Poisson Structures on moduli spaces of Higgs bundles over stacky curves

1 Introduction

Symplectic or Poisson structures on moduli spaces over a complex algebraic curve X have been obtained in a variety of cases. A basic paradigm involves the G-character varieties, as moduli spaces of fundamental group representations into a connected complex reductive group G. These are finite-dimensional complex symplectic manifolds when X is compact, and the symplectic structure in this case was first conceived analytically using the method of symplectic reduction from infinite-dimensional spaces in the seminal work of Atiyah and Bott [2]. At the same time, a more topological approach to the natural symplectic structure on such spaces of fundamental group representations was proposed by Goldman [20] interpreting these structures in terms of the intersection pairing on the underlying topological surface. The method of symplectic reduction was next further developed by Hitchin in [22] to produce Kähler and hyperkähler structures on the moduli space of stable Higgs bundles via the non-abelian Hodge correspondence and their counterparts to moduli spaces of solutions to the self-duality equations.

In the case when the curve is noncompact, the symplectic structure generalizes to a Poisson structure. The noncompact case is actually equivalent to equipping X with a reduced effective divisor D. The symplectic leaves consist of equivalence classes of connections with fixed conjugacy class of local holonomy around each point of a fixed reduced effective divisor on X. A primary description of this theory appeared in the book of Atiyah [1], while in the article of Audin [3] a review of several approaches is presented. In the particular situation of parabolic vector bundles on X with trivial flags in the parabolic structure, Poisson structures on moduli spaces of twisted Higgs bundles were obtained independently by Bottacin [15] and Markman [28].

Note that from the point of view of the tame nonabelian Hodge correspondence as described by Simpson [35], parabolic Higgs bundles correspond to filtered local systems which are regarded as representations of the fundamental group π₁(X ∖ D). In the wild case, a nonabelian Hodge correspondence when the structure group is GL(n, ℂ) was obtained collectively from the works of Biquard–Boalch [5] and Sabbah [33]. For a general connected complex reductive group G, Boalch introduced wild character varieties to classify meromorphic G-connections with higher order poles, and constructed Poisson structures on them; see for instance [12; 13] and the survey article [11].

In this article, we exhibit a wider class of Poisson structures on moduli spaces of Higgs bundles over stacky curves, demonstrating an intrinsic property that justifies the existence of the Poisson structure. The principal method by which we shall obtain Poisson structures on the moduli spaces of interest is via the duals of Lie algebroids.

In [27], Logares and Martens considered moduli spaces 𝓟_α of α-semistable parabolic Higgs bundles over an algebraic curve. In fact, the open subset Pα0 ⊂ 𝓟_α of pairs involving a stable underlying parabolic bundle is a vector bundle over the moduli space 𝓝_α of stable parabolic bundles. Logares and Martens showed that the dual of this vector bundle is an Atiyah algebroid associated to a principal bundle over the space 𝓝_α, thus admitting a Poisson structure. On the other hand, there exists a bi-vector field on 𝓟_α which agrees with the Poisson bracket on Pα0 , thus establishing the Poisson structure on the entire 𝓟_α. It is important here that the Atiyah sequence of the algebroid naturally follows from the deformation theory of parabolic vector bundles, and a Serre duality map in hypercohomology plays a pivotal role in the definition of the Poisson bracket. These ideas provided important motivation for the development of the present work.

Let 𝓧 be a smooth projective Deligne–Mumford stack over ℂ and let X be the coarse moduli space of 𝓧. The moduli space of semistable G-Higgs bundles on 𝓧 was constructed by Simpson in [34], who showed that this is, in fact, a quasi-projective scheme. The moduli problem and also moduli spaces of Higgs bundles on Deligne–Mumford stacks have been more generally studied in [39] and [38] by the second author.

Suppose that 𝓧 is a stacky curve, which is a smooth projective Deligne–Mumford stack of dimension one. We are thus considering the moduli space 𝓜_H(𝓧, α) of stable Higgs bundles over 𝓧 with a fixed parabolic structure α, as well as 𝓜_H(𝓧, G) and 𝓜_H(𝓧, G, α), the relative moduli spaces of pairs where the structure group of the underlying bundles is determined by a complex reductive algebraic group G. We show that the moduli space 𝓜_H(𝓧, α) is a Lie algebroid over the tangent space of the moduli space of stable bundles over X, thus implying the main theorem of this article:

In the course of developing the proof of this result, we highlight the importance of certain short exact sequences (Atiyah sequences) that arise. This opens the way for generalizing the above theorem in several directions. We first show similarly that the moduli space 𝓜_H(𝓧, G, α) for a fixed faithful representation G ↪ GL(V) is also equipped with a Poisson structure:

The notion of stability we consider here does not depend on the choice of a faithful representation G ↪ GL(V). An alternative notion of (semi)stability for parabolic principal G-bundles and parabolic G-Higgs bundles is considered in the more recent work of Biquard, García-Prada and Mundet i Riera [6]. Furthermore, it is proven by the authors recently that the stability condition considered in this paper is equivalent to Ramanathan’s stability condition of the corresponding logahoric Higgs torsor; see [26].

Note that in the special case of a root stack and a parabolic structure when all parabolic weights are rational, there is an alternative description of parabolic bundles as orbifold bundles; see [7; 19; 25; 29]. Therefore, our theorems provide an orbifold version of the result of Bottacin [15] and Markman [28] in the case of simple pole divisors, as was first conjectured by Logares and Martens [27, § 5.2].

Let 𝓜͠ be a moduli stack, and suppose that 𝓜 is a fine moduli space of 𝓜͠. We have 𝓜͠ ≅ Hom(–, 𝓜), therefore a Poisson structure on 𝓜 will induce a Poisson structure on 𝓜͠. A theory of Poisson structures on stacks is needed here, and this was introduced in the dissertation of Waldron [40]. Based on the theory of Poisson structures on stacks, we show that the moduli stack 𝓜͠_H(𝓧, α) has a Poisson structure as a stack (Corollary 4.5).

Next, we show that the moduli space 𝓜_H(𝓧, 𝓛, α) of stable 𝓛-twisted Higgs bundles over 𝓧 with fixed parabolic structure α is Poisson, subject to the existence of a certain short exact sequence for any stable bundle 𝓕 ∈ 𝓜(𝓧, α); the precise statement is the following:

If X is a smooth projective curve, if D̅ = p₁ + … + p_k is a reduced effective divisor on X and r̅ = (r₁, …, r_k) is a k-tuple of positive integers, let X_D̅, r̅} denote the corresponding root stack and consider the natural map π : 𝓧 → X from the root stack 𝓧 to its coarse moduli space X. Denote by 𝔻 the reduced divisor of π^–1(D̅). The correspondence between parabolic bundles on (X, D̅) and bundles on 𝓧 := X_D̅,r̅ implies the correspondence in the stability conditions as in [14, Remarque 10] and, more precisely, in moduli spaces 𝓜(𝓧, α) ≅ 𝓜^par(X, α). It is natural to extend this correspondence to Higgs bundles [9; 8; 29], thus having 𝓜_H(𝓧, α) ≅ MHspar (X, α).

Let now 𝓛 be a line bundle over 𝓧 and denote by L the parabolic line bundle over (X, D̅) corresponding to 𝓛, that is, a line bundle over X together with a collection of flags L_pi ⊃ {0} with a real weight 0 ≤ α_pi < 1 for each point p_i ∈ D̅ where i = 1, …, k. There is a one-to-one correspondence between 𝓛-twisted Higgs bundles on 𝓧 and L-twisted parabolic bundles on (X, D̅); see [24, § 5]. We have the following proposition:

Under these considerations, we find that the short exact sequence

in Theorem 1.3 can be translated in the language of parabolic bundles by taking 𝓛 = ω_𝓧(𝔻) to

where F is the corresponding parabolic bundle of 𝓕. With respect to the above observation, Theorem 1.3 gives an alternative proof to [27] on the existence of a Poisson structure on MHpar (X, α), the moduli space of stable parabolic Higgs bundles with parabolic structure α on (X, D̅), where X is an irreducible smooth curve and D̅ is a reduced effective divisor.

The method via Lie algebroids used in this article provides the construction of Poisson structures on a wide class of Higgs bundle moduli spaces over stacky curves. In order to further investigate completely integrable systems embedded as symplectic leaves of these Poisson moduli spaces, an explicit construction of a canonical moment map would be required giving a Hamiltonian G-stack, so that the symplectic leaves are well-defined via an appropriate Marsden–Weinstein symplectic reduction theorem for symplectic stacks. We hope to explicitly demonstrate this more direct approach in a future article.

2 Preliminaries

In this preliminary section, we collect the necessary background on stacks that we shall need for our purposes, and we also consider parabolic bundles and compare them to bundles on root stacks. In this paper, all stacks and schemes are defined over ℂ. The interested reader may refer to [8; 31] for further background on the material covered in § 2.1–2.3. A good reference for § 2.4 is [34, § 4, § 5]. In § 2.5, we give the definition of parabolic structures of G-bundles, which depends on a fixed faithful representation G ↪ GL(V).

2.1 Deligne–Mumford stacks

A Deligne–Mumford stack 𝓧 is an algebraic stack such that there exists a surjective étale morphism U → 𝓧, where U is a ℂ-scheme. The data (U, u) is called a chart of 𝓧, where u is surjective. If u : U → 𝓧 is an étale morphism (not necessarily surjective), the pair (U, u) is called a local chart of 𝓧. Let (U, u) and (V, v) be two charts of 𝓧. A morphism of charts (U, u) and (V, v) is a morphism f_uv : (U, u) → (V, v) of schemes such that the following diagram commutes:

If 𝓧 is locally of finite type and with finite diagonal, then there exists a coarse moduli space X (as an algebraic space) of 𝓧; see [31, Theorem 11.1.2]. Denote by π : 𝓧 → X the natural morphism.

Definition 2.1

A Deligne–Mumford stack 𝓧 is smooth and projective if it satisfies the following conditions:

1. there exists a surjective étale morphism Y → 𝓧 such that Y is a smooth projective variety;

2. 𝓧 can be written as a global quotient [Y∖Γ], where Γ is a finite group;

3. 𝓧 has a coarse moduli space X, which is a smooth projective variety,

and we say that 𝓧 is a smooth projective Deligne–Mumford stack. Furthermore, if 𝓧 is connected and of dimension one, we say that 𝓧 is a stacky curve.

The idea of smooth projective Deligne–Mumford stacks is introduced by Simpson to establish a stacky version of the nonabelian Hodge correspondence [34, Theorem 5.4]. Compared to Simpson’s definition, we also require that the coarse moduli space X is smooth and projective for the purposes of this paper.

Definition 2.2

Let 𝓧 be a smooth projective Deligne–Mumford stack. A Cartier divisor D on 𝓧 is defined on each chart (U, u) as follows: there is a Cartier divisor D_u on U such that if f_uv : (U, u) → (V, v) is a morphism of charts, then fuv∗ (D_v) = D_u. We say a Cartier divisor D has normal crossings if for each chart (U, u), the divisor D_u has normal crossings.

3 Deformation theory on moduli spaces of Higgs bundles over Deligne–Mumford stacks

In this section, we review some results on the deformation theory of moduli spaces of Higgs bundles over Deligne–Mumford stacks, which will help us calculate the tangent space of the moduli spaces we are interested in. In § 3.1, we define all the moduli spaces we consider in this paper and in § 3.2 we review the deformation theory on those moduli spaces. In § 3.3, we review the Grothendieck duality of coherent sheaves over Deligne–Mumford stacks, while in § 3.4 we restrict to stacky curves and apply the results from § 3.2 and § 3.3 to construct a morphism T^*(𝓜_H(𝓧)) → T(𝓜_H(𝓧)), which will be used in order to construct a Poisson structure on 𝓜_H(𝓧) later on in § 4.

4 Lie algebroids and Poisson structures

The principal method by which we shall obtain Poisson structures on our moduli spaces of interest is via the duals of Lie algebroids. Poisson structures are usually defined on smooth varieties. Nonetheless, Poisson structures on stacks were first conceived in the dissertation of Waldron [40]. Although the author only considers differential stacks in [40], the approach can be carried over to the algebraic setting naturally. In § 4.1, we shall review the definitions and some properties of Lie algebroids and Poisson structures on smooth varieties, as well as on stacks; we refer to [40, § 7.2] and [23, § 1] for a complete overview. In § 4.2 and § 4.3, we prove the main theorems of this section that the moduli spaces 𝓜_H(𝓧, α) and 𝓜_H(𝓧, G, α) admit a Poisson structure (Theorems 4.3 and 4.4). In § 4.4, we consider the stack 𝓜͠_H(𝓧, α) of Higgs bundles on 𝓧 by abuse of notation. As an application of Theorem 4.3, we next show that 𝓜͠_H(𝓧, α) also admits a Poisson structure.

4.2 Poisson structure on 𝓜_H(𝓧, α)

Let 𝓧 = [Y/Γ] be a stacky curve over ℂ and let X be the coarse moduli space of 𝓧. In this section, we will prove the main theorem of this paper.

Theorem 4.3

The moduli space 𝓜_H(𝓧, α) of stable Higgs bundles over 𝓧 with fixed parabolic structure α admits a Poisson structure.

Proof

The strategy of the proof is to show that 𝓜_H(𝓧, α) is a Lie algebroid over T𝓜(X), the tangent space of the moduli space of stable bundles over X, by constructing a map

MH(X,α)→TM(X),

which will play the role of an anchor map. Note that (3.7) gives precisely such a map. Following the approach as in [27, § 3], we first restrict to MH0 (𝓧, α), the moduli space of stable Higgs bundles (𝓕, Φ) where the underlying bundle 𝓕 is a stable bundle. Since this moduli space is a dense open subset of 𝓜_H(𝓧, α), if there is a Poisson structure on MH0 (𝓧, α) with the bi-vector field induced by (3.7), then the anchor map is used to give a Poisson structure on 𝓜_H(𝓧, α). In the sequel, we will construct a Poisson structure on MH0 (𝓧, α), of which the associated bi-vector field is given by (3.7).

By the discussion in § 2.4, let 𝓧 be a root stack X_D̅,r̅. Denote by π : 𝓧 → X the natural morphism; note that the dimension of 𝓧 is one. The divisor D̅ = p₁ + … + p_n is a sum of distinct points and let q_i be the corresponding point of p_i in 𝓧. Note that π^–1(p_i) = r(p_i)q_i. Let us denote by 𝔻 = q₁ + … + q_n the divisor in 𝓧.

Let 𝓕 be a locally free sheaf on 𝓧 with parabolic structure α. Given q ∈ 𝔻, denote by α(q) the parabolic structure of π_*𝓕 around p = π(q) ∈ D̅. Let P_q be the parabolic group of GL_n(ℂ) corresponding to α(q) as we discussed in § 2.5. Let P_q = L_qN_q be the Levi decomposition of P, where L_q is the Levi factor and N_q is a unipotent group. Denote by 𝔩_q, 𝔫_q the Lie algebras of L_q and N_q respectively. Define 𝓕′ := π^* π_* 𝓕. Since the stacky curve 𝓧 is defined over ℂ, the functor π_* is exact. Thus, 𝓕′ is a locally free sheaf on 𝓧. We have the following exact sequence

0→End(F)→End(F′)→∏q∈Dnq⊗Oq→0. (4.1)

This induces a long exact sequence

0→End(F)→End(F′)→H0(X,∏q∈Dnq⊗Oq)→Ext1(F,F)→Ext1(F′,F′)→H1(X,∏q∈Dnq⊗Oq)→0.

Note that the last term H¹(𝓧, ∏_q∈𝔻 𝔫_q ⊗ 𝓞_q) is trivial, and thus we have a short exact sequence

0→Ad→Ext1(F,F)→Ext1(F′,F′)→0, (4.2)

where Ad is the kernel of the surjective map Ext¹(𝓕, 𝓕) → Ext¹(𝓕′, 𝓕′). Remember that 𝓕 is a stable bundle and that we are working over the field ℂ. Therefore, 𝓕 is simple and the functor

π∗:Coh(X)→Coh(X)

is exact. The exactness of the functor π_* implies that π_* 𝓕 is stable, and so is 𝓕′. Therefore, End(𝓕′) ≅ ℂ. Now we go back to the term Ad. If 𝓕 is stable, then

Ad≅H0(X,∏q∈Dnq⊗Oq),

which is supported over q ∈ 𝔻. Over each point q ∈ 𝔻, Ad_q is isomorphic to the Lie algebra 𝔫_q. Therefore, Ad_q is the adjoint representation of the unipotent group N_q.

In the short exact sequence (4.2), the third term Ext¹(𝓕′, 𝓕′) is isomorphic to the tangent space of the moduli space of stable bundles over X at the point 𝓕′, thus

Ext1(F′,F′)≅TF′(M(X)).

With respect to this isomorphism, we have a natural map

Ext1(F,F)→TF′(M(X)).

Now we consider the tangent bundles on 𝓜(𝓧, α) and 𝓜(X)

Let ℱ, ℱ′ be the universal bundles on 𝓜(𝓧, α) × 𝓧 and 𝓜(X) × X respectively. Therefore, we have

0→End(F)→End(F′)→∏q∈Dnq⊗Oν1−1(q)→0

and

0→Ad→R1(μ1)∗End(F)→R1(μ1)∗End(F′)→0.

Clearly, the term

R1(μ1)∗End(F)≅TM(X,α)

is the tangent bundle of 𝓜(𝓧, α), and

R1(μ1)∗End(F′)≅TM(X)

is the tangent bundle of 𝓜(X).

For the rest of the proof, we prove the following two statements

1. (R¹(μ₁)_* 𝓔nd(ℱ))^* is isomorphic to MH0 (𝓧, α) as bundles over 𝓜(𝓧, α), and

2. R¹ (μ₁)_* 𝓔nd(ℱ) is a Lie algebroid over 𝓜(X),

which shall imply that the moduli space MH0 (𝓧, α) admits a Poisson structure by Theorem 4.1.

To prove the first statement, we work locally on a point 𝓕 ∈ 𝓜(𝓧, α). The space Ext¹(𝓕, 𝓕) is the tangent space of 𝓜(𝓧, α) at the point 𝓕, and we have

H1(X,End(F))≅Ext1(F,F)≅TF(M(X,α)).

By Grothendieck duality, see § 3.3, the following isomorphism holds:

H1(X,End(F))≅H0(X,End(F)⊗ωX)∗.

This isomorphism tells us that

TF∗(M(X,α))≅H0(X,End(F)⊗ωX),

which finishes the proof of the first statement. For every Higgs field Φ ∈ H⁰(𝓧, 𝓔nd(𝓕) ⊗ ω_𝓧) we have

TΦTF∗(M(X,α))≅H1(End(F)→End(F)⊗ωX)≅T(F,Φ)(MH0(X,α)).

For the second statement, note that in summary we have the following exact sequence

0→Ad→TM(X,α)→TM(X)→0,

which is an Atiyah sequence as studied in § 4.1. Therefore, the second statement also holds and this finishes the proof of the theorem.□

4.3 Poisson structure on 𝓜_H(𝓧, G, α)

Let G ↪ GL(V) be a fixed faithful representation. In this subsection, we will prove that there exists a Poisson structure on 𝓜_H(𝓧, G, α).

Theorem 4.4

The moduli space 𝓜_H(𝓧, G, α) of stable G-Higgs bundles over a stacky curve 𝓧 with fixed parabolic structure α admits a Poisson structure.

Proof

The proof is similar to the one for the moduli space 𝓜_H(𝓧, α). We work on the open dense subset MH0 (𝓧, G, α), where the underlying principal G-bundles are also stable. Denote by 𝓜(X, G) the moduli space of stable principal G-bundles on X and consider the tangent bundles

Let ℰ and ℰ′ be the universal bundles on 𝓜(𝓧, G, α) × 𝓧 and 𝓜(X, G) × X, respectively. Denote by ℰ(V) and ℰ′(V) the associated bundles with respect to G ↪ GL(V). Around a point q ∈ 𝔻, let P_q be the corresponding parabolic group in GL(V) with respect to ℰ(V). Denote by P_q = L_qN_q the Levi factorization. With the same discussion as in the proof of Theorem 4.3, we have

0→End(E(V))→End(E′(V))→∏q∈Dnq⊗Oν1−1(q)→0.

Let Pq′,Lq′andNq′ be the pre-images of P_q, L_q and N_q in G respectively via the fixed faithful representation. Therefore, we have

0→End(E)→End(E′)→∏q∈Dnq′⊗Oν1−1(q)→0,

where nq′ is the Lie algebra of Nq′. This short exact sequence induces the following one:

0→Ad→R1(μ1)∗End(E)→R1(μ1)∗End(E′)→0.

Clearly, we have

R1(μ1)∗End(E)≅TM(X,G,α)

and

R1(μ1)∗End(E′)≅TM(X,G),

the tangent bundle of 𝓜(X, G). Therefore, R¹(μ₁)_* 𝓔nd(ℰ) is a Lie algebroid over 𝓜(X, G). At the same time, the tangent space of (R¹ (μ₁)_* 𝓔nd(ℰ))^* is isomorphic to the tangent space of MH0 (𝓧, G, α), in other words

T(R1(μ1)∗End(E))∗≅TMH0(X,G,α).

Therefore, the moduli space of G-Higgs bundles 𝓜_H(𝓧, G, α) has a Poisson structure.□

5 Poisson structure on the moduli space of stable 𝓛-twisted Higgs bundles over stacky curves

In this section, we work on 𝓜_H(𝓧, 𝓛, α), the moduli space of stable 𝓛-twisted Higgs bundles on a stacky curve 𝓧 with fixed parabolic structure α. We prove that a Poisson structure exists over 𝓜_H(𝓧, 𝓛, α) under certain conditions.

Let 𝓧 = [U/Γ] be a stacky curve. Denote by X the coarse moduli space of 𝓧. Let α be a parabolic structure on X, and denote by D̅ = p₁ + … + p_k ∈ X the divisor with respect to α. Let 𝔻 = q₁ + … + q_k ∈ 𝓧 be the corresponding divisor on 𝓧, where q_i is the point corresponding to p_i.

Proof

Analogously to the proof of Theorem 4.3, we only have to work with MH0 (𝓧, 𝓛, α), the moduli space of 𝓛-twisted stable Higgs bundles over 𝓧, such that the underlying locally free sheaf is stable. If the short exact sequence (5.1) exists, we then obtain a long exact sequence

0→Hom(F⊗L,F⊗ωX)→End(F)→H0(X,n)→Ext1(F⊗L,F⊗ωX)→Ext1(F,F)→H1(X,n)→0.

By our assumption that the support of 𝔫 is contained in 𝔻, the last term H¹(𝓧, 𝔫) is trivial. This implies a short exact sequence

0→Ad→Ext1(F⊗L,F⊗ωX)→Ext1(F,F)→0, (5.2)

where Ad is the kernel of the map Ext¹(𝓕 ⊗ 𝓛, 𝓕 ⊗ ω_𝓧) → Ext¹(𝓕, 𝓕). Let ℱ be the universal bundle on 𝓜(𝓧, α) × 𝓧 and consider

The short exact sequence (5.2) induces the following sequence for universal bundles:

0→Ad→R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX)→R1(μ1)∗End(F)→0. (5.3)

With respect to our assumption about 𝔫, the short exact sequence (5.3) gives a Lie algebroid structure on R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX). Note that

R1(μ1)∗End(F)≅TM(X,α).

Therefore, R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX) is a Lie algebroid over T𝓜(𝓧, α). This implies that there is a Poisson structure on (R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX))∗. We only have to prove that

R1(μ1)∗End(F⊗ν1∗L,F⊗ν1∗ωX)∗≅MH0(X,L,α),

which would imply the result.

By Grothendieck duality, see § 3.3, we obtain

Ext1(F⊗L,F⊗ωX)∗≅H0(X,End(F)⊗L).

Equivalently,

R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX)≅(μ1)∗Hom(F,F⊗ν1∗L).

Therefore, R1(μ1)∗Hom(F⊗ν1∗L,F⊗ν1∗ωX)∗ is isomorphic to MH0 (𝓧, 𝓛, α) and this finishes the proof of the theorem.□

6 Relation with the work of Logares–Martens

In this section, we compare the results of Sections 4 and 5 to the main result of Logares and Martens [27]. In the special case of a root stack and a parabolic structure when all parabolic weights are rational, we show that the moduli space of stable orbifold G-Higgs bundles has a Poisson structure.

Let X be a smooth projective curve. Let D̅ = p₁ + … + p_k be a reduced effective divisor on X, and let r̅ = (r₁, …, r_k) be a k-tuple of positive integers. We denote by X_D̅,r̅ the corresponding root stack. There is a natural map π : X_D̅,r̅ → X; we write 𝔻 := π^–1(D̅). By the discussion in § 2.5, we know that there is a correspondence between parabolic bundles on (X, D̅) and bundles on 𝓧 := X_D̅,r̅. Moreover, we have the following equivalence in the language of tensor categories from [14]:

As discussed in § 2.5, this equivalence implies the correspondence in stability as in [14, Remarque 10]. More precisely, we have 𝓜(𝓧, α) ≅ 𝓜^par(X, α). It is natural to extend this correspondence to Higgs bundles as in [9; 8; 29], thus giving 𝓜_H(𝓧, α) ≅ MHspar (X, α). Now let 𝓛 be a line bundle on 𝓧 and let π : 𝓧 → X be the map from the root stack 𝓧 to its coarse moduli space X. Denote by L the corresponding parabolic line bundle of 𝓛 on (X, D̅). There is a one-to-one correspondence between 𝓛-twisted stable Higgs bundles on 𝓧 and L-twisted stable parabolic bundles on (X, D̅); see [24, § 5].

In conclusion, we have the following proposition.

Note that an orbifold is always a root stack. Thus, we have the following corollary to Theorem 4.4:

In [27, § 3.2.2], the following short exact sequence is used to construct a Poisson structure on MHpar (X, α)