Filtered Stokes G-local Systems in Nonabelian Hodge Theory on Curves

The study of the nonabelian Hodge correspondence on noncompact curves begins with Simpson [Sim90], where he introduced filtered regular Higgs bundles, filtered regular $D_{X}$-modules, and filtered local systems along with corresponding stability conditions to establish a one-to-one correspondence among them. This correspondence is famously known as the tame nonabelian Hodge correspondence, where tameness characterizes a polynomial growth condition of flat sections or the regularity of meromorphic connections. Under this framework, filtered local systems are local systems on a noncompact curve with additional structures called weighted filtrations (also called parabolic structures). They correspond to fundamental group representations satisfying certain compatibility conditions determined by weights (these conditions are trivial in some sense). The existence of weights makes the stability of filtered local systems not equivalent to the irreducibility of the corresponding fundamental group representations. In a similar way, Biquard–Boalch [BB04] generalized Simpson’s correspondence to a broader framework beyond tameness, called wildness, where meromorphic connection can be irregular. They established a one-to-one correspondence between (poly)stable filtered irregular Higgs bundles and (poly)stable filtered irregular $D_{X}$-modules under a “very good” condition. Although this work is known as the (unramified) wild nonabelian Hodge correspondence, it does not touch the objects from the Betti side.

To establish a comprehensive wild nonabelian Hodge correspondence, a crucial step involves figuring out the correct objects on the Betti side, which not only correspond to filtered irregular $D_{X}$-modules but also preserves stability conditions. Classically, the Riemann–Hilbert correspondence connects $D_{X}$-modules with local systems. The same idea holds in the wild case and it is well-known that there is a one-to-one correspondence between connections with irregular singularities (or called irregular $D_{X}$-modules) and Stokes local systems [Sib77, Mal78, Mal83, LR94], which is also known as the irregular Riemann–Hilbert correspondence. Moreover, this correspondence was studied systematically in a great generality by Boalch for reductive groups as the structure group [Boa14, Boa18, Boa21]. Inspired by previous works, the authors introduced (Betti) weights to Stokes local systems, which are called filtered Stokes local systems and regarded as the objects on the Betti side, and define its stability condition. Then, an (unramified) wild nonabelian Hodge correspondence at the level of categories was established [HS23a]. Furthermore, such a correspondence also holds for complex reductive groups as the structure groups. For the case of trivial Betti weights, filtered Stokes local systems from the Betti side reduce to Stokes local systems. In this case, Boalch constructed the moduli space of Stokes local systems by identifying Stokes local systems with representations of the fundamental groupoid of irregular curves, which is now known as the wild character variety. However, as the tame case Simpson ever observed, the stability of filtered Stokes $G$-local systems no longer aligns with the irreducibility of the corresponding representations, which is different from the case of trivial Betti weights investigated by Boalch–Yamakawa [BY23]. Consequently, this leads to the fact that a construction of the moduli spaces of filtered Stokes $G$-local systems becomes challenging.

In this paper, we provide a construction of this moduli space in detail, for both unramified and ramified cases and study some specific examples. Moreover, the (unramified) wild nonabelian Hodge correspondence considered in [HS23a] holds at the level of moduli spaces.

We introduce some notations first. Let $G$ be a connected complex reductive group with a given maximal torus $T$. Let $X$ be a connected smooth projective algebraic curve over $\mathbb{C}$ with a collection $\boldsymbol{D}$ of finite points. Denote by $X_{\boldsymbol{D}}:=X\backslash\boldsymbol{D}$ the punctured curve. We also fix a collection of irregular types $\boldsymbol{Q}=\{Q_{x},x\in\boldsymbol{D}\}$ and a collection of weights $\boldsymbol{\theta}=\{\theta_{x},x\in\boldsymbol{D}\}$ labelled by points in $\boldsymbol{D}$, where a weight is regarded as a rational cocharacter ${\rm Hom}(\mathbb{G}_{m},T)\otimes_{\mathbb{Z}}\mathbb{Q}$.

To construct the moduli space of filtered Stokes $G$-local systems, we first relate Stokes $G$-local systems to Stokes $G$-representations, and study the irregular Riemann–Hilbert correspondence for reductive groups $G$ on $X_{\boldsymbol{D}}$ in both unramified and ramified cases in §2 (Theorem 2.14 and 2.17). The proof is similar to [Boa14, Appendix], where the author studied the unramified case. Moreover, Hohl and Jakob recently proved the correspondence (Theorem 2.14) in a different way via Tannakian categories and we refer the reader to [HJ24, §3] for more details.

In §3, we review the stability conditions on filtered Stokes $G$-local systems (Definition 3.4), which is the stability condition considered in the wild nonabelian Hodge correspondence [HS23a], and construct the moduli space of filtered Stokes $G$-local systems. We would like to point out that King’s result [Kin94] inspires us to relate the stability condition of filtered Stokes $G$-local systems to the stability condition in the sense of GIT (Proposition 3.9), which help us to construct the moduli sapce.

In §4, we study some examples of filtered Stokes $G$-local systems together with their moduli spaces. Here is a summary of the results.

1. (1)

   When the weights $\boldsymbol{\theta}$ are trivial, the moduli space $\mathcal{M}_{\rm B}(X_{\boldsymbol{D}},G,\boldsymbol{Q},\boldsymbol{\theta})$ is exactly the wild character variety, which has been studied in [Boa14, BY15, BY23, DDP18, HMW19].

2. (2)

   When the irregular types $\boldsymbol{Q}$ are trivial, the moduli space $\mathcal{M}_{\rm B}(X_{\boldsymbol{D}},G,\boldsymbol{Q},\boldsymbol{\theta})$ is the moduli space of filtered $G$-local systems [HS23b], which is the Betti moduli space in the tame nonabelian Hodge correspondence.

3. (3)

   We equip Eguchi–Hanson space with distinct weights and find a filtered Stokes local system that is stable but not irreducible. This example shows that the Betti moduli spaces in the wild nonabelian Hodge correspondence may not be the wild character varieties.

4. (4)

   We calculate the corresponding Stokes $G$-representations of the classical Airy equation and show that the moduli space, where it lies, is a single point. Therefore, the corresponding irregular singular connection of the Airy equation is both rigid and physically rigid. This result is also obtained in [HJ24, Theorem 1.2.1] recently. Moreover, we prove that in the case of $G={\rm SL}_{2}(\mathbb{C})$ and a ramified irregular type $Q$, the moduli space $\mathcal{M}_{\rm B}(X_{\boldsymbol{D}},{\rm SL}_{2}(\mathbb{C}),Q,\theta)$ is always isomorphic to the wild character variety and does not depend on the weight (Proposition 4.4). This result can be generalized to $G={\rm SL}_{n}(\mathbb{C})$ in a certain extent (Remark 4.5).

In §5, we construct the Betti moduli space (Corollary 5.1) in the (unramified) wild nonabelian Hodge correspondence for principal bundles on curves given by the authors in [HS23a, Theorem in §1], and then we show the correspondence holds at the level of moduli spaces (Theorem 5.2). Moreover, the corresponding moduli space is shown admitting a hyperKälher structure (Theorem 5.3).

Acknowledgments. The authors would like to thank Konstantin Jakob, Yichen Qin and Xiaomeng Xu for helpful discussions. Pengfei Huang acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 101018839) and Deutsche Forschungsgemeinschaft (DFG, Projektnummer 547382045). Hao Sun is partially supported by National Key R$\&$D Program of China (No. 2022YFA1006600).

We fix some notations first

	$\displaystyle R=\mathbb{C}\{z\},\quad\quad\widehat{R}=\mathbb{C}[\![z]\!],$
	$\displaystyle K=\mathbb{C}(\!\{z\}\!),\quad\widehat{K}=\mathbb{C}(\!(z)\!).$

Sometimes we add subscript ‘z’ or ‘w’ to emphasize the local coordinate, for instance, $R_{z}=\mathbb{C}\{z\}$ and $R_{w}=\mathbb{C}\{w\}$. Let $\mathbb{D}={\rm Spec}\,R$ and $\widehat{\mathbb{D}}={\rm Spec}\,\widehat{R}$ be the disc and formal disc, respectively, and then

	$\displaystyle\mathbb{D}^{\ast}:\text{ punctured disc},$
	$\displaystyle\widehat{\mathbb{D}}^{\ast}:\text{ formal punctured disc}.$

Let $G$ be a connected complex reductive group with a given maximal torus $T$. Let $\mathfrak{g}$ (resp. $\mathfrak{t}$) be the Lie algebra of $G$ (resp. $T$). Denote by $\mathcal{R}$ the set of roots.

Let $V$ be a $G$-bundle on $\mathbb{D}^{\ast}$ with a $G$-connection $\nabla$ and a $G$-connection in this paper is always assumed to be algebraic. To simplify the terminology, such a pair $(V,\nabla)$ is also called a $G$-connection. With respect to the local coordinate ‘z’, we write $\nabla$ as

$\displaystyle\nabla=d+A(z)dz,$

where $d$ is the exterior differential and $A(z)\in\mathfrak{g}(K)$ is the connection form. Note that the connection form $A(z)$ of $\nabla$ is in $\mathfrak{g}(K)$ because we are working on connections with irregular singularities. Two $G$-connections $\nabla_{1}=d+A_{1}(z)dz$ and $\nabla_{2}=d+A_{2}(z)dz$ are (gauge) equivalent if there exists $g\in G(K)$ such that $g\circ\nabla_{1}=\nabla_{2}$, i.e.

$\displaystyle A_{2}(z)=-g^{\prime}\cdot g^{-1}+gA_{1}(z)g^{-1}.$

Moreover, we say that $\nabla_{1}$ and $\nabla_{2}$ are formally (gauge) equivalent if there exists an element $g\in G(\widehat{K})$ such that $g\circ\nabla_{1}=\nabla_{2}$.

Now given a $G$-connection $\nabla=d+A(z)dz$, we consider the following set

$\displaystyle\widehat{G}(\nabla):=\{g\in G(\widehat{K})\,|\,g\circ\nabla\emph{ is a $G$-connection}\}.$

Equivalently, this set can be regarded as

$\displaystyle\widehat{G}(\nabla)=\{g\in G(\widehat{K})\,|\,-g^{\prime}\cdot g^{-1}+gA(z)g^{-1}\in\mathfrak{g}(K)\}.$

Clearly,

$\displaystyle G(K)\subseteq\widehat{G}(\nabla)\subseteq G(\widehat{K}).$

The quotient set $\widehat{G}(\nabla)/G(K)$ classifies all $G$-connections which are formally equivalent to $\nabla$. Moreover, taking an arbitrary element $\nabla^{\prime}\in\widehat{G}(\nabla)$, we have $\widehat{G}(\nabla)=\widehat{G}(\nabla^{\prime})$, and thus $\widehat{G}(\nabla)/G(K)=\widehat{G}(\nabla^{\prime})/G(K)$, which implies that the quotient does not depend on the choice of representatives.

When $G={\rm GL}_{n}(\mathbb{C})$, the Malgrange–Sibuya isomorphism theorem [Mal78, Mal83, Sib77] describes $\widehat{G}(\nabla)/G(K)$ as a non-abelian cohomological set. The argument can be applied to complex reductive groups in the same way. We briefly review the setup and state the result. For convenience, let $\mathbb{D}=\mathbb{C}$ and denote by $\widetilde{\mathbb{D}}\rightarrow\mathbb{D}$ the real oriented blowup of $\mathbb{D}$ at $0\in\mathbb{D}$, i.e. $\widetilde{\mathbb{D}}=S^{1}\times[0,+\infty)$. Let $U$ be an open subset of $S^{1}$, and we define an open subset

$\displaystyle\widetilde{U}=\{(\rho,\theta)\in\widetilde{\mathbb{D}}\,|\,\rho>0,\,\theta\in U\}$

of $\widetilde{\mathbb{D}}$. Now, we define a nonabelian sheaf $\Lambda_{\nabla}$ on $S^{1}$, of which a germ $g$ at $\theta\in S^{1}$ is an (holomorphic) element in $G(\mathcal{O}(\widetilde{U}))$ such that

• •

  $g$ is asymptotic to the identity on $\widetilde{U}$ at $0$;

• •

  $g\circ\nabla=\nabla$.

We refer the reader to [Mal83, §3] and [LR94, I.2] for more details about the construction of this sheaf. With a similar argument as in the case of $G={\rm GL}_{n}(\mathbb{C})$, we have the following Malgrange–Sibuya theorem for reductive groups:

To give a more precise description of the cohomology $H^{1}(S^{1},\Lambda_{\nabla})$, we first introduce irregular types. An element in the following form

$\displaystyle Q(z)=q_{-n}z^{-n/r}+\dots+q_{-1}z^{-1/r}$

for some positive integers $n$ and $r$, where $q_{-i}\in\mathfrak{t}$, is called an irregular type. Under the substitution $z=w^{r}$, we have

$\displaystyle Q(w)=q_{-n}w^{-n}+\dots+q_{-1}w^{-1}.$

The substitution $z=w^{r}$ can be regarded as choosing a covering $\mathbb{D}\rightarrow\mathbb{D}$. An irregular type is called unramified if $r=1$, and ramified if $r\geq 2$.

It has been proven that any $G$-connection is of a certain irregular type.

Now we fix a $G$-connection $\nabla=d+A(z)dz$ with irregular type $Q$. In the case of $G={\rm GL}_{n}(\mathbb{C})$, Loday-Richaud gives a constructive description of $H^{1}(S^{1},\Lambda_{\nabla})$ from unipotent Lie groups [LR94]. The arguments can be applied to complex reductive groups as well. In the following, we only give the statement and refer the reader to [Boa14, BY15] for the description.

Given a root $\alpha\in\mathcal{R}$, it determines a meromorphic function $q_{\alpha}(z):=\alpha(Q(z))$. A direction $d\in S^{1}$ is an anti-Stokes direction (supported by $\alpha$) if the meromorphic function $\exp(q_{\alpha}(z))$ has maximal decay as $z$ goes to zero in the direction. Denote by $\mathbb{A}$ the set of all anti-Stokes directions with respect to the given irregular type $Q$. Given an anti-Stokes direction $d\in\mathbb{A}$, let $\mathcal{R}(d)\subseteq\mathcal{R}$ be the subset of roots supporting $d$, and for each $\alpha\in\mathcal{R}$, let $U_{\alpha}:=\exp(\mathfrak{g}_{\alpha})\subseteq G$ be the corresponding unipotent subgroup. Denote by $\mathbb{S}{\rm to}_{d}$ the image of the product map $\prod_{\alpha\in\mathcal{R}(d)}U_{\alpha}\rightarrow G$, which is a unipotent group [LR94, I.4.8]. We define

$\displaystyle\mathbb{S}{\rm to}(Q):=\prod_{d\in\mathbb{A}}\mathbb{S}{\rm to}_{d}.$

In [BY15], the authors define a local system $\mathcal{I}$ on $S^{1}$, of which sections over sectors are functions in the form $Q=\sum_{i=1}^{n}q_{-i}z^{-i/r}$, where $q_{-i/r}\in\mathbb{C}$ and $n,r\in\mathbb{N}$. The sheaf $\mathcal{I}$ can be regarded as a vast disjoint union of circle coverings of $S^{1}$, and each component of $\mathcal{I}$ (i.e. an element in $\pi_{0}(\mathcal{I})$) is a covering of $S^{1}$. Given a point $p\in S^{1}$, the fiber $\mathcal{I}_{p}$ is a free $\mathbb{Z}$-module. We define a pro-tori $\mathcal{T}_{p}:={\rm Hom}(\mathcal{I}_{p},\mathbb{C}^{*})$, where a pro-tori is an inverse limit of torus. We obtain a system $\mathcal{T}$ of pro-tori over $S^{1}$, of which $\mathcal{T}_{p}$ is the fiber at $p\in S^{1}$.

Boalch–Yamakawa proved the following equivalence of categories.

Recall that $\widetilde{\mathbb{D}}$ is the real oriented blow-up of $\mathbb{D}$ at zero, and the zero point is usually denoted by $x$. Denote by $\partial=S^{1}$ the boundary circle. We draw a concentric circle (a halo) $\partial^{\prime}$ on $\widetilde{\mathbb{D}}$, and denote by $\mathbb{H}$ the region between $\partial$ and $\partial^{\prime}$. In other words, $\mathbb{H}$ is regarded as a tubular neighbourhood of $\partial$ with another boundary circle $\partial^{\prime}$. We puncture $\partial^{\prime}$ at $\#\mathbb{A}$ many distinct points and denote them by $\{x_{d},\,d\in\mathbb{A}\}$. According to the anti-Stokes directions, we require that all the $\#\mathbb{A}_{x}$ auxiliary small cilia between each anti-Stokes direction and its nearby puncture do not cross (see the following picture for example).

Let $X$ be a connected smooth projective algebraic curve over $\mathbb{C}$. Let $\boldsymbol{D}$ be a given collection of finitely many distinct points on $X$, which is also regarded as a reduced effective divisor on $X$, and denote by $X_{\boldsymbol{D}}:=X\backslash\boldsymbol{D}$ the punctured curve, which is also called a noncompact curve. Let $\widetilde{X}$ be the real oriented blow-up of $X$ at each puncture $x\in\boldsymbol{D}$. It is equivalent to consider that $\widetilde{X}$ is obtained from $X$ by replacing each puncture $x\in\boldsymbol{D}$ by an oriented boundary circle $\partial_{x}$, of which points are considered to be oriented directions emanating from $x$. Now we equip each puncture $x\in\boldsymbol{D}$ with an irregular type $Q_{x}$, and denote by $\boldsymbol{Q}=\{Q_{x},\,x\in\boldsymbol{D}\}$ the collection with irregular types. For each $x\in\boldsymbol{D}$, let $\mathbb{A}_{x}$ be the set of anti-Stokes directions of $Q_{x}$. Then, we draw a concentric circle (a halo) $\partial^{\prime}_{x}$ on $\widetilde{X}$ near $\partial_{x}$. Denote by $\mathbb{H}_{x}$ the region between $\partial_{x}$ and $\partial^{\prime}_{x}$, which is a tubular neighbourhood of $\partial_{x}$. Then, we puncture $\partial^{\prime}_{x}$ at $\#\mathbb{A}_{x}$ distinct points according to anti-Stokes directions such that all the $\#\mathbb{A}_{x}$ auxiliary small cilia between each anti-Stokes direction and its nearby puncture do not cross. Denote by $\{x_{d},\,d\in\mathbb{A}_{x}\}$ the collection of punctures with respect to point $x\in\boldsymbol{D}$. Finally, let $X_{\boldsymbol{Q}}\hookrightarrow\widetilde{X}$ be the punctured surface obtained in the above way, which is called the irregular curve given by $\boldsymbol{Q}$.

If we do not fix a specific irregular type, then we have the following correspondence, which is regarded as the $G$-version of the classical irregular Riemmann–Hilbert correspondence of Deligne, Malgrange, Sibuya, Loday-Richaud [LR94, Mal83, Sib77]. Moreover, Hohl and Jakob give a different proof of Theorem 2.14 via Tannakian categories [HJ24, §3].

Now we will introduce the fundamental groupoid of $X_{\boldsymbol{Q}}$ and show that the category of Stokes $G$-local systems with irregular type $\boldsymbol{Q}$ on $X_{\boldsymbol{D}}$ is equivalent to the category of specific $G$-representations of the fundamental groupoid of $X_{\boldsymbol{Q}}$. We first introduce two sets $H(\partial)$ and $H$ determined by an irregular type. Given an irregular type $Q$, denote by $H$ the centralizer of $Q=q_{n}z^{-n/r}+\dots+q_{-1}z^{-1/r}$ in $G$. More precisely,

$\displaystyle H=\{k\in G\,|\,[k,q_{-i}]=0\text{ for each $i$.}\}$

Denote by $H(\partial)\subseteq G$ the subset of formal monodromies given by $Q$. In the unramfied case, $H=H(\partial)$. Boalch–Yamakawa proved the following result:

Let $\mathrm{Hom}(\Pi,G)$ be the space of $G$-representations of $\Pi$. An element (point) $\rho\in\mathrm{Hom}(\Pi,G)$ is called a Stokes $G$-representation with irregular type $\boldsymbol{Q}$ on $X_{\boldsymbol{D}}$ if for each $x\in\boldsymbol{D}$ and $d\in\mathbb{A}_{x}$, we have $\rho(\gamma_{x})\in H(\partial_{x})$, which is the set of formal monodromies given by $Q_{x}$, and $\rho(\gamma_{x,d})\in\mathbb{S}\mathrm{to}_{d}$. Denote by $\mathrm{Hom}_{\mathbb{S}}(\Pi,G)$ the space of all Stokes $G$-representations with irregular type $\boldsymbol{Q}$, which is a smooth affine variety. Here is an equivalent description of $\mathrm{Hom}_{\mathbb{S}}(\Pi,G)$ with respect to generators and relations of $\Pi$. For each $x\in\boldsymbol{D}$, we define

$\displaystyle\mathcal{A}(Q_{x})=H(\partial_{x})\times\mathbb{S}{\rm to}(Q_{x}).$

We consider the closed subvariety

$\displaystyle{\rm Hom}_{\mathbb{S}}(\Omega,G):=\left((G\times G)^{g}\times\prod_{x\in\boldsymbol{D}}(G\times\mathcal{A}(Q_{x}))\right)\subseteq{\rm Hom}(\Omega,G).$

Given a $G$-representation $\rho:\Omega\rightarrow G$, we use the following notations

$\displaystyle a_{i}=\rho(\alpha_{i}),\ b_{i}=\rho(\beta_{i}),\ \rho(\gamma_{x})=h_{x},\ \rho(\gamma_{x,d})=S_{x,d},\ \rho(\gamma_{0x})=c_{x}.$

Then ${\rm Hom}_{\mathbb{S}}(\Pi,G)$ includes all data

$\displaystyle((a_{i},b_{i})_{1\leq i\leq g},(c_{x},h_{x},S_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}})\in{\rm Hom}_{\mathbb{S}}(\Omega,G)$

such that

$\displaystyle\left(\prod_{i=1}^{g}[a_{i},b_{i}]\right)\cdot\left(\prod_{x\in\boldsymbol{D}}(c_{x}^{-1}h_{x}(\prod_{d\in\mathbb{A}_{x}}S_{x,d})c_{x})\right)=\mathrm{id}.$

Recall that $H_{x}$ is the stabilizer of $Q_{x}$ for $x\in\boldsymbol{D}$. We define

$\displaystyle\boldsymbol{H}:=\prod_{x\in\boldsymbol{D}}H_{x},\ \ \boldsymbol{H}(\partial):=\prod_{x\in\boldsymbol{D}}H(\partial_{x}).$

There is a $(G\times\boldsymbol{H})$-action on ${\rm Hom}_{\mathbb{S}}(\Pi,G)$ given as follows:

	$\displaystyle(g,(k_{x}$	$\displaystyle)_{x\in\boldsymbol{D}})\cdot((a_{i},b_{i})_{1\leq i\leq g},(c_{x},h_{x},S_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}):=$
		$\displaystyle\hskip 50.00008pt((ga_{i}g^{-1},gb_{i}g^{-1})_{1\leq i\leq g},(k_{x}c_{x}g^{-1},k_{x}h_{x}k^{-1}_{x},k_{x}S_{x,d}k_{x}^{-1})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}).$

Under this action, ${\rm Hom}_{\mathbb{S}}(\Pi,G)$ becomes a (twsited) quasi-Hamiltonian $(G\times\boldsymbol{H})$-space with moment map

	$\displaystyle\mu:{\rm Hom}_{\mathbb{S}}(\Pi,G)$	$\displaystyle\to G\times\boldsymbol{H}(\partial),$
	$\displaystyle\rho$	$\displaystyle\mapsto(\prod_{x\in\boldsymbol{D}}\Big{(}c_{x}^{-1}h_{x}(\prod_{d\in\mathbb{A}_{x}}S_{x,d})c_{x}\Big{)},(h_{x}^{-1})_{x\in\boldsymbol{D}}).$

As a result, the quotient $\mathcal{M}_{\rm B}(X_{\boldsymbol{D}},G,\boldsymbol{Q}):={\rm Hom}_{\mathbb{S}}(\Pi,G)/\!\!/(G\times\boldsymbol{H})$, which is called wild character variety, exhibits a structure of an algebraic Poisson variety with symplectic leaves [Boa14, BY15]. Two Stokes $G$-representations with irregular type $\boldsymbol{Q}$ are isomorphic if they are in the same $(G\times\boldsymbol{H})$-orbit.

In the previous subsection, we follow Boalch’s idea to construct the fundamental groupoid $\Pi$ of $X_{\boldsymbol{Q}}$ with respect to a collection of base points $\boldsymbol{b}=\{b_{0},b_{x}\,x\in\boldsymbol{D}\}$, and then in the definition of the fundamental groupoid $\Pi$, it has a path (a generator) $\gamma_{0x}$ connecting $b_{0}$ and $b_{x}$ for each $x\in\boldsymbol{D}$. In the following, we will define a fundamental group of $X_{\boldsymbol{Q}}$ with respect to a single base point and give an equivalent description of the space of Stokes $G$-representations.

We define a free group $\Omega^{\prime}$ with generators

1. (1)

   $\alpha^{\prime}_{1},\beta^{\prime}_{1},\dots,\alpha^{\prime}_{g},\beta^{\prime}_{g}$;

2. (2)

   $\gamma^{\prime}_{x}$ for each $x\in\boldsymbol{D}$;

3. (3)

   $\gamma^{\prime}_{x,d}$ for each $d\in\mathbb{A}_{x}$.

Adding a relation

($\ast^{\prime}$)

$$\left(\prod_{i=1}^{g}[\alpha^{\prime}_{i},\beta^{\prime}_{i}]\right)\cdot\left(\prod_{x\in\boldsymbol{D}}\mu^{\prime}_{x}\right)={\rm id},$$

where

$\displaystyle\mu^{\prime}_{x}=\gamma^{\prime}_{x}\cdot\left(\prod_{d\in\mathbb{A}_{x}}\gamma^{\prime}_{x,d}\right),$

we obtain a group $\Pi^{\prime}$. There is a morphism $\Omega^{\prime}\rightarrow\Omega$ (resp. $\Pi^{\prime}\rightarrow\Pi$)

$\displaystyle\alpha^{\prime}_{i}\mapsto\alpha_{i},\ \beta^{\prime}_{i}\mapsto\beta_{i},\ \gamma^{\prime}_{x}\mapsto\gamma_{0x}^{-1}\gamma_{x}\gamma_{0x},\ \gamma^{\prime}_{x,d}\mapsto\gamma_{0x}^{-1}\gamma_{x,d}\gamma_{0x},$

which induces one ${\rm Hom}(\Omega,G)\rightarrow{\rm Hom}(\Omega^{\prime},G)$ (resp. ${\rm Hom}(\Pi,G)\rightarrow{\rm Hom}(\Pi^{\prime},G)$). Therefore, the group $\Pi^{\prime}$ can be regarded as the fundamental group of $X_{\boldsymbol{Q}}$ with respect to a given base point $b_{0}$. Given a $G$-representation $\rho^{\prime}:\Omega^{\prime}\rightarrow G$, we introduce the following notations

$\displaystyle a^{\prime}_{i}=\rho^{\prime}(\alpha^{\prime}_{i}),\ b^{\prime}_{i}=\rho^{\prime}(\beta^{\prime}_{i}),\ h^{\prime}_{x}=\rho^{\prime}(\gamma^{\prime}_{x}),\ S^{\prime}_{x,d}=\rho^{\prime}(\gamma^{\prime}_{x,d}).$

Consider the group $\prod_{x\in\boldsymbol{D}}G_{x}$, where $G_{x}:=G$. We define an action

$\displaystyle(\prod_{x\in\boldsymbol{D}}G_{x})\times{\rm Hom}(\Omega^{\prime},G)\rightarrow{\rm Hom}(\Omega^{\prime},G)$

via

$\displaystyle(g_{x})_{x\in\boldsymbol{D}}\cdot((a^{\prime}_{i},b^{\prime}_{i})_{1\leq i\leq g},(h^{\prime}_{x},S^{\prime}_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}):=((a^{\prime}_{i},b^{\prime}_{i})_{1\leq i\leq g},(g^{-1}_{x}h^{\prime}_{x}g_{x},g^{-1}_{x}S^{\prime}_{x,d}g_{x})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}).$

Consider the fiber product

$\displaystyle\left((\prod_{x\in\boldsymbol{D}}G_{x})\times{\rm Hom}(\Omega^{\prime},G)\right)\times_{{\rm Hom}(\Omega^{\prime},G)}\left((G\times G)^{g}\times\prod_{x\in\boldsymbol{D}}\mathcal{A}(Q_{x})\right),$

where $\left((G\times G)^{g}\times\prod_{x\in\boldsymbol{D}}\mathcal{A}(Q_{x})\right)\hookrightarrow{\rm Hom}(\Omega^{\prime},G)$ is the natural inclusion. The fiber product is a closed subvariety of $(\prod_{x\in\boldsymbol{D}}G_{x})\times{\rm Hom}(\Omega^{\prime},G)$, and it includes all points

$\displaystyle((g_{x})_{x\in\boldsymbol{D}},((a^{\prime}_{i},b^{\prime}_{i})_{1\leq i\leq g},(h^{\prime}_{x},S^{\prime}_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}))\in(\prod_{x\in\boldsymbol{D}}G_{x})\times{\rm Hom}(\Omega^{\prime},G)$

such that

$\displaystyle g^{-1}_{x}h^{\prime}_{x}g_{x}\in H(\partial_{x}),\ g^{-1}_{x}S^{\prime}_{x,d}g_{x}\in\mathbb{S}{\rm to}_{d}$

for each $x\in\boldsymbol{D}$ and $d\in\mathbb{A}_{x}$. Then, we define

$\displaystyle{\rm Hom}_{\mathbb{S}}(\Omega^{\prime},G):=\left(((\prod_{x\in\boldsymbol{D}}G_{x})\times{\rm Hom}(\Omega^{\prime},G))\times_{{\rm Hom}(\Omega^{\prime},G)}((G\times G)^{g}\times\prod_{x\in\boldsymbol{D}}\mathcal{A}(Q_{x}))\right)\bigg{|}_{{\rm Hom}(\Omega^{\prime},G)}.$

Clearly, ${\rm Hom}_{\mathbb{S}}(\Omega^{\prime},G)$ is a locally closed subset and includes all points

$\displaystyle((a^{\prime}_{i},b^{\prime}_{i})_{1\leq i\leq g},(h^{\prime}_{x},S^{\prime}_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}})\in{\rm Hom}(\Omega^{\prime},G)$

satisfying the condition that for each $x\in\boldsymbol{D}$, there exists $g_{x}\in G$ such that

$\displaystyle g^{-1}_{x}h^{\prime}_{x}g_{x}\in H(\partial_{x}),\ g^{-1}_{x}S^{\prime}_{x,d}g_{x}\in\mathbb{S}{\rm to}_{d}.$

Adding the relation ($\ast^{\prime}$ ‣ 2.3), we obtain a closed subvariety ${\rm Hom}_{\mathbb{S}}(\Pi^{\prime},G)\hookrightarrow{\rm Hom}_{\mathbb{S}}(\Omega^{\prime},G)$. Furthermore, the natural $G$-action on ${\rm Hom}(\Omega^{\prime},G)$ given by conjugation

$\displaystyle g\cdot((a^{\prime}_{i},b^{\prime}_{i})_{1\leq i\leq g},(h^{\prime}_{x},S^{\prime}_{x,d})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}}):=((ga^{\prime}_{i}g^{-1},gb^{\prime}_{i}g^{-1})_{1\leq i\leq g},(gh^{\prime}_{x}g^{-1},gS^{\prime}_{x,d}g^{-1})_{x\in\boldsymbol{D},d\in\mathbb{A}_{x}})$

induces a $G$-action on ${\rm Hom}_{\mathbb{S}}(\Pi^{\prime},G)$.