Spin(8, ℂ)-Higgs bundles fixed points through spectral data

Álvaro Antón-Sancho

doi:10.1515/math-2025-0192

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Spin(8, ℂ)-Higgs bundles fixed points through spectral data

Álvaro Antón-Sancho

Veröffentlicht/Copyright: 5. September 2025

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Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

Let X be a compact Riemann surface of genus g ≥ 2 . The geometry of the moduli space ℳ ( Spin ( 8 , C ) ) of Spin ( 8 , C ) -Higgs bundles over X is of great interest both in algebraic geometry and mathematical physics. Consequently, several works have studied subvarieties of fixed points of this moduli space, especially those arising from the automorphism induced by the action of triality. In this work, fixed points of automorphisms of ℳ ( Spin ( 8 , C ) ) induced by outer automorphisms of Spin ( 8 , C ) are studied. This is done by giving explicit descriptions of the spectral data of these fixed points induced by the Hitchin fibration, under certain technical conditions that must be required. Specifically, it is proved that stable Spin ( 8 , C ) -Higgs bundles that admit nontrivial automorphisms reduce their structure group to a subgroup isomorphic to SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) . Subsequently, the spectral data of these reductions are described and the manner in which the outer automorphisms of Spin ( 8 , C ) act on them is analyzed. The final application of these results allows a description of the mentioned fixed points through their spectral data.

Keywords: spectral data; Hitchin fibration; automorphism; fixed point; group action

MSC 2010: 14H10; 14H60; 57R57

1 Introduction

Let X be a compact Riemann surface and G be a complex reductive Lie group with Lie algebra g . Higgs bundles, as introduced by Hitchin [1] and Simpson [2], consist of pairs ( E , φ ) , where E is a principal G -bundle over X and φ is a holomorphic global section of the adjoint bundle of E tensored by the canonical line bundle K over X . The moduli space ℳ ( G ) of polystable G -Higgs bundles is an algebraic variety of complex dimension 2 dim G ( g − 1 ) that parametrizes isomorphism classes of such objects and contains the moduli space of principal G -bundles as a subvariety.

The algebraic structure underlying Higgs bundles is revealed through Hitchin’s construction of the integrable system [3]. Building on Chevalley’s theorem [4] on invariant polynomials, the Hitchin fibration provides a proper map from ℳ ( G ) to the direct sum of spaces of holomorphic differentials, whose fibers can be described through spectral curves and associated line bundles, collectively known as spectral data.

The study of automorphisms of moduli spaces of Higgs bundles has emerged as a central topic in the field. The classification of automorphisms for vector Higgs bundles was established by Baraglia [5], who identified four generating families: duality, tensorization by line bundles, pullback by curve automorphisms, and scalar multiplication of the Higgs field. Recently, Fringuelli [6] extended this analysis to principal G -bundles for semisimple complex Lie groups, proving that the automorphism group is generated by three types: those induced by outer automorphisms of G , tensorization by principal bundles with structure group in the center of G , and pullback by curve automorphisms. This result provides the foundation for understanding automorphisms in the more general setting of G -Higgs bundles, though the complete classification in this case remains an open problem.

The analysis of fixed points under these automorphisms has attracted considerable attention, with most studies focusing on automorphisms induced by outer automorphisms of the structure group. When G is a classical simple complex Lie group admitting a nontrivial outer automorphism σ , principal G -bundles fixed by σ admit reductions of structure group to the subgroup of fixed points [7]. These techniques extend to G -Higgs bundles [8], though primarily for involutive cases. Other approaches have addressed different types of automorphisms, such as tensorization by line bundles [9] and combinations of outer automorphisms with other actions [10–12].

Among simple complex Lie groups, Spin ( 8 , C ) occupies a unique position due to the phenomenon of triality, where three fundamental eight-dimensional representations are interchangeable under the action of the symmetric group S 3 on the Dynkin diagram. This exceptional symmetry, combined with its role as a double cover of SO ( 8 , C ) , makes Spin ( 8 , C ) -Higgs bundles particularly significant in the Langlands program and Donaldson-Thomas theory [13,14]. The previous work on fixed points of Spin ( 8 , C ) -Higgs bundles has focused on the triality automorphism and its combinations with other actions [7,12], consistently seeking reductions of structure group to fixed point subgroups.

The present work departs from previous approaches by characterizing fixed points through their spectral data rather than through structure group reductions. This perspective, first introduced by Schaposnik [15] as a new method to understand G -Higgs bundles through their spectral data, offers a more direct connection to the integrable system structure and provides new insights into the geometric properties of these special points in the moduli space. The analysis begins with stable Spin ( 8 , C ) -Higgs bundles admitting nontrivial automorphisms, proving that such bundles always admit reductions to SL ( 2 , C ) 4 . Following and extending the construction of Bradlow et al. [16] for SO ( 4 , C ) , the spectral data of SO ( 8 , C ) -Higgs bundles with such reductions are characterized. Finally, by analyzing the action of all outer automorphisms of Spin ( 8 , C ) on these spectral data, complete descriptions of the spectral data corresponding to fixed points under each nontrivial outer automorphism are obtained, extending beyond the triality case considered in the previous literature.

The structure of this article is organized as follows. Section 2 introduces G -Higgs bundles and the Hitchin fibration, and this framework is specialized in Section 3 to SO ( 8 , C ) -Higgs bundles, whose spectral data are described. Section 4 establishes the result showing that a stable Spin ( 8 , C ) -Higgs bundle admitting nontrivial automorphisms reduces its structure group to SL ( 2 , C ) 4 . Section 5 develops the results that provide the specific form of the spectral data of the SO ( 8 , C ) -Higgs bundle arising from a stable Spin ( 8 , C ) -Higgs bundle with nontrivial automorphisms. Building upon these foundations, Section 6 presents and proves a result that describe the spectral data of the SO ( 8 , C ) -Higgs bundles induced by fixed points of each of the outer automorphisms of the group Spin ( 8 , C ) . The author concludes with a discussion of the main results and their implications.

2 G -Higgs bundles and the Hitchin fibration

Let X be a compact Riemann surface of genus g ≥ 2 and G be a reductive complex Lie group with Lie algebra g . The notion of G -Higgs bundle over X was first introduced by Hitchin [1].

Definition 1

A G -Higgs bundle over a compact Riemann surface X of genus g ≥ 2 for a reductive complex Lie group G is a pair ( E , φ ) where E is a principal G -bundle over X , and φ ∈ H 0 ( X , ad ( E ) ⊗ K ) , where ad ( E ) = E × ad g denotes the adjoint bundle associated with E , and K is the canonical line bundle over X .

The group G is called the structure group of the G -Higgs bundle ( E , φ ) and φ is called its Higgs field. An isomorphism between G -Higgs bundles ( E , φ ) and ( E ′ , φ ′ ) is an isomorphism of principal G -bundles f : E → E ′ whose induced homomorphism between the adjoint bundles tensored by the canonical bundle K , ad ( f ) ⊗ I K : ad ( E ) ⊗ K → ad ( E ′ ) ⊗ K , satisfies φ ′ = ( ad ( f ) ⊗ I K ) ( φ ) . Furthermore, given a subgroup H of G , a reduction of structure group of a G -Higgs bundle ( E , φ ) is an H -Higgs bundle ( E ′ , φ ) such that E ′ is a reduction of structure group of E to H and φ takes values in ad ( E ′ ) .

The notion of G -Higgs bundle generalizes that of the vector Higgs bundle (or simply Higgs bundle), which is a pair ( E , φ ) , where E is a vector bundle over X and φ is a homomorphism E → E ⊗ K . This was first introduced by Hitchin [1] and coincides with the notion of GL ( n , C ) -Higgs bundle, where n is the rank of E .

Suitable notions of stability and polystability can be given to construct the moduli space of G -Higgs bundles over X , which parametrizes isomorphism classes of polystable G -Higgs bundles [17]. It is an algebraic variety whose smooth locus is the open subset of stable and simple G -Higgs bundles, where simple means that the bundles have no automorphisms other tfhan those induced by the action of the center of the structure group.

For the purposes of this work, it is sufficient to consider structure groups that are semisimple complex Lie groups. When, in addition, the structure group is a subgroup of a group of matrices, it is possible to give an interpretation of G -Higgs bundles in terms of vector bundles.

The case of particular interest for this work is when G = SO ( 2 n , C ) . A SO ( 2 n , C ) -Higgs bundle is a pair ( E , φ ) , where E is a rank 2 n vector bundle with trivial determinant that is equipped with a globally defined nondegenerate holomorphic quadratic form Q : E → O X , where O X is the trivial line bundle over X , and φ : E → E ⊗ K is a traceless symmetric homomorphism satisfying Q ( φ ( v ) , w ) + Q ( v , φ ( w ) ) = 0 for all local sections v , w of E . This orthogonality condition ensures that φ preserves the quadratic form in the appropriate sense. The bundle is stable if deg E ′ < deg E for every proper isotropic subbundle E ′ of E preserved by φ , and it is polystable if E = E ′ ⊕ E ′ ⊥ for some proper isotropic subbundle E ′ preserved by φ .

The construction of the Hitchin fibration relies on the theory of invariant polynomials. If ℳ ( G ) denotes the moduli space of G -Higgs bundles for a complex reductive Lie group G , r is the rank of G , p 1 , … , p r is a basis of the algebra of invariant polynomials of the Lie algebra g and d i = deg p i for each i = 1 , … , r , then the Hitchin map or Hitchin fibration [3] is the map

h G : ℳ ( G ) → ⊕ i = 1 r H 0 ( X , K d i )

defined by h G ( E , φ ) = ( p 1 ( φ ) , … , p r ( φ ) ) . The set ⊕ i = 1 r H 0 ( X , K d i ) is an affine space called the base of the Hitchin fibration.

For the group SO ( 2 n , C ) , the invariant polynomial degrees are d i = 2 i for i = 1 , … , n , corresponding to the fundamental invariants of the orthogonal Lie algebra. The Hitchin fibration becomes

h SO ( 2 n , C ) : ℳ ( SO ( 2 n , C ) ) → ⊕ i = 1 n H 0 ( X , K 2 i ) .

Given a SO ( 2 n , C ) -Higgs bundle ( E , φ ) , its image under h SO ( 2 n , C ) determines the characteristic polynomial of φ , which takes the form

det ( λ I − φ ) = λ 2 n + a 1 λ 2 n − 2 + a 3 λ 2 n − 4 + … + a n ,

where a i ∈ H 0 ( X , K 2 i ) for i = 1 , … , n . Note that only even powers of λ appear due to the traceless and symmetric nature of φ . The zero locus of this polynomial defines a 2 n -to-1 cover π : S → X known as the spectral curve, where λ is the tautological section of π * K .

The spectral curves are not necessarily smooth, but they are smooth for the generic fiber of the Hitchin fibration by Bertini’s theorem. An important feature of the SO ( 2 n , C ) case is that the spectral curve S comes equipped with a natural involution σ : S → S induced by the map λ ↦ − λ . This involution reflects the symmetry present in the characteristic polynomial and plays a crucial role in understanding the geometry of the fibers.

The general fiber of the Hitchin fibration h SO ( 2 n , C ) is closely related to Prym varieties. Given a smooth spectral curve S with its natural involution σ , the associated Prym variety is defined as follows:

Prym ( S , σ ) = { L ∈ Jac ( S ) : σ * L ≅ L − 1 } .

This is the connected component of the kernel of the norm map Nm : Jac ( S ) → Jac ( X ) induced by the covering π : S → X . The Prym variety has a dimension g − 1 + n ( g − 1 ) , where g is the genus of X , and it parametrizes line bundles on S that satisfy the anti-invariance condition with respect to the involution.

For any SO ( 2 n , C ) -Higgs bundle ( E , φ ) with smooth spectral curve, there exists a line bundle L over S such that L ∈ Prym ( S , σ ) and the original bundle can be recovered through the spectral correspondence. The pair ( S , L ) with L ∈ Prym ( S , σ ) constitutes the spectral data of the Higgs bundle. This establishes a bijective correspondence between stable SO ( 2 n , C ) -Higgs bundles with smooth spectral curve and points in the corresponding Prym variety, providing a concrete realization of the fibers of the Hitchin fibration in terms of well-understood algebraic objects.

The ramification divisor R of the covering π : S → X determines a line bundle [ R ] = K S ⊗ π * K − 1 over S . Since the spectral curve S lies in the total space of K , we have K S = π * K 2 n , and therefore, [ R ] = π * K 2 n − 1 . The anti-invariance property of line bundles in the Prym variety ensures compatibility with the involution structure, allowing for the reconstruction of the orthogonal bundle E from the spectral data through appropriate pushforward constructions.

3 Spectral data for SO ( 2 n , C ) -Higgs bundles

Let ( E , φ ) be an SO ( 2 n , C ) -Higgs bundle over the compact Riemann surface X , for n ≥ 2 . The associated characteristic polynomial is of the form

(1) λ 2 n + a 1 λ 2 n − 2 + … + a n − 1 λ 2 + r n 2 ,

where a i ∈ H 0 ( X , K 2 i ) for i = 1 , … , n − 1 and r n ∈ H 0 ( X , K n ) is the Pfaffian of the Higgs field φ . The induced spectral curve S admits a natural involution s , which acts by sending each λ to − λ , and it is smooth except for the fixed points of s , which form the zero locus of r n . Therefore, the spectral curve considered here is the desingularization S of S , which is equipped with the covering map π ¯ : S → X induced by the covering π : S → X , and also admits an involution σ ¯ without fixed points. The spectral curve S is generically smooth and both points ( a 1 , … , a n − 1 , r n ) and ( a 1 , … , a n − 1 , − r n ) define the same S , so its definition does not depend on the choice of the square root of r n 2 .

The Prym variety Prym ( S , S ⁄ s ¯ ) is composed of vector bundles with the trivial norm, where the norm is defined on the Picard group of S ⁄ s ¯ . Here, S ⁄ s ¯ denotes the quotient of the desingularized spectral curve S by the involution s ¯ , which is the extension of the original involution s to the desingularized curve. Consequently, a generic fiber of the Hitchin fibration can be understood as a connected component of Prym ( S , S ⁄ s ¯ ) , which is an abelian variety. The spectral data of an SO ( 2 n , C ) -Higgs bundle are then given by a pair ( S , L ) with L ∈ Prym ( S , S ⁄ s ¯ ) such that

E = π * L ⊗ ( K S ⊗ π * ( K * ) ) 1 2 .

Since the norm of L is trivial, there is an automorphism s ¯ * L ≅ L − 1 , from which the orthogonal structure of E is derived.

It is worth noting that, if L ∈ Prym ( S , S ⁄ s ¯ ) , then there exists a degree 0 or 1 line bundle N over S such that L ≅ N ⊗ s ¯ * ( N * ) [18, Lemma 1]. The structure of Prym varieties associated with double covers naturally exhibits two connected components, which arises from the geometric properties of the involution and the corresponding quotient map. This decomposition can be understood through the action of the involution on the Jacobian of the spectral curve, where elements of the Prym variety correspond to line bundles whose pullback under the involution is isomorphic to their inverse. The parity of the degree of N determines the two connected components that the Prym variety Prym ( S , S ⁄ s ¯ ) admits. Specifically, line bundles N of even degree and those of odd degree give rise to distinct components of the Prym variety, reflecting the topological obstruction present in the covering structure. This bipartition is fundamental to understanding the geometry of the Hitchin fibration, as each connected component corresponds to different types of spectral data and influences the moduli space structure of the underlying Higgs bundles.

4 Stable Spin ( 8 , C ) -Higgs bundles with nontrivial automorphisms

There are three different simple complex Lie groups that admit the simple complex Lie algebra so ( 8 , C ) of type D 4 as their Lie algebra: the simply connected group Spin ( 8 , C ) , the group SO ( 8 , C ) , and the centerless group PSO ( 8 , C ) , which is the quotient of SO ( 8 , C ) by its center. The group Spin ( 8 , C ) serves as the universal cover of SO ( 8 , C ) through a 2-to-1 covering map

(2) p : Spin ( 8 , C ) → SO ( 8 , C ) .

The center of Spin ( 8 , C ) is isomorphic to Z ⁄ ( 2 ) × Z ⁄ ( 2 ) , while the center of SO ( 8 , C ) is isomorphic to Z ⁄ ( 2 ) . This structure induces an exact sequence of groups given by the covering map p :

1 → Z ⁄ ( 2 ) → Spin ( 8 , C ) → SO ( 8 , C ) → 1 ,

which, in turn, gives rise to an exact sequence of locally constant sheaves of groups and therefore an exact sequence of cohomology sets. Given a compact Riemann surface X , this yields:

H 1 ( X , Spin ( 8 , C ) ) → H 1 ( X , SO ( 8 , C ) ) → ω 2 H 2 ( X , Z ⁄ ( 2 ) ) ≅ Z ⁄ ( 2 ) .

Since H 1 ( X , G ) parametrizes isomorphism classes of principal G -bundles over X for any semisimple complex Lie group G , the map ω 2 represents the second Stiefel-Whitney class of the principal SO ( 8 , C ) -bundles. Consequently, a principal SO ( 8 , C ) -bundle over X can be lifted to a principal Spin ( 8 , C ) -bundle if and only if its second Stiefel-Whitney class vanishes. The moduli space of principal SO ( 8 , C ) -bundles admits two distinct connected components, corresponding to the two possible values of the Stiefel-Whitney class. It is important to note that every principal Spin ( 8 , C ) -bundle gives rise to a principal SO ( 8 , C ) -bundle through map (2).

Moreover, if ( E , φ ) is a Spin ( 8 , C ) -Higgs bundle over X and E 0 is the principal SO ( 8 , C ) -bundle induced by E , then since the adjoint bundle ad ( E ) coincides with ad ( E 0 ) , the Higgs field φ can be viewed as a global section of ad ( E 0 ) ⊗ K . Therefore, ( E 0 , φ ) forms an SO ( 8 , C ) -Higgs bundle, which we call the SO ( 8 , C ) -Higgs bundle induced by ( E , φ ) .

The notions of stability and polystability for Spin ( 8 , C ) -Higgs bundles and their corresponding SO ( 8 , C ) -Higgs bundles coincide [7,12]. This means that a Spin ( 8 , C ) -Higgs bundle is stable (respectively polystable) if and only if the induced SO ( 8 , C ) -Higgs bundle is stable (respectively polystable). This correspondence gives rise to a map from the moduli space of Spin ( 8 , C ) -Higgs bundles to the moduli space of SO ( 8 , C ) -Higgs bundles, induced by the homomorphism p introduced in (2).

More precisely, this map embeds the moduli space of Spin ( 8 , C ) -Higgs bundles as a connected subvariety within the moduli space of SO ( 8 , C ) -Higgs bundles. Since every Spin ( 8 , C ) -bundle projects to an SO ( 8 , C ) -bundle with trivial second Stiefel-Whitney class, the image of this morphism lies entirely within the connected component of the SO ( 8 , C ) moduli space corresponding to ω 2 = 0 . However, the moduli space of Spin ( 8 , C ) -Higgs bundles does not exhaust this entire component, as there exist SO ( 8 , C ) -bundles with vanishing second Stiefel-Whitney class that do not lift to Spin ( 8 , C ) -bundles due to topological obstructions. The morphism therefore constitutes an inclusion into one of the two components of the SO ( 8 , C ) moduli space, but it is not surjective onto that component.

This section provides a characterization of stable Spin ( 8 , C ) -Higgs bundles over X that admit nontrivial automorphisms. Specifically, we prove that these bundles admit a reduction of structure group of the Spin ( 8 , C ) -Higgs bundle to a subgroup of the form

SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C )

of SO ( 8 , C ) which is, in turn, a subgroup of Spin ( 8 , C ) , through a homomorphism of groups SL ( 2 , C ) 4 → SO ( 8 , C ) defined by

(3) ( A , B , C , D ) ↦ ( A ⊗ B * ) ⊕ ( C ⊗ D * ) .

Proposition 1

Let ( E , φ ) be a stable Spin ( 8 , C ) -Higgs bundle over X that admits a nontrivial automorphism, and let E 0 be the principal SO ( 8 , C ) -bundle induced by E through the covering map p defined in (2). Then there exist rank 2 projectively nonisomorphic stable Higgs bundles ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , and ( E 4 , φ 4 ) with trivial determinant such that

( E 0 , φ ) = ( ( E 1 ⊗ E 2 * ) ⊕ ( E 3 ⊗ E 4 * ) , ( φ 1 ⊗ φ 2 * ) ⊕ ( φ 3 ⊗ φ 4 * ) ) .

Proof

Given a nontrivial automorphism f : ( E , φ ) → ( E , φ ) , whose existence is guaranteed by the hypothesis, it induces a nontrivial automorphism of the principal bundle E . For any choice of an element e in a fiber of E , there exists an element g e ∈ Spin ( 8 , C ) such that f ( e ) = e g e . The element g e must be semisimple, since the group of outer automorphisms of Spin ( 8 , C ) is finite, and g e is nontrivial, since f is a nontrivial automorphism.

The conjugacy class of g e in Spin ( 8 , C ) is independent of the particular choice of element e , since replacing e by e g for some g ∈ Spin ( 8 , C ) yields f ( e g ) = ( e g ) ( g − 1 g e g ) . Moreover, the conjugacy class of g e does not depend on the choice of the fiber of E over X , since its trace, in any representation of the group, is a constant function defined on the connected variety X .

Since the conjugacy class of g e is uniquely determined by the automorphism f , the centralizer of g e , denoted Z ( g e ) , is a well-defined subgroup of Spin ( 8 , C ) . Here, Z ( g e ) denotes the centralizer { h ∈ Spin ( 8 , C ) : h g e = g e h } , which is indeed a subgroup of Spin ( 8 , C ) that depends only on the conjugacy class of g e and hence is canonically associated with the automorphism f .

The subvariety F = { e ′ ∈ E : f ( e ′ ) = e ′ g e } of E is nonempty, since at least e ∈ F , and clearly defines a reduction of the structure group of ( E , φ ) to Z ( g e ) , as F is invariant under the action of Z ( g e ) and

( f ⊗ 1 K ) ( φ ( e ′ ) ) = φ ( f ( e ′ ) ) = φ ( e ′ ) g e

for e ′ ∈ F , since f is an automorphism of ( E , φ ) .

As proved in [12, Remark after Proposition 2.3], the image under the projection p defined in (2) of Z ( g e ) is the image of the map S ( GL ( 2 , C ) 4 ) → SO ( 8 , C ) defined by

( A , B , C , D ) ↦ ( A ⊗ B * ) ⊕ ( C ⊗ D * ) ,

where the image of S ( GL ( 2 , C ) 4 ) under the above map acts on C 2 ⊗ ( C 2 ) * ⊕ C 2 ⊗ ( C 2 ) * = C 8 in the natural way:

( A , B , C , D ) ( v 1 ⊗ v 2 * ⊕ v 3 ⊗ v 4 * ) = ( A v 1 ⊗ B * v 2 * ) ⊕ ( C v 3 ⊗ D * v 4 * ) .

Note that the reduction of structure group of E to S ( GL ( 2 , C ) 4 ) defines four vector bundles, all of which have the same determinant bundle, as proved in [12, Proposition 2.3]. The choice of a square root of this common determinant bundle is made to simplify the notation and does not affect the essential structure of the decomposition, since the resulting vector bundles are determined up to tensoring with a common line bundle. By choosing a square root of this common determinant bundle, we may assume that all four vector bundles have trivial determinant. Therefore, this determines a reduction of structure group to SL ( 2 , C ) 4 . The final decomposition of ( E 0 , φ ) is canonical and independent of the auxiliary choices made in the construction, including the choice of the square root of the determinant bundle.

The homomorphism SL ( 2 , C ) 2 → SO ( 4 , C ) given by ( A , B ) ↦ A ⊗ B * , where A ⊗ B * acts on the quadratic space C 2 ⊗ ( C 2 ) * , yields the announced form of the image of ( E , φ ) under the map p defined in (2). The stability of the vector Higgs bundles ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , and ( E 4 , φ 4 ) follows from the stability of ( E , φ ) .

Moreover, ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , and ( E 4 , φ 4 ) are pairwise projectively nonisomorphic. Suppose, for the sake of contradiction, that ( E 1 , φ 1 ) and ( E 2 , φ 2 ) are projectively isomorphic. Then there exists a line bundle L of degree zero over X such that E 1 ≅ E 2 ⊗ L and E 1 ⊗ E 2 * ≅ L ⊗ ad ( E 1 ) . However, E 1 is an isotropic subbundle of ad ( E 1 ) invariant under the action of φ 1 ⊗ φ 1 * , so ( ad ( E 1 ) , φ 1 ⊗ φ 1 * ) is strictly polystable as an SO ( 4 , C ) -Higgs bundle. Therefore, ( E 1 ⊗ E 2 * , φ 1 ⊗ φ 2 * ) cannot be stable as an SO ( 4 , C ) -Higgs bundle, which contradicts the stability of ( E 0 , φ ) derived from the stability of ( E , φ ) . Thus, ( E 1 , φ 1 ) and ( E 2 , φ 2 ) are projectively nonisomorphic.□

Remark 1

It is worth noting that the center of SL ( 2 , C ) 4 can be identified with the center of Spin ( 8 , C ) , with its elements being:

( I , I , I , I ) , ( I , − I , − I , I ) , ( I , − I , I , − I ) , ( I , I , − I , − I ) ,

so SL ( 2 , C ) 4 can be viewed as a subgroup of Spin ( 8 , C ) .

5 The map SL ( 2 , C ) 4 → SO ( 8 , C ) and spectral data

The aim of this section is to describe the spectral data of an SO ( 8 , C ) -Higgs bundle that admits the reduction of structure group established in Proposition 1 in terms of the spectral data of an SL ( 2 , C ) 4 -Higgs bundle.

The composition of the isogeny

SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) → SO ( 4 , C ) × SO ( 4 , C )

given by ( A , B , C , D ) ↦ A ⊗ B * ⊕ C ⊗ D * with the natural inclusion

SO ( 4 , C ) × SO ( 4 , C ) → SO ( 8 , C )

defines a morphism of groups

(4) ℱ : SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) → SO ( 8 , C )

introduced in (3). The action of the group SL ( 2 , C ) × SL ( 2 , C ) on C 2 ⊗ ( C 2 ) * = C 4 given by ( A , B ) ( v 1 ⊗ v 2 * ) = ( A v 1 , B * v 2 * ) , together with the symmetric, nondegenerate bilinear form with which C 4 is equipped, ensures that the map (4) is well defined. Moreover, the kernel of this morphism ℱ consists of quadruples ( A , B , C , D ) , where each component is ± I . Indeed, there is a well-defined morphism from the subgroup of GL ( 2 , C ) 2 consisting of matrices ( A , B ) with det A = det B to SO ( 4 , C ) , which preserves the determinant and whose kernel is a subgroup isomorphic to Z ⁄ ( 2 ) × Z ⁄ ( 2 ) [19, Chapter 30]. At the level of Lie algebras, the derivative of ℱ , d ℱ : sl ( 2 , C ) 4 → so ( 8 , C ) , acts as follows:

d ℱ ( A 0 , B 0 , C 0 , D 0 ) = ( A 0 ⊗ I + I ⊗ B 0 t , C 0 ⊗ I + I ⊗ D 0 t ) .

The morphism of groups ℱ defined in (4) induces the following morphism from the moduli space of SL ( 2 , C ) 4 -Higgs bundles to the moduli space of SO ( 8 , C ) -Higgs bundles, which we also denote by ℱ :

(5) ℱ ( ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , ( E 4 , φ 4 ) ) = E 1 ⊗ E 2 * ⊕ E 3 ⊗ E 4 * , φ 1 ⊗ I + I ⊗ φ 2 t 0 0 φ 3 ⊗ I + I ⊗ φ 4 t .

In the next result, we provide a specific description of the spectral data of an SO ( 8 , C ) -Higgs bundle that satisfies the conditions of Proposition 1 in terms of the spectral data of the reduction of structure group to SL ( 2 , C ) 4 that it admits. The proof closely follows the program established by [15,16], extends their construction to SO ( 8 , C ) , and also discusses whether the bundles lift to Spin ( 8 , C ) -bundles through Proposition 1 in terms of the spectral data constructed.

Proposition 2

Let ( E i , φ i ) be SL ( 2 , C ) -Higgs bundles over X for i = 1, 2, 3, 4 , and let ( Y i , L i ) be their corresponding spectral data, where Y i is the spectral curve associated with ( E i , φ i ) with 2-to-1 covering map π i : Y i → X and L i ∈ Prym ( Y i , X ) is the corresponding line bundle over Y i . Let Y = Y 1 × X Y 2 ∐ Y 3 × X Y 4 , let q i : Y → Y i be the corresponding projection map for each i, and let ℒ be the line bundle over Y defined by

q 1 * L 1 ⊗ ( q 2 * L 2 * ⊗ [ R 2 ] − 1 ) ∐ q 3 * L 3 ⊗ ( q 4 * L 4 * ⊗ [ R 4 ] − 1 ) ,

where [ R i ] denotes the line bundle defined by the ramification divisor R i in Y i , for i = 2 , 4. Then the pair ( Y , ℒ ) is the spectral data of an SO ( 8 , C ) -Higgs bundle. Moreover, ( Y , ℒ ) is the spectral data of the SO ( 8 , C ) -Higgs bundle

ℱ ( ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , ( E 4 , φ 4 ) ) ,

where ℱ is defined in (5).

Proof

We first show that ( Y , ℒ ) is the spectral data of an SO ( 8 , C ) -Higgs bundle. To verify this, it suffices to show that ( Y 1,2 , ℒ 1,2 ) is the spectral data of an SO ( 4 , C ) -Higgs bundle, where Y 1,2 = Y 1 × X Y 2 and ℒ 1,2 = q 1 * L 1 ⊗ ( q 2 * L 2 * ⊗ [ R 2 ] − 1 ) (the same proof applies to Y 3 × X Y 4 and ℒ 3,4 = q 3 * L 3 ⊗ ( q 4 * L 4 * ⊗ [ R 4 ] − 1 ) ). This would imply that the disjoint union ℒ 1,2 ∐ ℒ 3,4 is a line bundle over the curve defined as the disjoint union Y = Y 1 × X Y 2 ∐ Y 3 × X Y 4 of the statement. Moreover, the projection over X is the direct sum E 1 ⊗ E 2 * ⊕ E 3 ⊗ E 4 * , as desired, with the Higgs field being the direct sum φ 1 ⊗ I + I ⊗ φ 2 t ⊕ φ 3 ⊗ I + I ⊗ φ 4 t . Thus, the constructed pair ( Y , ℒ 1,2 ∐ ℒ 3,4 ) would be, as desired, the spectral data of ℱ ( ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , ( E 4 , φ 4 ) ) .

For the case of SO ( 4 , C ) , we follow the construction given in [16, Propositions 13 and 15]. Let λ i 2 + a i = 0 be the equation that defines the spectral curve Y i , where a i ∈ H 0 ( X , K 2 ) , for i = 1 , 2. Note that for each i , λ i is the tautological section of K , whose total space can be viewed as the fiber product K × X K . Also note that Y 1,2 is a subvariety of the fiber product K × X K and therefore of K ⊕ K . It follows that ( λ 1 , λ 2 ) is the tautological section of K ⊕ K and that Y 1,2 is smooth for a generic pair ( a 1 , a 2 ) .

Let Y 1,2 ¯ be the image of Y 1,2 under the fiberwise addition K ⊕ K → K , so Y 1,2 ¯ is a subset of the total space of K . It is clear that the fiberwise addition defines a map Y 1,2 → Y 1,2 ¯ that is an isomorphism on the smooth locus of Y 1,2 ¯ . The curve Y 1,2 ¯ is a 4-to-1 cover of X . It typically possesses singularities located at the zeros of a 1 − a 2 . More precisely, when a 1 ≠ a 2 generically, the curve Y 1,2 ¯ is singular at points where the difference a 1 − a 2 vanishes, which occurs at a finite number of points on X for generic choices of the Higgs fields. In the special case where a 1 = a 2 , the curve degenerates and exhibits a different type of singularity structure. Equations λ = λ 1 + λ 2 , λ 1 2 + a 1 = 0 , and λ 2 2 + a 2 = 0 define Y 1,2 ¯ .

Therefore, the equation

(6) λ 4 + 2 ( a 1 + a 2 ) λ 2 + ( a 1 − a 2 ) 2 = 0

defines Y 1,2 ¯ . This is precisely the equation of a spectral curve of an SO ( 4 , C ) -Higgs bundle, particularizing the general expression given in (1).

The involution s defined by λ ↦ − λ induces an involution of Y 1,2 given by s 1,2 = ( s 1 , s 2 ) (here, s 1 and s 2 refer to the same involution s but preserving Y 1 and Y 2 , respectively). The involution s 1,2 generically has no fixed points since the fixed points of s i are the zeros of a i and the zeros of a 1 and those of a 2 are generically distinct. Moreover, the involution of Y 1,2 ¯ induced by s 1,2 has fixed points at the branch locus. Note also that s i * L i ≅ L i * , since L i ∈ Prym ( Y i , X ) for i = 1,2 , and s 2 * [ R 2 ] ≅ [ R 2 ] * , since [ R 2 ] is the line bundle defined by the ramification locus of s 2 . Then s 1,2 sends ℒ 1,2 = q 1 * L 1 ⊗ ( q 2 * L 2 * ⊗ [ R 2 ] − 1 ) to its dual. This proves that ℒ 1,2 ∈ Prym ( Y 1,2 , Y 1,2 ⁄ s 1,2 ) and therefore that ℒ 1,2 is the line bundle of the spectral data of an SO ( 4 , C ) -Higgs bundle.

The aforementioned proves that ( Y 1,2 , ℒ 1,2 ) is the spectral data of an SO ( 4 , C ) -Higgs bundle. Moreover, (6) is the equation of the spectral curve associated with

( E 1,2 , φ 1,2 ) = ( E 1 ⊗ E 2 * , φ 1 ⊗ I + I ⊗ φ 2 t ) .

Let π 1,2 be the covering map Y 1,2 → X . Then

( π 1,2 ) * ( ℒ 1,2 ) = ( π 1 ) * ( L 1 ) ⊗ ( π 2 ) * ( L 2 * ⊗ [ R 2 ] − 1 ) = E 1 ⊗ E 2 * ⊗ K − 1 ,

and then

E 1,2 = ( π 1,2 ) * ( ℒ 1,2 ⊗ ( K Y 1,2 ⊗ π 1,2 * K * ) 1 ⁄ 2 ) ,

since K Y 1,2 ⊗ π 1,2 * K * is defined by the ramification divisor R of Y 1,2 , the ramification divisors R i of Y i ( i = 1, 2 ) satisfy [ R i ] = π i * K and, then,

K Y 1,2 ⊗ π 1,2 * K * = [ R ] = q 1 * [ R 1 ] ⊗ q 2 * [ R 2 ] = ( π 1,2 * K ) 2 ,

so E 1,2 = ( π 1,2 ) * ( ℒ 1,2 ) ⊗ K = E 1 ⊗ E 2 * . This finally proves the result.□

Remark 2

It should be noticed that the SO ( 8 , C ) -Higgs bundle

( E 0 , φ 0 ) = ℱ ( ( E 1 , φ 1 ) , ( E 2 , φ 2 ) , ( E 3 , φ 3 ) , ( E 4 , φ 4 ) ) ,

where ℱ is defined in (5), cannot lift to a Spin ( 8 , C ) -Higgs bundle. Indeed, as explained earlier, Prym ( Y , Y ⁄ s ) has two connected components, where s denotes the involution of Y induced by the involutions s i for i = 1, 2, 3, 4 (with the notation of Proposition 2). Any line bundle ℳ over Y satisfying the condition that Nm ( ℳ ) is trivial must be of the form ℳ ≅ N ⊗ s * ( N * ) for a degree 0 or 1 line bundle N over Y [18, Lemma 1]. These two possibilities for the degree of N determine the two connected components of Prym ( Y , Y ⁄ s ) . Furthermore, ( E 0 , φ 0 ) lifts to a Spin ( 8 , C ) -Higgs bundle ( E , φ ) if and only if the degree of the line bundle N induced by the spectral line bundle ℒ is 0. The connected component of Prym ( Y , Y ⁄ s ) of spectral data that correspond to SO ( 8 , C ) -Higgs bundles that lift to Spin ( 8 , C ) -Higgs bundles corresponds to the connected component of the moduli space of SO ( 8 , C ) -Higgs bundles (of the two components that it admits) consisting of Higgs bundles whose second Stiefel-Whitney class is zero.

6 Spectral data and fixed points of automorphisms of the moduli space of Spin ( 8 , C ) -Higgs bundles

Since Spin ( 8 , C ) is the simply connected complex Lie group whose Lie algebra is so ( 8 , C ) , of type D 4 , and the group of symmetries of D 4 is isomorphic to the group of permutations on three elements, this is also the group Out ( Spin ( 8 , C ) ) of outer automorphisms of Spin ( 8 , C ) . A choice of an element of order 3 among such outer automorphisms, τ , determines what is called the triality automorphism.

In general, if G is a semisimple complex Lie group and X is a compact Riemann surface, the group Out ( G ) of outer automorphisms of G acts on the moduli space of G -Higgs bundles over X in such a way that each nontrivial outer automorphism of G induces a nontrivial automorphism of the moduli space [7]. Specifically, if ( E , φ ) is a G -Higgs bundle and ρ ∈ Out ( G ) is a nontrivial outer automorphism, then ρ ( E ) is defined to be the principal G -bundle whose total space is the same as that of E , but which is equipped with a right action of G on it given by e ⋅ g = e f ρ ( g ) − 1 for g ∈ G and e ∈ ρ ( E ) , where f ρ is a choice of an automorphism of G that represents ρ . In addition, the Higgs field of ρ ( E ) is the same as that of E but transformed by the derivative d f ρ , that is, ρ ( φ ) = d f ρ ⊗ I K ( φ ) , where K denotes the canonical line bundle over X . The isomorphism class of ( ρ ( E ) , ρ ( φ ) ) is independent of the choice of the representative f ρ of ρ [7].

When G = Spin ( 8 , C ) , the aforementioned defines five different nontrivial automorphisms of the moduli space of Spin ( 8 , C ) -Higgs bundles over X , which are induced, respectively, by the five different nontrivial outer automorphisms of Spin ( 8 , C ) . Two of them have order 3 (those corresponding to the triality automorphism τ and its square τ 2 ) and the rest are involutions. The following result describes the spectral data of the SO ( 8 , C ) -Higgs bundle induced by a fixed point of the action of the triality automorphism of Spin ( 8 , C ) or of an outer involution of Spin ( 8 , C ) .

Proposition 3

Let ( E , φ ) be a stable Spin ( 8 , C ) -Higgs bundle over X which admits a nontrivial automorphism, let ( E 0 , φ 0 ) be the induced SO ( 8 , C ) -Higgs bundle, and let γ ∈ Out ( Spin ( 8 , C ) ) be an outer automorphism. Since Out ( Spin ( 8 , C ) ) ≅ S 3 is the symmetric group on three elements, which can also be viewed as the dihedral group of order 6, one can classify the action of γ on the center Z ( Spin ( 8 , C ) ) ≅ ( Z ⁄ 2 Z ) 2 according to whether γ corresponds to a three-cycle (triality) or a transposition (outer involution).

The center of Spin ( 8 , C ) has three nontrivial elements that can be represented as ( I , − I , − I , I ) , ( I , − I , I , − I ) , and ( I , I , − I , − I ) under the natural identification with ( Z ⁄ 2 Z ) 2 . The action of Out ( Spin ( 8 , C ) ) permutes these three elements, and this permutation completely determines the form of the spectral data for Higgs bundles fixed by γ .

Then ( E , φ ) is fixed by the action of γ on the moduli space of Spin ( 8 , C ) -Higgs bundles if and only if the spectral data of ( E 0 , φ 0 ) takes one of the following forms:

Case 1 (Triality): When γ acts as a three-cycle on the center elements, the spectral data are ( Y , ℒ ) where Y = Y ′ × X Y ″ ∐ Y ″ × X Y ″ for some 2-to-1 spectral covers Y ′ and Y ″ of X, and

ℒ = ( q ′ ) * L ⊗ ( q 2 ″ ) * M ∐ ( q 3 ″ ) * M ⊗ ( q 4 ″ ) * M

for certain line bundles L → Y ′ and M → Y ″ , where q ′ : Y → Y ′ and q i ″ : Y → Y ″ are the natural projections.

Case 2 (Outer involution): When γ acts as a transposition on the center elements, the spectral data are ( Y , ℒ ) where Y and ℒ take one of three possible forms corresponding to the three distinct transpositions in S 3 . Specifically, if γ fixes one center element and exchanges the other two, then Y = Y 1 ′ × X Y 2 ′ ∐ Y ″ × X Y ″ with

ℒ = q 1 ′ * L 1 ⊗ q 2 ′ * L 2 * ⊗ [ R 2 ] − 1 ∐ q ″ * M ⊗ q ″ * M * ,

or Y = Y ′ × X Y 1 ″ ∐ Y 1 ″ × X Y 2 ″ with

ℒ = q ′ * L ⊗ q 1 ″ * M 1 ∐ q 1 ″ * M 1 ⊗ q 2 ″ * M 2 * ⊗ [ R 2 ] − 1 ,

or Y = Y ′ × X Y 1 ″ ∐ Y 2 ″ × X Y 1 ″ with

ℒ = q ′ * L ⊗ q 1 ″ * M 1 * ⊗ [ R 1 ] − 1 ∐ q 2 ″ * M 2 ⊗ q 1 ″ * M 1 * ⊗ [ R 1 ] − 1 ,

where all Y i are appropriate 2-to-1 covers of X, the line bundles L i and M j are defined on the corresponding covers, the q ⋅ denote the natural projection maps, and [ R i ] represents the line bundle induced by the ramification divisor R i .

Proof

Since ( E , φ ) is stable with a nontrivial automorphism, Proposition 1 shows that ( E 0 , φ 0 ) admits a reduction of structure to SL ( 2 , C ) 4 , so its spectral data can be expressed as a quadruple ( Y i , L i ) for i = 1 , 2, 3, 4 by Proposition 2.

The key observation is that the action of any γ ∈ Out ( Spin ( 8 , C ) ) ≅ S 3 on the moduli space is completely determined by how it permutes the three nontrivial center elements. Once this permutation is identified, the constraints on the spectral data follow automatically from the requirement that the Higgs bundle be fixed under the action.

For the triality case, γ cyclically permutes all three center elements

( I , − I , − I , I ) ↦ ( I , − I , I , − I ) ↦ ( I , I , − I , − I ) ↦ ( I , − I , − I , I ) .

This forces Y 2 , Y 3 , and Y 4 to be isomorphic, as well as L 2 , L 3 , and L 4 . Setting Y ′ = Y 1 , Y ″ = Y 3 , L = L 1 , and M = L 3 , the spectral data take the required form in Case 1.

For the outer involution cases, γ fixes one center element and exchanges the other two. The three distinct ways to choose which element to fix correspond to the three transpositions in S 3 , yielding the three forms given in Case 2. The specific forms of Y and ℒ in each subcase arise directly from the constraint that the corresponding spectral covers and line bundles must be preserved under the relevant permutation of indices.

Conversely, if the spectral data have any of the prescribed forms, then the reconstruction of ( E 0 , φ 0 ) via the pushforward along the covering map π : Y → X automatically yields a Higgs bundle that is invariant under the corresponding outer automorphism since the spectral data itself encode this symmetry.□

Remark 3

Note that the group of outer automorphisms of Spin ( 8 , C ) , which is isomorphic to the symmetric group on three elements, is generated by the triality automorphism τ and a nontrivial outer involution σ . There are two elements of order 3 among the nontrivial outer automorphisms, τ and τ 2 , whose respective actions on the moduli space of Spin ( 8 , C ) -Higgs bundles leave the same fixed points. There are also three nontrivial involutions, σ , σ τ , and σ τ 2 , whose respective Spin ( 8 , C ) -Higgs bundle fixed points are described, under the technical conditions of the proposition, in Proposition 3.

7 Conclusion

Let X be a compact Riemann surface of genus g ≥ 2 . In this work, the moduli space ℳ ( Spin ( 8 , C ) ) of Spin ( 8 , C ) -Higgs bundles over X has been considered. The group of outer automorphisms of Spin ( 8 , C ) , isomorphic to the symmetric group on three elements, acts on ℳ ( Spin ( 8 , C ) ) by inducing a family of automorphisms of the moduli space, the order of each one being equal to the order of the corresponding outer automorphism. It has been proved that any stable Spin ( 8 , C ) -Higgs bundle that admits a nontrivial automorphism (that is, an automorphism not coming from the center of the structure group) admits a reduction of structure group to a subgroup of the form SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) . This is an auxiliary result that allows one to associate a collection of four SL ( 2 , C ) -Higgs bundles to each fixed point. After that, a result has been proved that establishes the specific form that the spectral data of an SO ( 8 , C ) -Higgs bundle that admits a reduction of structure group to SL ( 2 , C ) 4 must have. This allows for describing the spectral data of the SO ( 8 , C ) -Higgs bundle induced by a stable Spin ( 8 , C ) -Higgs bundle that admits nontrivial automorphisms. Finally, the previous results have been applied to describe the spectral data of the orthogonal Higgs bundles given by the fixed points of the different automorphisms of ℳ ( Spin ( 8 , C ) ) defined by outer automorphisms of Spin ( 8 , C ) . All this allows for an understanding of the fixed points of the aforementioned automorphisms through their spectral data, which leads to an essential deepening in their understanding and provides a methodological novelty in their study and description.

Funding information: The author states no funding involved.
Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] N. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 59–126, DOI: https://doi.org/10.1112/plms/s3-55.1.59. 10.1112/plms/s3-55.1.59Suche in Google Scholar

[2] C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918. 10.1090/S0894-0347-1988-0944577-9Suche in Google Scholar

[3] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114, DOI: https://doi.org/10.1215/S0012-7094-87-05408-1. 10.1215/S0012-7094-87-05408-1Suche in Google Scholar

[4] C. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), no. 4, 778–782. 10.2307/2372597Suche in Google Scholar

[5] D. Baraglia, Classification of the automorphism and isometry groups of Higgs bundle moduli spaces, Proc. Lond. Math. Soc. 112 (2016), no. 5, 827–854, DOI: https://doi.org/10.1112/plms/pdw014. 10.1112/plms/pdw014Suche in Google Scholar

[6] R. Fringuelli, Automorphisms of moduli spaces of principal bundles over a smooth curve, Internat. J. Math. 35 (2024), no. 10, 2450036, DOI: https://doi.org/10.1142/S0129167X24500368. 10.1142/S0129167X24500368Suche in Google Scholar

[7] Á. Antón-Sancho, Automorphisms of order three of the moduli space of Spin(8,C)-Higgs bundles, Hokkaido Math. J. 47 (2018), no. 2, 387–426, DOI: https://doi.org/10.14492/hokmj/1529308825. 10.14492/hokmj/1529308825Suche in Google Scholar

[8] O. Garcia-Prada and S. Ramanan, Involutions and higher order automorphisms of Higgs bundle moduli spaces, Proc. Lond. Math. Soc. 119 (2019), no. 3, 681–732, DOI: https://doi.org/10.1112/plms.12242. 10.1112/plms.12242Suche in Google Scholar

[9] J.-M. Hwang and S. Ramanan, Hecke curves and Hitchin discriminant, Ann. Sci. Éc. Norm. Supér. 37 (2004), no. 4, 801–817, DOI: https://doi.org/10.1016/j.ansens.2004.07.001. 10.1016/j.ansens.2004.07.001Suche in Google Scholar

[10] A. Anton-Sancho, F4 and PSp(8,C)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface, Open Math. 20 (2022), 1–11, DOI: https://doi.org/10.1515/math-2022-0543. 10.1515/math-2022-0543Suche in Google Scholar

[11] A. Anton-Sancho, Fixed points of automorphisms of the vector bundle moduli space over a compact Riemann surface, Mediterr. J. Math. 21 (2024), 20, DOI: https://doi.org/10.1007/s00009-023-02559-z. 10.1007/s00009-023-02559-zSuche in Google Scholar

[12] A. Anton-Sancho, Triality and automorphisms of principal bundles moduli spaces, Adv. Geom. 24 (2024), no. 3, 421–435. Suche in Google Scholar

[13] R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 166 (2006), no. 1, 103–143, DOI: https://doi.org/10.1007/s00222-012-0373-8. 10.1007/s00222-012-0373-8Suche in Google Scholar

[14] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press, Cambridge, 2012, DOI: https://doi.org/10.1017/CBO9781139248563. 10.1017/CBO9781139248563Suche in Google Scholar

[15] L. P. Schaposnick, Spectral Data for G-Higgs Bundles, PhD Thesis, 2013, arXiv:1301.1981v1 [math.DG]. Suche in Google Scholar

[16] S. Bradlow, L. Branco, and L. P. Schaposnik, Orthogonal Higgs bundles with singular spectral curves, Comm. Anal. Geom. 28 (2020), no. 8, 1895–1931, DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a6. 10.4310/CAG.2020.v28.n8.a6Suche in Google Scholar

[17] C. T. Simpson, Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. 75 (1992), 5–95, DOI: https://doi.org/10.1007/BF02699491. 10.1007/BF02699491Suche in Google Scholar

[18] D. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. Éc. Norm. Supér. 4 (1971), no. 4, 181–192. 10.24033/asens.1209Suche in Google Scholar

[19] D. Bump, Lie Groups, Springer, New-York, 2013, DOI: https://doi.org/10.1007/978-1-4614-8024-2. 10.1007/978-1-4614-8024-2Suche in Google Scholar

Received: 2024-08-02

Revised: 2025-06-22

Accepted: 2025-07-30

Published Online: 2025-09-05

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https://doi.org/10.1515/math-2025-0192

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spectral data; Hitchin fibration; automorphism; fixed point; group action

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