parameters of Gross–Pitaevskii equations in order to model
aspects of the non-equilibrium behavior of holographic superfluids.

Unlike the quenches that we study in this manuscript, PhysRevLett.127.101601 ;
Yan:2022jfc  investigated inhomogeneous setups where space-derivatives are
non-zero which increases the complexity of the problem significantly. In
addition, in our paper we explicitly aim to study behavior near or even
exactly at the critical point which means that our results should be ideally
suited for such a phenomenological description, since e.g. the Ginzburg-Landau
equation is usually seen as a series expansion around vanishing order-
parameter, where higher order terms in the free energy are dropped.

We now postulate the phenomenological equation

$\displaystyle\left[\partial_{t}-iC_{1}\left(\mathcal{A}_{t}(t)-\rho+C_{5}|\Psi(t)|^{2}\right)\right]\Psi(t)\equiv-(C_{2}+iC_{3})\left[|\Psi(t)|^{2}-C_{4}(\rho-\rho_{c})\right]\Psi(t),$
(10)

where again $\Psi=\phi e^{i\psi}$,
$\rho,\rho_{c},\psi,\mathcal{A}_{t},C_{i}\in\mathbb{R}$, $\phi>0$, and we
manifestly neglect any terms including spatial derivatives or higher orders of
$\Psi$ or $\rho-\rho_{c}$. The parameter $\rho$ is assumed to be constant in
time, and $C_{1}$ is merely the charge of the complex field which in our case
has the value $+1$.

The right hand side of the equation is similar to the variation of the free
energy that would appear in the Ginzburg-Landau equation, multiplied with a
complex prefactor which is inspired by the dissipative Gross-Pitaevski
equations used in PhysRevLett.127.101601 ; Yan:2022jfc , see also
RevModPhys.74.99 ; doi:10.1137/0142054 . The left hand side is essentially a
gauge-covariant time-derivative plus some extra terms whose relevance will
become clear shortly. The gauge covariance is necessary in order to respect
the background gauge-invariance (without dynamical photon) described in
Maeda:2010br  which arises as a consequence of large gauge transformations
which do not fall off towards the boundary and hence change the boundary
values of the bulk fields such as $\mathcal{A}_{t}$.

Because (10) contains complex factors, we can split it into a real and
imaginary part when trying to solve it (after dividing by $e^{i\psi}$ on both
sides). Besides the trivial case $\phi=0$ which we will ignore, the real part
of this equation only depends on $\phi(t)$ and has the non-trivial exact
solution

$\displaystyle\phi(t)=\sqrt{\frac{C_{4}(\rho-\rho_{c})}{1-\left(1-\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}\right)e^{-2C_{2}C_{4}t(\rho-\rho_{c})}}}$
(11)

with $\phi(0)=\phi_{0}$. This solution can then be plugged into the imaginary
part of (10) in order to obtain an algebraic equation that can be solved for
the gauge-invariant expression $\dot{\psi}-C_{1}\mathcal{A}_{t}$ (see appendix
B). Obtaining a unique solution for both $\psi(t)$ and $\mathcal{A}_{t}(t)$
requires some kind of gauge fixing condition.

As a simple consistency check, looking for non-trivial ($\phi\neq 0$) static
solutions yields

$\displaystyle\phi$ $\displaystyle=\sqrt{C_{4}(\rho-\rho_{c})}$ (12)
$\displaystyle\mathcal{A}_{t}$ $\displaystyle=\rho-C_{5}\phi^{2}$ (13)
$\displaystyle=\text{$\rho$}-C_{4}C_{5}(\rho-\rho_{c})$ (14)

when $C_{4}(\rho-\rho_{c})>0$, i.e. we observe the expected formation of the
superfluid phase, and the deviation of the chemical potential
$\mu=\mathcal{A}_{t}$ from its value $\mu=\rho$ in the normal phase.

In the non-equilibrium case, the late time behavior of (11) and the more
cumbersome solution for $\dot{\psi}-C_{1}\mathcal{A}_{t}$ depend crucially on
whether we are in the superfluid phase where

$\displaystyle\phi(t)-\sqrt{C_{4}(\rho-\rho_{c})}\propto
e^{-2C_{2}C_{4}(\rho-\rho_{c})t}+...$ (15)

or in the normal phase where

$\displaystyle\phi(t)\propto e^{C_{2}C_{4}(\rho-\rho_{c})t}+...\ .$ (16)

The crucial insight is that we can determine the phenomenological parameters
$C_{i}$ by comparing our results so far to the properties of the static
solution in the superfluid phase and the late time quasinormal mode like
falloff towards this static solution after a non-critical quench. As explained
in appendix B, this allows to numerically fix the parameters (normalized to
$\bar{T}$) of the model to222The dimensionless ratios are
$C_{2}{\bar{T}}^{3},\,C_{3}{\bar{T}},\,C_{4}/{\bar{T}}^{2},\,C_{5}{\bar{T}}^{3}.$

$\displaystyle\begin{tabular}[]{lllllll}$C_{2}$&$\approx$&0.03018&$C_{3}$&$\approx$&0.09308\\\
$C_{4}$&$\approx$&4.09192&$C_{5}$&$\approx$&0.14967\end{tabular}\ .$ (19)

Once this determination has been made, the model (10) is able to make
predictions for the behavior of the system at earlier times after a non-
critical quench (via the full expression (11)) as well as the exactly critical
quenches where at late times $\rho=\rho_{c}$. Let’s consider the latter case
first. In this limit, (11) simplifies to

$\displaystyle\phi(t)=\frac{1}{\sqrt{2C_{2}t+\frac{1}{\phi_{0}^{2}}}}\approx\frac{4.07}{t^{1/2}}+...$
(20)

and additionally we find

$\displaystyle\dot{\psi}-C_{1}(\mathcal{A}_{t}-\rho_{c})=\frac{C_{1}C_{5}+C_{3}}{2C_{2}t+\frac{1}{\phi_{0}^{2}}}\approx\frac{0.94}{t}+...\
.$ (21)

Even though these quantities will be small at late times, it is important to
note that these solutions depend on the non-linear features of equation (10)
and could not be obtained through a linearized ansatz. Comparing the
predictions (20) and (21) with our results given in equations (5), (6), and
(9), we find excellent agreement.

Let us now return to the issue of near-critical quenches, by which we mean
quenches that both start and end near the critical point, but not exactly at
it. We know that at very late times, after such a quench the system will be
characterized by an exponential falloff towards the new equilibrium, however
via the solution (11), equation (10) allows us to describe the behavior of the
system already at much earlier times. A representative example of a near-
critical quench is depicted in figure 2. In the curve for $\phi(t)$, we can
see that before the equilibrium is reached at very late times, there is a long
intermediate stretch of time in which the condensate appears to fall off in a
power law-like manner. In particular, after a quench that brings the system
infinitesimally close to the critical point, the system will initially react
as if it was relaxing exactly to the critical point, and only after what we
call ”handover-timescale”

$\displaystyle t_{\text{ho}}\sim\frac{1}{\rho-\rho_{c}}$ (22)

the system will notice that it is not at the critical point, and the power-law
behavior gives way to an exponential falloff towards a small but finite
condensate. Of course, this handover timescale is identical to the relaxation
timescale of the system close to the critical point. This phenomenological
behavior is exactly encapsulated in equation (11), see also appendix B and
appendix C for further information.

As the parameters $C_{i}$ are given in (19) and the value $\rho$ is determined
by the choice of quench, the only parameter that needs to be determined in
order to compare our analytical prediction with the numerical result is
$\phi_{0}$. We could hence use (11) as a model with one free parameter that
needs to be fitted to the data. However, as we can see in figure 2, in
contrast to $\dot{\psi}-\mathcal{A}_{t}$, $\phi(t)$ does not change
significantly during and immediately after the quench. Neither would a sudden
change be predicted by (11). Hence, as we know the initial state before the
quench exactly, we can simply set $\phi_{0}\equiv\phi(0)$, even though
formally equations (10) and (11) only become valid after the quench, when
$\rho$ is constant. As can be seen in figure 2, this trick is sufficient to
obtain a very good match between numerical data and analytical solution.

Figure 2:  Comparison between numerical (solid blue) and analytical (dashed
orange) results for a near-critical quench, with
$\rho_{\text{initial}}=$4.06626 and $\rho_{\text{final}}=$4.06373. The top
frame shows $\phi(t)$, while the bottom frame shows
$\dot{\psi}(t)-C_{1}\mathcal{A}_{t}(t)$. For the times after the quench
($t\gtrsim 10$), numerical and analytical curves agree very well. The dot
signifies the handover-time scale (22), while the dash-dotted red line shows
the critical solution (20). The dotted purple line shows the approximation
$\phi(t)\approx(2C_{2}t+1/\phi_{o}^{2})^{-1/2}e^{C_{4}(\rho-\rho_{c})(2C_{2}t+1/\phi_{0}^{2})/4}$
derived in appendix C.

##  4 Discussion

In this paper, we have studied the relaxation of a holographic superconductor
close or exactly at the critical point, both from a bulk and a boundary
perspective.

In the bulk, our methods of choice were extensive numerical simulations over
long ranges of time. We demonstrated that for critical quenches, in addition
to the expected power law falloff of the modulus $|\langle\mathcal{O}\rangle|$
of the order parameter, its complex phase undergoes rotations which are
periodic on a logarithmic time axis, leading to a discrete scale invariance of
the solution. Furthermore, we observed that the power law falloff
characteristic for the critical quenches is approximately observable even in
the non-critical quenches for an intermediate time-period before the onset of
the late-time exponential falloff at a handover-timescale (22).

On the field theory side, we postulated a phenomenological Gross–Pitaevskii-
like equation to describe the behavior of the system. This equation can be
solved analytically, and its parameters can be fixed by comparing to the
numerical data on the late-time exponential falloff after non-critical
quenches. In particular, we can fix the parameters of the equation with
information about the static equilibrium states and linearized fluctuations
about them (quasinormal modes (QNMs)) which are computationally much easier to
obtain than the non-linear time evolution. The constants obtained from the QNM
data in the superfluid and normal fluid phase match which is a non-trivial
check of our suggested equation (10). In our study we also successfully
applied and tested a novel concept about computing the amplitudes of QNM
excitations developed in Arean:2021tks  leading also to some novel insights
about the linear response behavior.

Once the parameters are fixed, the phenomenological equation allows to predict
with good accuracy both the behavior after exactly critical quenches, as well
as the behavior after near-critical quenches for intermediate time-scales. It
should be pointed out that both in the bulk and on the boundary, our results
are intimately tied to the non-linearities of the respective equations of
motion, and could not be studied with a simple linearized ansatz.
Interestingly, the papers Mitman:2022qdl ; Cheung:2022rbm  have recently
commented on the limitations of linearized ansätze in the study of black hole
ringdowns.

There are multiple possible directions for further research. Three obvious
generalisations would be to include a possible $x$-dependence (similar to
PhysRevLett.127.101601 ; Yan:2022jfc ; Adams:2012pj ; Sonner:2014tca ), to
study holographic superconductors in different dimensions, and to turn on
backreaction on the metric in the bulk. Preliminary results indicate the
possibility to generalize our phenomenological equation (10) to a finite rate
of superflow within the parameter regime of second order phase transitions up
to the tricritical point. We hope to revisit these ideas in a future
publication.

###### Acknowledgements.

We would like to thank Matteo Baggioli, Carlo Ewerz, Adrien Florio, Thomas
Gasenzer, Karl Landsteiner, Hesam Soltanpanahi, and Derek Teaney for useful
discussions. The work of MF (until the 31st of August 2022), SG and SM was
supported through the grants CEX2020-001007-S and PGC2018-095976-B-C21, funded
by MCIN/AEI/10.13039/ 501100011033 and ERDF A way of making Europe. Since
September 1st 2022, the work of MF was supported by the Polish National
Science Centre (NCN) grant 2021/42/E/ST2/00234. SG was in part supported by
the “Juan de la Cierva- Formación” program (grant number FJC2020-044057-I)
funded by MCIN/AEI/10.13039/501100011033 and NextGenerationEU/PRTR. SG is
supported by the U.S. Department of Energy, Office of Science grants No. DE-
FG88ER41450. SM was further supported by an FPI-UAM predoctoral fellowship.

##  Appendix A Numerical methods

In this section, we give a brief overview of the numerical methods used to
solve the partial differential equations numerically.

Within numerical holography, pseudo-spectral methods are widely applied to
find highly accurate solutions to boundary value problems in terms of elliptic
partial differential equations or ordinary differential equations. However,
for initial value problems of hyperbolic partial differential equations space
and time are typically treated differently. The spatial dependence is usually
discretized by means of a (pseudo)-spectral method which is combined with an
explicit 4th order Runge-Kutta scheme or Adams-Bashforth method to evolve the
solution in time Chesler:2013lia . Within a fully spectral scheme, we
discretize time and space with (pseudo)-spectral methods yielding a highly
implicit and accurate time evolver.

The basic idea of (pseudo)-spectral methods Boyd1989ChebyshevAF  is that the
unknown functions $u(x)$ which is the solution to the differential equation

$\mathcal{L}_{x}u(x)=g(x),$ (23)

where $\mathcal{L}_{x}$ is a differential operator, may be approximated by a
finite number $N$ of basis polynomials $\phi_{i}(x)$

$u(x)\approx u_{N}(x)=\sum_{i=0}^{N-1}c_{i}\phi_{i}(x).$ (24)

To find the solution, we require that the residuum $R=\mathcal{L}_{x}u_{N}-g$
vanishes exactly on the chosen set of discrete grid points. Note that for the
exact solution, the residuum vanishes identically. For a given choice of grid
points and basis polynomials, the derivatives are replaced by discrete
matrices acting on the whole domain.

Fully spectral algorithms have been employed within (asymptotically flat)
numerical relativity in Hennig:2008af ; Hennig:2012zx ; Petri2014ASM ;
PanossoMacedo:2014dnr ; Hennig:2020rns  and in the context of holography in
Ammon:2016fru ; Grieninger:2016xue . Let us outline the recipe for our
numerical algorithm (see also the appendix of Grieninger:2016xue ).

###  A.1 Numerical algorithm

We are looking for a solution at time $t=t_{\text{final}}$ to the initial
value problem at $t=t_{\text{initial}}$.

  1. 1.

At $t=t_{\text{initial}}$, we may obtain the initial configuration by solving
the static set of ordinary differential equations subject to the boundary
conditions $\varphi^{\prime}(t_{\text{initial}},0)=0$ and
$\varphi^{\prime\prime}(t_{\text{initial}},0)=|\langle\mathcal{O}_{\text{initial}}\rangle|$
and $A_{t}(t_{\text{initial}},1)=0$. As discretization of the radial direction
$u\in[0,1]$, we chose Gauss-Lobatto grid points
$u_{j}=\frac{1}{2}\left(1+\cos(\pi j/N_{u})\right)$, where $j\in[0,N_{u}-1]$.

  2. 2.

To evolve in time, we decompose the time interval
$(t_{\text{initial}},t_{\text{final}}]=(t_{\text{initial}},t_{1}]\cup(t_{1},t_{2}]\cup...\cup(t_{p},t_{\text{final}}]$
in $p+1$ subintervals in the spirit of multi-domain decompositions. Note that
the different time intervals may have different sizes which we set by a
adaptive step control depending on how much the solution changes on the
interval.

  3. 3.

For a given initial solution, we introduce auxiliary functions
$h(t,u)=h_{\text{in}}(t=t_{i},u)+(t-t_{i})\,h_{\text{aux}}(t,u)$ on each
subinterval $(t_{i},t_{j}],\,t_{j}>t_{i}$.

  4. 4.

The radial coordinate is discretized by the Chebyshev-Lobatto (CL) grid and to
discretize the time coordinate, we chose a right-sided Chebyshev-Radau (rCR)
grid (for some generic time interval $t\in(t_{i},t_{j}]$)

$\displaystyle u_{j}=\frac{1}{2}\left(1+\cos\left(\frac{\pi
j}{N_{u}}\right)\right),$ (25) $\displaystyle
t_{k}=\frac{1}{2}\left[(t_{j}-t_{i})+(t_{j}-t_{i})\cos\left(\frac{2\pi
k}{2N_{t}+1}\right)\right],$ (26)

where $j=0,\ldots,N_{u}-1$, and $k=0,\ldots,N_{t}$. Note that the right-sided
Chebychev-Radau grid does not include the initial slice of the interval where
we already know the solution.

  5. 5.

Replace all derivatives by their discrete versions given by the derivative
matrices $D$: $\partial_{u}\to D^{\text{CL}},\partial_{t}\to D^{\text{rCR}}$
and discretize the equations of motion on the square spanned by the discrete
spectral coordinates and impose the desired boundary.

  6. 6.

Solve the corresponding non-linear system with a Newton-Raphson method.

  7. 7.

Use the solution on the final slice $t_{j}$ as new initial solution in the
next step.

Typically, we use $N_{u}=40$ or $N_{u}=50$ in combination with $N_{t}=14$. We
monitored that the constraint equation is satisfied better than $10^{-15}$
during the time evolution. Note that we additionally fix $A_{t}(t,1)=0$. The
numerical algorithm is implemented in Mathematica.

The numerical methods used to compute the initial configurations, background
solutions and QNMs are the same codes as used in Ammon:2021pyz .

###  A.2 Quench profiles

For spontaneous $U(1)$ symmetry breaking, charge conservation imposes
$\dot{\rho}(t)=0$ in the absence of external sources. However, our goal is to
study quenches from the superfluid phase with $\rho=\rho_{\text{initial}}$ to
the critical point $\rho=\rho_{\text{crit}}$ (or close to it for some
$\rho=\rho_{\text{final}}$). Since we work in the probe limit, we have to
introduce an external source in order to change $\rho$ to our desired final
value. We could achieve this by breaking the $U(1)$ explicitly with a scalar
source as done in Bhaseen:2012gg , or we consider some generic external source
that changes $\rho$ directly by considering the null fluid (nf) current
Fernandez-Pendas:2019rkh

$2\kappa^{2}\,J_{(\text{nf})}^{u}=\frac{\dot{\rho}}{\sqrt{-g}}.$ (27)

which may be achieved by coupling

$S_{\text{nf}}=\int_{\mathcal{M}}dx^{4}\sqrt{-g}A_{\mu}j^{\mu}_{\text{ext}}\,,$
(28)

to the action. This leads to a covariantly conserved external current

$j_{\text{ext}}^{\mu}=\dfrac{\dot{\rho}}{\sqrt{-g}}\delta^{\mu}_{u}\,.$ (29)

which allows us to change the electric charge at will. Technically, the
external source also drives the $T^{tt}$ component of the energy-momentum
tensor. However, in the large $q$ expansion of the probe limit this
contribution is subleading and we can neglect it similarly to the backreaction
of the matter fields onto the geometry.

To quench our system, we chose to manipulate $\rho$ with the external source
and quench it to its final value. Note that in the late time behavior, the
external source is switched off and we verified that the late time behavior is
independent of the quench profile. Concretely, we performed quenches with

$\rho(t)=\rho_{\text{initial}}+\frac{1}{2}(\rho_{\text{final}}-\rho_{\text{initial}})(1+\tanh[\Omega(t-t_{s})]);$
(30)

where $\Omega=10$ is the rapidity and $t_{s}=1.5$ is the center of the quench.

##  Appendix B Determination of parameters

The parameters $C_{4},C_{5}$ of equation (10) can be determined by fitting
equations (12),(14) to the behavior of the static holographic superconductor
close to the critical point. One way to determine $C_{2},C_{3}$ is to compare
the predicted late time behavior after a non-critical quench into the
superfluid phase to numerical data of the non-linear time evolution. With the
parameters fixed in this way, (10) then allows to make genuine predictions for
both exactly critical quenches, and for the behavior of non-critical quenches
at early and intermediate times.

Giving some more details than in eq. (15), we find that for $t\gg 1$, quenches
in the superfluid phase will relax as

$\displaystyle\phi(t)=\sqrt{C_{4}}\sqrt{\rho-\rho_{c}}+\frac{\sqrt{C_{4}}}{2}\sqrt{\rho-\rho_{c}}\left(1-\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}\right)e^{-2C_{2}C_{4}t(\rho-\rho_{c})}+...$
(31) $\displaystyle\dot{\psi}(t)-C_{1}\mathcal{A}_{t}(t)=$ $\displaystyle-
C_{1}\rho+C_{1}C_{4}C_{5}(\rho-\rho_{c})$
$\displaystyle-\frac{C_{4}(C_{1}C_{5}-C_{3})}{\phi_{0}^{2}}\left(C_{4}(\rho-\rho_{c})-\phi_{0}^{2}\right)(\rho-\rho_{c})e^{-2C_{2}C_{4}t(\rho-\rho_{c})}+...$
(32)

while in the normal phase we would find

$\displaystyle\phi(t)=\sqrt{\frac{C_{4}(\rho_{c}-\rho)}{1-\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}}}e^{-C_{2}C_{4}t(\rho_{c}-\rho)}+...$
(33)
$\displaystyle\dot{\psi}(t)-C_{1}\mathcal{A}_{t}(t)=-C_{1}\rho+C_{3}C_{4}(\rho-\rho_{c})+\frac{C_{4}\phi_{0}^{2}(\rho-\rho_{c})(C_{1}C_{5}-C_{3})}{C_{4}(\rho-\rho_{c})-\phi_{0}^{2}}e^{-2C_{2}C_{4}t(\rho_{c}-\rho)}+...\
.$ (34)

Clearly, $C_{2}$ can be determined by fitting the half-life time of the
predicted late time exponential falloff in the superfluid phase to the
numerical data of the non-linear time evolution. The amplitude of this
exponential falloff itself is of course dependent on $\phi_{0}$ for each of
these functions, however the ratio between the amplitudes

$\displaystyle\frac{\text{Amplitude}_{\phi}}{\text{Amplitude}_{\dot{\psi}(t)-C_{1}\mathcal{A}_{t}(t)}}=\frac{1}{2\sqrt{C_{4}(\rho-\rho_{c})}}\frac{1}{C_{1}C_{5}-C_{3}}$
(35)

is independent of $\phi_{0}$ and can be used to obtain a unique value of
$C_{3}$.

However, it is not necessary to perform the non-linear time evolution to fit
the constants (19). In the following, we explain how all of them may be
obtained within linear response theory and from the static solutions in the
context of holography. The two constants $C_{4}$ and $C_{5}$ may simply be
obtained by constructing the static solutions near the phase transition and
fit the condensate and chemical potential, respectively, to the deviation of
$\rho$ from its critical value $\rho_{c}$ according to eqs. (12) and (14). The
constant $C_{2}$ may be obtained from the QNMs at zero wavevector in the
superfluid phase (15) or in the normal phase (16). To probe the QNMs we
consider linearized fluctuations of a complex scalar
$\delta\bar{\Psi}=\delta\Psi e^{-i\omega t}$ and the gauge field
$\delta\bar{a}_{t}=\delta a_{t}e^{-i\omega t}$. Note that we may decompose the
scalar fluctuations about a state with zero background phase according to
$(\langle\mathcal{O}_{\text{eq}}\rangle+\langle\delta\mathcal{O}\rangle)e^{i\delta\psi}=(\langle\mathcal{O}_{\text{eq}}\rangle+\langle\delta\mathcal{O}\rangle)(1+i\delta\psi)=(\langle\mathcal{O}_{\text{eq}}\rangle+\text{Re}(\delta\Psi)+i\,\text{Im}(\delta\bar{\Psi})$,
with $\text{Re}(\delta\Psi)=\langle\delta\mathcal{O}\rangle)$ and
$\text{Im}(\delta\bar{\Psi})=\langle\mathcal{O}_{\text{eq}}\rangle\,\delta\psi$.

Let us focus on the normal phase first and consider linearized solutions about
the static normal phase solution with
$\langle\mathcal{O}\rangle=0,\,\rho<\rho_{c}$. The corresponding QNM
responsible for the relaxation to equilibrium is the pair of massive scalar
modes. Close to the phase transition the QNMs in the normal phase behave (to
lowest order in $\rho-\rho_{c}$) like

$\omega_{\pm}=-(\pm 0.38087-0.12348i)\,(\rho-\rho_{c})$ (36)

According to eq. (16), we can read of the constant $C_{2}$ (since we already
know $C_{4}$ from the static solution) from the imaginary part of the QNM (36)
leading to the value indicated in (19). Similarly, the real part determines
the constant $C_{3}=-\text{Re}(\omega_{+})/C_{4}$ as may be seen from eq.(34).
Since the QNM comes as a pair, the sign is seemingly not determined. However,
it is possible to reconstruct which sign belongs to fluctuations of
$\delta\Psi$ and $\delta\bar{\Psi}$. In order to determine the QNMs we solve
the fluctuation equations as generalized eigenvalue problem of the form
$(\bm{A}\omega-\bm{B})\,\bm{x}=0$, where $\bm{A}$ and $\bm{B}$ are
differential operators of a non-hermitian Sturm-Liouville problem (see
Arean:2021tks  for more details). Usually, only the eigenvalues $\omega$ are
of interest since they correspond to the QNM frequencies. However, it is also
possible to examine the eigenvector $\bm{x}$ corresponding to the eigenvalue
$\omega$. In our case, we observe that fluctuations with $\omega=\omega_{+}$
are carried by $\bm{x}=\\{\delta\Psi,0\\}$ while $\omega=\omega_{-}$ is
carried by $\bm{x}=\\{0,\delta\bar{\Psi}\\}$.

As an independent, non-trivial check of our proposed equation, we now compute
the constants $C_{2}$ and $C_{3}$ from the QNMs in the superfluid phase. To
compute $C_{3}$, we need information about the relative amplitudes of the
fluctuations supporting the QNM responsible for equilibration. Only recently,
the authors of Arean:2021tks  suggested a method to compute the relative
contributions of boundary operators to a certain QNM excitation from the
aforementioned eigenvectors. Here, we want to dissect the so called
“amplitude” or Higgs mode. At zero wave vector, this pseudo-diffusive mode is
driving the system to equilibrium in the superfluid phase Amado:2013xya . More
recently, the dynamics of this mode was discussed in terms of a linearized
bulk analysis in Donos:2022xfd . Close to equilibrium, we find for the QNM
frequency to lowest order in $\rho-\rho_{c}$ (and at zero wave vector)

$\omega_{\text{Ampl}}=-0.2469\,i\,(\rho-\rho_{c}).$ (37)

According to eq. (16), we may extract $C_{2}$ from this information leading to
the same numerical value as computed in the normal phase. Employing the
techniques developed in Arean:2021tks , we can extract the expectation values
of the operators carrying this QNM excitation from the corresponding
eigenvector. Intriguingly, we find that the gauge fluctuations have
expectation value zero and the mode is solely carried by the scalar
fluctuations. Close to the critical point we thus find to lowest order in
$\rho-\rho_{c}$ (and at zero wave vector) that

$\displaystyle\frac{\text{Amplitude}_{\langle\delta\mathcal{O}\rangle}}{\text{Amplitude}_{\langle\delta\dot{\psi}\rangle-\langle\delta
a_{t}\rangle}}=\frac{17.67}{2\,\sqrt{C_{4}(\rho-\rho_{c})}}.$ (38)

Once $C_{5}$ is fixed from the background data, eq. (38) determines the value
of $C_{3}$ in accordance with our data from the normal phase. Note that this
is a non-trivial and independent check of the numerical values we obtained for
$C_{2}$ and $C_{3}$ thus confirming the prediction of our suggested model.

##  Appendix C Analysis of intermediate time behavior

We will now give an analysis of the behavior at intermediate time scales
predicted by the solution (11) as well as the corresponding solution

$\displaystyle\dot{\psi}-C_{1}\mathcal{A}_{t}=-C_{1}\rho-
C_{3}C_{4}(\rho-\rho_{c})+\frac{C_{4}(C_{3}+C_{1}C_{5})(\rho-\rho_{c})}{1-e^{-2C_{2}C_{4}t(\rho-\rho_{c})}\left(1-\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}\right)}.$
(39)

First of all, we notice that
$\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}=\frac{\phi(t\rightarrow\infty)^{2}}{\phi(0)^{2}}$.
As we have been interested exclusively in quenches that lead to a decay of the
condensate, we will assume $\frac{C_{4}(\rho-\rho_{c})}{\phi_{0}^{2}}<1$.
Hence the bracket depending on $\phi_{0}$ in (11) and (39) is positive and can
be absorbed in the exponent as a shift $t\rightarrow t+t_{0}$ in the time
coordinate. As we would like to ignore such shifts, we will from now on take
the limit $\phi_{0}\rightarrow\infty$. This limit is technically unphysical,
as equation (10) is only expected to be reliable for small values of $\phi$,
however it simplifies equations (11) and (39) and their analysis considerably.
The results of this analysis should then also hold for realistic settings up
to shifts on the time axis.

We already discussed the very late time behavior in equations (31) and (32),
seeing an exponential falloff towards equilibrium at $t\gg 1$. Now, we turn
our attention to earlier times. For this, we define a map
$\mathcal{M}(y(t))\equiv\frac{\dot{y}(t)\times t}{y(t)}$. This has the benefit
that it easily allows us to analyse and distinguish the qualitative behavior
of functions, as e.g.

$\displaystyle\mathcal{M}(At^{a})$ $\displaystyle=a,$ (40)
$\displaystyle\mathcal{M}(Ae^{at})$ $\displaystyle=at,$ (41)
$\displaystyle\mathcal{M}(At^{b}e^{at})$ $\displaystyle=b+at.$ (42)

For our solutions (11) and (39), we find

$\displaystyle\mathcal{M}(\phi(t))$
$\displaystyle=-\frac{1}{2}+\frac{1}{2}C_{2}C_{4}(\rho-\rho_{c})t+...,$ (43)
$\displaystyle\mathcal{M}(\dot{\psi}-C_{1}(\mathcal{A}_{t}-\rho_{c}))$
$\displaystyle=-1+\frac{C_{2}(-C_{3}C_{4}+C_{1}(-2+C_{4}C_{5}))}{C_{3}+C_{1}C_{5}}(\rho-\rho_{c})t+...\
.$ (44)

Assuming all coefficients $C_{i}$ to be roughly of order 1, this demonstrates
that the solutions for non-critical quenches exhibit the same kind of power
law behavior as the critical quenches until $(\rho-\rho_{c})t$ is of order 1,
which establishes the handover timescale (22). See figure 3 for an
illustration. More precisely, we could have said that for early times
$\phi(t)$ can be approximated as

$\displaystyle\phi(t)\propto
t^{-1/2}e^{\frac{1}{2}C_{2}C_{4}(\rho-\rho_{c})t},$ (45)

however for $t\ll t_{\text{ho}}$ the exponential function will deviate from 1
only slightly.

Figure 3:  The top frame shows $\phi(t)$ (using (11)) and the bottom frame
shows $|\dot{\psi}-C_{1}(\mathcal{A}_{t}-\rho_{c})|$(using (39)) for
$\phi_{0}\rightarrow\infty$ and $\rho=\rho_{c}+2^{a}$ with $a$ varying from
$a=-20$ (purple) to $a=0$ (red) in steps of $4$. The black lines represent the
exactly critical solutions $\rho=\rho_{c}$. The dots placed on each curve
signify the handover timescale $t_{\text{ho}}$, which clearly describes well
in an order of magnitude manner until what timescale the solutions are well
approximated by the critical solutions (20) and (21).

## References

  * (1) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295].
  * (2) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563].
  * (3) C. P. Herzog, P. K. Kovtun and D. T. Son, Holographic model of superfluidity, Phys. Rev. D 79 (2009) 066002 [arXiv:0809.4870].
  * (4) C. P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975].
  * (5) S. S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977].
  * (6) O. Domenech, M. Montull, A. Pomarol, A. Salvio and P. J. Silva, Emergent Gauge Fields in Holographic Superconductors, JHEP 08 (2010) 033 [arXiv:1005.1776].
  * (7) K. Maeda, M. Natsuume and T. Okamura, On two pieces of folklore in the AdS/CFT duality, Phys. Rev. D 82 (2010) 046002 [arXiv:1005.2431].
  * (8) P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49 (Jul, 1977) 435–479.
  * (9) K. Maeda, M. Natsuume and T. Okamura, Dynamic critical phenomena in the AdS/CFT duality, Phys. Rev. D 78 (2008) 106007 [arXiv:0809.4074].
  * (10) K. Maeda, M. Natsuume and T. Okamura, Universality class of holographic superconductors, Phys. Rev. D 79 (2009) 126004 [arXiv:0904.1914].
  * (11) M. Natsuume, Critical phenomena in the AdS/CFT duality, Prog. Theor. Phys. Suppl. 186 (2010) 491–497 [arXiv:1006.4930].
  * (12) C. Ewerz, T. Gasenzer, M. Karl and A. Samberg, Non-Thermal Fixed Point in a Holographic Superfluid, JHEP 05 (2015) 070 [arXiv:1410.3472].
  * (13) Z. Cai, C. Hubig and U. Schollwöck, Universal long-time behavior of aperiodically driven interacting quantum systems, Phys. Rev. B 96 (Aug, 2017) 054303.
  * (14) C. De Grandi, V. Gritsev and A. Polkovnikov, Quench dynamics near a quantum critical point, Phys. Rev. B 81 (Jan, 2010) 012303.
  * (15) S. S. Gubser and S. S. Pufu, The Gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960].
  * (16) P. M. Chesler, A. M. Garcia-Garcia and H. Liu, Defect Formation beyond Kibble-Zurek Mechanism and Holography, Phys. Rev. X 5 (2015), no. 2 021015 [arXiv:1407.1862].
  * (17) J. Sonner, A. del Campo and W. H. Zurek, Universal far-from-equilibrium Dynamics of a Holographic Superconductor, Nature Commun. 6 (2015) 7406 [arXiv:1406.2329].
  * (18) M. J. Bhaseen, J. P. Gauntlett, B. D. Simons, J. Sonner and T. Wiseman, Holographic Superfluids and the Dynamics of Symmetry Breaking, Phys. Rev. Lett. 110 (2013), no. 1 015301 [arXiv:1207.4194].
  * (19) A. del Campo, F. J. Gómez-Ruiz, Z.-H. Li, C.-Y. Xia, H.-B. Zeng and H.-Q. Zhang, Universal statistics of vortices in a newborn holographic superconductor: beyond the Kibble-Zurek mechanism, JHEP 06 (2021) 061 [arXiv:2101.02171].
  * (20) Z.-H. Li, H.-Q. Shi and H.-Q. Zhang, Holographic topological defects in a ring: role of diverse boundary conditions, JHEP 05 (2022) 056 [arXiv:2111.15230].
  * (21) M. Ammon, S. Grieninger, A. Jimenez-Alba, R. P. Macedo and L. Melgar, Holographic quenches and anomalous transport, JHEP 09 (2016) 131 [arXiv:1607.06817].
  * (22) D. Arean, M. Baggioli, S. Grieninger and K. Landsteiner, A holographic superfluid symphony, JHEP 11 (2021) 206 [arXiv:2107.08802].
  * (23) D. Sornette, Discrete-scale invariance and complex dimensions, Physics Reports 297 (1998), no. 5 239–270.
  * (24) J. Erdmenger, M. Flory, M.-N. Newrzella, M. Strydom and J. M. S. Wu, Quantum Quenches in a Holographic Kondo Model, JHEP 04 (2017) 045 [arXiv:1612.06860].
  * (25) E. W. Hirschmann and D. M. Eardley, Universal scaling and echoing in the gravitational collapse of a complex scalar field, Phys. Rev. D 51 (Apr, 1995) 4198–4207.
  * (26) I. S. Aranson and L. Kramer, The world of the complex ginzburg-landau equation, Rev. Mod. Phys. 74 (Feb, 2002) 99–143.
  * (27) T. Tsuneto, Superconductivity and Superfluidity.  Cambridge University Press, 1998.
  * (28) P. Wittmer, C.-M. Schmied, T. Gasenzer and C. Ewerz, Vortex motion quantifies strong dissipation in a holographic superfluid, Phys. Rev. Lett. 127 (Aug, 2021) 101601.
  * (29) Y.-K. Yan, S. Lan, Y. Tian, P. Yang, S. Yao and H. Zhang, Towards an effective description of holographic vortex dynamics, arXiv:2207.02814.
  * (30) P. S. Hagan, Spiral waves in reaction-diffusion equations, SIAM Journal on Applied Mathematics 42 (1982), no. 4 762–786.
  * (31) K. Mitman et. al., Nonlinearities in black hole ringdowns, arXiv:2208.07380.
  * (32) M. H.-Y. Cheung et. al., Nonlinear effects in black hole ringdown, arXiv:2208.07374.
  * (33) A. Adams, P. M. Chesler and H. Liu, Holographic Vortex Liquids and Superfluid Turbulence, Science 341 (2013) 368–372 [arXiv:1212.0281].
  * (34) P. M. Chesler and L. G. Yaffe, Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes, JHEP 07 (2014) 086 [arXiv:1309.1439].
  * (35) J. P. Boyd, Chebyshev and Fourier Spectral Methods.  Dover Publications Inc., 2003.
  * (36) J. Hennig and M. Ansorg, A Fully Pseudospectral Scheme for Solving Singular Hyperbolic Equations on Conformally Compactified Space-Times, J. Hyperbol. Diff. Equat. 6 (2009), no. 01 161–184 [arXiv:0801.1455].
  * (37) J. Hennig, Fully pseudospectral time evolution and its application to 1+1 dimensional physical problems, J. Comput. Phys. 235 (2013) 322–333 [arXiv:1204.4220].
  * (38) J. Pétri, A spectral method in space and time to solve the advection-diffusion and wave equations in a bounded domain, arXiv:1401.1373.
  * (39) R. Panosso Macedo and M. Ansorg, Axisymmetric fully spectral code for hyperbolic equations, J. Comput. Phys. 276 (2014) 357–379 [arXiv:1402.7343].
  * (40) J. Hennig and R. Panosso Macedo, Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background, Class. Quant. Grav. 38 (2021), no. 13 135006 [arXiv:2012.02240].
  * (41) S. Grieninger, Holographic quenches and anomalous transport, arXiv:1711.08422.
  * (42) M. Ammon, D. Arean, M. Baggioli, S. Gray and S. Grieninger, Pseudo-spontaneous $U(1)$ symmetry breaking in hydrodynamics and holography, JHEP 03 (2022) 015 [arXiv:2111.10305].
  * (43) J. Fernández-Pendás and K. Landsteiner, Out of equilibrium chiral magnetic effect and momentum relaxation in holography, Phys. Rev. D 100 (2019), no. 12 126024 [arXiv:1907.09962].
  * (44) I. Amado, D. Arean, A. Jimenez-Alba, K. Landsteiner, L. Melgar and I. S. Landea, Holographic Type II Goldstone bosons, JHEP 07 (2013) 108 [arXiv:1302.5641].
  * (45) A. Donos and C. Pantelidou, Higgs/amplitude mode dynamics from holography, JHEP 08 (2022) 246 [arXiv:2205.06294].