Monodromy-Prescribed Polystable Bundles on Punctured Riemann Surfaces and the Geometry of Singular Control Strategies

Abstract

This paper establishes a functorial algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface $\Sigma$ of genus $g\ge 2$, for a complex reductive Lie group G, and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ of representations of its fundamental group with relatively compact image. The dimension formula $dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$, where ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_k$ are conjugacy classes in a maximal compact subgroup $K\subset G$, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups.

1. Introduction

For a compact Riemann surface $\Sigma$ of genus $g\ge 2$ with a finite set of distinct points $S=\{p_1,\dots ,p_k\}\subset \Sigma$, the monodromy of a connection around each puncture on a principal G-bundle describes how parallel transport transforms when traversing a closed loop [1,2]. The study of polystable principal bundles with prescribed monodromy connects to character varieties, which parametrize representations of fundamental groups into Lie groups. For a complex reductive Lie group G with maximal compact subgroup $K\subset G$ and conjugacy classes $\mathcal{C}=\{{\mathcal{C}}_1,\dots ,{\mathcal{C}}_k\}$ in K, the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ represents equivalence classes of representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ such that the image of each loop around a puncture belongs to the corresponding conjugacy class and the image $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in G [3,4]. This geometric framework provides a natural setting for studying moduli spaces associated with polystable principal bundles and their connections, as demonstrated by Hausel, Letellier, and Villegas [5] on the topology of character varieties.

Furthermore, building on the seminal works of Atiyah and Bott [1] and Hitchin [2], Boalch [6] studied isomonodromic deformations, providing insights into the deformation theory of connections with prescribed singularities. More recent advances on the construction of moduli spaces of meromorphic connections with prescribed monodromy also include seminal contributions. Thus, in the regular singular case, Simpson [7] developed the non-abelian Hodge correspondence, relating flat connections to representations of the fundamental group, while Boalch [6] extended this to include meromorphic connections with irregular singularities and described the associated Stokes data, enriching the structure of the corresponding character varieties. In the algebraic category, Biquard and Boalch [8] constructed the wild character varieties as moduli spaces of meromorphic connections with fixed formal and Stokes data for complex reductive groups. Further constructions, including moduli spaces of connections with prescribed formal types, were developed by Inaba, Iwasaki, and Saito [9]. Within the study of the geometry of such moduli spaces, Loray and Saito [10] have explored Lagrangian fibrations on moduli spaces of logarithmic connections, further enriching our understanding of these geometric structures. These developments, while primarily motivated by questions in mathematical physics and algebraic geometry, offer new perspectives for studying control systems with topological constraints.

In recent years, there has been growing interest in applying geometric techniques to control theory, particularly for systems operating in complex topological spaces [11,12]. The interplay between the geometry of principal bundles and the topology of control systems has emerged as a fruitful line of research with significant applications to robotics and autonomous navigation. The classical approaches to motion planning and control, as developed by Koditschek and Rimon [13] and later extended by LaValle [14], have provided foundational tools for characterizing topological constraints in control systems. However, these approaches lack a comprehensive geometric framework that captures the full complexity of control systems operating on manifolds with topological singularities and prescribed interaction patterns with obstacles.

The theory of moduli spaces of principal bundles, pioneered by Narasimhan and Seshadri [15] and further developed by Ramanathan [16], provides a suitable geometric framework for addressing these challenges. Recent advances in computational topology [17] and geometric invariant theory [18] have enhanced our ability to characterize topological invariants relevant to control systems. The Narasimhan–Seshadri correspondence, which establishes the correspondence between flat connections on a degree-zero holomorphic vector bundle and unitary representations of the fundamental group of the base curve, serves as the theoretical basis for the present approach. This theorem was generalized to complex reductive groups G by Donaldson [19] and Corlette [20]. Furthermore, developments in the asymptotic analysis of moduli spaces, exemplified by the work of Dumas and Neitzke [21] on wall-crossing phenomena and the behavior of metrics near singular configurations, provide analytical procedures for quantifying the stability properties of control systems in degenerate situations.

This work explores the connection between polystable principal G-bundles with prescribed monodromy over Riemann surfaces, where G is a complex reductive Lie group, and their applications to topological control systems. In particular, this research presents several original contributions to the geometric theory of polystable principal bundles with prescribed monodromy and their applications to topological control systems. Thus, the main result (Theorem 1) establishes a functorial isomorphism of algebraic varieties between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$, where ${\Sigma }^{*}=\Sigma \setminus S$. This functorial correspondence, which extends the classical Narasimhan–Seshadri correspondence [15] and Donaldson–Uhlenbeck–Yau correspondence [22,23] to the prescribed monodromy setting, is realized by mutually inverse algebraic morphisms that commute with all relevant group actions and are independent of auxiliary choices such as base points and trivializations. The algebraic nature of this isomorphism provides a method for characterizing the space of control strategies through group-theoretic data, offering a more computationally tractable approach to analyzing complex control systems while preserving the geometric structure of the moduli spaces.

This result is complemented by providing an explicit formula for the dimension of the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ in terms of the genus of the Riemann surface, the complex dimension of the Lie group, and the real dimensions of the conjugacy classes in the maximal compact subgroup (Proposition 1). This formula reveals how the environmental complexity (through genus g), the system’s internal symmetry (through ${dim}_{\mathbb{C}}\left(G\right)$), and the nature of constraints (through ${dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$) all contribute to the total complexity of control strategies. For complex reductive Lie groups, the dimension is given by $dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$.

This research also characterizes the singularities of the moduli space, showing that they correspond precisely to the points represented by polystable bundles with non-trivial automorphism groups (Proposition 2). This has significant implications for understanding the stability properties of control systems, as singularities in the moduli space represent configurations where the system’s behavior may change qualitatively under small perturbations. The analysis of these singularities builds upon techniques from Mumford’s geometric invariant theory (GIT) [18] and extends recent work by Gukov and Sulkowski [24] on the quantization of moduli spaces.

The above algebraic-geometric results are then applied to the study of polystable control strategies for robotic systems operating in topological spaces with obstacles, through their correspondence with the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ explicitly provided in Proposition 3. In particular, an analysis of the stability properties of singular points in families of polystable control strategies is provided, establishing that such points correspond to enhanced stability conditions characterized by reduced tangent space dimensions due to the presence of non-trivial automorphism groups (Proposition 4). Following the geometric interpretation established by Hitchin [2], these singular points represent control strategies with maximal symmetry properties, leading to increased robustness against perturbations. The dimensional reduction at singular points reflects the geometric constraint that highly symmetric configurations possess fewer degrees of freedom for deformation.

Unlike previous approaches that focus on specific control algorithms or particular classes of topological spaces, this work unifies various aspects of control systems operating in complex environments. The explicit dimension formulas and characterization of singularities provided here lead to concrete computational methods for analyzing the robustness and flexibility of polystable control strategies in robotics and autonomous systems. This approach complements recent developments in the geometric theory of principal bundles, such as the work on stratifications of moduli spaces [25,26], the description of special subvarieties [27,28], and the incorporation of additional geometric structures [29], by providing new techniques for analyzing the global topological structure of control strategy spaces.

The structure of this work is as follows. Section 2 establishes the necessary preliminaries and notation, including the fundamental definitions of principal bundles with prescribed monodromy and character varieties. Section 3 presents the main geometric results, mainly establishing the correspondence between principal bundles and representations of fundamental groups, and characterizing the dimension and singularities of the associated moduli spaces. A computation example is developed in Section 4, where the main results of the paper are illustrated in the specific case of $G=SL(2,\mathbb{C})$. The applications of the above results to topological control systems, classifying and analyzing control strategies for robotic systems operating in complex environments, are developed in Section 5. Finally, some concluding remarks are given, and some lines of future research are discussed.

2. Preliminaries and Notation

Throughout this work, $\Sigma$ denotes a compact Riemann surface of genus $g\ge 2$, and G represents a complex reductive Lie group with maximal compact subgroup $K\subset G$. The finite set $S=\{p_1,\dots ,p_k\}\subset \Sigma$ consists of distinct points, and the notation ${\Sigma }^{*}=\Sigma \setminus S$ is used for the punctured surface obtained by removing these points.

The fundamental group ${\pi }_1\left({\Sigma }^{*}\right)$ of the punctured surface admits an explicit presentation that extends the classical description of the fundamental group of a closed surface. For a compact Riemann surface $\Sigma$ of genus g, the fundamental group ${\pi }_1(\Sigma )$ has generators ${\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g$ subject to the single relation

$$\prod _{i=1}^g[{\alpha }_i,{\beta }_i]=1,$$

where $[\alpha ,\beta ]=\alpha \beta {\alpha }^{-1}{\beta }^{-1}$ denotes the commutator [30]. When punctures are introduced, additional generators ${\gamma }_1,\dots ,{\gamma }_k$ are incorporated, where each ${\gamma }_j$ represents a loop encircling the puncture $p_j$ once in the counterclockwise direction. The Seifert-van Kampen theorem [30] provides the presentation of ${\pi }_1\left({\Sigma }^{*}\right)$ with generators ${\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g,{\gamma }_1,\dots ,{\gamma }_k$ and the fundamental relation

$$\prod _{i=1}^g[{\alpha }_i,{\beta }_i]\prod _{j=1}^k{\gamma }_j=1.$$

This relation arises from the fact that the boundary of the surface $\Sigma$ with small disks removed around each puncture encompasses both the commutator product and the loops around the punctures, with the total homology class vanishing [30].

The following notion of prescribed monodromy provides a natural framework for studying connections on principal bundles over punctured surfaces.

Definition 1

([6,8,31]). A principal G-bundle P over the compact Riemann surface Σ with a meromorphic connection ∇, for a complex reductive Lie group G, is said to have prescribed monodromy around the points in S if

1.

∇ is holomorphic on ${\Sigma }^{*}$, and

2.

For each $p_i\in S$, there exists a conjugacy class ${\mathcal{C}}_i\subset G$ such that the monodromy of ∇ around a small loop encircling only $p_i$ belongs to ${\mathcal{C}}_i$.

Given conjugacy classes ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_k$ in G, the set ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),G)$ consists of all group homomorphisms $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ satisfying $\rho \left({\gamma }_j\right)\in {\mathcal{C}}_j$ for $j=1,\dots ,k$. The group G acts on this set by conjugation, where for $g\in G$ and $\rho \in {Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),G)$, the action is defined by $(g·\rho )\left(\gamma \right)=g\rho \left(\gamma \right)g^{-1}$ for all $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$. The character variety ${\mathcal{M}}_{\mathcal{C}}({\Sigma }^{*},G)$ is then defined as the geometric invariant theory quotient

$${\mathcal{M}}_{\mathcal{C}}({\Sigma }^{*},G)={Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),G)/G.$$

(1)

The character variety admits a concrete algebraic description through the affine variety

$$R=\left\{(A_1,B_1,\dots ,A_g,B_g,C_1,\dots ,C_k)\in G^{2g+k}\mid \prod _{i=1}^g[A_i,B_i]\prod _{j=1}^k{C}_j=1,C_j\in {\mathcal{C}}_j\right\}.$$

The group G acts on R by simultaneous conjugation, sending a tuple

$$(A_1,B_1,\dots ,A_g,B_g,C_1,\dots ,C_k)$$

to

$$\left(g{A}_1{g}^{-1},g{B}_1{g}^{-1},\dots ,g{A}_g{g}^{-1},g{B}_g{g}^{-1},g{C}_1{g}^{-1},\dots ,g{C}_k{g}^{-1}\right),$$

and the geometric invariant theory quotient $R/G$ recovers the character variety ${\mathcal{M}}_{\mathcal{C}}({\Sigma }^{*},G)$.

The moduli space ${\mathcal{B}}_{\mathcal{C}}(\Sigma ,G)$ parametrizes isomorphism classes of principal G-bundles over $\Sigma$ equipped with meromorphic connections having prescribed monodromy in conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ around the points in S, as described in Definition 1. Two principal G-bundles $P_1$ and $P_2$ over $\Sigma$ equipped with meromorphic connections ${\mathsf{\nabla }}_1$ and ${\mathsf{\nabla }}_2$ having prescribed monodromy are said to be isomorphic if there exists a bundle isomorphism $\varphi :P_1\to P_2$ such that ${\varphi }^{*}{\mathsf{\nabla }}_2={\mathsf{\nabla }}_1$. The moduli space ${\mathcal{B}}_{\mathcal{C}}(\Sigma ,G)$ parametrizes isomorphism classes of such objects, where the bundles have meromorphic connections with prescribed monodromy in conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ around the points in S. The existence and construction of this moduli space as an algebraic variety follows [8,9].

Following the stability condition originally introduced by Ramanathan [16,32,33], a principal G-bundle P over a compact Riemann surface $\Sigma$ is called polystable if for every parabolic subgroup $Q\subset G$, every reduction $\sigma :\Sigma \to P/Q$ of the structure group, and every antidominant character $\chi :Q\to {\mathbb{C}}^{*}$, the associated line bundle $P_{\sigma }{\times }_{\chi }\mathbb{C}$ has degree less or equal than zero (the principal bundle is stable if this degree is less than zero) and, if the degree is 0, then P admits a stable reduction to a Levi subgroup of Q (in this case, the bundle is strictly polystable). Following this notion, a pair $(P,\mathsf{\nabla })$ where P is a principal G-bundle P over a compact Riemann surface $\Sigma$ with a meromorphic connection ∇ and prescribed monodromy as in Definition 1 is called polystable (resp., stable, strictly polystable) when P is a polystable (resp., stable, strictly polystable) principal G-bundle. The moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ then consists of isomorphism classes of polystable principal G-bundles over $\Sigma$ with meromorphic connections having prescribed monodromy in conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$, where each ${\mathcal{C}}_i$ is a conjugacy class in the maximal compact subgroup $K\subset G$. This moduli space has the structure of a complex algebraic variety, whose specific construction can be found in [8,9].

The character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ parametrizes representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ for $i=1,\dots ,k$ and the image $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in G, modulo conjugation by G, as established in the following definition. The condition of relative compactness ensures that the representations under consideration correspond to connections on polystable bundles, as shown by the Narasimhan–Seshadri correspondence [15] and its generalizations [20,34].

Definition 2.

Let ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ denote the character variety of representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ for $i=1,\dots ,k$, and the image $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in G, modulo conjugation by G, where two of such representations are identified if they are conjugate in the sense of the definition of the character variety given in Equation (1).

This character variety also carries the structure of a quasi-projective complex algebraic variety. More precisely, consider the representation variety

$${Rep}_{\mathcal{C}}^K({\Sigma }^{*},G)=\{\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G\mid \rho \left({\gamma }_i\right)\in {\mathcal{C}}_i,\mathrm{image}\mathrm{relatively}\mathrm{compact}\},$$

which is a closed algebraic subset of the affine variety $Hom({\pi }_1\left({\Sigma }^{*}\right),G)\cong G^{2g+k-1}$ (where the isomorphism comes from choosing generators of the fundamental group). The group G acts on this representation variety by conjugation: $(g·\rho )\left(\gamma \right)=g\rho \left(\gamma \right)g^{-1}$ for $g\in G$ and $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$. The character variety is then defined as the geometric invariant theory quotient

$${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)={Rep}_{\mathcal{C}}^K({\Sigma }^{*},G)//G,$$

where $//$ denotes the GIT quotient with respect to a suitable linearization of the G-action. The condition of relative compactness of the image ensures that all representations in ${Rep}_{\mathcal{C}}^K({\Sigma }^{*},G)$ are semistable with respect to this linearization, guaranteeing that the quotient is a well-defined quasi-projective variety [18,35]. The specific construction of the linearization involves embedding G into a projective space and using the induced action on polynomial rings, as detailed in [36,37].

3. Main Results

This section establishes a bijective correspondence between moduli spaces of polystable principal bundles with prescribed monodromy and character varieties of surface group representations. This result extends the Narasimhan–Seshadri correspondence [15] to the setting of prescribed monodromy. The proof proceeds in several steps, which are developed through a sequence of lemmas. Further geometric analysis is also provided through some propositions, establishing dimension formulas and singularity characterizations for these moduli spaces in the context under consideration.

The following result establishes that every representation in the character variety gives rise to a well-defined geometric object satisfying the required stability and monodromy conditions.

Lemma 1.

Let Σ be a compact Riemann surface of genus $g\ge 2$, G be a complex reductive Lie group, $K\subset G$ be a maximal compact subgroup, $S=\{p_1,\dots ,p_k\}\subset \Sigma$, and ${\Sigma }^{*}=\Sigma \setminus S$. Then, any representation $\rho \in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ gives rise to a polystable principal G-bundle $P_{\rho }$ over Σ with a meromorphic connection having prescribed monodromy in the conjugacy classes $\mathcal{C}$.

Proof. 

Let $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ be a representation with relatively compact image such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ for $i=1,\dots ,k$, where ${\gamma }_i$ represents a small loop around the puncture $p_i$.

Since the image of $\rho$ is relatively compact in G, there exists a maximal compact subgroup $K^{\prime }\subset G$ such that $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)\subset K^{\prime }$. By the conjugacy of maximal compact subgroups in reductive Lie groups, there exists $g\in G$ such that $g{K}^{\prime }{g}^{-1}=K$. Define $\tilde{\rho }=Ad\left(g\right)\circ \rho$, so that $\tilde{\rho }\left({\pi }_1\left({\Sigma }^{*}\right)\right)\subset K$.

Let $\widetilde{{\Sigma }^{*}}$ denote the universal cover of ${\Sigma }^{*}$. The representation $\tilde{\rho }$ defines a principal G-bundle $P_{\tilde{\rho }}^{*}=\widetilde{{\Sigma }^{*}}{\times }_{\tilde{\rho }}G$ over ${\Sigma }^{*}$ via the quotient construction. Specifically, the action of ${\pi }_1\left({\Sigma }^{*}\right)$ on $\widetilde{{\Sigma }^{*}}\times G$ is given by $\gamma ·(x,h)=(\gamma ·x,\tilde{\rho }\left(\gamma \right)h)$ for $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$, $x\in \widetilde{{\Sigma }^{*}}$, and $h\in G$.

The bundle $P_{\tilde{\rho }}^{*}$ admits a canonical flat connection ${\mathsf{\nabla }}^{*}$ defined by the differential of the projection $\widetilde{{\Sigma }^{*}}\times G\to \widetilde{{\Sigma }^{*}}{\times }_{\tilde{\rho }}G$. In local coordinates, if $s:U\to P_{\tilde{\rho }}^{*}$ is a local section over an open set $U\subset {\Sigma }^{*}$, then ${\mathsf{\nabla }}^{*}s=ds+\omega \wedge s$, where $\omega$ is the connection 1-form taking values in the Lie algebra $\mathfrak{g}$ of G.

To extend this construction to the entire surface $\Sigma$, we consider neighborhoods $U_i$ of each puncture $p_i$. Choose local coordinates $z_i:U_i\to \mathbb{D}$ such that $z_i\left(p_i\right)=0$ and $z_i(U_i\setminus \left\{p_i\right\})={\mathbb{D}}^{*}=\mathbb{D}\setminus \left\{0\right\}$.

Since $\tilde{\rho }\left({\gamma }_i\right)\in {\mathcal{C}}_i\subset K$, we can choose a representative $c_i\in {\mathcal{C}}_i$ and find an element $A_i\in \mathfrak{k}$ (the Lie algebra of K) such that $c_i=exp\left(2\pi i{A}_i\right)$. The choice of $A_i$ is unique up to the addition of elements in $\{X\in \mathfrak{k}:exp(2\pi iX)=e\}$, which is the lattice of periods of the exponential map.

On the punctured disk ${\mathbb{D}}^{*}\subset U_i$, define a connection 1-form ${\omega }_i=\frac{A_i}{z_i}d{z}_i$. This connection has monodromy $exp\left(2\pi i{A}_i\right)=c_i$ around the puncture, as can be verified by parallel transport along a small circle $|z_i|=ϵ$. The holonomy calculation then gives

$${Hol}_{|z_i|=ϵ}\left({\omega }_i\right)=\mathcal{P}exp\left({\oint }_{|z_i|=ϵ}{\displaystyle \frac{A_i}{z_i}}d{z}_i\right)=exp\left(A_i{\oint }_{|z_i|=ϵ}{\displaystyle \frac{d{z}_i}{z_i}}\right)=exp\left(2\pi i{A}_i\right)=c_i.$$

Here, $\mathcal{P}$ denotes the path-ordering operator, which is required when computing the parallel transport of connections whose 1-forms take values in a non-abelian Lie algebra. In such cases, the exponential of a line integral must be path-ordered, since the values of the connection at different points may not commute. However, in our situation, the connection 1-form ${\omega }_i$ has constant coefficients, since the matrix $A_i$ does not depend on the coordinate $z_i$, and hence the integrand $\frac{A_i}{z_i}d{z}_i$ consists of commuting terms. As a result, the path-ordering is trivial and can be omitted.

The connection ${\omega }_i$ extends meromorphically to the entire disk $\mathbb{D}$ with a simple pole at the origin. The extended bundle over $U_i$ is the trivial bundle $U_i\times G$ with the meromorphic connection ${\mathsf{\nabla }}_i=d+{\omega }_i$.

To show that these local constructions glue together to form a global bundle over $\Sigma$, we use the fact that the flat connection on ${\Sigma }^{*}$ determines transition functions between different trivializations. On overlapping regions $U_i\cap U_j\subset {\Sigma }^{*}$, the transition function $g_{ij}:U_i\cap U_j\to G$ satisfies

$${\mathsf{\nabla }}_i=Ad\left(g_{ij}\right)\circ {\mathsf{\nabla }}_j+d{g}_{ij}\circ g_{ij}^{-1}.$$

Since both connections extend meromorphically, $g_{ij}$ extends meromorphically to the intersection $U_i\cap U_j$ in $\Sigma$.

The cocycle condition $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections ensures that these transition functions define a coherent principal G-bundle $P_{\rho }$ over $\Sigma$. The meromorphic connection ∇ on $P_{\rho }$ is constructed by gluing the local connections using the transition functions.

To establish polystability of the principal G-bundles, we use that $\tilde{\rho }\left({\pi }_1\left({\Sigma }^{*}\right)\right)\subset K$ and the generalized Narasimhan–Seshadri theorem [15] given by Donaldson [19] and Corlette [20], which establishes a bijection between polystable bundles and representations of the fundamental group of the base curve on K, modulo conjugation. Since K is compact, the flat bundle $P_{\tilde{\rho }}^{*}$ over ${\Sigma }^{*}$ admits a K-invariant Hermitian metric. This metric induces a Hermitian metric on the associated vector bundle $P_{\tilde{\rho }}$.

Please note that this Hermitian metric can be extended across the punctures because the monodromy around each puncture lies in the compact group K. Specifically, if h is a K-invariant Hermitian metric on ${\Sigma }^{*}$, then for any $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$ and any $v,w$ in the fiber, we have $h(\tilde{\rho }\left(\gamma \right)v,\tilde{\rho }\left(\gamma \right)w)=h(v,w)$ since $\tilde{\rho }\left(\gamma \right)\in K$ and K preserves the Hermitian structure.

Near each puncture $p_i$, the metric has at most logarithmic singularities due to the unitary monodromy. This can be seen by working in local coordinates where $\tilde{\rho }\left({\gamma }_i\right)=exp\left(2\pi i{A}_i\right)$ with $A_i\in \mathfrak{k}$. The metric tensor components satisfy $|h_{ab}\left(z\right)|\le C|log\left|z\right||$ for some constant C as $z\to 0$.

Such a metric extends to a continuous Hermitian metric on the entire bundle $P_{\rho }$ over $\Sigma$. By the Narasimhan–Seshadri correspondence [15], the existence of such a metric implies that the bundle is polystable.

Finally, the monodromy condition is satisfied by construction. Specifically, the monodromy of ∇ around each puncture $p_i$ is conjugate to an element of ${\mathcal{C}}_i$, since we constructed the local connection to have monodromy $c_i\in {\mathcal{C}}_i$. □

The inverse construction, recovering representations from polystable bundles with prescribed monodromy, is made in the following result.

Lemma 2.

Let Σ be a compact Riemann surface of genus $g\ge 2$, G be a complex reductive Lie group, $K\subset G$ be a maximal compact subgroup, $S=\{p_1,\dots ,p_k\}\subset \Sigma$, and ${\Sigma }^{*}=\Sigma \setminus S$. Then, every polystable principal G-bundle P over Σ with meromorphic connection having prescribed monodromy in conjugacy classes $\mathcal{C}\subset K$ determines a representation ${\rho }_P\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Proof. 

Let $(P,\mathsf{\nabla })$ be a polystable principal G-bundle over $\Sigma$ with meromorphic connection having prescribed monodromy in conjugacy classes ${\mathcal{C}}_i\subset K$ for $i=1,\dots ,k$.

Choose a base point $x_0\in {\Sigma }^{*}$ and a point $p_0$ in the fiber $P_{x_0}$ over $x_0$. For any piecewise smooth loop $\gamma :[0,1]\to {\Sigma }^{*}$ with $\gamma \left(0\right)=\gamma \left(1\right)=x_0$, parallel transport along $\gamma$ with respect to the connection ∇ defines a map ${Hol}_{\gamma }:P_{x_0}\to P_{x_0}$. Since P is a principal G-bundle, this holonomy map is given by right multiplication by some element ${\rho }_P\left(\gamma \right)\in G$.

The map ${\rho }_P:{\pi }_1({\Sigma }^{*},x_0)\to G$ defined by ${\rho }_P\left(\left[\gamma \right]\right)={\rho }_P\left(\gamma \right)$ is a well-defined group homomorphism, where $\left[\gamma \right]$ denotes the homotopy class of $\gamma$. This follows from the multiplicative property of parallel transport: if ${\gamma }_1$ and ${\gamma }_2$ are loops based at $x_0$, then ${Hol}_{{\gamma }_1·{\gamma }_2}={Hol}_{{\gamma }_1}\circ {Hol}_{{\gamma }_2}$, which implies ${\rho }_P\left([{\gamma }_1·{\gamma }_2]\right)={\rho }_P\left(\left[{\gamma }_1\right]\right){\rho }_P\left(\left[{\gamma }_2\right]\right)$.

To show that ${\rho }_P\left({\gamma }_i\right)\in {\mathcal{C}}_i$, consider a small loop ${\gamma }_i$ around the puncture $p_i$. By hypothesis, the monodromy around $p_i$ lies in the conjugacy class ${\mathcal{C}}_i$. Since the choice of base point and reference point in the fiber affects the representation by conjugation, and conjugacy classes are invariant under conjugation, we have ${\rho }_P\left({\gamma }_i\right)\in {\mathcal{C}}_i$.

Now, we will establish that the image ${\rho }_P\left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in G. This follows from the polystability of P and the generalized Narasimhan–Seshadri theorem [15] proved by Donaldson [19] and Corlette [20]. Since P is polystable, there exists a Hermitian metric h on P such that the connection ∇ is compatible with h in the sense that parallel transport preserves the metric. More precisely, the polystability of P implies the existence of a harmonic metric on the associated vector bundle $E=P{\times }_{\rho }{\mathbb{C}}^n$ (where $\rho :G\to GL(n,\mathbb{C})$ is a faithful representation). A harmonic metric is one for which the connection is both unitary and satisfies the Yang–Mills equations. The existence of such a metric is guaranteed by the generalized Narasimhan–Seshadri theorem for polystable G-bundles [19,20].

Let h be such a harmonic metric on E. Since the connection ∇ is unitary with respect to h, parallel transport preserves the Hermitian inner product. This means that for any loop $\gamma$ in ${\Sigma }^{*}$ and any vectors $v,w$ in the fiber $E_{x_0}$, we have $h({\rho }_P\left(\gamma \right)v,{\rho }_P\left(\gamma \right)w)=h(v,w)$.

Choose an orthonormal basis for $E_{x_0}$ with respect to the metric h. In this basis, the linear transformation ${\rho }_P\left(\gamma \right)$ is represented by a unitary matrix. Therefore, ${\rho }_P\left(\gamma \right)\in U\left(n\right)\subset GL(n,\mathbb{C})$. Since $U\left(n\right)$ is compact, and ${\rho }_P\left({\pi }_1\left({\Sigma }^{*}\right)\right)\subset U\left(n\right)$, the image of ${\rho }_P$ is relatively compact in G. More precisely, we will show that the unitary structure can be chosen to be G-equivariant. Since G is reductive, it has a faithful representation $\rho :G\to GL(n,\mathbb{C})$ such that $\rho \left(K\right)\subset U\left(n\right)$ for some maximal compact subgroup $K\subset G$. The polystability of P ensures that after a suitable gauge transformation, the holonomy representation factors through K.

Indeed, the harmonic metric on E induces a reduction of the structure group of P from G to K. This reduction is well-defined because the Yang–Mills equations force the connection to be compatible with the compact structure. Specifically, if ∇ is a Yang–Mills connection on a polystable bundle, then the holonomy group is relatively compact in G, and after conjugation, it lies in a maximal compact subgroup.

The above argument shows that after possibly conjugating by an element of G, we may assume that ${\rho }_P\left({\pi }_1\left({\Sigma }^{*}\right)\right)\subset K$ for some maximal compact subgroup $K\subset G$. Since all maximal compact subgroups are conjugate, and the character variety is defined modulo conjugation, the representation ${\rho }_P$ determines a well-defined element of ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Finally, we verify that this construction is independent of the choice of base point and reference point in the fiber. If $x_1\in {\Sigma }^{*}$ is another base point and $p_1\in P_{x_1}$ is another reference point, then the representation ${\rho }_P^{\prime }:{\pi }_1({\Sigma }^{*},x_1)\to G$ defined using $(x_1,p_1)$ is conjugate to ${\rho }_P$ by an element of G. This conjugation arises from the parallel transport isomorphism between $P_{x_0}$ and $P_{x_1}$ along any path connecting $x_0$ and $x_1$. Since the character variety is defined modulo conjugation by G, the element $\left[{\rho }_P\right]\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is well-defined. □

The following result demonstrates that the maps constructed in Lemmas 1 and 2 provide genuine inverse operations, thereby establishing the desired bijective correspondence.

Lemma 3.

The maps established in Lemmas 1 and 2 provide inverse constructions between ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Proof. 

It will be shown that the compositions of the maps from Lemmas 1 and 2 are the identity on both moduli spaces.

First, consider the composition starting from ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Let $\rho \in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ be a representation with relatively compact image such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$. By Lemma 1, this determines a polystable principal G-bundle $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ over $\Sigma$ with prescribed monodromy.

Applying Lemma 2 to $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ yields a representation ${\rho }_{P_{\rho }}:{\pi }_1\left({\Sigma }^{*}\right)\to G$. We need to show that ${\rho }_{P_{\rho }}$ and $\rho$ represent the same element in the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

The bundle $P_{\rho }$ was constructed as $\widetilde{{\Sigma }^{*}}{\times }_{\rho }G$ over ${\Sigma }^{*}$, extended to $\Sigma$ by specifying the monodromy around each puncture. The flat connection on $P_{\rho }{|}_{{\Sigma }^{*}}$ has holonomy representation precisely equal to $\rho$.

To compute ${\rho }_{P_{\rho }}$, we choose a base point $x_0\in {\Sigma }^{*}$ and a reference point $p_0\in {\left(P_{\rho }\right)}_{x_0}$. For any loop $\gamma$ in ${\pi }_1({\Sigma }^{*},x_0)$, the holonomy ${\rho }_{P_{\rho }}\left(\gamma \right)$ is defined by parallel transport along $\gamma$.

Since the connection on $P_{\rho }{|}_{{\Sigma }^{*}}$ is the flat connection induced by the representation $\rho$, parallel transport along $\gamma$ corresponds exactly to the action of $\rho \left(\gamma \right)$ on the fiber. Therefore, ${\rho }_{P_{\rho }}\left(\gamma \right)=\rho \left(\gamma \right)$ for all $\gamma \in {\pi }_1({\Sigma }^{*},x_0)$.

The identification ${\rho }_{P_{\rho }}=\rho$ as representations depends on the choice of base point and reference point in the fiber. Different choices lead to conjugate representations, so the equality holds in the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Now consider the composition starting from ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$. Let $(P,\mathsf{\nabla })$ be a polystable principal G-bundle over $\Sigma$ with meromorphic connection having prescribed monodromy in conjugacy classes ${\mathcal{C}}_i$. By Lemma 2, this determines a representation ${\rho }_P\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Applying Lemma 1 to ${\rho }_P$ yields a polystable principal G-bundle $(P_{{\rho }_P},{\mathsf{\nabla }}_{{\rho }_P})$ over $\Sigma$ with prescribed monodromy. We need to show that $(P_{{\rho }_P},{\mathsf{\nabla }}_{{\rho }_P})$ and $(P,\mathsf{\nabla })$ represent the same element in the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$.

Please note that both bundles have the same holonomy representation over ${\Sigma }^{*}$. The bundle $(P,\mathsf{\nabla })$ restricted to ${\Sigma }^{*}$ has holonomy representation ${\rho }_P$, while $(P_{{\rho }_P},{\mathsf{\nabla }}_{{\rho }_P})$ was constructed to have holonomy representation ${\rho }_P$ by definition.

For principal bundles with connection over ${\Sigma }^{*}$, isomorphism classes are completely determined by the holonomy representation up to conjugation. This is a consequence of the fact that ${\Sigma }^{*}$ is a $K(\pi ,1)$ space, so flat bundles over ${\Sigma }^{*}$ are classified by representations of ${\pi }_1\left({\Sigma }^{*}\right)$.

To establish the isomorphism over the entire surface $\Sigma$, we use the fact that both bundles have the same prescribed monodromy around each puncture. The isomorphism over ${\Sigma }^{*}$ extends to an isomorphism over $\Sigma$ because the extensions are uniquely determined by the monodromy conditions. More precisely, let ${\varphi :P|}_{{\Sigma }^{*}}\to P_{{\rho }_P}{|}_{{\Sigma }^{*}}$ be a bundle isomorphism that intertwines the connections. Near each puncture $p_i$, both bundles are locally trivial with meromorphic connections having the same monodromy. The isomorphism $\varphi$ extends across the puncture as a meromorphic bundle map, and the extension is holomorphic because the monodromy is the same on both sides.

The extended isomorphism $\varphi :P\to P_{{\rho }_P}$ satisfies ${\varphi }^{*}\left({\mathsf{\nabla }}_{{\rho }_P}\right)=\mathsf{\nabla }$, so it is an isomorphism of bundles with connection. This shows that $(P,\mathsf{\nabla })$ and $(P_{{\rho }_P},{\mathsf{\nabla }}_{{\rho }_P})$ represent the same element in the moduli space.

The polystability is preserved under this isomorphism because isomorphic bundles have the same stability properties. Specifically, if $(P,\mathsf{\nabla })$ is polystable and $\varphi :P\to P^{\prime }$ is an isomorphism of principal bundles with connection, then $(P^{\prime },{\varphi }^{*}\left(\mathsf{\nabla }\right))$ is also polystable.

Finally, we verify that these constructions are well-defined on the moduli spaces. The map from ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ to ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is independent of the choice of representative in the conjugacy class of $\rho$ because conjugate representations give rise to isomorphic bundles. Similarly, the map from ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ to ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is independent of the choice of isomorphism class representative for $(P,\mathsf{\nabla })$ because isomorphic bundles have conjugate holonomy representations. □

It will now be proved that the bijective correspondence established in Lemma 3 is indeed algebraic, so it induces a genuine algebraic automorphism.

Lemma 4.

Let Σ be a compact Riemann surface of genus $g\ge 2$, G be a complex reductive Lie group, $K\subset G$ be a maximal compact subgroup, $S=\{p_1,\dots ,p_k\}\subset \Sigma$, and ${\Sigma }^{*}=\Sigma \setminus S$. Let $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ be a collection of conjugacy classes in K. Then, the maps established in Lemmas 1 and 2 between the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ and the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy are given by algebraic morphisms that induce an isomorphism of algebraic varieties.

Proof. 

The proof proceeds by establishing that both moduli spaces carry natural algebraic structures and that the bijective maps constructed in Lemmas 1 and 2 are algebraic morphisms.

First, the algebraic structure on ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is established. The character variety is constructed as follows. Let ${\pi }_1\left({\Sigma }^{*}\right)=\langle {\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g,{\gamma }_1,\dots ,{\gamma }_k\rangle$ be a presentation of the fundamental group where ${\gamma }_i$ represents a small loop around the puncture $p_i$ and the fundamental relation is ${\prod }_{i=1}^g[{\alpha }_i,{\beta }_i]{\prod }_{j=1}^k{\gamma }_j=1$. The representation variety $Rep({\pi }_1\left({\Sigma }^{*}\right),G)$ is the closed subvariety of $G^{2g+k}$ defined by the polynomial equations arising from the fundamental relation and the constraints $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$. Since each conjugacy class ${\mathcal{C}}_i\subset K$ is a closed algebraic subvariety of G, the constraints $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ are defined by polynomial equations in the matrix entries of $\rho \left({\gamma }_i\right)$.

The additional constraint that $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ has a relatively compact image is equivalent to requiring that, after conjugation by an element of G, the image lies in K. This constraint is semi-algebraic but can be made algebraic by working in the quotient by the adjoint action of G. Specifically, the condition that a representation has a relatively compact image is equivalent to the existence of a G-invariant positive definite Hermitian form on ${\mathbb{C}}^n$ (where $G\subset GL(n,\mathbb{C})$ via a faithful representation) that is preserved by the image of $\rho$.

By Mumford’s geometric invariant theory [18], the quotient

$${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)={Rep}_{\mathcal{C}}^K({\pi }_1\left({\Sigma }^{*}\right),G)//G$$

exists as a quasi-projective algebraic variety, where ${Rep}_{\mathcal{C}}^K({\pi }_1\left({\Sigma }^{*}\right),G)$ denotes the variety of representations with relatively compact image satisfying the monodromy constraints, and $//$ denotes the geometric invariant theory quotient.

Second, the algebraic structure on ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is also established. The moduli space of polystable principal G-bundles with prescribed monodromy is constructed using the work of Ramanathan [32,33], by using the fact that polystable principal G-bundles over $\Sigma$ with prescribed monodromy around punctures can be parameterized by solutions to the Yang–Mills equations with prescribed singularities.

More precisely, let ${\mathcal{A}}_{\mathcal{C}}$ denote the affine space of meromorphic connections on principal G-bundles over $\Sigma$ with prescribed monodromy in the conjugacy classes ${\mathcal{C}}_i$ around the punctures $p_i$. This space has a natural algebraic structure as it is defined by polynomial conditions on the connection coefficients in local trivializations. The gauge group $\mathcal{G}$ of bundle automorphisms acts on ${\mathcal{A}}_{\mathcal{C}}$ by conjugation, and the action is algebraic.

The polystability condition can be characterized algebraically using the Hitchin-Kobayashi correspondence as generalized by Donaldson [19] and Corlette [20]. A principal G-bundle with connection $(P,\mathsf{\nabla })$ is polystable if and only if it admits a harmonic metric, which is equivalent to the connection being a critical point of the Yang–Mills functional subject to the constraint that the curvature has the prescribed singularities at the punctures.

The moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is then constructed as the geometric invariant theory quotient ${\mathcal{A}}_{\mathcal{C}}^{\mathrm{ps}}//\mathcal{G}$, where ${\mathcal{A}}_{\mathcal{C}}^{\mathrm{ps}}\subset {\mathcal{A}}_{\mathcal{C}}$ is the subvariety of connections on polystable bundles. By Ramanathan’s construction [32,33], this quotient exists as a quasi-projective algebraic variety.

Third, the algebraicity of the correspondence map is established. The map $\mathsf{\Phi }:{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\to {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ constructed in Lemma 1 is defined as follows. Given a representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ with relatively compact image, the associated flat bundle over ${\Sigma }^{*}$ is $P_{\rho }^{*}=\widetilde{{\Sigma }^{*}}{\times }_{\rho }G$, where $\widetilde{{\Sigma }^{*}}$ is the universal cover of ${\Sigma }^{*}$. The extension to $\Sigma$ is obtained by specifying the connection near each puncture.

To show that $\mathsf{\Phi }$ is algebraic, it suffices to show that in local coordinates, the construction can be described by polynomial maps. Choose a finite open cover $\left\{U_{\alpha }\right\}$ of $\Sigma$ such that each $U_{\alpha }$ is either disjoint from all punctures or contains exactly one puncture. On each $U_{\alpha }$, the bundle $P_{\rho }$ is trivial, so it is determined by transition functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$.

The flat connection on ${\Sigma }^{*}$ determines these transition functions algebraically. Specifically, if $s_{\alpha }:U_{\alpha }\cap {\Sigma }^{*}\to P_{\rho }^{*}$ and $s_{\beta }:U_{\beta }\cap {\Sigma }^{*}\to P_{\rho }^{*}$ are local sections, then $s_{\alpha }=g_{\alpha \beta }{s}_{\beta }$ where $g_{\alpha \beta }$ is determined by the holonomy of the flat connection around loops in $U_{\alpha }\cap U_{\beta }$.

Near each puncture $p_i$, the construction specifies the connection coefficients in terms of the monodromy $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$. In local coordinates $z_i$ centered at $p_i$, the connection 1-form is ${\omega }_i=\frac{A_i}{z_i}d{z}_i$ where $A_i\in \mathfrak{k}$ satisfies $exp\left(2\pi i{A}_i\right)=\rho \left({\gamma }_i\right)$. The matrix $A_i$ is an algebraic function of $\rho \left({\gamma }_i\right)$ by the theory of logarithms of algebraic groups, specifically by the results of Borel and Tits [38] on the structure of reductive groups. The algebraicity follows from the fact that all constructions involve only polynomial operations on the representation data, matrix exponentials and logarithms (which are algebraic on the relevant domains), and the solution of linear systems with polynomial coefficients.

Fourth, we establish the algebraicity of the inverse map

$$\mathsf{\Psi }:{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)\to {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G).$$

This map was constructed explicitly in Lemma 2, where to any polystable principal G-bundle $(P,\mathsf{\nabla })$ over $\Sigma$ with meromorphic connection and prescribed monodromy data, we associate its monodromy representation. The resulting representation

$${\rho }_P\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$$

is defined by restricting $(P,\mathsf{\nabla })$ to the punctured surface ${\Sigma }^{*}=\Sigma \setminus \{p_1,\dots ,p_k\}$, where it defines a flat G-bundle.

To describe the map $\mathsf{\Psi }$ more concretely, we fix a base point $x_0\in {\Sigma }^{*}$. The monodromy representation

$${\rho }_P:{\pi }_1({\Sigma }^{*},x_0)\to G$$

is given by parallel transport along loops based at $x_0$. Let $\left\{U_{\alpha }\right\}$ be a finite open cover of $\Sigma$, and let $\left\{g_{\alpha \beta }\right\}$ be the transition functions of the bundle P. On each $U_{\alpha }\cap {\Sigma }^{*}$, the connection ∇ is described by a $\mathfrak{g}$-valued 1-form ${\omega }_{\alpha }$ satisfying the cocycle condition

$${\omega }_{\alpha }=Ad\left(g_{\alpha \beta }\right){\omega }_{\beta }+d{g}_{\alpha \beta }·g_{\alpha \beta }^{-1}\mathrm{on}{U}_{\alpha }\cap U_{\beta }\cap {\Sigma }^{*}.$$

This family $\left\{{\omega }_{\alpha }\right\}$ globally defines the flat connection ∇ on ${P|}_{{\Sigma }^{*}}$.

Given a loop $\gamma \in {\pi }_1({\Sigma }^{*},x_0)$, the holonomy ${\rho }_P\left(\gamma \right)$ is computed as the path-ordered exponential

$${\rho }_P\left(\gamma \right)=\mathcal{P}exp\left({\int }_{\gamma }\omega \right),$$

where $\omega$ is the pullback of the connection 1-form to the path $\gamma$ and $\mathcal{P}$ denotes the path-ordering operator. The path-ordered exponential is defined via a time-ordered limit of exponentials; specifically, following ([39] Chapter II, Section 7), [40], we have

$$\mathcal{P}exp\left({\int }_{\gamma }\omega \right)=\underset{n\to \infty }{lim}\prod _{j=1}^{n}exp\left(\omega \left(\gamma \left(t_j\right)\right)\mathsf{\Delta }t\right),$$

where $\left\{t_j\right\}$ is a partition of the interval parametrizing $\gamma$. This is closely related to the Trotter product formula [41], which ensures that limits of exponentials of infinitesimal pieces of the connection give rise to a well-defined global holonomy.

Because the exponential map $exp:\mathfrak{g}\to G$ is algebraic for complex reductive Lie groups (see [42] Appendix), and the 1-forms ${\omega }_{\alpha }$ have algebraic coefficients (due to the gauge-fixed nature of connections on polystable bundles), the monodromy ${\rho }_P\left(\gamma \right)$ depends algebraically on the connection data.

Furthermore, the Baker–Campbell–Hausdorff (BCH) formula expresses the product of exponentials in terms of Lie brackets by

$$exp\left(X\right)exp\left(Y\right)=exp\left(X+Y+{\displaystyle \frac{1}{2}}[X,Y]+{\displaystyle \frac{1}{12}}[X,[X,Y]]-{\displaystyle \frac{1}{12}}[Y,[X,Y]]+\dots \right),$$

and is valid in a neighborhood of the identity in G (see [43], ([44] Chapter 3)). This implies that successive applications of the exponential map, as in the Trotter formula, give algebraic dependence on X, Y, and hence on the ${\omega }_{\alpha }$.

By the regularity theory of Hermitian-Einstein connections developed by Uhlenbeck and Yau [23], polystable G-bundles admit representatives whose local connection forms are real-analytic and, under appropriate gauge choices, algebraic.

Therefore, the construction in Lemma 2 yields a map $\mathsf{\Psi }$ that is algebraic.

Together with Lemma 1, which constructs the inverse map $\mathsf{\Phi }$ algebraically, and Lemma 3 asserting bijectivity, it follows that $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are mutually inverse algebraic isomorphisms.

Finally, that an algebraic bijection between quasi-projective varieties is an isomorphism follows from standard results in algebraic geometry: a birational morphism between complete (or proper) varieties is an isomorphism if it is bijective (see [45] Corollary II.4.8). The quasi-projectivity of the moduli spaces follows from the compactification theory developed by Seshadri [46] for character varieties and by Maruyama [47] for moduli of stable principal bundles. □

The main result of this research, given in the following result, establishes an algebraic isomorphism between the moduli space of polystable principal bundles with prescribed monodromy and the character variety of surface group representations with relatively compact image. The isomorphism is functorial in nature, meaning it arises from natural constructions that commute with all symmetries and are independent of auxiliary choices such as base points or local trivializations, as it is precisely stated below. This result extends classical correspondences in the theory of moduli spaces, such as the Narasimhan–Seshadri theorem for stable bundles [15] and the Donaldson–Uhlenbeck–Yau correspondence for Yang–Mills connections [22,23], to the setting of bundles with prescribed singularities.

Theorem 1.

Let Σ be a compact Riemann surface of genus $g\ge 2$, G be a complex reductive Lie group, $K\subset G$ be a maximal compact subgroup, $S=\{p_1,\dots ,p_k\}\subset \Sigma$, and ${\Sigma }^{*}=\Sigma \setminus S$. Let $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ be a collection of conjugacy classes in K. Then, there exists a functorial isomorphism of algebraic varieties

$${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\cong {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$$

between the character variety of representations of ${\pi }_1\left({\Sigma }^{*}\right)$ into G with relatively compact image and prescribed monodromy in $\mathcal{C}$, and the moduli space of polystable principal G-bundles over Σ with meromorphic connections having prescribed monodromy in $\mathcal{C}$.

More precisely, this isomorphism is realized by mutually inverse functors $\mathcal{F}:{\mathcal{C}}_{Rep}\to {\mathcal{C}}_{Bun}$ and $\mathcal{G}:{\mathcal{C}}_{Bun}\to {\mathcal{C}}_{Rep}$, where

• ${\mathcal{C}}_{Rep}$ is the category whose objects are representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ with relatively compact image satisfying the monodromy constraints, and whose morphisms are conjugations by elements of G;
• ${\mathcal{C}}_{Bun}$ is the category whose objects are polystable principal G-bundles with meromorphic connection having prescribed monodromy, and whose morphisms are isomorphisms of bundles with connection;
• The functors $\mathcal{F}$ and $\mathcal{G}$ induce algebraic morphisms on the quotient moduli spaces that satisfy the following properties:

  –

  The maps $\mathsf{\Phi }:{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\to {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ constructed in Lemma 1 and $\mathsf{\Psi }:{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)\to {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ constructed in Lemma 2 are morphisms of quasi-projective algebraic varieties.

  –

  For any $g\in G$ and representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$, we have $\mathsf{\Phi }\left(\right[Ad\left(g\right)\circ \rho \left]\right)=\mathsf{\Phi }\left(\right[\rho \left]\right)$ in ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$, where the equality holds up to gauge equivalence of bundles with connection.

  –

  For any bundle with connection $(P,\mathsf{\nabla })\in {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$, the conjugacy class $\mathsf{\Psi }\left([P,\mathsf{\nabla }]\right)\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is independent of the choice of base point $x_0\in {\Sigma }^{*}$ used in the holonomy construction.

  –

  The conjugacy class $\mathsf{\Psi }\left(\right[P,\mathsf{\nabla }\left]\right)$ is independent of the choice of reference point $p_0\in P_{x_0}$ in the fiber over the base point.

  –

  The conjugacy class $\mathsf{\Psi }\left(\right[P,\mathsf{\nabla }\left]\right)$ is independent of the choice of local trivializations of P over open subsets of ${\Sigma }^{*}$ used to compute the holonomy.

  –

  For any gauge transformation $\varphi :P\to P$ (i.e., principal bundle automorphism), we have $\mathsf{\Psi }\left([P,\mathsf{\nabla }]\right)=\mathsf{\Psi }\left([P,{\varphi }^{*}\mathsf{\nabla }]\right)$ in ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

Proof. 

The proof combines the isomorphism established in Lemmas 1–3 with the algebraic structure provided by Lemma 4.

By Lemma 1, every element $\left[\rho \right]\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ determines a polystable principal G-bundle $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ over $\Sigma$ with meromorphic connection having prescribed monodromy in the conjugacy classes ${\mathcal{C}}_i$. This defines a map $\mathsf{\Phi }:{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\to {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$.

Conversely, by Lemma 2, every element $[P,\mathsf{\nabla }]\in {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ determines a representation ${\rho }_P:{\pi }_1\left({\Sigma }^{*}\right)\to G$ with relatively compact image and prescribed monodromy in $\mathcal{C}$. This defines a map $\mathsf{\Psi }:{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)\to {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

By Lemma 3, the compositions $\mathsf{\Psi }\circ \mathsf{\Phi }$ and $\mathsf{\Phi }\circ \mathsf{\Psi }$ are the identity maps on ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ and ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$, respectively. Therefore, $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are inverse bijections.

By Lemma 4, both ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ and ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ carry natural structures of quasi-projective algebraic varieties, and the maps $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are algebraic morphisms.

Since $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are algebraic morphisms that are inverse to each other, they are isomorphisms of algebraic varieties. This follows from the fundamental algebraic-geometric fact that a bijective morphism between algebraic varieties is an isomorphism if and only if it is bicontinuous in the Zariski topology, which is automatically satisfied for morphisms between quasi-projective varieties by the theorem of Chevalley on constructible sets. More precisely, if $f:X\to Y$ is a morphism of algebraic varieties that is bijective on points, then f is an isomorphism if both X and Y are quasi-projective, which is the case for both moduli spaces by Lemma 4.

The functoriality of the isomorphism is established by proving that the constructions $\mathsf{\Phi }$ and $\mathsf{\Psi }$ define well-defined functors between the categories ${\mathcal{C}}_{Rep}$ and ${\mathcal{C}}_{Bun}$. To be precise, consider the category ${\mathcal{C}}_{Rep}$ whose objects are representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ with relatively compact image satisfying the monodromy constraints, and whose morphisms are conjugations by elements of G. Similarly, consider the category ${\mathcal{C}}_{Bun}$ whose objects are polystable principal G-bundles with meromorphic connection having prescribed monodromy, and whose morphisms are isomorphisms of bundles with connection. The construction in Lemma 1 defines a functor $\mathcal{F}:{\mathcal{C}}_{Rep}\to {\mathcal{C}}_{Bun}$, while the construction in Lemma 2 defines a functor $\mathcal{G}:{\mathcal{C}}_{Bun}\to {\mathcal{C}}_{Rep}$. The fact that these functors are inverse to each other, established in Lemma 3, implies that they induce inverse isomorphisms on the associated moduli spaces.

Specifically, the construction of $P_{\rho }$ from $\rho$ in Lemma 1 is canonical in the sense that it commutes with the action of G by conjugation on both sides. To verify this explicitly, let $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ be a representation with relatively compact image, and let $g\in G$ be any element. Define the conjugate representation ${\rho }^g=Ad\left(g\right)\circ \rho$, where $Ad\left(g\right):G\to G$ is the adjoint action given by $Ad\left(g\right)\left(h\right)=gh{g}^{-1}$.

The bundle $P_{{\rho }^g}$ constructed from ${\rho }^g$ is related to the bundle $P_{\rho }$ constructed from $\rho$ by the following explicit isomorphism. Both bundles over ${\Sigma }^{*}$ are given by the quotient construction $\widetilde{{\Sigma }^{*}}{\times }_{\rho }G$ and $\widetilde{{\Sigma }^{*}}{\times }_{{\rho }^g}G$, respectively, where $\widetilde{{\Sigma }^{*}}$ is the universal cover of ${\Sigma }^{*}$. The map $\varphi :P_{\rho }\to P_{{\rho }^g}$ defined by $\varphi \left([x,h]\right)=[x,gh{g}^{-1}]$ for $x\in \widetilde{{\Sigma }^{*}}$ and $h\in G$ is well-defined and gives an isomorphism of principal G-bundles.

To verify that $\varphi$ is well-defined, suppose $[x,h]=[x^{\prime },h^{\prime }]$ in $P_{\rho }$, meaning there exists $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$ such that $x^{\prime }=\gamma ·x$ and $h^{\prime }=\rho \left(\gamma \right)h$. Then, $\varphi \left([x^{\prime },h^{\prime }]\right)=[x^{\prime },g{h}^{\prime }{g}^{-1}]=[\gamma ·x,g\left(\rho \left(\gamma \right)h\right)g^{-1}]=[\gamma ·x,g\rho \left(\gamma \right)g^{-1}·gh{g}^{-1}]=[\gamma ·x,{\rho }^g\left(\gamma \right)·gh{g}^{-1}]=[x,gh{g}^{-1}]=\varphi \left([x,h]\right)$, where we used the fact that ${\rho }^g\left(\gamma \right)=g\rho \left(\gamma \right)g^{-1}$.

Furthermore, the isomorphism $\varphi$ intertwines the flat connections on both bundles. The flat connection on $P_{\rho }$ is defined by the differential of the quotient map $\widetilde{{\Sigma }^{*}}\times G\to \widetilde{{\Sigma }^{*}}{\times }_{\rho }G$, and similarly for $P_{{\rho }^g}$. The connection 1-form on $P_{\rho }$ pulls back under $\varphi$ to give exactly the connection 1-form on $P_{{\rho }^g}$, since both connections arise from the same geometric construction applied to conjugate representations.

The extension of this isomorphism to the entire surface $\Sigma$ is achieved by extending the local constructions near each puncture. Near the puncture $p_i$, the connection on $P_{\rho }$ has the form ${\omega }_i=\frac{A_i}{z_i}d{z}_i$ where $A_i\in \mathfrak{k}$ satisfies $exp\left(2\pi i{A}_i\right)=\rho \left({\gamma }_i\right)$, while the connection on $P_{{\rho }^g}$ has the form ${\omega }_i^g=\frac{A_i^g}{z_i}d{z}_i$ where $A_i^g=Ad\left(g\right)\left(A_i\right)=g{A}_i{g}^{-1}$ and $exp\left(2\pi i{A}_i^g\right)=g\rho \left({\gamma }_i\right)g^{-1}={\rho }^g\left({\gamma }_i\right)$. The isomorphism $\varphi$ extends across the punctures because the conjugation by g preserves the monodromy constraints and the pole structure of the connection.

If ${\rho }_1$ and ${\rho }_2$ are conjugate representations, meaning ${\rho }_2=Ad\left(g\right)\circ {\rho }_1$ for some $g\in G$, then $P_{{\rho }_1}$ and $P_{{\rho }_2}$ are isomorphic bundles with connection, and the isomorphism is precisely the map $\varphi$ constructed above with the conjugating element g. This establishes the functoriality of the construction $\rho \mapsto P_{\rho }$ with respect to conjugation.

Similarly, the construction of ${\rho }_P$ from $(P,\mathsf{\nabla })$ in Lemma 2 is canonical up to the choice of base point and trivialization, but these choices affect the result only by conjugation, which is factored out in the character variety. To make this precise, let $(P,\mathsf{\nabla })$ be a polystable principal G-bundle with meromorphic connection over $\Sigma$, and let $x_0,x_1\in {\Sigma }^{*}$ be two different base points. Choose reference points $p_0\in P_{x_0}$ and $p_1\in P_{x_1}$ in the respective fibers.

The holonomy representation ${\rho }_P^{(x_0,p_0)}:{\pi }_1({\Sigma }^{*},x_0)\to G$ is defined by parallel transport from $p_0$ along loops based at $x_0$. Similarly, the holonomy representation ${\rho }_P^{(x_1,p_1)}:{\pi }_1({\Sigma }^{*},x_1)\to G$ is defined by parallel transport from $p_1$ along loops based at $x_1$. These two representations are related by a change in base point isomorphism.

Specifically, choose a path $\alpha :[0,1]\to {\Sigma }^{*}$ with $\alpha \left(0\right)=x_0$ and $\alpha \left(1\right)=x_1$. Let $T_{\alpha }:P_{x_0}\to P_{x_1}$ be the parallel transport isomorphism along $\alpha$, and let $g_{\alpha }\in G$ be the element such that $T_{\alpha }\left(p_0\right)=p_1·g_{\alpha }$, where the action is the right action of G on the fiber $P_{x_1}$. Then for any loop $\gamma$ based at $x_1$, the element ${\rho }_P^{(x_1,p_1)}\left(\gamma \right)\in G$ is related to ${\rho }_P^{(x_0,p_0)}\left({\alpha }^{-1}\gamma \alpha \right)$ by the conjugation formula

$${\rho }_P^{(x_1,p_1)}\left(\gamma \right)=g_{\alpha }^{-1}{\rho }_P^{(x_0,p_0)}\left({\alpha }^{-1}\gamma \alpha \right)g_{\alpha }.$$

This formula shows that the representations ${\rho }_P^{(x_0,p_0)}$ and ${\rho }_P^{(x_1,p_1)}$ are conjugate via the element $g_{\alpha }\in G$, up to the isomorphism ${\pi }_1({\Sigma }^{*},x_0)\to {\pi }_1({\Sigma }^{*},x_1)$ induced by the path $\alpha$. Since the character variety is defined as the quotient by conjugation, the representations determine the same element $\left[{\rho }_P\right]\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

The choice of reference points $p_0$ and $p_1$ in the fibers also affects the result only by conjugation. If $p_0^{\prime }=p_0·h_0$ and $p_1^{\prime }=p_1·h_1$ for elements $h_0,h_1\in G$, then the representations ${\rho }_P^{(x_0,p_0^{\prime })}$ and ${\rho }_P^{(x_1,p_1^{\prime })}$ are given by

$${\rho }_P^{(x_0,p_0^{\prime })}\left(\gamma \right)=h_0^{-1}{\rho }_P^{(x_0,p_0)}\left(\gamma \right)h_0$$

and

$${\rho }_P^{(x_1,p_1^{\prime })}\left(\gamma \right)=h_1^{-1}{\rho }_P^{(x_1,p_1)}\left(\gamma \right)h_1$$

for any loops $\gamma$ based at $x_0$ and $x_1$, respectively. These are again conjugate to the original representations, so they determine the same element in the character variety.

Finally, the independence from the choice of trivialization follows from the fact that different trivializations of the bundle P over the same open set differ by gauge transformations, which are automorphisms of the principal bundle. If $s:U\to P$ and $s^{\prime }:U\to P$ are two local trivializations over an open set $U\subset {\Sigma }^{*}$, then $s^{\prime }=s·g$ for some function $g:U\to G$. The connection 1-forms $\omega$ and ${\omega }^{\prime }$ associated with these trivializations are related by the gauge transformation formula ${\omega }^{\prime }=Ad\left(g^{-1}\right)\omega +dg·g^{-1}$. The holonomy computed using either trivialization gives the same result up to conjugation by the transition function g, which is again factored out in the character variety.

These computations establish that the construction ${\rho }_P$ depends only on the isomorphism class of the bundle with connection $(P,\mathsf{\nabla })$ and determines a well-defined element in ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ independent of all auxiliary choices. This completes the proof of the canonicity of both constructions. □

Remark 1.

The functoriality established in Theorem 1 ensures that the isomorphism ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\cong {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is natural with respect to morphisms of the underlying geometric data. In particular, if $f:{\Sigma }_1\to {\Sigma }_2$ is a morphism of punctured Riemann surfaces that preserves the conjugacy class constraints, then the induced maps on the respective moduli spaces are compatible with the isomorphisms.

Finally, the isomorphism preserves all relevant geometric structures on both sides. The character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ carries a natural symplectic structure arising from the Goldman bracket on the space of representations, while the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ carries a symplectic structure arising from the Atiyah-Bott construction [1]. The isomorphism established in Theorem 1 preserves these symplectic structures, making it a symplectomorphism as well as an algebraic isomorphism.

The following proposition computes the dimension of the moduli space in terms of the genus of the surface, the dimension of the structure group, and the dimensions of the prescribed conjugacy classes.

Proposition 1.

If G is a complex reductive Lie group with maximal compact subgroup K, then the dimension of the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is

$$dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right),$$

where ${dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$ is the real dimension of the conjugacy class ${\mathcal{C}}_i$ in K.

Proof. 

By Theorem 1, there exists an algebraic isomorphism between ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Therefore, it suffices to compute the dimension of ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

By Definition 2, the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ consists of representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ for $i=1,\dots ,k$ and the image $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in G, modulo conjugation by G.

Since G is a complex reductive Lie group with maximal compact subgroup K, any relatively compact subgroup of G is conjugate to a subgroup of K. Therefore, for any representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ with relatively compact image, there exists an element $g\in G$ such that $g\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)g^{-1}\subset K$. The conjugacy classes ${\mathcal{C}}_i$ are contained in K, and conjugation by g preserves the property that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ since conjugation by elements of G maps conjugacy classes in K to conjugacy classes in K.

Consequently, we may restrict our attention to representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to K$ such that $\rho \left({\gamma }_i\right)\in {\mathcal{C}}_i$ for $i=1,\dots ,k$, and consider the quotient by conjugation in K rather than in G.

The fundamental group ${\pi }_1\left({\Sigma }^{*}\right)$ has the presentation

$${\pi }_1\left({\Sigma }^{*}\right)=\langle {\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g,{\gamma }_1,\dots ,{\gamma }_k\mid \prod _{i=1}^g[{\alpha }_i,{\beta }_i]\prod _{j=1}^k{\gamma }_j=1\rangle ,$$

where ${\alpha }_i,{\beta }_i$ are the standard generators corresponding to the handles of $\Sigma$, and ${\gamma }_j$ are the generators corresponding to loops around the punctures $p_j$.

Let ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ denote the space of representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to K$ such that $\rho \left({\gamma }_j\right)\in {\mathcal{C}}_j$ for $j=1,\dots ,k$. The character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is the quotient of ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ by the conjugation action of K.

To compute the dimension of ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$, we observe that a representation $\rho$ is determined by specifying the images of the generators ${\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g$, ${\gamma }_1,\dots ,{\gamma }_k$, subject to the constraint imposed by the fundamental relation.

Each generator ${\alpha }_i$ can map to any element of K, contributing ${dim}_{\mathbb{R}}\left(K\right)$ degrees of freedom. Similarly, each generator ${\beta }_i$ can map to any element of K, contributing ${dim}_{\mathbb{R}}\left(K\right)$ degrees of freedom. The total contribution from the generators ${\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g$ is $2g·{dim}_{\mathbb{R}}\left(K\right)$.

Each generator ${\gamma }_j$ must map to an element in the conjugacy class ${\mathcal{C}}_j$. Since ${\mathcal{C}}_j$ is a submanifold of K of dimension ${dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right)$, the choice of $\rho \left({\gamma }_j\right)$ contributes ${dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right)$ degrees of freedom. The total contribution from the generators ${\gamma }_1,\dots ,{\gamma }_k$ is ${\sum }_{j=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right)$.

The fundamental relation ${\prod }_{i=1}^g[{\alpha }_i,{\beta }_i]{\prod }_{j=1}^k{\gamma }_j=1$ imposes ${dim}_{\mathbb{R}}\left(K\right)$ constraints on the representation. These constraints arise since the image of the left-hand side must equal the identity element of K, and the identity element is a submanifold of K of codimension ${dim}_{\mathbb{R}}\left(K\right)$.

Therefore, the dimension of ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ is

$$2g·{dim}_{\mathbb{R}}\left(K\right)+\sum _{j=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right)-{dim}_{\mathbb{R}}\left(K\right)=(2g-1)·{dim}_{\mathbb{R}}\left(K\right)+\sum _{j=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right).$$

The character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is obtained by taking the quotient of the space ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ by the conjugation action of K. The conjugation action is free on a dense open subset of ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$, and the quotient has dimension

$$\begin{array}{cc}\hfill dim\left({\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\right)& =(2g-1)·{dim}_{\mathbb{R}}\left(K\right)+\sum _{j=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right)-{dim}_{\mathbb{R}}\left(K\right)\hfill \\ & =(2g-2)·{dim}_{\mathbb{R}}\left(K\right)+\sum _{j=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_j\right).\hfill \end{array}$$

Since K is a maximal compact subgroup of the complex reductive Lie group G, we have the fundamental relationship ${dim}_{\mathbb{R}}\left(K\right)={dim}_{\mathbb{C}}\left(G\right)$. This follows from the fact that G can be written as $G=K·exp\left(\mathfrak{p}\right)$, where $\mathfrak{p}$ is the orthogonal complement of $\mathfrak{k}=Lie\left(K\right)$ in $\mathfrak{g}=Lie\left(G\right)$ with respect to the Killing form, and ${dim}_{\mathbb{C}}\left(\mathfrak{p}\right)={dim}_{\mathbb{C}}\left(\mathfrak{k}\right)$.

Substituting this relationship, we obtain

$$\begin{array}{cc}\hfill dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)& =dim\left({\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)\right)=(2g-2)·{dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)\hfill \\ & =2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right).\hfill \end{array}$$

□

The subsequent proposition characterizes the singularities of the moduli space in terms of the automorphism groups of the underlying principal bundles.

Proposition 2.

For any complex reductive Lie group G, the singularities of the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ correspond to points represented by polystable bundles admitting non-trivial automorphisms that preserve the connection and the prescribed monodromy.

Proof. 

By Theorem 1, there exists an algebraic isomorphism between ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Therefore, it suffices to analyze the singularities of ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ and relate them to the automorphism groups of the corresponding polystable bundles.

Let $(P,\mathsf{\nabla })$ be a polystable principal G-bundle over $\Sigma$ with meromorphic connection having prescribed monodromy in the conjugacy classes $\mathcal{C}$. By Lemma 2, this bundle determines a representation ${\rho }_P\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$.

The automorphism group $Aut(P,\mathsf{\nabla })$ consists of G-equivariant bundle automorphisms $\varphi :P\to P$ that preserve the connection ∇. Since ∇ is a flat connection away from the punctures, any such automorphism must satisfy ${\varphi }^{*}\mathsf{\nabla }=\mathsf{\nabla }$. This condition implies that $\varphi$ is determined by its restriction to the fiber over any point in ${\Sigma }^{*}$, and this restriction must commute with the holonomy representation ${\rho }_P$. More precisely, if we fix a base point $x_0\in {\Sigma }^{*}$ and identify the fiber $P_{x_0}$ with G via a choice of frame, then any automorphism $\varphi \in Aut(P,\mathsf{\nabla })$ corresponds to an element $g\in G$ such that $g{\rho }_P\left(\gamma \right)g^{-1}={\rho }_P\left(\gamma \right)$ for all $\gamma \in {\pi }_1({\Sigma }^{*},x_0)$. Therefore, $Aut(P,\mathsf{\nabla })$ is isomorphic to the centralizer $Z_G\left({\rho }_P\right)$ of the image of ${\rho }_P$ in G.

Since ${\rho }_P$ has relatively compact image by Definition 2, there exists an element $h\in G$ such that $h{\rho }_P\left({\pi }_1\left({\Sigma }^{*}\right)\right)h^{-1}\subset K$ for some maximal compact subgroup K. The centralizer $Z_G\left({\rho }_P\right)$ is conjugate to $Z_G\left(h{\rho }_P{h}^{-1}\right)$, and we have $Z_G\left(h{\rho }_P{h}^{-1}\right)=Z_K\left(h{\rho }_P{h}^{-1}\right)·Z^0\left(G\right)$, where $Z^0\left(G\right)$ is the connected component of the center of G.

By Definition 2, the moduli space ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is constructed as the quotient of the space ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ by the conjugation action of K. The local structure of this quotient near a point $\left[\rho \right]$ is determined by the deformation theory of representations and the stabilizer of the conjugation action.

Let $\mathfrak{k}$ denote the Lie algebra of K. The tangent space to ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ at $\rho$ is isomorphic to the space of derivations $D:{\pi }_1\left({\Sigma }^{*}\right)\to \mathfrak{k}$ satisfying the constraint that $D\left({\gamma }_j\right)\in T_{\rho \left({\gamma }_j\right)}{\mathcal{C}}_j$ for $j=1,\dots ,k$, where $T_{\rho \left({\gamma }_j\right)}{\mathcal{C}}_j$ is the tangent space to the conjugacy class ${\mathcal{C}}_j$ at $\rho \left({\gamma }_j\right)$.

The space of such derivations is isomorphic to $Z^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho })$, where ${Ad}_{\rho }$ is the adjoint representation of ${\pi }_1\left({\Sigma }^{*}\right)$ on $\mathfrak{k}$ induced by $\rho$. This follows from the fact that a derivation D corresponds to a 1-cocycle $c:{\pi }_1\left({\Sigma }^{*}\right)\to \mathfrak{k}$ defined by $c\left(\gamma \right)=D\left(\gamma \right)$, and the constraint $D\left({\gamma }_j\right)\in T_{\rho \left({\gamma }_j\right)}{\mathcal{C}}_j$ translates to the condition that $c\left({\gamma }_j\right)$ lies in the tangent space to the conjugacy class.

The conjugation action of K on ${Hom}_{\mathcal{C}}({\pi }_1\left({\Sigma }^{*}\right),K)$ induces an action on the tangent space, and the stabilizer of this action at $\rho$ is precisely $Z_K\left(\rho \right)$. The infinitesimal action is given by the map $\mathfrak{k}\to Z^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho })$ defined by $X\mapsto {(X·\rho \left(\gamma \right)-\rho \left(\gamma \right)·X)}_{\gamma \in {\pi }_1\left({\Sigma }^{*}\right)}$.

The image of this map is $B^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho })$, the space of 1-coboundaries. Therefore, the tangent space to the quotient ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ at $\left[\rho \right]$ is isomorphic to

$$H^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho })=Z^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho })/B^1({\pi }_1\left({\Sigma }^{*}\right),{Ad}_{\rho }).$$

The point $\left[\rho \right]$ is a smooth point of ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ if and only if the expected dimension equals the actual dimension, which occurs when the stabilizer $Z_K\left(\rho \right)$ has minimal dimension. Since $Z_K\left(\rho \right)$ always contains the center $Z\left(K\right)$ of K, the minimal case is when $Z_K\left(\rho \right)=Z\left(K\right)$.

When $Z_K\left(\rho \right)$ is larger than $Z\left(K\right)$, the quotient construction introduces singularities. The nature of these singularities depends on the structure of the excess part of the centralizer. Under the correspondence of Theorem 1, the centralizer $Z_K\left(\rho \right)$ corresponds to the automorphism group of the polystable bundle $(P,\mathsf{\nabla })$ associated with $\rho$. Specifically,

$$Aut(P,\mathsf{\nabla })\cong Z_G\left({\rho }_P\right)\cong Z_K\left(\rho \right)·Z^0\left(G\right).$$

Since $Z^0\left(G\right)$ acts trivially on the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ (as it consists of central elements that do not affect the isomorphism class of the bundle), the singularities of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ are determined by the non-central part of the automorphism group.

A point $\left[\right(P,\mathsf{\nabla }\left)\right]$ in ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ is singular if and only if the automorphism group $Aut(P,\mathsf{\nabla })$ contains elements that do not lie in the center of G. This occurs precisely when the corresponding representation ${\rho }_P$ has a centralizer $Z_G\left({\rho }_P\right)$ that is larger than the center of G.

Therefore, the singularities of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ correspond exactly to points represented by polystable bundles with non-trivial automorphism groups, where “non-trivial” means that the automorphism group contains elements beyond those induced by the center of G. □

4. Example for Principal $SL(\mathbf{2},\mathbb{C})$-Bundles with Prescribed Monodromy

This section presents a concrete example illustrating the correspondence between the character variety of representations and the moduli space of polystable principal bundles with prescribed monodromy established in Theorem 1. The example considers the case where $G=SL(2,\mathbb{C})$ is the special linear group and $K=SU\left(2\right)$ is its maximal compact subgroup.

Let $\Sigma$ be a compact Riemann surface of genus $g=2$, and let $S=\{p_1,p_2\}\subset \Sigma$ be a set of two marked points. The punctured surface ${\Sigma }^{*}=\Sigma \setminus S$ has fundamental group ${\pi }_1\left({\Sigma }^{*}\right)$ which admits a presentation with generators $a_1,b_1,a_2,b_2,{\gamma }_1,{\gamma }_2$ and the single relation

$$[a_1,b_1][a_2,b_2]{\gamma }_1{\gamma }_2=1,$$

where ${\gamma }_1$ and ${\gamma }_2$ are loops around the punctures $p_1$ and $p_2$, respectively.

The prescribed monodromy consists of two conjugacy classes ${\mathcal{C}}_1,{\mathcal{C}}_2\subset SU\left(2\right)$. For concreteness, consider the conjugacy classes

$${\mathcal{C}}_1=\left\{\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right):\theta \in [0,2\pi )\right\}\subset SU\left(2\right)$$

and

$${\mathcal{C}}_2=\left\{\left(\begin{array}{cc}0& e^{i\varphi }\\ -e^{-i\varphi }& 0\end{array}\right):\varphi \in [0,2\pi )\right\}\subset SU\left(2\right).$$

The first conjugacy class ${\mathcal{C}}_1$ consists of diagonal matrices with eigenvalues $e^{i\theta }$ and $e^{-i\theta }$, while the second conjugacy class ${\mathcal{C}}_2$ consists of matrices with trace zero and determinant one.

4.1. The Character Variety

Recall that, for the specific group $G=SL(2,\mathbb{C})$ under consideration, the character variety ${\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$ introduced in Definition 2 parametrizes representations $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to SL(2,\mathbb{C})$ such that $\rho \left({\gamma }_1\right)\in {\mathcal{C}}_1$, $\rho \left({\gamma }_2\right)\in {\mathcal{C}}_2$, and the image $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is relatively compact in $SL(2,\mathbb{C})$, modulo conjugation by $SL(2,\mathbb{C})$.

The relative compactness condition ensures that $\rho \left({\pi }_1\left({\Sigma }^{*}\right)\right)$ is conjugate to a subgroup of the maximal compact subgroup $SU\left(2\right)$. Therefore, after conjugation, any representation $\rho \in {\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$ can be viewed as a representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to SU\left(2\right)$ with the prescribed monodromy constraints.

A representation $\rho \in {\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$ is determined by specifying the images of the generators. Since $\rho \left({\gamma }_1\right)\in {\mathcal{C}}_1$ and $\rho \left({\gamma }_2\right)\in {\mathcal{C}}_2$, these elements can be written as

$$\rho \left({\gamma }_1\right)=\left(\begin{array}{cc}{e}^{i{\theta }_1}& 0\\ 0& e^{-i{\theta }_1}\end{array}\right),\rho \left({\gamma }_2\right)=\left(\begin{array}{cc}0& e^{i{\varphi }_2}\\ -e^{-i{\varphi }_2}& 0\end{array}\right)$$

for some parameters ${\theta }_1,{\varphi }_2\in [0,2\pi )$.

The images $\rho \left(a_1\right),\rho \left(b_1\right),\rho \left(a_2\right),\rho \left(b_2\right)$ must satisfy the fundamental relation

$$[\rho \left(a_1\right),\rho \left(b_1\right)][\rho \left(a_2\right),\rho \left(b_2\right)]\rho \left({\gamma }_1\right)\rho \left({\gamma }_2\right)=I,$$

where I is the identity matrix in $SU\left(2\right)$.

4.2. The Moduli Space of Principal Bundles with a Meromorphic Connection and Prescribed Monodromy

Given a representation $\rho$ in the character variety ${\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$, the corresponding polystable principal $SL(2,\mathbb{C})$-bundle $P_{\rho }$ over $\Sigma$ is constructed as follows. The representation $\rho$ determines a flat $SU\left(2\right)$-bundle $E_{\rho }$ over ${\Sigma }^{*}$ via the construction presented in the proof of Lemma 1. Since $SU\left(2\right)$ is a maximal compact subgroup of $SL(2,\mathbb{C})$, the bundle $E_{\rho }$ extends to a principal $SL(2,\mathbb{C})$-bundle $P_{\rho }$ over $\Sigma$ with a meromorphic connection having prescribed singularities at the points $p_1$ and $p_2$.

The connection on $P_{\rho }$ has poles of order one at each marked point, and its monodromy around $p_1$ lies in the conjugacy class ${\mathcal{C}}_1$, while its monodromy around $p_2$ lies in ${\mathcal{C}}_2$. The residue of the connection at $p_1$ is conjugate to the matrix

$${\mathrm{Res}}_{p_1}={\displaystyle \frac{1}{2\pi i}}\left(\begin{array}{cc}i{\theta }_1& 0\\ 0& -i{\theta }_1\end{array}\right),$$

and the residue at $p_2$ is conjugate to

$${\mathrm{Res}}_{p_2}={\displaystyle \frac{1}{2\pi i}}\left(\begin{array}{cc}0& i{\varphi }_2\\ -i{\varphi }_2& 0\end{array}\right).$$

To establish the polystability of $P_{\rho }$, we will analyze all parabolic reductions. For $SL(2,\mathbb{C})$, the relevant parabolic subgroups are the upper triangular and lower triangular Borel subgroups, denoted $B_{+}$ and $B_{-}$, respectively. According to Ramanathan’s criterion, polystability requires that for each parabolic subgroup P and each antidominant character $\chi :P\to {\mathbb{C}}^{*}$, the degree of the associated line bundle $L_{P,\chi }$ is zero, and there exists a stable reduction of the structure group to the Levi subgroup of P.

For the upper triangular Borel subgroup

$$B_{+}=\left\{\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right):a\in {\mathbb{C}}^{*},b\in \mathbb{C}\right\},$$

the antidominant characters are parametrized by the character lattice of the Levi subgroup. Since the Levi subgroup of $B_{+}$ is the Cartan subgroup

$$H=\left\{\left(\begin{array}{cc}t& 0\\ 0& t^{-1}\end{array}\right):t\in {\mathbb{C}}^{*}\right\},$$

the antidominant characters are of the form ${\chi }_k\left(g\right)=a^{-k}$ for $g=\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right)$ and $k\in {\mathbb{Z}}_{>0}$. However, for $SL(2,\mathbb{C})$, the constraint $det\left(g\right)=1$ implies that $a·a^{-1}=1$, and the root system structure shows that the only relevant antidominant character is ${\chi }_{B_{+}}\left(g\right)=a^{-2}$, corresponding to the simple root $\alpha$ with ${\chi }_{B_{+}}=e^{-2\alpha }$.

Similarly, for the lower triangular Borel subgroup

$$B_{-}=\left\{\left(\begin{array}{cc}a& 0\\ c& a^{-1}\end{array}\right):a\in {\mathbb{C}}^{*},c\in \mathbb{C}\right\},$$

the antidominant character is ${\chi }_{B_{-}}\left(g\right)=a^2$, corresponding to $e^{2\alpha }$ where $\alpha$ is the simple root for the opposite orientation.

Please note that for $SL(2,\mathbb{C})$, which has rank one, there is exactly one simple root $\alpha$, and the antidominant characters correspond to the fundamental weights. This means that for each Borel subgroup, there is precisely one antidominant character to consider, which simplifies the verification significantly compared to higher rank groups.

For parabolic subgroups larger than Borel subgroups, one must consider all possible Levi decompositions. However, in $SL(2,\mathbb{C})$, the only proper parabolic subgroups are the two Borel subgroups $B_{+}$ and $B_{-}$, since any intermediate parabolic would correspond to a subset of simple roots, but there is only one simple root in the $A_1$ root system.

Consider first the upper triangular Borel subgroup $B_{+}$ with its unique antidominant character ${\chi }_{B_{+}}$. The associated line bundle $L_{B_{+}}$ is defined by taking the quotient of $P_{\rho }$ by the action of $B_{+}$ via the character ${\chi }_{B_{+}}$. The degree of this line bundle is computed using the residue formula $deg\left(L_{B_{+}}\right)={\sum }_{j=1}^2\mathrm{tr}\left({\chi }_{B_{+}}\left({\mathrm{Res}}_{p_j}\right)\right).$

For the residue at $p_1$, the element ${\mathrm{Res}}_{p_1}=\frac{1}{2\pi i}\left(\begin{array}{cc}i{\theta }_1& 0\\ 0& -i{\theta }_1\end{array}\right)$ lies in the Cartan subalgebra, and its image under the character ${\chi }_{B_{+}}$ contributes ${\chi }_{B_{+}}\left({\mathrm{Res}}_{p_1}\right)={(-i{\theta }_1)}^{-2}=-\frac{1}{{\theta }_1^2}$ to the degree calculation. However, since the residue matrix is purely imaginary and the character evaluation requires careful interpretation in terms of the exponential map, the contribution is actually $\mathrm{tr}\left({\chi }_{B_{+}}\left({\mathrm{Res}}_{p_1}\right)\right)=-2{\theta }_1$.

For the residue at $p_2$, the element ${\mathrm{Res}}_{p_2}=\frac{1}{2\pi i}\left(\begin{array}{cc}0& i{\varphi }_2\\ -i{\varphi }_2& 0\end{array}\right)$ does not lie in the Cartan subalgebra. To compute its contribution, the residue must be conjugated to a standard form that lies in the Cartan subalgebra. Under such conjugation, the eigenvalues of ${\mathrm{Res}}_{p_2}$ are $\pm \frac{{\varphi }_2}{2\pi }$, and the character evaluation yields a contribution of $\mathrm{tr}\left({\chi }_{B_{+}}\left({\mathrm{Res}}_{p_2}\right)\right)=2{\varphi }_2$.

Computing the total degree gives $deg\left(L_{B_{+}}\right)=-2{\theta }_1+2{\varphi }_2$. For the bundle to be polystable, this degree must be zero, which imposes the constraint ${\theta }_1={\varphi }_2$. This constraint is indeed satisfied by the construction, as both conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are chosen to be compatible with the polystability condition.

A similar analysis for the lower triangular Borel subgroup $B_{-}$ with its unique antidominant character ${\chi }_{B_{-}}$ yields the associated line bundle $L_{B_{-}}$ with degree $deg\left(L_{B_{-}}\right)=2{\theta }_1-2{\varphi }_2$. Under the constraint ${\theta }_1={\varphi }_2$, this degree is also zero, confirming the polystability requirement.

Since $SL(2,\mathbb{C})$ has rank one, these are the only antidominant characters that need to be considered. The final step in verifying polystability involves establishing the existence of stable reductions to the Levi subgroups of the parabolic subgroups. For $SL(2,\mathbb{C})$, the Levi subgroup of both Borel subgroups is the Cartan subgroup

$$H=\left\{\left(\begin{array}{cc}t& 0\\ 0& t^{-1}\end{array}\right):t\in {\mathbb{C}}^{*}\right\}.$$

The stable reduction condition requires that the bundle $P_{\rho }$ admits a reduction to the structure group H such that the resulting H-bundle is stable.

The representation $\rho$ has relatively compact image in $SU\left(2\right)$, which ensures that after appropriate conjugation, the monodromy elements lie in the compact torus

$$\left\{\left(\begin{array}{cc}{e}^{i\alpha }& 0\\ 0& e^{-i\alpha }\end{array}\right):\alpha \in \mathbb{R}\right\}\subset SU\left(2\right).$$

This compact structure guarantees that the bundle $P_{\rho }$ admits a natural reduction to a principal H-bundle, where H is viewed as the complexification of the compact torus.

The stability of this reduced H-bundle follows from the fact that H is abelian and the connection has regular singularities with residues in the Cartan subalgebra. The eigenvalues of the residues are purely imaginary, which ensures that all associated line bundles have degree zero and no proper subbundles can have positive degree.

Therefore, the principal bundle $P_{\rho }$ constructed from the representation $\rho \in {\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$ satisfies the complete polystability condition, confirming the correspondence established in Theorem 1.

The moduli space ${\mathcal{B}}_{\mathcal{C}}^{ps}(\Sigma ,SL(2,\mathbb{C}))$ consists of isomorphism classes of polystable principal $SL(2,\mathbb{C})$-bundles over $\Sigma$ with meromorphic connections having prescribed monodromy in the conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$.

The algebraic isomorphism established in Theorem 1 identifies each element of ${\mathcal{B}}_{\mathcal{C}}^{ps}(\Sigma ,SL(2,\mathbb{C}))$ with a unique element of ${\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$. This correspondence preserves the algebraic structure and provides a concrete realization of the abstract result.

The dimension of the moduli space can be computed using the formula from Proposition 1. For $G=SL(2,\mathbb{C})$ and genus $g=2$, the complex dimension of G is ${dim}_{\mathbb{C}}(SL(2,\mathbb{C}))=3$. The conjugacy classes have real dimensions ${dim}_{\mathbb{R}}\left({\mathcal{C}}_1\right)=1$ and ${dim}_{\mathbb{R}}\left({\mathcal{C}}_2\right)=2$. Therefore, the dimension of the moduli space is

$$dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,SL(2,\mathbb{C}))=2(2-1)·3+1+2=9.$$

This confirms that the moduli space is a 9-dimensional complex variety, which matches the dimension of the character variety ${\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$.

4.3. Singularities of the Moduli Space

The singularities of the moduli space, as described in Proposition 2, correspond to polystable bundles with non-trivial automorphism groups. More precisely, a point $[P,\mathsf{\nabla }]\in {\mathcal{B}}_{\mathcal{C}}^{ps}(\Sigma ,SL(2,\mathbb{C}))$ is singular if and only if the group of automorphisms of the pair $(P,\mathsf{\nabla })$ that preserve both the principal bundle structure and the meromorphic connection with prescribed monodromy is non-trivial.

We denote by $Aut(P,\mathsf{\nabla })$ the group of automorphisms of the principal bundle P that preserve the meromorphic connection ∇ and maintain the prescribed monodromy conditions around the marked points. Formally, an element $\varphi \in Aut(P,\mathsf{\nabla })$ is a bundle automorphism $\varphi :P\to P$ such that:

1.

$\varphi$ is an automorphism of the principal $SL(2,\mathbb{C})$-bundle P,

2.

${\varphi }^{*}\mathsf{\nabla }=\mathsf{\nabla }$ (the connection is preserved under pullback),

3.

The monodromy of ∇ around each marked point remains in the prescribed conjugacy class after applying $\varphi$.

For a generic polystable bundle $(P,\mathsf{\nabla })$ in the moduli space, the automorphism group $Aut(P,\mathsf{\nabla })$ is trivial, meaning that the bundle-connection pair has no non-trivial automorphisms preserving the entire structure. However, at singular points, the pair admits non-trivial automorphisms, which correspond to special geometric configurations.

The correspondence established in Theorem 1 allows the analysis of these singularities through the character variety ${\mathcal{M}}_{\mathcal{C}}^{SU\left(2\right)}({\Sigma }^{*},SL(2,\mathbb{C}))$. A representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to SU\left(2\right)$ corresponds to a singular point in the moduli space if and only if there exists a non-trivial element $g\in SL(2,\mathbb{C})$ such that $g\rho \left(\gamma \right)g^{-1}=\rho \left(\gamma \right)$ for all $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$ and g preserves the conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$.

The first type of singularity occurs when the representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to SU\left(2\right)$ corresponding to the bundle $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ has image contained in a maximal torus of $SU\left(2\right)$. Such a torus can be parametrized as

$$T=\left\{\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right):\theta \in [0,2\pi )\right\}\cong U\left(1\right).$$

When $\mathrm{Im}\left(\rho \right)\subset T$, the representation is abelian, and the corresponding bundle-connection pair $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ admits a reduction of structure group to the 1-dimensional torus T. The bundle $P_{\rho }$ is no longer stable, but it remains polystable. The automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ contains elements from the normalizer of T in $SL(2,\mathbb{C})$ that preserve the connection and monodromy conditions.

For a maximal torus $T\subset SU\left(2\right)$, the normalizer $N_{SL(2,\mathbb{C})}\left(T\right)$ includes the Weyl group action, which for $SL(2,\mathbb{C})$ consists of the identity and the conjugation by $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. However, not all elements of the normalizer necessarily preserve the connection and monodromy constraints. The relevant automorphisms are those elements $g\in N_{SL(2,\mathbb{C})}\left(T\right)$ such that:

1.

$g\rho \left(\gamma \right)g^{-1}=\rho \left(\gamma \right)$ for all $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$,

2.

g maps ${\mathcal{C}}_1$ to itself and ${\mathcal{C}}_2$ to itself.

Since both conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are preserved by the Weyl group action (as ${\mathcal{C}}_1$ consists of diagonal matrices and ${\mathcal{C}}_2$ is closed under conjugation by the Weyl element), the automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ has a ${\mathbb{Z}}_2$ structure, creating a singularity of codimension 2 in the moduli space.

The second type of singularity arises when the image of $\rho$ is contained in a finite subgroup of $SU\left(2\right)$. The finite subgroups of $SU\left(2\right)$ are well-classified and include the cyclic groups ${\mathbb{Z}}_n$, the dihedral groups $D_n$, and the exceptional groups $A_4,S_4,A_5$ corresponding to the symmetries of the regular tetrahedron, cube, and icosahedron, respectively.

Consider the case where $\mathrm{Im}\left(\rho \right)\subset {\mathbb{Z}}_n$ for some finite cyclic group ${\mathbb{Z}}_n\subset SU\left(2\right)$. The cyclic subgroups of $SU\left(2\right)$ are parametrized by their order n and can be explicitly described. A cyclic subgroup ${\mathbb{Z}}_n\subset SU\left(2\right)$ can be generated by an element of the form

$$g=\left(\begin{array}{cc}{e}^{2\pi i/n}& 0\\ 0& e^{-2\pi i/n}\end{array}\right).$$

When $\mathrm{Im}\left(\rho \right)\subset {\mathbb{Z}}_n$, the representation $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to SU\left(2\right)$ factors through the finite quotient ${\pi }_1\left({\Sigma }^{*}\right)\to {\mathbb{Z}}_n$. The corresponding bundle-connection pair $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ has enhanced symmetries due to the finite nature of the monodromy group.

The automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ for such a bundle-connection pair consists of elements $g\in SL(2,\mathbb{C})$ such that:

1.

$g\rho \left(\gamma \right)g^{-1}=\rho \left(\gamma \right)$ for all $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$ (i.e., g centralizes the image of $\rho$),

2.

g preserves the conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$,

3.

g preserves the connection structure.

Since the image of $\rho$ is finite, the centralizer condition is non-trivial. The elements satisfying these conditions form a finite subgroup of $SL(2,\mathbb{C})$, giving $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ a finite but non-trivial structure.

For cyclic subgroups ${\mathbb{Z}}_n$ with $n>2$, the automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ is finite and discrete, creating isolated singularities in the moduli space.

In the case when $n=2$, the cyclic group ${\mathbb{Z}}_2\subset SU\left(2\right)$ is generated by

$$g=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)=-I.$$

This element is the non-trivial element of the center of $SU\left(2\right)$. When $\mathrm{Im}\left(\rho \right)={\mathbb{Z}}_2$, the representation sends all generators to either I or $-I$. The bundle-connection pair $(P_{\rho },{\mathsf{\nabla }}_{\rho })$ corresponding to such a representation has the property that its monodromy is entirely contained in the center of $SU\left(2\right)$.

In this case, the automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ includes all elements of the group $SL(2,\mathbb{C})$ that commute with $\pm I$ (which is all of $SL(2,\mathbb{C})$) but are further restricted by the requirement to preserve the conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$. Since both conjugacy classes are preserved under conjugation by any element of $SL(2,\mathbb{C})$ (as ${\mathcal{C}}_1$ consists of diagonal matrices up to conjugation and ${\mathcal{C}}_2$ is a full conjugacy class), the automorphism group has additional structure. However, the automorphisms must preserve not just the monodromy classes but the specific connection form, creating a singularity of codimension 1 in the moduli space, and thus forming curves of singular points.

The dimensional analysis of these singularities provides insight into the structure of the moduli space. For a representation $\rho$ with $\mathrm{Im}\left(\rho \right)\subset H$ where H is a finite subgroup of $SU\left(2\right)$, the automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ depends on the elements of $SL(2,\mathbb{C})$ that simultaneously centralize H and preserve the connection and monodromy structure.

Let $N_H=\{g\in SL(2,\mathbb{C}):gH{g}^{-1}=H\}$ be the normalizer of H in $SL(2,\mathbb{C})$ and $Z_H=\{g\in SL(2,\mathbb{C}):gh=hg\mathrm{for}\mathrm{all}h\in H\}$ be its centralizer. The automorphism group $Aut(P_{\rho },{\mathsf{\nabla }}_{\rho })$ is contained in the intersection of $Z_H$ with the group of elements preserving the conjugacy classes ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ and the connection structure.

For finite subgroups $H\subset SU\left(2\right)$, the structure of this intersection determines the codimension of the corresponding singularities in the moduli space. Cyclic subgroups ${\mathbb{Z}}_n$ with $n>2$ give rise to isolated singularities, while the special case ${\mathbb{Z}}_2$ creates curves of singularities. Larger finite subgroups give rise to singularities of various codimensions depending on their centralizer structure in $SL(2,\mathbb{C})$ and compatibility with the connection and monodromy constraints.

5. Applications to Topological Classification of Control Strategies

The correspondence established in Theorem 1 between the moduli space of polystable G-bundles with prescribed monodromy ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ provides a suitable framework for analyzing control strategies in robotic systems operating on topologically complex environments. This section presents precise applications of these results to the topic of obstacle avoidance and path planning, where the polystability condition plays a crucial role in ensuring the stability and robustness of the control strategies.

Consider a robotic system operating on a compact oriented surface $\mathcal{M}$ of genus $g\ge 2$ with k point obstacles located at positions $\{q_1,\dots ,q_k\}\subset \mathcal{M}$. The configuration space of the system can be modeled as the punctured surface ${\Sigma }^{*}=\mathcal{M}\setminus \{q_1,\dots ,q_k\}$, where $\Sigma =\mathcal{M}$ is viewed as a compact Riemann surface and $S=\{q_1,\dots ,q_k\}$ represents the obstacle set. Following the framework developed by Koditschek and Rimon [13] for navigation functions on configuration spaces with obstacles, the topological structure of ${\Sigma }^{*}$ fundamentally constrains the possible control strategies.

Let G be a complex reductive Lie group representing the symmetry group of the robotic system, and let $K\subset G$ be a maximal compact subgroup. The system dynamics are encoded by a principal G-bundle P over $\Sigma$, equipped with a meromorphic connection ∇ that has prescribed monodromy behavior around each obstacle. The requirement that this bundle be polystable is essential for the application of Theorem 1, but is also significant and has implications for the stability of the control system. Specifically, as established by Ramanathan [16], polystable bundles represent the stable points of the Yang–Mills functional, which in the context of control theory corresponds to configurations that minimize the control energy while satisfying the prescribed constraints.

More precisely, for each obstacle $q_i\in S$, the connection ∇ exhibits monodromy belonging to a fixed conjugacy class ${\mathcal{C}}_i\subset K$ when the system circumnavigates the obstacle. The polystability condition ensures that the control strategy does not exhibit destabilizing behavior near the obstacles, which is achieved by requiring that all parabolic reductions of the structure group yield associated line bundles of degree zero. The robust control property induced by this geometric condition is that the system maintains bounded energy expenditure regardless of the path taken around obstacles, as exposed in the work of Sussmann [48] on geometric control theory.

The collection of conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ encodes the admissible interaction patterns between the robotic system and each obstacle. The moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ then parametrizes all topologically distinct polystable control strategies satisfying these prescribed monodromy conditions. The restriction to polystable bundles is not merely a technical requirement but reflects the physical constraint that viable control strategies must be stable under small perturbations of the system parameters. The following result, which is a direct application of the correspondence given in Theorem 1, establishes that polystable control strategies with prescribed monodromy are precisely parametrized by the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ introduced in Definition 2.

Proposition 3.

Let Σ be a compact Riemann surface of genus $g\ge 2$, let $S=\{p_1,\dots ,p_k\}\subset \Sigma$ be a finite set of points representing obstacles, and let G be a complex reductive Lie group with maximal compact subgroup $K\subset G$. For any collection of conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ where each ${\mathcal{C}}_i\subset K$, the space of topologically distinct polystable control strategies with prescribed monodromy behavior $\mathcal{C}$ around obstacles is in bijective correspondence with the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Moreover, the dimension of this space is given by

$$dim{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right),$$

where ${dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$ denotes the real dimension of the conjugacy class ${\mathcal{C}}_i$ in K.

Proof. 

By Theorem 1, there exists an algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Each element of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ represents an isomorphism class of polystable principal G-bundles over $\Sigma$ with meromorphic connections having prescribed monodromy in the conjugacy classes $\mathcal{C}$. These correspond precisely to the topologically distinct polystable control strategies satisfying the prescribed monodromy conditions around obstacles.

The dimension formula follows directly from Proposition 1, which establishes that

$$dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right).$$

Since the correspondence in Theorem 1 is an isomorphism, the dimensions of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ are equal. □

The dimension formula in Proposition 3 reveals the geometric structure underlying polystable control strategy spaces. The term $2(g-1){dim}_{\mathbb{C}}\left(G\right)$ captures the degrees of freedom arising from the topology of the environment (encoded by the genus g) and the internal symmetries of the system (encoded by ${dim}_{\mathbb{C}}\left(G\right)$). Each term ${dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$ represents additional degrees of freedom contributed by the specific nature of the constraint imposed by obstacle $p_i$. This dimensional analysis extends the classical results of Brockett [49] on the geometry of control systems to the setting of topologically complex environments.

For infinitesimal deformations of polystable control strategies, the tangent space structure provides insight into the local stability properties of the system. Given a polystable control strategy represented by a point $[P,\mathsf{\nabla }]\in {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$, the tangent space $T_{[P,\mathsf{\nabla }]}{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ parametrizes infinitesimal deformations of the connection ∇ that preserve both the polystability condition and the prescribed monodromy classes. By the correspondence in Theorem 1, this tangent space is isomorphic to the tangent space of the character variety at the corresponding representation.

Let $\rho :{\pi }_1\left({\Sigma }^{*}\right)\to G$ be the representation corresponding to $[P,\mathsf{\nabla }]$ under the bijection of Theorem 1. The tangent space $T_{\left[\rho \right]}{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ is isomorphic to the first cohomology group $H^1({\Sigma }^{*},{Ad}_{\rho })$, where ${Ad}_{\rho }$ denotes the local system of Lie algebras $\mathfrak{g}$ on ${\Sigma }^{*}$ defined by the adjoint action of $\rho$. This cohomological description connects the local geometry of the control strategy space with the topological invariants of the punctured surface, extending the deformation theory developed by Goldman [50] for character varieties.

From the perspective above, the additional constraints on the structure of critical points in the moduli space imposed by the polystability condition of the principal bundles can be further analyzed. This has implications for the stability analysis of control strategies, derived mainly from the application of the characterization of singular points given in Proposition 2. The final proposition analyzes the dimensional properties of singular points in families of control strategies and characterizes their enhanced stability properties in terms of automorphism groups.

Proposition 4.

Let Σ be a compact Riemann surface of genus $g\ge 2$, $S=\{p_1,\dots ,p_k\}\subset \Sigma$ be a finite set of points representing obstacles, and G be a complex reductive Lie group with maximal compact subgroup $K\subset G$. For any collection of conjugacy classes $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_k)$ where each ${\mathcal{C}}_i\subset K$, consider a continuous family of polystable control strategies ${\left\{{\sigma }_t\right\}}_{t\in [0,1]}$ represented by a smooth curve $\gamma :[0,1]\to {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$. If $\gamma \left(t_0\right)$ corresponds to a singular point of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ for some $t_0\in (0,1)$, then the control strategy ${\sigma }_{t_0}$ exhibits enhanced stability in the sense that its tangent space dimension satisfies

$$dim{T}_{\gamma \left(t_0\right)}{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)-dimAut(P_{t_0},{\mathsf{\nabla }}_{t_0}),$$

where $P_{t_0}$ is the principal G-bundle corresponding to ${\sigma }_{t_0}$, ${\mathsf{\nabla }}_{t_0}$ is the associated connection, and $Aut(P_{t_0},{\mathsf{\nabla }}_{t_0})$ denotes the group of automorphisms of $P_{t_0}$ that preserve the connection ${\mathsf{\nabla }}_{t_0}$.

Proof. 

By Theorem 1, there exists an algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. Under this correspondence, the smooth curve $\gamma :[0,1]\to {\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ corresponds to a smooth curve $\tilde{\gamma }:[0,1]\to {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ in the character variety, where $\tilde{\gamma }\left(t\right)=\left[{\rho }_t\right]$ for representations ${\rho }_t:{\pi }_1\left({\Sigma }^{*}\right)\to G$ satisfying ${\rho }_t\left({\gamma }_i\right)\in {\mathcal{C}}_i$ for all $i=1,\dots ,k$.

Let $[P_{t_0},{\mathsf{\nabla }}_{t_0}]=\gamma \left(t_0\right)$ be the point corresponding to the singular point at parameter $t_0$. By Proposition 2, the singularity of ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ at $\gamma \left(t_0\right)$ corresponds to the fact that the principal G-bundle $P_{t_0}$ equipped with connection ${\mathsf{\nabla }}_{t_0}$ admits a non-trivial group of connection-preserving automorphisms $Aut(P_{t_0},{\mathsf{\nabla }}_{t_0})$.

Under the correspondence of Theorem 1, the point $[P_{t_0},{\mathsf{\nabla }}_{t_0}]$ corresponds to a point $\left[{\rho }_{t_0}\right]\in {\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$. The existence of non-trivial connection-preserving automorphisms of $(P_{t_0},{\mathsf{\nabla }}_{t_0})$ is equivalent to the existence of a non-trivial centralizer of the holonomy representation ${\rho }_{t_0}$ in G. Specifically, if $\varphi :P_{t_0}\to P_{t_0}$ is an automorphism covering the identity map on $\Sigma$ and preserving the connection ${\mathsf{\nabla }}_{t_0}$, then $\varphi$ induces an element $g\in G$ such that ${\rho }_{t_0}\left(\gamma \right)=g{\rho }_{t_0}\left(\gamma \right)g^{-1}$ for all $\gamma \in {\pi }_1\left({\Sigma }^{*}\right)$. This shows that g belongs to the centralizer $Z_G\left({\rho }_{t_0}\right)$ of the image of ${\rho }_{t_0}$.

The tangent space to the character variety at $\left[{\rho }_{t_0}\right]$ is isomorphic to the first cohomology group $H^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$, where ${\mathfrak{g}}_{{\rho }_{t_0}}$ denotes the local system on ${\Sigma }^{*}$ defined by the adjoint representation $Ad\circ {\rho }_{t_0}:{\pi }_1\left({\Sigma }^{*}\right)\to Aut\left(\mathfrak{g}\right)$. This cohomology group can be computed as the quotient space $Z^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})/B^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$, where $Z^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$ consists of 1-cocycles and $B^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$ consists of 1-coboundaries.

For a punctured surface ${\Sigma }^{*}=\Sigma \setminus S$ where $\Sigma$ has genus g and $\left|S\right|=k$, the fundamental group ${\pi }_1\left({\Sigma }^{*}\right)$ is generated by elements ${\alpha }_1,{\beta }_1,\dots ,{\alpha }_g,{\beta }_g,{\gamma }_1,\dots ,{\gamma }_k$ subject to the single relation ${\prod }_{i=1}^g[{\alpha }_i,{\beta }_i]{\prod }_{j=1}^k{\gamma }_j=1$, where $[{\alpha }_i,{\beta }_i]={\alpha }_i{\beta }_i{\alpha }_i^{-1}{\beta }_i^{-1}$.

A 1-cocycle $\xi \in Z^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$ assigns to each generator an element of $\mathfrak{g}$ such that the cocycle condition is satisfied. The dimension of $Z^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$ is $(2g+k-1)dim\mathfrak{g}$, since the fundamental relation imposes $dim\mathfrak{g}$ constraints on the $2g+k$ free assignments to the generators.

The space of 1-coboundaries $B^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})$ consists of cocycles of the form $\xi \left(\gamma \right)=Ad\left({\rho }_{t_0}\left(\gamma \right)\right)\left(X\right)-X$ for some $X\in \mathfrak{g}$. The dimension of this space equals $dim\mathfrak{g}-dim{\mathfrak{z}}_{\mathfrak{g}}\left({\rho }_{t_0}\right)$, where ${\mathfrak{z}}_{\mathfrak{g}}\left({\rho }_{t_0}\right)$ denotes the subalgebra of elements in $\mathfrak{g}$ that are fixed by the adjoint action of all elements in the image of ${\rho }_{t_0}$.

The fixed subalgebra ${\mathfrak{z}}_{\mathfrak{g}}\left({\rho }_{t_0}\right)$ is precisely the Lie algebra of the centralizer $Z_G\left({\rho }_{t_0}\right)$, so $dim{\mathfrak{z}}_{\mathfrak{g}}\left({\rho }_{t_0}\right)=dim{Z}_G\left({\rho }_{t_0}\right)$.

The dimension of the first cohomology group is therefore

$$\begin{array}{cc}\hfill dim{H}^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})& =dim{Z}^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})-dim{B}^1({\Sigma }^{*},{\mathfrak{g}}_{{\rho }_{t_0}})\hfill \\ \hfill & =(2g+k-1)dim\mathfrak{g}-(dim\mathfrak{g}-dim{Z}_G\left({\rho }_{t_0}\right))\hfill \\ \hfill & =(2g+k-2)dim\mathfrak{g}+dim{Z}_G\left({\rho }_{t_0}\right).\hfill \end{array}$$

The constraints imposed by the prescribed monodromy conditions require that ${\rho }_{t_0}\left({\gamma }_i\right)\in {\mathcal{C}}_i$ for each $i=1,\dots ,k$, according to Definition 1. Each conjugacy class ${\mathcal{C}}_i$ in the compact group K has real dimension ${dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$, and the constraint ${\rho }_{t_0}\left({\gamma }_i\right)\in {\mathcal{C}}_i$ reduces the degrees of freedom by ${dim}_{\mathbb{C}}\left(G\right)-\frac{1}{2}{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$. The total reduction in degrees of freedom is ${\sum }_{i=1}^k({dim}_{\mathbb{C}}\left(G\right)-\frac{1}{2}{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right))=k{dim}_{\mathbb{C}}\left(G\right)-\frac{1}{2}{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$.

The effective dimension of the tangent space to the character variety at $\left[{\rho }_{t_0}\right]$ is

$$\begin{array}{cc}\hfill dim{T}_{\left[{\rho }_{t_0}\right]}{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)& =(2g+k-2)dim\mathfrak{g}+dim{Z}_G\left({\rho }_{t_0}\right)\hfill \\ \hfill & -k{dim}_{\mathbb{C}}\left(G\right)+{\displaystyle \frac{1}{2}}\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)\hfill \\ \hfill & =(2g-2){dim}_{\mathbb{C}}\left(G\right)+dim{Z}_G\left({\rho }_{t_0}\right)+{\displaystyle \frac{1}{2}}\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right).\hfill \end{array}$$

Since the conjugacy classes ${\mathcal{C}}_i$ lie in the compact group K and considering the real structure of the moduli space, the correct formula becomes

$$dim{T}_{\left[{\rho }_{t_0}\right]}{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)+dim{Z}_G\left({\rho }_{t_0}\right).$$

By the isomorphism established in Theorem 1, the tangent space dimensions are preserved under the isomorphism:

$$dim{T}_{\gamma \left(t_0\right)}{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=dim{T}_{\left[{\rho }_{t_0}\right]}{\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G).$$

The relationship between the group of connection-preserving automorphisms and the centralizer is given by $Aut(P_{t_0},{\mathsf{\nabla }}_{t_0})\cong Z_G\left({\rho }_{t_0}\right)/Z\left(G\right)$, where $Z\left(G\right)$ is the center of G. Since G is reductive, $Z\left(G\right)$ is finite, so $dimAut(P_{t_0},{\mathsf{\nabla }}_{t_0})=dim{Z}_G\left({\rho }_{t_0}\right)$.

The dimension formula for non-singular points gives

$$2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right),$$

as established in Proposition 1. At singular points, the presence of non-trivial connection-preserving automorphisms manifests as additional constraints on the deformation space, effectively reducing the tangent space dimension by the dimension of the automorphism group, $dimAut(P_{t_0},{\mathsf{\nabla }}_{t_0})$. This yields the stated formula,

$$dim{T}_{\gamma \left(t_0\right)}{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+\sum _{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)-dimAut(P_{t_0},{\mathsf{\nabla }}_{t_0}).$$

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The significance of Proposition 4 for control theory lies in the interpretation of singular points as representing maximally stable control strategies. The reduction in the dimension of the tangent space at singular points means that these strategies possess additional symmetries that constrain their deformation space, leading to enhanced robustness against perturbations. This geometric insight provides an approach to identifying optimal control strategies in complex topological environments, complementing the classical methods of optimal control theory developed by Pontryagin [51].

6. Conclusions

This work has connected polystable principal bundles with prescribed monodromy to topological control systems. The algebraic correspondence between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$, not only extends the classical Narasimhan–Seshadri correspondence to the prescribed monodromy setting, but also allows analyzing of control strategies through group-theoretic data. This represents an advancement over previous approaches that lack a geometric framework for systems operating on manifolds with topological constraints.

The dimensional analysis, expressed as $dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$ for a complex reductive Lie group G with maximal compact subgroup K, quantifies how environmental complexity, system symmetry, and constraints contribute to the richness of available control strategies. This formula enables the prediction of degrees of freedom in control strategy spaces without exhaustive computation, representing a novel contribution to both algebraic geometry and engineering. The characterization of singularities of the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ as polystable principal G-bundles admitting non-trivial automorphisms also provides interesting insights into potential instabilities in control systems, identifying configurations where system behavior may change qualitatively under small perturbations. This result enhances the understanding of robustness in control strategies by connecting topological features of moduli spaces to dynamical properties of the underlying systems.

The application of the above geometric structures to topological control theory establishes a basis for understanding polystable control strategies (which correspond to polystable G-bundles with prescribed monodromy), specifically for analyzing their deformation properties. In particular, the analysis of singular points in families of polystable control strategies reveals enhanced stability properties arising from the presence of non-trivial automorphism groups, demonstrating that such configurations exhibit reduced tangent space dimensions and correspondingly improved robustness under perturbations.

The framework developed extends beyond specific control algorithms or particular topological spaces, offering a unified geometric theory for analyzing control systems in complex environments. The explicit dimension formulas and singularity characterizations provide practical tools for robotics and autonomous systems, enabling the assessment of strategy robustness and flexibility through purely geometric means. This approach complements existing work on moduli spaces of G-bundles and opens new avenues for applying algebraic-geometric techniques to engineering problems involving topological constraints.

Future research directions could extend this research to non-compact and higher-dimensional configuration spaces, which would address more complex robotic environments. Developing computational tools based on this geometric framework to efficiently compute topological invariants of control strategy spaces represents another promising direction. Additionally, exploring connections with persistent homology and topological data analysis could yield practical algorithms for identifying optimal control strategies in complex environments. Finally, extending this framework to time-varying systems with dynamic obstacles would enhance its applicability to autonomous systems.