\begin{equation}\label{eqn:varphi_0_1}
f_i'(x)=f_i(x)+\sum_{j\in A(i)}a_{ij}(x) f_j(x)
\end{equation}

such that $df_i'(x)$ is $\rho(i)$-admissible. Say,

\begin{equation}
df_i'(x)=u_{i\rho(i)}(d) f_{\rho(i)}'(x)
\end{equation}

with $u_{i\rho(i)}(d)\in\field^*$, and

\begin{equation}\label{eqn:varphi_0_2}
f_{\rho(i)}'(x)=f_{\rho(i)}(x)+\sum_{\ell\in I(\rho(i))}a_{\rho(i) \ell}(x)f_{\ell}(x)
\end{equation}

for some $a_{\rho(i)\ell}(x)\in\field$. In addition, $f_i'(x)$ depends algebraically on $d$.

Now, $\varphi_0(d)$ is given by

\begin{equation}
\varphi_0(d)(f_i(x))=f_i'(x)
\end{equation}

for all $1\leq i\leq n_x$.

§.§.§ The ruling decomposition

Now, we can make the following

Let $\rho\in\mathrm{NR}(\Lambda)$ be a normal ruling, and $q$ be any degree $0$ crossing of $\pi_{xz}(\Lambda)$ with $x_q$ contained in a small open interval $(x,x')$ so that $q$ switches the strands $k,k+1$. Define an algebraic map

\[
r_q:\aug^{\rho}(\Lambda,\overrightarrow{*};\field)\rightarrow \field
\]

\begin{equation}
\end{equation}

where $\langle\varphi_0(d(x,\epsilon))(f_k(x)),f_{k+1}(x)\rangle$ denotes the coefficient of $f_{k+1}(x)$ in $\varphi_0(d(x,\epsilon))(f_k(x))$. And, $a_{kk+1}(x)$ is defined in (<ref>) or (<ref>), or $0$ otherwise.

The upshot is a decomposition for the augmentation variety:

There exists a canonical decomposition into locally closed subvarieties

\begin{equation}
\aug(\Lambda,\overrightarrow{*};\field)=\sqcup_{\rho\in\mathrm{NR}(\Lambda)}\aug^{\rho}(\Lambda,\overrightarrow{*};\field)
\end{equation}

and for each $\rho\in\mathrm{NR}(\Lambda)$, the natural map $\aug^{\rho}(\Lambda,\overrightarrow{*};\field)\ni\epsilon\mapsto (r_q(\epsilon))_{q\in S(\rho)\sqcup R(\rho)}$ induces a natural isomorphism

\begin{equation}
\overrightarrow{r}:\aug^{\rho}(\Lambda,\overrightarrow{*};\field)\isomorphic (\field^*)^{s(\rho)}\times \field^{r(\rho)}.
\end{equation}

In the working example with $\Lambda=\beta^>$ for the $2$-strand positive braid $\beta=\sigma_1^3$, as in Figure <ref>. Recall for any augmentation $\epsilon\in\aug(\beta^>,\overrightarrow{*};\field)$, the Morse complex at $x_i$ is given by $(C(x_i,\epsilon),d(x_i,\epsilon))$ with $C(x_i,\epsilon)=\oplus_{i=1}^4\field\cdot f_i(x)$, and $(f_1(x_i),f_2(x_i),f_3(x_i),f_4(x_i))=(e_2^+,e_1^+,e_1(x_i),e_2(x_i))$, where $e_2^+,e_1^+,e_1(x_i),e_2(x_i)$ are defined as in Example <ref>. In addition, $d(x_i,\epsilon)e_i^+=-e_i=-e_i(x_1)$. Denote $\alpha_i:=\epsilon(a_i)$ for $1\leq i\leq 3$, then one can check that

\[
-\langle d(x_2,\epsilon)e_1^+, e_1(x_2)\rangle=\alpha_1, -\langle d(x_3,\epsilon)e_1^+, e_1(x_3)\rangle=1+\alpha_1\alpha_2,
\]

where the coefficient $\langle d(x_i,\epsilon)f_k(x), f_{\ell}(x_i)\rangle\in\field$ is defined by $d(x_i,\epsilon)f_k(x_i)=\sum \langle d(x_i,\epsilon)f_k(x_i), f_{\ell}(x_i)\rangle f_{\ell}(x_i)$.

By the definition of $\aug^{\ruling}(\beta^>,\overrightarrow{*};\field)$ and the formula above, we in fact have

\begin{eqnarray*}
&&\aug^{\ruling_1}(\beta^>,\overrightarrow{*};\field)=\{\alpha_1\neq 0,1+\alpha_1\alpha_2=0,\alpha_1+\alpha_3+\alpha_1\alpha_2\alpha_3\neq 0\}\isomorphic \{(\alpha_1,\alpha_3)\in\field^*\times\field\}\isomorphic\field^*\times\field;\\
&&\aug^{\ruling_2}(\beta^>,\overrightarrow{*};\field)=\{\alpha_1=0, 1+\alpha_1\alpha_2\neq 0,\alpha_1+\alpha_3+\alpha_1\alpha_2\alpha_3\neq 0\}\isomorphic\{(\alpha_2,\alpha_3)\in\field\times\field^*\}\isomorphic \field^*\times\field;\\
&&\aug^{\ruling_3}(\beta^>,\overrightarrow{*};\field)=\{\alpha_1\neq 0,1+\alpha_1\alpha_2\neq 0,\alpha_1+\alpha_3+\alpha_1\alpha_2\alpha_3\neq 0\}\isomorphic (\field^*)^3.
\end{eqnarray*}

where $\ruling_j\in\mathrm{NR}(\beta^>)$ are defined as in Example <ref>. Let's also compute the maps $r_q=r_{q,\ruling}:\aug^{\ruling}(\beta^>,\overrightarrow{*};\field)\rightarrow\field$ for $q\in\{a_1,a_2,a_3\}$ and $\ruling\in\mathrm{NR}(\beta^>)=\{\ruling_1,\ruling_2,\ruling_3\}$. By Section <ref>, we have

* $\ruling=\ruling_1$:

* Suppose $q=a_1$, which is a switch of $\ruling_1$. We have $f_i'(x_1)=f_i(x_1)$ and $\varphi_0(d(x_1,\epsilon))=id$. Hence, by definition we have

\[
\]

* Suppose $q=a_2$, which is a departure of $\ruling_1$. We have $f_2'(x_2)=f_2(x_2)=e_1^+$ and $d(x_2,\epsilon)f_2'(x_2)=-e_1=-\epsilon(a_1)e_1(x_2)-e_2(x_2)=-\alpha_1f_3'(x_2)$,
where $f_3'(x_2)=e_1(x_2)+\alpha_1^{-1}e_2(x_2)=f_3(x_2)+\alpha_1^{-1}f_4(x_2)$. In particular, $a_{34}(x_2)=\alpha_1^{-1}$ and

\[
\]

for $\epsilon\in\aug^{\ruling_1}(\beta^>,\overrightarrow{*};\field)$, as desired.

* Suppose $q=a_3$, which is a return of $\ruling_1$. We have $f_1'(x_3)=f_1(x_3)=e_2^+$ and $d(x_3,\epsilon)f_1'(x_3)=-e_2=-e_1(x_2)=-\epsilon(a_2)e_1(x_3)-e_2(x_3)=-\alpha_2f_3'(x_3)$, where $f_3'(x_3)=e_1(x_3)+\alpha_2^{-1}e_2(x_3)=f_3(x_3)+\alpha_2^{-1}f_4(x_3)$. Notice that, $\epsilon\in\aug^{\ruling}(\beta^>,\overrightarrow{*};\field)$ implies that $1+\alpha_1\alpha_2=0$, so $\alpha_2\neq 0$. It follows that $a_{34}(x_3)=\alpha_2^{-1}$ and

\[
\]

* $\ruling=\ruling_2$:

* Suppose $q=a_1$, which is a departure of $\ruling_2$. As in ($1a$), we have

\[
\]

for $\epsilon\in\aug^{\ruling_2}(\beta^>,\overrightarrow{*};\field)$, as desired.

* Suppose $q=a_2$, which is a return of $\ruling_2$. We have $f_1'(x_2)=f_1(x_2)=e_2^+$ and $d(x_2,\epsilon)f_1'(x_2)=-e_2=-e_1(x_2)=-f_3'(x_2)$, where $f_3'(x_2)=f_3(x_2)=e_1(x_2)$. In particular, $a_{34}(x_2)=0$ and

\[
\]

* Suppose $q=a_3$, which is a switch of $\ruling_2$. We have $f_2'(x_3)=f_2(x_3)=e_1^+$ and $d(x_3,\epsilon)f_2'(x_3)=-e_1=-e_2(x_2)=-e_1(x_3)=-f_3'(x_3)$, where $f_3'(x_3)=f_3(x_3)=e_1(x_3)$. In particular, $a_{34}(x_3)=0$ and

\[
\]

for $\epsilon\in\aug^{\ruling_2}(\beta^,\overrightarrow{*};\field)$.

* $\ruling=\ruling_3$:

* Suppose $q=a_1$, which is a switch of $\ruling_3$. As in ($1a$), we have

\[
\]

* Suppose $q=a_2$, which is a switch of $\ruling_3$. As in ($1b$), we have

\[
\]

for $\epsilon\in\aug^{\ruling_3}(\beta^>,\overrightarrow{*};\field)$.

* Suppose $q=a_3$, which is a switch of $\ruling_3$. We have $f_2'(x_3)=f_2(x_3)=e_1^+$ and

\begin{eqnarray*}
\end{eqnarray*}

where $f_3'(x_3)=e_1(x_3)+(1+\alpha_1\alpha_2)^{-1}\alpha_1 e_2(x_3)=f_3(x_3)+(1+\alpha_1\alpha_2)^{-1}\alpha_1 f_4(x_3)$. In particular, $a_{34}(x_3)=(1+\alpha_1\alpha_2)^{-1}\alpha_1$ and

\[
\]

for $\epsilon\in\aug^{\ruling_3}(\beta^>,\overrightarrow{*};\field)$.

As a consequence, we see that all the above computations are compatible with Lemma <ref>.

§.§ The cell decomposition for the Betti moduli space

From now on, we take $\Lambda=\beta^>$ for a $n$-strand positive braid $\beta$, and $\field=\mathbb{C}$ for simplicity. As usual, denote $\modulistack_B(\field):=\modulistack_1(\mathbb{C}P^1,\infty,\beta^>;\field)\isomorphic [\aug(\beta^>,\overrightarrow{*};\field)/\mathbb{G}_m^n]$ and $\modulispace_B(\field):=\modulispace_1(\mathbb{C}P^1,\infty,\beta^>;\field)\isomorphic \mathrm{Spec}~\mathcal{O}(\aug(\beta^>,\overrightarrow{*};\field))^{\mathbb{G}_m^n}$.

Let $\beta$ be a $n$-strand positive braid so that $\beta^>$ is connected, then the Betti moduli space $\modulispace_B(\field)$ is a smooth affine and connected variety over $\field$, and admits a ruling decomposition into locally closed subvarieties

\[
\modulispace_B(\field)=\sqcup_{\ruling\in\NR(\beta^>)} \modulispace_B^{\ruling}(\field)
\]

such that

* For each $\ruling\in\NR(\beta^>)$, we have a natural isomorphism

\[
\modulispace_B^{\ruling}(\field)\isomorphic (\field^*)^{s(\ruling)-n+1}\times\field^{r(\ruling)}.
\]

* Let $\ruling_m\in\NR(\beta^>)$ be the normal ruling so that all the crossings of $\beta^>$ are switches, then $\ruling_m$ is the unique normal ruling such that $r(\ruling_m)=0$. In addition, $\modulispace_B^{\ruling_m}(\field)$ is open and dense in $\modulispace_B(\field)$, and

\[
\modulispace_B^{\ruling_m}(\field)\isomorphic (\field^*)^{s(\ruling_m)-n+1}=(\field^*)^d
\]

where $d=\mathrm{dim}_{\mathbb{\field}}\modulispace_B(\mathbb{\field})$.

Notice that under the identification $\modulispace_B(\field)\isomorphic\aug(\beta^>;\field)$ in Proposition <ref>, the proposition above essentially reduces to <cit.>.
But for completeness, we will give an alternative proof in the rest of this subsection. The strategy is to show that the torus action of $\mathbb{G}_m^n$ on $\aug(\beta^>,\overrightarrow{*};\field)$ is compatible with the ruling decomposition in Lemma <ref>.

The torus action of $\mathbb{G}_m^n$ on $\aug(\beta^>,\overrightarrow{*};\field)$ preserves $\aug^{\rho}(\beta^,\overrightarrow{*};\field)$, and under the isomorphism $\overrightarrow{r}:\aug^{\rho}(\beta^>,\overrightarrow{*};\field)\isomorphic (\field^*)^{s(\ruling)}\times\field^{r(\ruling)}$, we have

\begin{equation}
\end{equation}

Let $\overrightarrow{\lambda}\in\mathbb{G}_m^n$ and $\epsilon\in\aug^{\ruling}(\beta^>,\overrightarrow{*};\field)$.
Recall that $x_L$ (resp. $x_R$)is a generic $x$-coordinate to the right (resp. left) of all left (resp. right) cusps and to the left (resp. right) of all the crossings of $\beta^>$. For any generic $x\in [x_L,x_R]$,
let $(C(x,\epsilon),d(x,\epsilon))$ be the Morse complex at $x$ determined by $\epsilon$, then $C(x,\epsilon)=\oplus_{i=1}^{n_x}\field\cdot f_i(x,\epsilon)$. By (<ref>), we have

\begin{eqnarray*}
&&\langle f_1(x,\epsilon),\ldots,f_{n_x}(x,\epsilon)\rangle=\langle e_n^+(\epsilon),\ldots,e_1^+(\epsilon),e_1(x,\epsilon),\ldots,e_n(x,\epsilon)\rangle\\
&& d(x,\epsilon) e_i^+(\epsilon) = -e_i(\epsilon),~ d(x,\epsilon) e_i(\epsilon)=0.
\end{eqnarray*}

where $e_i(x_L,\epsilon)=e_i(\epsilon)$, and $e_i(x,\epsilon)$ is defined inductively as in (<ref>).
As in the proof of Lemma <ref>, we can use the identification $e_i(\overrightarrow{\lambda}\cdot\epsilon)=\lambda_i^{-1}e_i(\epsilon)$ and $e_i^+(\overrightarrow{\lambda}\cdot\epsilon)=\lambda_i^{-1}e_i^+(\epsilon)$, then

\[
\]

Equivalently, we have

\begin{equation}
\end{equation}

where $c(i,x)=n+1-i$ for $1\leq i\leq n$, and $c(i,x)=c_{i-n}(x)$ for $n+1\leq i\leq 2n$.

The relation between $d(x,\epsilon)$ and $d(x,\overrightarrow{\lambda}\cdot\epsilon)$ is given by

\begin{eqnarray}\label{eqn:differential_under_scaling_of_augmentations}
&&d(x,\epsilon)e_i(\overrightarrow{\lambda}\cdot\epsilon)=d(x,\overrightarrow{\lambda}\cdot\epsilon) e_i(\overrightarrow{\lambda}\cdot\epsilon).\nonumber
\end{eqnarray}

Recall that $\varphi_0(d(x,\epsilon))$ in Lemma <ref> is uniquely determined by

\begin{eqnarray*}
&&\varphi_0(d(x,\epsilon))(f_i(x,\epsilon))=f_i'(x,\epsilon), 1\leq i\leq 2n\\
&& f_i'(x,\epsilon)=f_i(x,\epsilon) + \sum_{j\in A(i)} a_{ij}(x,\epsilon) f_j(x,\epsilon), 1\leq i\leq n\\
&& d(x,\epsilon) f_i'(x,\epsilon) = u_{i\ruling(i)}(x,\epsilon) f_{\ruling(i)}'(x,\epsilon), 1\leq i\leq n\\
&& f_i'(x,\epsilon) = f_i(x,\epsilon) + \sum_{j>i} a_{ij}(x,\epsilon) f_j(x,\epsilon), n+1\leq i\leq 2n.
\end{eqnarray*}

where $a_{ij}(x,\epsilon)\in\field$ and $u_{i\ruling(i)}(x,\epsilon)\in\field^*$.
It then follows that

\begin{eqnarray*}
&&\varphi_0(d(x,\epsilon))(\lambda_{c(i,x)}^{-1}f_i(x,\epsilon))=\lambda_{c(i,x)}^{-1}f_i'(x,\epsilon), 1\leq i\leq 2n\\
&& \lambda_{c(i,x)}^{-1}f_i'(x,\overrightarrow{\lambda}\cdot\epsilon)=\lambda_{c(i,x)}^{-1}f_i(x,\epsilon) + \sum_{j\in A(i)} (\lambda_{c(i,x)}^{-1}a_{ij}(x,\epsilon)\lambda_{c(j,x)}) (\lambda_{c(j,x)}^{-1}f_j(x,\epsilon)), 1\leq i\leq n\\
&& d(x,\overrightarrow{\lambda}\cdot\epsilon) (\lambda_{c(i,x)}^{-1}f_i'(x,\epsilon))\\
&=&d(x,\epsilon) \lambda_{c(i,x)}^{-1}f_i'(x,\epsilon) = (\lambda_{c(i,x)}^{-1}u_{i\ruling(i)}(x,\epsilon)\lambda_{c(\ruling(i),x)}) (\lambda_{c(\ruling(i),x)}^{-1}f_{\ruling(i)}'(x,\epsilon)), 1\leq i\leq n\\
&&  \lambda_{c(i,x)}^{-1}f_i'(x,\epsilon) =  \lambda_{c(i,x)}^{-1}f_i(x,\epsilon) + \sum_{j>i}  (\lambda_{c(i,x)}^{-1}a_{ij}(x,\epsilon) \lambda_{c(j,x)}) (\lambda_{c(j,x)}^{-1}f_j(x,\epsilon)), n+1\leq i\leq 2n.
\end{eqnarray*}

where the first identity in the third equation follows from the second equation and Equation (<ref>). In particular, the third equation implies that $\overrightarrow{\lambda}\cdot\epsilon\in\aug^{\rho}(\beta^>,\overrightarrow{*};\field)$.

Since $f_i(x,\overrightarrow{\lambda}\cdot\epsilon)=\lambda_{c(i,x)}^{-1}f_i(x,\epsilon)$, it follows from the definition of $\varphi_0(d,\overrightarrow{\lambda}\cdot\epsilon)$ and the equations above that

\begin{eqnarray}
&&f_i'(x,\overrightarrow{\lambda}\cdot\epsilon)=\lambda_{c(i,x)}^{-1}f_i'(x,\epsilon), 1\leq i\leq 2n\\
&&u_{i\ruling(i)}(x,\overrightarrow{\lambda}\cdot\epsilon)=\lambda_{c(i,x)}^{-1}u_{i\ruling(i)}(x,\epsilon)\lambda_{c(\ruling(i),x)}, 1\leq i\leq n\\
&&\varphi_0(d,\overrightarrow{\lambda}\cdot\epsilon) f_i(x,\overrightarrow{\lambda}\cdot\epsilon)=f_i'(x,\overrightarrow{\lambda}\cdot\epsilon)=\varphi_0(d,\epsilon) f_i(x,\overrightarrow{\lambda}\cdot\epsilon).
\end{eqnarray}

Now, if $q$ is a crossing of $\beta^>$ with $x_q$ contained in a small open interval $(x,x')$, and $q$ switches the ends $k,k+1$ of $\beta^>|_{(x,x')}$, then by definition we have

\[
\]

In addition, the above implies that

\[
r_q(\overrightarrow{\lambda}\cdot\epsilon)=-\lambda_{c(k,x)}^{-1}a_{kk+1}(x,\epsilon)\lambda_{c(k+1,\epsilon)} - \lambda_{r(q)}^{-1}\epsilon(q)\lambda_{c(q)}=\lambda_{r(q)}^{-1}r_q(\epsilon)\lambda_{c(q)},
\]

as desired.

In the Lemma above, observe that

* The diagonal $\mathbb{G}_m\xrightarrow[]{\Delta}\mathbb{G}_m^n$ acts trivially on $(\field^*)^{s(\ruling)}\times\field^{r(\ruling)}$.

* The same formula induces a torus action of $\mathbb{G}_m^n$ on $(\field^*)^{s(\ruling)}$.

Let $\beta$ be a $n$-strand positive braid so that $\beta^>$ is connected, and $\ruling\in\mathrm{NR}(\beta^>)$ be any normal ruling, then the torus action of $\mathbb{G}_m^n$ on $(\field^*)^{s(\ruling)}$ has constant stabilizer $\mathbb{G}_m\xrightarrow[]{\Delta}\mathbb{G}_m^n$, i.e. the induced action of $\mathbb{G}_m^n/\mathbb{G}_m$ on $(\field^*)^{s(\ruling)}$ is free. In particular, we always have $s(\ruling)-n+1\geq 0$.

Denote the standard coordinates of $(\field^*)^{s(\ruling)}$ by $\overrightarrow{r}=(r_q)_{q\in S(\ruling)}$. Given any $\overrightarrow{\lambda}\in\mathbb{G}_m^n$ and any $\overrightarrow{r}=(r_q)\in(\field^*)^{s(\ruling)}$ such that $\overrightarrow{\lambda}\cdot \overrightarrow{r} =\overrightarrow{r}$, we need to show that $\overrightarrow{\lambda}\in\mathbb{G}_m$.

By Remark <ref>, the normal ruling $\ruling$ of $\beta^>$ determines a finite set of eyes $\{e_1,\ldots,e_n\}$. Say, $e_i$ contains the $i$-th innermost left cusp. Draw a graph $G(\ruling)$ as follows: the  vertices are the eyes $e_1,\ldots,e_n$; for each switch $q$ of $\ruling$, it's contained in the boundaries of exactly two eyes, say $e_i, e_j$, then we add en edge between $e_i, e_j$.
We claim that the assumption $\beta^>$ is connected implies that so is the graph $G(\ruling)$. This follows immediately from the observation that, the union of the boundaries of the eyes contained in each connected component of $G(\ruling)$ comes from a finite union of connected components of $\Lambda$.

Now, let $q\in S(\ruling)$ be any switch, say $q$ connects $e_i, e_j$. Let's show that $\lambda_i=\lambda_j$.

Let $P_{i,q}$ (resp. $P_{j,q}$) the unique path contained in the lower path of $\partial e_i$ (resp. $\partial e_j$), which goes from the left cusp of $e_i$ (resp. $e_j$) to $q$. Label the arcs of $\beta^>$ contained in $P_{i,q}$ by $A_1,\ldots,A_k$, from left to right. Define $c(A_{\ell})\in\{1,2,\ldots,n\}$ so that $A_{\ell}$ is contained in $C_{c(A_{\ell})}$, then $c(A_1)=i$, and $c(A_k)\in\{r(q),c(q)\}$. For each $1\leq\ell<k$, $A_i$ meets $A_{i+1}$ at a switch $q_i$, then $r(q_i)=c(A_i), c(q_i)=c(A_{i+1})$ or $r(q_i)=c(A_{i+1}), c(q_i)=c(A_i)$. Now, the assumption $\overrightarrow{\lambda}\cdot \overrightarrow{r} =\overrightarrow{r}$ implies that

\[
\lambda_{r(q_i)}^{-1}r_{q_i}\lambda_{c(q_i)} = r_{q_i} \Leftrightarrow \lambda_{r(q_i)}=\lambda_{c(q_i)},
\]

i.e. $\lambda_{c(A_i)}=\lambda_{c(A_{i+1})}$. It follows that $\lambda_i=\lambda_{c(A_1)}=\ldots=\lambda_{c(A_k)}$. Notice that the assumption applied to $r_q$ also implies that $\lambda_{r(q)}=\lambda_{c(q)}$, then $\lambda_i=\lambda_{c(A_k)}=\lambda_{r(q)}=\lambda_{c(q)}$. Similarly, we have $\lambda_j=\lambda_{r(q)}=\lambda_{c(q)}$. Thus, we have $\lambda_i=\lambda_j$, as desired.

In other words, we have shown that for any two vertex $e_i, e_j$ of the graph $G(\ruling)$, if $e_i, e_j$ is connected by an edge, then $\lambda_i=\lambda_j$. However, the graph $G(\ruling)$ is connected, this implies that $\lambda_i=\lambda_j$ for all $1\leq i, j\leq n$. That is, $\overrightarrow{\lambda}$ is contained in the diagonal $\mathbb{G}_m$.

Now, we're ready to prove Proposition <ref>.

The part of connectedness and smoothness is Proposition <ref>.

For each normal ruling $\ruling\in\mathrm{NR}(\beta^>)$, define

\[
\modulistack_B^{\ruling}:=[\aug^{\ruling}(\beta^>,\overrightarrow{*};\field)/\mathbb{G}_m^n]
\]

\[
\modulispace_B^{\ruling}:=\mathrm{Spec}~[\mathcal{O}(\aug^{\ruling}(\beta^>,\overrightarrow{*};\field))^{\mathbb{G}_m^n}].
\]

By Lemma <ref> and Lemma <ref>, we obtain a decomposition of the moduli stack $\modulistack_B\isomorphic[\aug(\beta^>,\overrightarrow{*};\field)/\mathbb{G}_m^n]$ into locally closed sub-stacks:

\begin{equation}
\modulistack_B=\sqcup_{\ruling\in\mathrm{NR}(\beta^>)}\modulistack_B^{\ruling}.
\end{equation}

Passing to the good moduli spaces, we then obtain a decomposition of the Betti moduli space $\modulispace_B\isomorphic \mathrm{Spec}~[\mathcal{O}(\aug(\beta^>,\overrightarrow{*};\field))^{\mathbb{G}_m^n}]$ into locally closed subvarieties:

\begin{equation}
\modulispace_B=\sqcup_{\ruling\in\mathrm{NR}(\beta^>)}\modulispace_B^{\ruling}.
\end{equation}

Again by Lemma <ref> and Lemma <ref>, we have

\[
\modulispace_B^{\ruling}\isomorphic \mathrm{Spec}~[\mathcal{O}((\field^*)^{s(\ruling)}\times \field^{r(\ruling)})^{\mathbb{G}_m^n}]\isomorphic (\field^*)^{s(\ruling)-n+1}\times\field^{r(\ruling)},
\]

where the last isomorphism follows immediately from Lemma <ref>. This shows $(1)$.

It suffices to show $(2)$. For any $\ruling\in\mathrm{NR}(\beta^>)$ so that $\ruling\neq\ruling_m$, then not all the crossings are switches of $\ruling$, i.e. $s(\ruling)<s(\ruling_m)$. Notice that by <cit.>, the index $s(\ruling)+2r(\ruling)$ is a constant $C(\beta^>)$ independent of $\ruling\in\mathrm{NR}(\beta^>)$. Then $r(\ruling)>r(\ruling_m)=0$, which shows the uniqueness property of $\ruling_m$.

In addition, we have $\mathrm{dim}_{\mathbb{C}}\modulispace_B^{\ruling}(\mathbb{C})=s(\ruling)-n+1+r(\ruling)=\frac{1}{2}s(\ruling)+\frac{1}{2}C(\beta^>)-n+1<\mathrm{dim}_{\mathbb{C}}\modulispace_B^{\ruling_m}(\mathbb{C})=s(\ruling_m)-n+1$, this shows that $d:=\mathrm{dim}_{\mathbb{C}}\modulispace_B(\mathbb{C})=s(\ruling_m)-n+1$.

Finally, we show that the inclusion $\aug^{\ruling_m}(\beta^>,\overrightarrow{*};\field)\subset\aug(\beta^>,\overrightarrow{*};\field)$ is open and dense, then the open and dense property in the proposition will follow.
Label the crossings of $\beta^>$ from left to right by $q_1,\ldots, q_N$. Observe that $r_{q_1}$ is a regular function on the whole variety $\mathcal{U}_0:=\aug(\beta^>,\overrightarrow{*};\field)$, then

\[
\mathcal{U}_1:=\{\epsilon\in\mathcal{U}_0 | r_{q_1}(\epsilon)\neq 0\}
\]

is a distinguished Zariski open subset of $\mathcal{U}_0$.
Notice that by Lemma <ref>, $\aug(\beta^>,\overrightarrow{*};\field)$ is connected and smooth, then so is $\mathcal{U}_1$ and the inclusion $\mathcal{U}_1\subset \mathcal{U}_0$ is open and dense. Next, observe that $r_{q_2}$ is a regular function on the whole $\mathcal{U}_1$, then by the same argument we have that

\[
\mathcal{U}_2:=\{\epsilon\in \mathcal{U}_1 | r_{q_2}(\epsilon)\neq 0\}
\]

is connected and smooth, and the inclusion $\mathcal{U}_2\subset\mathcal{U}_1$ is open and dense. Now, by induction for $1\leq i\leq N$, we have

\[
\mathcal{U}_i:=\{\epsilon\in \mathcal{U}_{i-1} | r_{q_i}(\epsilon)\neq 0\}
\]

is connected and smooth, and the inclusion $\mathcal{U}_2\subset\mathcal{U}_1$ is open and dense.
Therefore, the inclusion

\[
\aug^{\ruling_m}(\beta^>,\overrightarrow{*};\field)=\mathcal{U}_N\subset\ldots\subset\mathcal{U}_0=\aug(\beta^>,\overrightarrow{*};\field)
\]

is open and dense.

§ THE DUAL BOUNDARY COMPLEX OF THE BETTI MODULI SPACE

§.§ Background on dual boundary complexes

We firstly review the basic concepts, properties, and tools concerning the dual boundary complex of a complex algebraic variety.

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$, by Hironaka's resolution of singularities, one can obtain a smooth compactification $\overline{X}$ with simple normal crossing divisor $D$ at infinity. We may also assume the intersections of the irreducible components of $D$ are connected. By encoding the information of the intersections of the irreducible components of $D$, we get a simplicial complex, called dual boundary complex $\mathbb{D}\boundary X$.

More precisely, let $D=\cup_{i=1}^m D_i$ with $D_i$'s the irreducible components.

The dual boundary complex $\mathbb{D}\boundary X$ is the dual complex of $D$: The 0-simplices are $1,2,\ldots,m$, with $i$ corresponds to the irreducible component $D_i$. $\mathbb{D}\boundary X$ contains the simplex $[i_1,i_2,\ldots,i_k]$ if and only if $D_{i_1}\cap D_{i_2}\cap\ldots\cap D_{i_k}$ is non-empty.

The definition of $\mathbb{D}\boundary X$ is justified by the following

The homotopy type of $\mathbb{D}\partial X$ is an invariant of $X$, and is independent of the choice of the compactification $\overline{X}$.

[Toy examples]

* For $X=\mathbb{A}^1$, we can take $\overline{X}=\mathbb{C}P^1$ with $D=\{\infty\}$. So, $\mathbb{D}\boundary(\mathbb{A}^1)=*$.

* For $X=\mathbb{G}_m$, we can take $\overline{X}=\mathbb{C}P^1$ with $D=\{0\}\sqcup\{\infty\}$. So, $\mathbb{D}\boundary(\mathbb{G}_m)=S^0$.

* For $X=\mathbb{A}^1\times\mathbb{G}_m$, we can take $\overline{X}=\mathbb{C}P^1\times\mathbb{C}P^1$, with

\[D=(D_1=\{\infty\}\times\mathbb{C}P^1)\cup (D_2=\mathbb{C}P^1\times\{0\})\cup (D_3=\mathbb{C}P^1\times\{\infty\}).
\]

So $\mathbb{D}\boundary(\mathbb{A}^1\times\mathbb{G}_m)=[1,2]\cup_{\{1\}}[1,3]$ is the union of two 1-simplices along a common 0-simplex, in particular contractible. Also, Observe that $\mathbb{D}\boundary(\mathbb{A}^1\times\mathbb{G}_m)=\{1\}\join\{2,3\}\isomorphic (\mathbb{D}\boundary\mathbb{A}^1)\join(\mathbb{D}\boundary\mathbb{G}_m)$ is the join of the dual boundary complexes of the factors of $X$.

* For $X=\mathbb{G}_m\times\mathbb{G}_m$, we can take $\overline{X}=\mathbb{C}P^1\times\mathbb{C}P^1$, with

\[
D=(D_1=\{0\}\times\mathbb{C}P^1)\cup(D_2=\{\infty\}\times \mathbb{C}P^1)\cup (D_3=\mathbb{C}P^1\times\{0\})\cup (D_4=\mathbb{C}P^1\times\{\infty\}).
\]

So $\mathbb{D}\boundary(\mathbb{G}_m\times\mathbb{G}_m)=[1,3]\cup[1,4]\cup[2,3]\cup[2,4]=\{1,2\}\join\{3,4\}\isomorphic S^1$, which is also the join $(\mathbb{D}\boundary\mathbb{G}_m)\join(\mathbb{D}\boundary\mathbb{G}_m)$.

It's then easy to see that $\mathbb{D}\boundary \mathbb{G}_m^N\isomorphic S^{N-1}$, which also follows immediately from Lemma <ref> below.

The previous example has already illustrated the compatible property of dual boundary complexes with products:

Let $X,Y$ be smooth quasi-projective varieties over $\mathbb{C}$, then there's a natural homeomorphism

\[
\mathbb{D}\boundary(X\times Y)\isomorphic (\mathbb{D}\boundary X) \join (\mathbb{D}\boundary Y)
\]

where `$\join$' stands for `join'. In particular, if $X=\mathbb{A}^1$, then $\mathbb{D}\boundary(\mathbb{A}^1\times Y)\homotopic *$ is contractible.

The following lemma then implies that, by removing a smaller piece of the form $\mathbb{A}^1\times Y$, the homotopy type of $\mathbb{D}\boundary X$ is unchanged:

Let $X$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$. If $Z\subset X$ is a smooth irreducible closed sub-variety of smaller dimension with complement $U$, and $\mathbb{D}\boundary Z\homotopic *$ is contractible. Then the natural map $\mathbb{D}\boundary X\rightarrow\mathbb{D}\boundary U$ is a homotopy equivalence.

By an inductive procedure of removing smaller pieces of the form $\mathbb{A}^1\times Y$, one can further simplify the problem of determining the homotopy type of the dual boundary complex $\mathbb{D}\boundary X$:

Let $X$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$. Let $U$ be a non-empty open subset, with the complement $Z=X\setminus U$ admitting a decomposition into finitely many locally closed sub-varieties of the form $Z_j\isomorphic \mathbb{A}^1\times Y$. If there is a total order on the indices such that $\cup_{j\leq a}Z_j$ is closed for all $a$. Then we obtain a homotopy equivalence $\mathbb{D}\boundary X\homotopic\mathbb{D}\boundary U$.

Notice that in the above statement, we no longer require that $Z_j$ is smooth. The reason is that, if some $Z_j$ is singular, we can further decompose $Z_j$ in to finitely many smooth pieces. Then an inductive procedure reduces the proof essentially to Lemma <ref>.
As in <cit.>, Lemma <ref> will be the key tool in the proof of our main Theorem <ref>.

§.§ The dual boundary complex of the Betti moduli space

Finally, we come to our main theorem, i.e. the homotopy type conjecture <ref> holds for the Betti moduli space $\modulispace_B(\mathbb{C})$ considered in this article:

Let $\beta$ be a $n$-strand positive braid so that the rainbow closure $\beta^>$ (equivalently, the cylindrical closure $(\beta\Delta^2)^{\circ}$) is connected. Then the Betti moduli space $\modulispace_B(\mathbb{C}):=\modulispace_1(\mathbb{C}P^1,\infty,(\beta\Delta^2)^{\circ};\mathbb{C})\isomorphic \modulispace_1(\mathbb{C}P^1,\infty,\beta^>;\mathbb{C})$ is a connected and smooth affine variety over $\mathbb{C}$, and the dual boundary complex of $\modulispace_B(\mathbb{C})$ is homotopy equivalent to $S^{\mathrm{dim}_{\mathbb{C}}\modulispace_B(\mathbb{C})-1}$.

By Lemma <ref> and Proposition <ref>, it suffices to show the homotopy equivalence. The strategy is to use the cell decomposition for $\modulispace_B$ (Proposition <ref>) and apply Lemma <ref>.

Let $X:=\modulispace_B(\mathbb{C})$, $U:=\modulispace_B^{\ruling_m}\isomorphic (\mathbb{C}^*)^d$, and $Z:=X\setminus U$. Here, $d:=\mathrm{dim}_{\mathbb{C}}\modulispace_B(\mathbb{C})$. For each $1\leq i<d$, define a locally closed subvariety $Z_i$ of $X$ to be the disjoint union of $\modulispace_B^{\ruling}(\mathbb{C})$ of dimension $i$, then we obtain a decomposition $Z=\sqcup_{1\leq i\leq d-1} Z_i$. By Proposition <ref>, each $Z_i$ is of the form $\mathbb{A}^1\times Y_i$, and $\cup_{j\leq a}Z_j$ is closed for all $1\leq a\leq d-1$. Hence, the conditions of Lemma <ref> are satisfied. It then follows that we have a homotopy equivalence

\[
\mathbb{D}\partial\modulispace_B(\mathbb{C})\sim \mathbb{D}\partial (U=(\mathbb{C}^*)^d)=S^{d-1}.
\]

This finishes the proof.

Let $\beta$ be a $n$-strand positive braid so that $\beta^>$ is connected, then the complex dimension $d$ of $\modulispace_B(\mathbb{C})$ is even.

Reason: For each normal ruling $\ruling\in\mathrm{NR}(\beta^>)$, recall that the filling surface $S_{\ruling}$ is the disjoint union of the eyes determined by $\ruling$, glued along the switches via half-twisted strips <cit.>. Observe that $S_{\ruling}$ is an orientable compact surface with boundary $
\beta^>$. Say, $S_{\ruling}$ is of genus $g(\ruling)$. Then the Euler characteristic of $S_{\ruling}$ is $\chi(\ruling)=2-2g(\ruling)-1$.
On the other hand, if follows from the gluing construction of $S_{\ruling}$ that $\chi(\ruling)=n-s(\ruling)$. Thus, we obtain $s(\ruling)-n+1=2g(\ruling)$, and the complex dimension of $\modulispace_B(\mathbb{C})$ is

\[
\]

Indeed, motivated by the non-abelian Hodge theory for wild character varieties mentioned in the introduction, it's natural to expect that the Betti moduli space $\modulispace_B(\mathbb{C})$ (equivalently, the augmentation variety $\aug(\beta^>;\mathbb{C})$) is hyperkähler and log Calabi-Yau (see <cit.> for related results). It's also expected that $\modulispace_B(\mathbb{C})$ is a cluster variety. This makes it very interesting to study the Hodge theory of and mirror symmetry for $\modulispace_B(\mathbb{C})$. We hope to return to this subject in the future.
For example, in the working example with $n=2$ and $\beta=\sigma_1^3$, the Betti moduli space $\modulispace_B(\mathbb{C})$ is the $A_2$-cluster variety <cit.>, and satisfies all the properties above, and is expected to be self-mirror.

§ IRREGULAR RIEMANN-HILBERT CORRESPONDENCE OVER CURVES

Let $C$ be a compact Riemann surface with punctures (or marked points) $\{p_1,p_2,\ldots,p_k\}$. The main related case in this article is $C=\mathbb{C}P^1$ with puncture $\{p_1=\infty\}$. We're interested in integrable meromorphic connections $(H,\nabla)$ on $C$ with prescribed (possibly irregular) singularities along the punctures.

In the regular case, by taking monodromy of connections, the Riemann-Hilbert correspondence states that there's an equivalence of categories between the (rank $r$) integrable meromorphic connections on $C$ with regular singularities along the punctures $\{p_1,p_2,\ldots,p_k\}$, and (rank $r$) locally constant sheaves on $C\setminus\{p_1,p_2,\ldots,p_k\}$, in other words, (rank $r$) representation of the fundamental group of $C\setminus\{p_1,p_2,\ldots,p_k\}$. Passing to moduli spaces, we then obtain a complex analytic isomorphism between a de Rham moduli space $\modulispace_{\mathrm{DR}}$ of regular connections, and a Betti moduli space $\modulispace_{\mathrm{B}}$ of locally constant sheaves (or fundamental group representations), i.e. character varieties.
The isomorphism is not algebraic, since taking the monodromy of a connection involves an exponential.

The regular Riemann-Hilbert correspondence admits a generalization to the irregular case, by specifying the formal types of irregular singularities of the connections at the punctures on the one hand, and encoding the Stokes data of the solution sheaves on the other. This is termed as irregular Riemann-Hilbert correspondence, about which we review some details here.

Fix a puncture $p_i$, let $u$ be the local coordinate near $p_i$ with $u(p_i)=0$. Let $(H,\nabla)$ be a rank $r$ holomorphic bundle with meromorphic connection on $C$ as above, hence $\nabla$ is holomorphic on a punctured disk at $p_i$.

$\mathbb{C}\{u\}:=\text{the algebra of convergent power series in $u$}$, and $\mathbb{C}\{u\}[u^{-1}]:= \text{the fraction field}$;

$\mathbb{C}[[u]]:=\text{the algebra of formal power series in $u$}$, and $\mathbb{C}((u)):=\text{the fraction field}$.

§.§ Formal classification of irregular connections

Let $(\mathscr{M},\nabla)$ be the germ of a rank $r$ meromorphic bundle with connection, termed as meromorphic connection over $\mathbb{C}\{u\}[u^{-1}]$. That is, $\mathscr{M}$ is a finite dimensional $\mathbb{C}\{u\}[u^{-1}]$-vector space, and
$\nabla:\mathscr{M}\rightarrow\mathbb{C}\{u\}[u^{-1}]du\otimes_{\mathbb{C}\{u\}[u^{-1}]}\mathscr{M}$ is a $\mathbb{C}$-linear map satisfying the Leibniz rule. By a holomorphic extension of $(\mathscr{M},\nabla)$, we mean the germ of a rank $r$ holomorphic bundle with meromorphic connection $(\mathscr{H},\nabla)$ such that $(\mathscr{M},\nabla)=(\mathscr{H},\nabla)|_{\mathbb{C}\{u\}[u^{-1}]}$, i.e. $\mathscr{M}=\mathscr{H}\otimes_{\mathbb{C}\{u\}}\mathbb{C}\{u\}[u^{-1}]$.

In our case, let $(\mathscr{H},\nabla):=(H,\nabla)|_{\mathbb{C}\{u\}}$ be the germ of $(H,\nabla)$ at $p_i$. Then $(\mathscr{M},\nabla)=(\mathscr{H},\nabla)|_{\mathbb{C}\{u\}[u^{-1}]}$, and( $\mathscr{H},\nabla)$ defines a holomorphic extension of $(\mathscr{M},\nabla)$.

Taking a trivialization $\mathscr{M}\isomorphic \mathbb{C}\{u\}[u^{-1}]^r$, the connection $\nabla$ can be written as:

\begin{eqnarray}\label{eqn:meromorphic_connection_local_model}
\nabla=d+A_n\frac{du}{u^n}+\ldots+A_2\frac{du}{u^2}+(A_1\frac{du}{u}+B(u)du).
\end{eqnarray}

where $A_i\in\mathrm{gl}_r(\mathbb{C})$, and $B(u)\in\mathrm{gl}_r(\mathbb{C}\{u\})$ is holomorphic. Recall that $\nabla$ has a regular singularity at $0$ if and only if $A_i=0$ for $i>1$, otherwise we say $\nabla$ has an irregular singularity at $0$.

For any $f\in t^{-1}\mathbb{C}[t^{-1}]$, denote $\mathscr{E}^f:=(\mathbb{C}\{t\},d-df)$, a rank $1$ holomorphic bundle with meromorphic connection.
The formal classification of $(\mathscr{M},\nabla)$ is given by the Turrittin-Levelt formal decomposition theorem:

There exists a finite base change $\pi_N:\mathrm{Spec}\mathbb{C}\{t\}\rightarrow\mathrm{Spec}\mathbb{C}\{u\}$ with $\pi_N(t):=t^N=u$, distinct polynomials $g_i\in t^{-1}\mathbb{C}[t^{-1}], 1\leq i\leq s$, and $\mathbb{C}\{t\}[t^{-1}]$-vector spaces $\mathscr{R}_i$ with meromorphic connections $\nabla_i$ having a regular singularity at $t=0$, such that

\begin{eqnarray}\label{eqn:Turrittin-Levelt}
\pi_N^*(\mathscr{M},\nabla)\otimes_{\mathbb{C}\{t\}[t^{-1}]}\mathbb{C}((t))\isomorphic [\oplus_{i=1}^s\mathscr{E}^{g_i}\otimes_{\mathbb{C}\{t\}[t^{-1}]} (\mathscr{R}_i,\nabla_i)]\otimes_{\mathbb{C}\{t\}[t^{-1}]}\mathbb{C}((t))
\end{eqnarray}

\[\nabla_i=d-C_i\frac{dt}{t}\]

with respect to a trivialization $\mathscr{R}_i\isomorphic \mathbb{C}\{t\}[t^{-1}]^{r_i}$, such that:

\begin{eqnarray}\label{eqn:Jordan_normal_form}
\text{$C_i$ is of Jordan normal form, and for any two eigenvalues $\lambda,\mu\in\mathbb{C}$, $\lambda-\mu\in\mathbb{Z}\Rightarrow \lambda=\mu$.}
\end{eqnarray}

Moreover, the formal decomposition is unique in the following sense:

the positive integer $N$ and the polynomials $g_i$ with multiplicity $r_i$ for all $i$, are uniquely determined;

for each $i$, the Jordan normal form $C_i$ satisfying (<ref>) is uniquely determined, up to permuting the Jordan blocks and adding to each block an integer multiple.

Now, we can make the following
Let $(\mathscr{M},\nabla)$ be a meromorphic connection over $\mathbb{C}\{u\}[u^{-1}]$, with its Turrittin-Levelt formal decomposition as in the theorem.
The formal type of irregular singularity of $(\mathscr{M},\nabla)$ is simply the set of polynomials $\{g_i\in u^{-\frac{1}{N}}\mathbb{C}[u^{-\frac{1}{N}}]\}$.

Given $(\mathscr{M},\nabla)$ with the Turrittin-Levelt formal decomposition as in the theorem, there's a group action of $C_N=\mathbb{Z}/N$ on the set $\{(g_i,r_i),[C_i]\}$.

Here, $C_N=\langle\sigma\rangle$ is the Galois group of $\pi_N:\mathrm{Spec}\mathbb{C}\{t\}\rightarrow\mathrm{Spec}\mathbb{C}\{u\}$, in which $\sigma$ acts by $\sigma\cdot t=\zeta t$ with $\zeta=e^{\frac{2\pi i}{N}}$ the $N$-th primitive root of unity. And, $[C_i]$ denotes the equivalence class of $C_i$ up to permuting the Jordan blocks and adding to each block an integer multiple.

$\pi_N\circ\sigma=\pi_N$ implies that

\begin{eqnarray*}
\pi_N^*(\mathscr{M},\nabla)\otimes_{\mathbb{C}\{t\}[t^{-1}]}\mathbb{C}((t))\isomorphic [\oplus_{i=1}^s\mathscr{E}^{\sigma^*g_i}\otimes_{\mathbb{C}\{t\}[t^{-1}]} (\sigma^*\mathscr{R}_i,\sigma^*\nabla_i)]\otimes_{\mathbb{C}\{t\}[t^{-1}]}\mathbb{C}((t)),
\end{eqnarray*}

and with respect to the trivialization $\sigma^*\mathscr{R}_i\isomorphic\sigma^*(\mathbb{C}\{t\}[t^{-1}]^{r_i})\isomorphic\mathbb{C}\{t\}[t^{-1}]^{r_i}$, we have $\sigma^*\nabla_i=d-C_i\frac{d(\sigma^*t)}{\sigma^*t}=d-C_i\frac{dt}{t}$. It follows from the uniqueness of the Turrittin-Levelt formal decomposition that

\[
\{(\sigma^*(g_i,r_i),[C_i]),1\leq i\leq s\}=\{(g_i,r_i,[C_i]),1\leq i\leq s\}.
\]

In other words, the group $C_N=\langle\sigma\rangle$ acts on $\{(g_i,r_i,[C_i])\}$.

§.§ Birkhoff extension

Let $(\mathscr{M},\nabla)$ be the germ of a meromorphic bundle with connection as above.

The functor $\mathscr{R}_0$ of taking germ at $0\in\mathbb{C}P^1$ is an equivalence of categories from

\[
(\text{finite rank algebraic vector bundles with connections on $\mathbb{A}^1\setminus\{0\}$ with regular singularity at $\infty$})
\]

\[
(\text{finite dimensional $\mathbb{C}\{u\}[u^{-1}]$-vector spaces with meromorphic connections}).
\]

Denote by $\mathscr{B}_0$ a fixed inverse functor of $\mathscr{R}_0$. It's called the algebraization or Birkhoff extension functor.

The proof essentially uses regular Riemann-Hilbert correspondence. We sketch the main ideas for the inverse construction of the functor $\mathscr{R}_0$, i.e. Birkhoff extension.

Given any $(\mathscr{M},\nabla)$, say a $r$-dimensional $\mathbb{C}\{u\}[u^{-1}]$-vector space with meromorphic connection, we want to construct $\mathscr{B}_0(\mathscr{M},\nabla)$.

Step 1: Take a basis for $\mathscr{M}$, with respect to which we can write the connection $\nabla$ as in (<ref>).
Recall $B(u)\in\mathrm{gl}_r(\mathbb{C}\{u\})$ is a convergent power series in $u$ with matrix coefficients, with some positive radius $\epsilon$ of convergence. Then the same formula defines an extension of $(\mathscr{M},\nabla)$, again denoted by $(\mathscr{M},\nabla)$, to the punctured disk $D_{\epsilon}^*$ of radius $\epsilon$ centered at $0$.

Step 2: Take the local system $\mathscr{M}^{\nabla}$ of local $\nabla$-horizontal sections on $D_{\epsilon}^*$, then we have identification $(\mathscr{M},\nabla)\isomorphic(\mathscr{O}_{D_{\epsilon}^*}\otimes_{\mathbb{C}}\mathscr{M}^{\nabla}, d\otimes 1)$. Notice that the data of the local system $\mathscr{M}^{\nabla}$ on $D_{\epsilon}^*$ is the same as that of representations of the fundamental group $\pi_1(D_{\epsilon}^*,*)\isomorphic\mathbb{Z}$, with $*$ some fixed base point. The latter is identical to $\pi_1(\mathbb{A}^1\setminus\{0\},*)$, so $\mathscr{M}^{\nabla}$ extends to a local system $M^{\nabla}$ on $\mathbb{A}^1\setminus\{0\}$. Now, $(M,\nabla):=(\mathscr{O}_{\mathbb{A}^1\setminus\{0\}}\otimes_{\mathbb{C}}M^{\nabla},d\otimes 1)$ gives a holomorphic extension of $(\mathscr{M},\nabla)$ to $\mathbb{A}^1\setminus\{0\}$.

Step 3: The local system $M^{\nabla}$ defines a monodromy $T$ (au automorphism of $\mathbb{C}^r$) near $\infty$. By a local study of regular connections (regular Riemann-Hilbert correspondence), $T$ corresponds to a holomorphic bundle with connection $(\mathscr{H},\nabla)$ on a small disk $D_{\infty}$ centered at $\infty$, whose restriction to $D_{\infty}^*=D_{\infty}\setminus\{\infty\}$ coincides with $(M,\nabla)|_{D_{\infty}^*}$. Morally, $(\mathscr{H},\nabla)$ is defined via:

\[
\nabla=d+\frac{\mathrm{log}T}{2\pi i}\frac{du}{u}.
\]

This is essentially <cit.>.

Step 4: Now, by gluing $(M,\nabla)$ and $(\mathscr{H},\nabla)$ over $D_{\infty}^*$, we obtain a holomorphic bundle with connection $(M,\nabla)$ on $\mathbb{A}^1\setminus\{0\}$, with regular singularity at $\infty$. By applying a theorem of GAGA type, we can make $(M,\nabla)$ algebraic. This gives a Birkhoff extension of the germ $(\mathscr{M},\nabla)$ we started with.

Finally, the uniqueness of the construction above follows from fixing a choice of $\mathrm{log}T$ (a priori, multi-valued) systematically.

We remark that, for our purpose in this article, we really only need the first step.

§.§ The main asymptotic existence theorem

Let $(\mathscr{M},\nabla)$ be a rank $r$ meromorphic connection over $\mathbb{C}\{u\}[u^{-1}]$.
Take the Birkhoff extension $(M,\nabla)=\mathscr{B}_0(\mathscr{M},\nabla)$. Let $D^2\subset \mathbb{C}P^1$ be a disk centered at $0$. The sheaf of local $\nabla$-horizontal sections of $M^{\mathrm{an}}$ defines a locally constant sheaf $\sol$ on $D^2\setminus\{0\}$. Take the Turrittin-Levelt formal decomposition of $(\mathscr{M},\nabla)$ as in (<ref>),
with formal type of irregular singularity $\tau=\{N\in\mathbb{N};(g_i,r_i)_{i=1}^s:g_i\in t^{-1}\mathbb{C}[t^{-1}] \text{ distinct, and } r_i\in\mathbb{N}\}$, where $t=u^{\frac{1}{N}}$.

The formal decomposition implies that, with respect to the corresponding formal basis of $\mathscr{M}((t))=\mathscr{M}\otimes_{\mathbb{C}\{t\}[t^{-1}]}\mathbb{C}((t))$, the columns of

\begin{eqnarray*}
\mathrm{diag}(e^{g_i}t^{C_i}:1\leq i\leq s)
\end{eqnarray*}

give $r$ linearly independent formal $\nabla$-horizontal sections near $u=0$, denoted by $(\hat{S_1},\ldots,\hat{S}_s)$. Here $\hat{S}_i=(\hat{S_i}^1,\ldots, \hat{S_i}^{r_i})$ consists of $r_i$ sections corresponding to the $r_i$ columns of the block $e^{g_i}t^{C_i}$.
Specifically, let $(\vec{e}_1(t),\ldots,\vec{e}_s(t))$ be the formal basis, where $\vec{e}_i(t)=(e_i^1(t),\ldots,e_i^{r_i}(t))$ with $e_i^j(t)\in\mathscr{M}((t))=\mathscr{H}((t))$. Here, $(\mathscr{H},\nabla)$ is any holomorphic extension of $(\mathscr{M},\nabla)$.
Then $\hat{S_i}=\vec{e}_i(t)e^{g_i}t^{C_i}$. Equivalently, $e^{-g_i}\hat{S}_it^{-\mathrm{D}_i}=\vec{e}_i(t)t^{\mathrm{N}_i}$, where $C_i=\mathrm{D}_i+\mathrm{N}_i$ with $\mathrm{D}_i=\mathrm{diag}(\lambda_i^1,\ldots,\lambda_i^{r_i})$ the semisimple part and $\mathrm{N}_i$ the nilpotent part. Hence, we can write

\begin{eqnarray}
\hat{S_i}^j=e^{g_i(t)}t^{\lambda_i^j}\sum_{n=n_{i,j}}^{\infty}\sum_{h=0}^{r_i-1}a_{i,h,n}^jt^n(\mathrm{log}t)^h,\quad a_{i,h,n}^j\in\mathscr{H}.
\end{eqnarray}

The main asymptotic existence theorem states that, one can lift the formal sections to honest local $\nabla$-horizontal sections over any small sector, with the same asymptotics. More precisely, we have

For any $\varphi_0\in S^1$, there exists a neighborhood $V_{\epsilon}=(\varphi_0-\epsilon,\varphi_0+\epsilon)$ of $\varphi_0$ and some $0<\delta\ll1$ such that,
for any choices of branches of $t=u^{\frac{1}{N}}$ and $\mathrm{log}(t)=\frac{1}{N}\mathrm{log}(u)$ over $\mathrm{Sec}(V_{\epsilon})$, there exists a basis of $r$ sections $\{S_i^j:1\leq i\leq s, 1\leq j\leq r_i\}$ of $\sol$ over $Sec(V_{\epsilon})\cap D_{\delta}^2$, such that
$S_i^j\sim \hat{S_i}^j$. That is,

\begin{eqnarray}\label{eqn:asymptotic_estimate}
||S_i^j(u)-\hat{S}_{i,L}^j(u)||=|e^{g_i(t)}t^{\lambda_i^j}|\cdot o(|t|^L)=e^{\mathrm{Re}(g_i(t))}\cdot o(|t^{\lambda_i^j+L}|), u\rightarrow 0 \text{ in $\mathrm{Sec}(V_{\epsilon})\cap D_{\delta}^2$},
\end{eqnarray}

where $\hat{S}_{i,L}^j$ is the finite partial sum

\[
\hat{S}_{i,L}^j(u):=e^{g_i}t^{\lambda_i^j}\sum_{n\leq L}\sum_{h=0}^{r_i-1}a_{i,h,n}^jt^n(\mathrm{log}t)^h.
\]

And, $||\cdot||$ is any Hermitian norm on the holomorphic bundle $\mathscr{H}$ induced from a trivialization $\mathscr{H}\isomorphic\mathbb{C}\{t\}^r$.

For simplicity, we write $S_i:=(S_i^1,\ldots, S_i^{r_i})$, which consists of $r_i$ sections.

For each $i$ and any small open $V\subset S_u^1$, the set of all the branches of $g_i$ over $\mathrm{Sec}(V)$, denoted by $\{g_i^k\}$, is identical to the orbit $C_N\cdot g_i$ in Corollary <ref>, where in the later we fix a choice of the branches of $t=u^{\frac{1}{N}}$ over $\mathrm{Sec}(V)$.

§.§ Stokes data

As above, let $(\mathscr{M},\nabla)$ be a meromorphic connection over $\mathbb{C}\{u\}[u^{-1}]$.
Take the Birkhoff extension $(M,\nabla)=\mathscr{B}_0(\mathscr{M},\nabla)$, and solution sheaf $\sol$ over a disk $D^2\subset \mathbb{C}P^1$ centered at $0$.
Let $\pi:\tilde{D^2}\rightarrow D^2$ be the real oriented blow-up at $0$ and $j:D^2\setminus\{0\}\hookrightarrow \tilde{D^2}$ be the natural inclusion, then define $\mathcal{S}:=j_*\sol|_{\pi^{-1}(0)=S^1}$.

There's a natural local filtration $\{\mathcal{S}_{\leq f}\}_{f\in{\Del}}$ by subsheaves on $\mathcal{S}$, termed as Stokes filtration. Here, $\Del$ is Deligne's sheaf of “growth rates”. $\{\mathcal{S}_{\leq f}\}_{f\in{\Del}}$ means that, for any open subset $U\subset S^1$ and $f\in\Del(U)$, $\mathcal{S}_{\leq f}\subset\mathcal{S}|_U$ defines a sub-sheaf, which induces a stalk-wise increasing filtration on $\mathcal{S}_{\varphi}, \varphi\in S^1$. More precisely,

* Deligne's sheaf of “growth rates”: $\Del$ is a locally constant sheaf of $\infty$-dimensional complex vector spaces on $S^1$ such that, for any open subset $U$ of $S^1$, let $\mathrm{Sec}(U)$ to be the sector in $D^2$ induced by $U$, i.e. $\mathrm{Sec}(U):=\{re^{i\varphi}\in D^2:r>0,\varphi\in U\}$, then

\[
\Del(U):=\{f\in u^{-\frac{1}{N}}\mathbb{C}[u^{-\frac{1}{N}}]: \text{ for some $N\in\mathbb{N}$ and some branch $u^{\frac{1}{N}}$ defined on $\mathrm{Sec}(U)$}\}.
\]

In addition, $\Del$ is stalk-wise naturally (partially) ordered: for any $f\neq g$ in $\Del(U)$ and $\varphi\in U$, we can write

\[f-g=c_au^a+(\text{higher order terms})\]

for some $a\in\mathbb{Q}_{<0}$ and $c_a\neq 0$. We say $f<_{\varphi}g$ if and only if $\mathrm{Re}(c_ae^{i\varphi a})<0$.

* For any $f\in\Del(U)$, define $\mathcal{S}_{\leq f}\subset\mathcal{S}|_U$ stalk-wise as follows: for any $\varphi_0\in U\subset S^1$, set

\begin{eqnarray*}
(\mathcal{S}_{\leq f})_{\varphi_0}&:=&\{s\in\mathcal{S}_{\varphi_0}=\sol(\mathbb{R}_+e^{i\varphi_0}\cap D^2): ||(e^{-f}\cdot s)(re^{i\varphi})||=O(r^{-N}) (r\rightarrow 0),\\
&&\text{ uniformly in a neighborhood of $\varphi_0$, and for some $N\gg 0$}\}.
\end{eqnarray*}

where $||\cdot||$ is any Hermitian norm on the meromorphic bundle $\mathscr{M}$ induced from a trivialization $\mathscr{M}\isomorphic\mathbb{C}\{u\}[u^{-1}]^r$. In other words, $s\in(\mathcal{S}_{\leq f})_{\varphi_0}$ if and only $e^{-f}\cdot s$ has moderate (i.e. at most polynomial) growth in all directions near $\varphi_0$.

By the ordering on $\Del_{\varphi}$, $\{\mathcal{S}_{\leq f}\}_{f\in\Del_{\varphi}}$ then defines an increasing filtration of $\mathcal{S}_{\varphi}$. The data of stalk-wise filtration $\{\mathcal{S}_{\leq f}\}_{f\in\Del}$ defined above is called the Stokes filtration of $\mathcal{S}$, and we say $\mathcal{S}$ is a $\Del$-filtered sheaf.

It turns out, we can encode the data of the Stokes filtration by a simpler object, called Stokes Legendrian.
\begindefinition/proposition[Stokes Legendrian]
Take the Turrittin-Levelt formal decomposition of $(\mathscr{M},\nabla)$ as in (<ref>),
with formal type of irregular singularity $\tau=\{g_i\in t^{-1}\mathbb{C}[t^{-1}], t=u^{\frac{1}{N}}\}$.

* Fix $0<\epsilon\ll 1$, for each $i$, we define a real-valued (analytic) function $\mathrm{Re}^t_{\epsilon}(g_i):S_t^1\rightarrow\mathbb{R}$ via $\mathrm{Re}^t_{\epsilon}(g_i)(\theta):=\mathrm{Re}(g_i(\epsilon^{\frac{1}{N}} e^{i\theta}))$, which can be regarded as a multi-valued function $\mathrm{Re}^u_{\epsilon}(g_i)=\mathrm{Re}^t_{\epsilon}(g_i)\circ\pi_N^{-1}$ on $S_u^1$ via $\pi_N:S_t^1\rightarrow S_u^1: \pi_N(t)=t^N$, with local branches $\mathrm{Re}^u_{\epsilon}(g_i^k)$ (for $g_i^k$'s in Remark <ref>). Notice that, as a multi-valued function on $S_u^1$, we always have $\mathrm{Re}^u_{\epsilon}(g_i)=\mathrm{Re}^u_{\epsilon}(\sigma^*g_i)$, for $\sigma$ in Corollary <ref>.

The union of the graphs of $\mathrm{Re}^u_{\epsilon}(g_i)$ in $S_u^1\times\mathbb{R}_z$ is called the Stokes diagram of $(\mathscr{M},\nabla)$.

The Stokes Legendrian of $(\mathscr{M},\nabla)$ is the Legendrian link $\Lambda$ in $J^1S_u^1\times\mathbb{R}_z\isomorphic T^{\infty,-}(S_u^1\times\mathbb{R}_z)$ whose front projection is the Stokes diagram. By definition, the Stokes Legendrian $\Lambda$ (up to Legendrian isotopy) depends only on the formal type $\tau$.

* By the main asymptotic existence theorem, the Stokes filtration is determined by a local system of finite sets $\Del_{(\mathscr{M},\nabla)}\subset\Del$. For any (small) open $U\subset S^1$, set

\[
\Del_{(\mathscr{M},\nabla)}(U):=\{g_i\in u^{-\frac{1}{N}}\mathbb{C}[u^{-\frac{1}{N}}]\}
\]

where we fix a branch of $u^{\frac{1}{N}}$ over $U$ as in Theorem <ref>. Notice that by Remark <ref>, $\Del_{(\mathscr{M},\nabla)}(U)$ contains all the branches $\{g_i^k\}$ of $g_i$ over $U$, for each $i$.
By theorem <ref>, each $g_i$ determines a locally constant sub-sheaf $\mathcal{S}_{g_i}$ of $\mathcal{S}|_U$, spanned by the local sections $S_i^j$. Then for any $f\in\Del(U)$ and $\varphi\in U$, have

\[
(\mathcal{S}_{\leq f})_{\varphi}=\oplus_{g_i\leq_{\varphi}f}\mathcal{S}_{g_i}.
\]

* Via the projection $S^1\times\mathbb{R}_z\rightarrow S^1\isomorphic\pi^{-1}(0)$, we can regard $\mathcal{S}$ as a locally constant sheaf on $S^1\times\{z\gg0\}$. Fix $0<\epsilon\ll 1$, define a sheaf $\mathcal{S}^{\epsilon}$ on $S^1\times\mathbb{R}_z$, termed as Stokes sheaf of $(\mathscr{M},\nabla)$,  via

\begin{eqnarray}
\mathcal{S}^{\epsilon}_{\varphi,z}:=\oplus_{\mathrm{Re}^u_{\epsilon}(g_i)(\varphi)<z}\mathcal{S}_{g_i}.
\end{eqnarray}

Here, by fixing a branch of $u^{\frac{1}{N}}$ near $\varphi$ as above, we're fixing a branch of $\mathrm{Re}^u_{\epsilon}(g_i):S_u^1\rightarrow\mathbb{R}_z$.
Equivalently, the stalk of $\mathcal{S}^{\epsilon}$ at $(\varphi,z)$ consists of local sections of the solution sheaf $\sol$ over a small sector near $\varphi$, which grow at most polynomially faster than $e^{g_i^k}$, for some $g_i^k$ with $\mathrm{Re}(g_i^k)(\epsilon e^{i\varphi})<z$.

Then $\mathcal{S}^{\epsilon}|_{\{z\gg 0\}}=\mathcal{S}$, and the local $\Del_{(\mathscr{M},\nabla)}$-filtration $\{\mathcal{S}_{\leq g_i}\}_{g_i\in\Del_{(\mathscr{M},\nabla)}}$ is equivalent to the property that, the micro-support at infinity of $\mathcal{S}^{\epsilon}$ is equal to the Stokes Legendrian $\Lambda$ defined above.

* Moreover, passing to the associated local $\Del_{(\mathscr{M},\nabla)}$-graded local system of $(\mathcal{S},\{\mathcal{S}_{\leq g_i}\})$ over $S_u^1$

\[
\mathrm{Gr_f\mathcal{S}}_{\varphi}:=(\mathcal{S}_{\leq f})_{\varphi}/\oplus_{g_i <_{\varphi} f}(\mathcal{S}_{\leq g_i})_{\varphi},
\]

amounts to taking the microlocal monodromy of the Stokes sheaf $\mathcal{S}^{\epsilon}$ on $S_u^1\times\mathbb{R}_z$, i.e. a local system over the Stokes Legendrian $\Lambda$.

\enddefinition/proposition

In fact we can be more concrete. Let $(\mathscr{M}^{\land}=\mathscr{M}\otimes_{\mathbb{C}\{u\}[u^{-1}]}\mathbb{C}((u)),\nabla)$ be the formal completion. Take the Turrittin-Levelt formal decomposition of $(\mathscr{M}^{\land},\nabla)$ as in (<ref>), remembering the group action of $C_N$.
The connected components of the Stokes Legendrian $\Lambda$ of $(\mathscr{M},\nabla)$ are in one-to-one correspondence with the orbits $\{C_N\cdot g_i\}$.
The connected component corresponding to $C_N\cdot g_i$, denoted by $\Lambda_{[i]}$, has front projection the graph of the (multi-valued) function $\mathrm{Re}^u_{\epsilon}(g_i)$. In particular, $\Lambda_{[i]}$ is a $n_i$-to-$1$ cover of $S_u^1$ via the projection $\Lambda_{[i]}\hookrightarrow J^1S^1\rightarrow S^1$, with $n_i:=|C_N\cdot g_i|$. And, $N=\mathrm{lcm}(n_1,\ldots,n_s)$.

Alternatively, the microlocal monodromy of $\mathcal{S}^{\epsilon}$ over $\Lambda_{[i]}$ is a rank $r_i$ local system, obtained as follows: the graph of $\mathrm{Re}^t_{\epsilon}(g_i)$ defines a Legendrian knot $\Lambda^t_i\hookrightarrow J^1S_t^1$. It's naturally identified with $S_t^1$, and equipped with an action of $\mathrm{Stab}_{C_N}(g_i)=<\sigma^{n_i}>\isomorphic\mathbb{Z}/(N/n_i)$. The quotient by the group is the natural map $\Lambda^t_i\rightarrow \Lambda_{[i]}$, in particular a $(N/n_i)$-to-$1$ cover.

Now, by taking local horizontal sections, the regular factor $(\mathscr{R}_i,\nabla_i)$ in the Turrittin-Levelt formal decomposition gives a well-defined local system of rank $r_i$ on $\Lambda_i^t$. The local system has monodromy $\exp(2\pi iC_i)$. The uniqueness of the decomposition implies that $\mathrm{Stab}_{C_N}(g_i)$ preserves $(\mathscr{R}_i,\nabla_i)$, hence the local system is $\mathrm{Stab}_{C_N}(g_i)$-equivariant, equivalently, descends to a local system of rank $r_i$ on $\Lambda_{[i]}$ as desired.
The monodromy $T_i$ of the latter satisfies $T_i^{N/n_i}=\exp(2\pi iC_i)$. We see that $T_i$ is not uniquely determined by $[C_i]$, but by $[C_i]$ together with the group action of $\mathrm{Stab}_{C_N}(g_i)$ on the local system over $\Lambda_i^t$ determined by $[C_i]$.

As a consequence, fix a formal type $\tau=\{g_i\in u^{-\frac{1}{N}}\mathbb{C}[u^{-\frac{1}{N}}]\}$, with associated Stokes Legendrian $\Lambda$, we obtain a commutative diagram of functors

\[
\begin{tikzcd}
\left(
\text{
\minipage[c]{2.5in}
meromorphic connections $(\mathscr{M},\nabla)$ over $\mathbb{C}\{u\}[u^{-1}]$ with prescribed formal type $\tau$
\endminipage
\right)
\arrow{r}{\mathrm{St}}\arrow{d}{F}
\Sh_{\Lambda}(S_u^1\times\mathbb{R}_z,\{z=-\infty\})\arrow{d}{\mumon}\\
\left(
\text{
\minipage[c]{2.5in}
formal connections $(\mathscr{M}^{\land}, \nabla)$ over $\mathbb{C}((u))$ with prescribed formal type $\tau$
\endminipage
\right)
\arrow{r}{\overline{\mathrm{St}}}
\end{tikzcd}
\]

where $\mathrm{St}$ sends $(\mathscr{M},\nabla)$ to its Stokes sheaf $\mathcal{S}^{\epsilon}$, $F$ is the formal completion. $\Sh_{\Lambda}(S_u^1\times\mathbb{R}_z,\{z=-\infty\})$ is the full sub-category of $\Sh_{\Lambda}(S_u^1\times\mathbb{R}_z)$ whose objects have zero stalks at $z\ll 0$, and $\mumon$ is the microlocal monodromy.
The definition of $\overline{\mathrm{St}}$ needs some attention:

* Given a formal connection $(\mathscr{M}^{\land},\nabla)$, we lift it to a meromorphic connection $(\mathscr{M},\nabla)$ <cit.>, and apply to $(\mathscr{M},\nabla)$ the “associated graded local system” construction in Definition <ref>.(4), then $\overline{\mathrm{St}}(\mathscr{M}^{\land},\nabla):=\mathrm{Gr}\circ\mathrm{St}(\mathscr{M},\nabla)$, which is just the commutativity of the diagram. The definition is independent of the meromorphic lift <cit.>.

* Alternatively, we can define $\overline{\mathrm{St}}$ directly via the Turrittin-Levelt formal decomposition of $(\mathscr{M}^{\land},\nabla)$ equipped with the group action of $C_N$, described as in the previous paragraph.

Note: here by a formal connection $(\mathscr{M}^{\land},\nabla)$ with prescribed formal type $\tau$, we mean its formal type of irregular singularity $\tau(\mathscr{M}^{\land},\nabla)$ is a subset of $\tau$.

The local form of irregular Riemann-Hilbert correspondence now states:

The functors $\mathrm{St}$ and $\overline{\mathrm{St}}$ are equivalences of categories.

§.§ Irregular Riemann-Hilbert correspondence

Finally, we come back to our global setting and state the irregular Riemann-Hilbert correspondence for curves. Let $C$ be a Riemann surface with punctures $p_1,\ldots,p_k$. Fix a formal type of irregular singularity $\tau_i$ at each $p_i$. Let $\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})$ be the category of meromorphic connections on $C$ with prescribed formal irregular singularity $\tau_i$ at $p_i$. Notice that the case $\tau_i=\{0\}$ corresponds to the regular singularity at $p_i$.

Let $\Lambda_i$ be the Stokes Legendrian associated to $\tau_i$. Take the real oriented blow-up $\pi_i:\mathrm{Bl}_{p_i}(C)\rightarrow C$ at each $p_i$, and glue $S^1\times\mathbb{R}_z$ into $\mathrm{Bl}_{p_i}(C)$ along the identification $S^1\times\{\infty\}\isomorphic\pi^{-1}(p_i)$. The resulting surface can again be identified with the punctured surface $C\setminus\{p_i\}$. Now, $\Lambda_i\hookrightarrow J^1S^1=T^*S^1\times\mathbb{R}_z$ becomes a Legendrian link in $T^{\infty}C$ whose front projection lives near $p_i$ via the identification $S^1\times\mathbb{R}_z\isomorphic D_{\delta}^*(p_i)$, where $D_{\delta}^*(p_i)$ is a small punctured disk in $C$ centered at $p_i$. Then recall that $\mathcal{C}_{\mathrm{B}}(C,\{p_i\},\{\Lambda_i\})$ is the full sub-category of $\Sh_{\cup\Lambda_i}(C)$ of constructible sheaves whose stalk at each $p_i$ vanishes.

Let $(M,\nabla)$ be any meromorphic connection in $\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})$, take the solution sheaf $\sol$ of local $\nabla$-horizontal sections and apply the Stokes sheaf construction to $\sol$ at each $p_i$, the previous subsection shows that we then obtain a global Stokes sheaf $\mathcal{S}^{\epsilon}\in\mathcal{C}_{\mathrm{B}}(C,\{p_i\},\{\Lambda_i\})$. As in the local case, this gives a well-defined global Stokes sheaf functor $\mathrm{St}:\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})\rightarrow\mathcal{C}_{\mathrm{B}}(C,\{p_i\},\{\Lambda_i\})$, which fits into the following commutative diagram of functors:

\begin{eqnarray}
\begin{tikzcd}
\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})\arrow{r}{\mathrm{St}}\arrow{d}{\prod_iF_i} & \mathcal{C}_{\mathrm{B}}(C,\{p_i\},\{\Lambda_i\})\arrow{d}{\prod_i\mumon_i}\\
\prod_i\left(
\text{
\minipage[c]{2.5in}
formal connections $(\mathscr{M}_i^{\land}, \nabla)$ over $\mathbb{C}((u))$ with prescribed formal type $\tau_i$
\endminipage
\right)
\arrow{r}{\prod_i\overline{\mathrm{St}}_i}
\end{tikzcd}
\end{eqnarray}

Now the global form of irregular Riemann-Hilbert correspondence over curves states:

The global Stokes sheaf functor $\mathrm{St}$ is an equivalence of categories.

By definition, $\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})$ is a fiber product of categories:

\[
\mathcal{C}_{\mathrm{dR}}(C,\{p_i\},\{\tau_i\})=\mathcal{C}_{\mathrm{dR}}(C,\{p_i\})\times_{\prod_i\mathcal{L}oc(S^1(p_i))}
\prod_i\left(
\text{
\minipage[c]{2.5in}
formal connections $(\mathscr{M}_i^{\land}, \nabla)$ over $\mathbb{C}((u))$ with prescribed formal type $\tau_i$
\endminipage
\right)
\]

where $\mathcal{C}_{\mathrm{dR}}(C,\{p_i\})$ is the category of meromorphic connections on $C$ with regular singularities at each $p_i$, and $S^1(p_i)$ is the boundary circle of a small closed disk $\overline{D}_{\delta}(p_i)$ centered at $p_i$ whose interior contains the front projection of $\Lambda_i$.

Similarly, we have a fiber product of categories:

\[
\mathcal{C}_{\mathrm{B}}(C,\{p_i\},\{\Lambda_i\})=\mathcal{C}_{\mathrm{B}}(C,\{p_i\})\times_{\prod_i\mathcal{L}oc(S^1(p_i))} \prod_i\Sh_{\Lambda_i}(D^*_{\delta}(p_i))_0
\]

where $\mathcal{C}_{\mathrm{B}}(C,\{p_i\})$ is just the category of local systems on $C\setminus\{p_i\}$, and $\Sh_{\Lambda_i}(D^*_{\delta}(p_i))_0=\Sh_{\Lambda_i}(S^1\times\mathbb{R}_z)_0$ under the identification $D^*_{\delta}(p_i)\isomorphic S^1\times\mathbb{R}_z$ mentioned above.

Now, notice that the global Stokes sheaf functor $\mathrm{St}$ is compatible with the two fiber products in the obvious sense. Then, the fact that $\mathrm{St}$ is an equivalence in the theorem follows form the regular Riemann-Hilbert correspondence

\[
\sol:\mathcal{C}_{\mathrm{dR}}(C,\{p_i\})\xrightarrow[]{\sim}\mathcal{C}_{\mathrm{B}}(C,\{p_i\})
\]

and the local form of irregular Riemann-Hilbert correspondence in Theorem <ref>:

\[
\prod_i\overline{St}_i:\prod_i\left(
\text{
\minipage[c]{2.5in}
formal connections $(\mathscr{M}_i^{\land}, \nabla)$ over $\mathbb{C}((u))$ with prescribed formal type $\tau_i$
\endminipage
\right)
\xrightarrow[]{\sim}
\prod_i\Sh_{\Lambda_i}(D^*_{\delta}(p_i))_0.
\]

§.§ Example

Take $C=\mathbb{C}P^1$ with one puncture $p=\infty$ and let $z$ be the standard complex coordinate on $\mathbb{C}P^1$. Fix $M=\mathcal{O}_{\mathbb{C}P^1}^{\oplus 2}$ to be the trivial rank 2 holomorphic bundle on $\mathbb{C}P^1$. Let $\nabla:M\rightarrow M\otimes\Omega_{\mathbb{C}P^1}^1$ be the meromorphic connection whose matrix form is

\begin{eqnarray*}
\nabla=d-Adz,\quad A=\left(\begin{array}{cc}
0 & 1\\
z^n & 0
\end{array}\right)
\end{eqnarray*}

This corresponds to the linear ODE $f''-z^nf=0$ (generalized Airy equation), via $
\left(\begin{array}{c}
\end{array}\right)'=A\left(\begin{array}{c}
\end{array}\right)$. Here, $'=\frac{d}{dz}$.
We want to determine the associated Stokes Legendrian at the (irregular) singularity $\infty$. We firstly take the coordinate $x=z^{-1}$ centered at $\infty$ so that $x(\infty)=0$, and define the differential operator $\delta=x\frac{d}{dx}=-z\frac{d}{dz}$, then the linear ODE becomes $Lf=0$ with $L=\delta^2+\delta-x^{-(n+2)}$. We can use Newton polygons <cit.> to take a formal decomposition of $L$ over some finite field extension of $\mathbb{C}((x))$.

important line/.style=very thick,
dashed line/.style=dashed, thin,
dot/.style=circle,fill=black,minimum size=0pt,inner sep=0pt,
outer sep=0pt,
important dot/.style= circle,fill=black,minimum size=4pt,inner sep=0pt,
outer sep=-1pt,

(-.5,0) coordinate (es) – (1.5,0) coordinate (ee);
[important dot,label=above right:$2$] at (1,0)();
[important dot,label=above right:$1$] at (.5,0)();

(0,.5) coordinate (es)–(0,-2) coordinate (ee);
[dot,label=left:-1] at (0,-.5)();
[dot,label=left:-2] at (0,-1)();
[important dot,label=below left:-3] at (0,-1.5)();

[important line]
(1,0) coordinate (es) – (1,.5) coordinate (ee);

[important line]
(1,0) coordinate (es) – (0,-1.5) coordinate (es);

[important line]
(0,-1.5) coordinate (es) – (-.5,-1.5) coordinate (ee);

[dashed line]
(-0.1,-1.5) coordinate (es) – (1,.15) coordinate (ee);

[dashed line]
(-0.2,-1.5) coordinate (es) – (1,.3) coordinate (ee);

[dashed line]
(-0.3,-1.5) coordinate (es) – (1,.45) coordinate (ee);

$n=1$: Newton polygon for $L=\delta^2+\delta-x^{-3}$.

The Newton polygon $\mathrm{N}(L)$ of $L$ is the convex hull of the second quadrants with vertices $(i,j)$ corresponding to the monomials $x^j\delta^i$ in $L$. In the case $n=1$, it's given by the shaded region in Figure <ref>. $\mathrm{N}(L)$ has 2 extremal points $\{(n_1,m_1)=(0,-(n+2)),(n_2,m_2)=(2,0)\}$ with $0\leq n_1<n_2$. $L$ has a unique positive slope $k_1=\frac{m_2-m_1}{n_2-n_1}=\frac{n+2}{2}$. Write $\frac{b}{a}:=\frac{n+2}{2}$ with $a>0, \mathrm{gcd}(a,b)=1$. Say, $n$ is odd, then $(a,b)=(2,n+2)$. We make the field extension $\mathbb{C}((t))\supset\mathbb{C}((x))$ with $t^a=t^2=x$, and define $\Delta:=t^b\delta=t^{(n+2)}\delta$. We can rewrite $L=t^{-2b}(\Delta^2+(1-b/2)t^b\Delta-1)$, so $\tilde{L}:=t^{2b}L=\Delta^2+(1-b/2)t^b\Delta-1=(\Delta-1)(\Delta+1) (\mathrm{mod}~ t)$ (in fact true $\mathrm{mod~} t^b$). A version of Hensel's lemma (<cit.>) or a direct calculation shows that

\[
\tilde{L}=(\Delta-1+t^b(\frac{2-b}{4}+\ldots))(\Delta+1+t^b(\frac{2-b}{4}+\ldots))
\]

where `$\ldots$' means some formal power series in $t\mathbb{C}[[t]]$. Hence, another simple calculation gives

\[
\]

By solving $(\delta+t^{-(n+2)}+\frac{2-(n+2)}{4}+\ldots)f_-=0$, we then obtain a formal solution to $Lf_-=0$, of the form

\[
f_-=\exp(\frac{2}{n+2}x^{-\frac{n+2}{2}})x^{\frac{n}{4}}\sum_{m\geq 0}a_m^-x^{\frac{m}{2}}.
\]

Similarly, using $\Delta^2-1=(\Delta+1)(\Delta-1)$, one obtains another factorization of $L$, through which one immediately find another formal solution to $Lf=0$, of the form

\[
f_+=\exp(-\frac{2}{n+2}x^{-\frac{n+2}{2}})x^{\frac{n}{4}}\sum_{m\geq 0}a_m^+x^{\frac{m}{2}}.
\]

The exponents $\{\pm\frac{2}{n+2}x^{-\frac{n+2}{2}}\}$ in $f_-,f_+$ are the $g_i's$ in the Turrittin-Levelt formal decomposition of $(M,\nabla)$ near $\infty$, i.e. the formal type of the irregular singularity at $\infty$. Now, it's direct to see that the Stokes Legendrian of $(M,\nabla)$ near $\infty$ has front diagram the $(2,n+2)$-braid in $\mathbb{C}P^1\setminus\{\infty\}$ encircling $\infty$.