Triality and Non-Abelian Spectral Data for $\mathbf{Spin}\mathbf{(}\mathbf{8}\mathbf{,}\mathbb{C}\mathbf{)}$-Higgs Bundles

Abstract

In this research, we study the geometry of the moduli space of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of $\mathrm{Spin}(8,\mathbb{C})$. We establish a characterization of triality invariance, proving that a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle admits a reduction to the exceptional group $G_2$ if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting $G_2$-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from $G_2$-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset.

1. Introduction

Given a complex reductive Lie group, a G-Higgs bundle over a compact Riemann surface X, as introduced by Hitchin in their seminal work [1] and further developed by Simpson [2,3,4], consists of a holomorphic principal G-bundle together with a Higgs field valued in the adjoint bundle twisted by the canonical bundle K. The moduli space of such objects carries several geometric structures, encoded in the Hitchin fibration, which realizes it as an algebraically completely integrable system [2,3,4].

For classical groups, the geometry of Higgs bundles is well understood through the theory of spectral curves. When $G=\mathrm{GL}(n,\mathbb{C})$, Hitchin showed that the Higgs field determines a spectral curve in the total space of the canonical bundle, and the correspondence between stable Higgs bundles and spectral data allows us to understand the moduli space. This spectral correspondence was extended to orthogonal and symplectic groups by Hitchin [1] and further developed by Beauville, Narasimhan, and Ramanan [5], who established the connection between spectral curves and Prym varieties.

The case of singular spectral curves has attracted the attention of researchers in algebraic geometry in recent years. While generic Higgs bundles correspond to smooth spectral curves (in the sense that Higgs bundles with smooth spectral curves form a Zariski open dense subset of the moduli space via Bertini’s theorem [6]), understanding the behavior of the moduli space near singular loci is crucial for understanding their geometry. Schaub [7] initiated the systematic study of Higgs bundles with singular spectral curves, introducing the notion of spectral data for certain classes of singularities. More recently, Bradlow, Branco, and Schaposnik [8] developed a theory concerning non-abelian spectral data for orthogonal Higgs bundles with nodal spectral curves, showing that a singular locus supports rich geometric structures beyond the classical abelian theory.

Among the classical groups, $\mathrm{Spin}(8,\mathbb{C})$ occupies a special position due to its triality automorphism, an outer automorphism of order three that permutes the three eight-dimensional irreducible representations: the vector representation and the two spinor representations. This phenomenon, discovered by Cartan [9] and studied extensively in the context of the Lie theory [10], has no analogue for other spin groups. This triality automorphism is connected to the exceptional group $G_2$, which arises as the fixed-point subgroup of triality in $\mathrm{Spin}(8,\mathbb{C})$. The geometry of $G_2$-Higgs bundles has been investigated by Hitchin [11], who established several remarkable results on the associated spectral curves and integrable systems.

Despite the theory of orthogonal Higgs bundles developed in [8] and the role of triality in the geometry of $\mathrm{Spin}(8,\mathbb{C})$, the interaction between these two aspects remains unexplored. Several fundamental questions require systematic investigation: How does triality automorphism act geometrically on the space of spectral curves? What is the geometric meaning of triality invariance for spectral data? Do non-abelian structures compatible with triality exist on singular spectral curves? How does triality interact with the stratification of the moduli space by singularity type?

A recent work [12,13] clarified the action of outer automorphisms on moduli spaces of principal bundles and initiated the study of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles in the context of the Hitchin integrable system. Schaposnik and Schulz [14] investigated triality for homogeneous polynomials in the Hitchin base, providing the foundation for understanding how triality acts on the space of spectral curves. Most recently, in [15], explicit descriptions of spectral data for fixed points of outer automorphisms of $\mathrm{Spin}(8,\mathbb{C})$ acting on the moduli space have been provided, including a detailed study of orthogonal Higgs bundles arising as fixed points. This work establishes foundational techniques for understanding the spectral geometry of triality-invariant configurations.

The purpose of the present paper is to extend the theory of orthogonal Higgs bundles with singular spectral curves developed in [8] from $\mathrm{SO}(4,\mathbb{C})$ to $\mathrm{Spin}(8,\mathbb{C})$. The present work builds upon and extends the Bradlow–Branco–Schaposnik framework [8] in several ways:

(1)

While ref. [8] studied $\mathrm{SO}(4,\mathbb{C})$-Higgs bundles, which have rank 4 and no exceptional outer automorphisms, we work with $\mathrm{Spin}(8,\mathbb{C})$, which has rank 8 and admits a unique triality automorphism of order 3. This automorphism has no analogue in orthogonal groups.

(2)

The triality automorphism introduces a ${\mathbb{Z}}_3$-action on the moduli space, leading to a novel stratification. We show that the discriminant locus decomposes into fixed points (corresponding to $G_2$-structures) and free ${\mathbb{Z}}_3$-orbits, a phenomenon absent in [8].

(3)

The triality-invariant sublocus connects to the exceptional Lie group $G_2$. This connection allows us to characterize $G_2$-reductions through spectral data, complementing the analysis in [11].

(4)

We prove that non-abelian spectral data can occur in free triality orbits, demonstrating that the non-abelian phenomena of [8] persist in the presence of triality symmetry.

The first main contribution of this paper characterizes triality-invariant spectral curves geometrically. Specifically, in Theorem 2, we prove that a spectral curve is invariant under the induced action of triality if and only if the corresponding $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle admits a reduction in structure group to $G_2$. This establishes a precise connection between the exceptional group $G_2$ and the fixed-point locus of triality on the moduli space, complementing the spectral data approach developed in [15].

We also analyze the singular locus of the Hitchin fibration under triality action. Theorem 3 provides a decomposition of the discriminant locus into two strata: a fixed locus, corresponding to $G_2$-Higgs bundles with singular spectral curves, and a free locus, consisting of orbits of size three under the ${\mathbb{Z}}_3$-action induced by triality. This stratification is further refined in Proposition 1, which describes the geometric structure of each stratum.

In addition, we establish the existence of genuinely non-abelian spectral data compatible with triality. Theorem 4 shows that there exist $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles with nodal spectral curves in free triality orbits that support non-abelian spectral data, similar to [8]. This demonstrates that the non-abelian phenomena discovered for $\mathrm{SO}(4,\mathbb{C})$ persist in the presence of triality symmetry.

In this paper, we introduce a quantitative measure of triality symmetry breaking. Specifically, in Remark 16, we define the triality defect ${\delta }_{\tau }(E,\mathrm{\Phi })$ as the dimension of the quotient of the Prym variety by its triality-invariant sublocus. This invariant takes values in the range $[0,63g-63]$ for a Riemann surface of genus g, with ${\delta }_{\tau }=0$ characterizing $G_2$-Higgs bundles. In Proposition 3, we prove that Higgs bundles with ${\delta }_{\tau }>0$ form a Zariski open dense subset of the smooth locus, showing that perfect triality symmetry is a special, non-generic phenomenon. The term non-generic here means that the triality-invariant locus ${\mathcal{M}}_0$ corresponds to the condition $b_4=c_4$, which defines a proper closed subset of codimension $7g-7$ in the space of Hitchin invariants, and hence has a measure of zero.

This paper is organized as follows. Section 2 develops the preliminary material on Higgs bundles, spectral curves, and the triality automorphism of $\mathrm{Spin}(8,\mathbb{C})$. We recall the basic definitions from [1,3], the spectral correspondence from [5], the extension to singular curves from [7,8], and the structure of triality from [9,10]. In Section 3, we develop the main results: the characterization of triality-invariant curves (Theorem 2), the classification of singular orbits (Theorem 3), the existence of non-abelian triality-compatible data (Theorem 4), and the action on Prym varieties (Theorem 5). In Section 4, concrete examples are presented. Section 5 provides the application of the main results to the establishment of a novel stratification of the smooth locus of the Hitchin fibration using the triality defect mentioned above. We conclude with a discussion of the main conclusions, open problems, and directions for future research.

2. Preliminaries

Let X be a compact connected Riemann surface of genus $g\ge 2$, and let $K=K_X$ denote its canonical bundle. We recall the fundamental notion introduced by Hitchin [1].

Definition 1

([1]). Let G be a complex reductive Lie group with Lie algebra $\mathfrak{g}$. A G-Higgs bundle over X is a pair $(E,\mathrm{\Phi })$ where

(1) 

E is a holomorphic principal G-bundle over X;

(2) 

$\mathrm{\Phi }\in H^0(X,E\left(\mathfrak{g}\right)\otimes K)$ is a holomorphic section, called the Higgs field, where $E\left(\mathfrak{g}\right)=E{\times }_G\mathfrak{g}$ is the adjoint bundle.

The notion of stability for G-Higgs bundles was developed by Ramanathan [16] and Hitchin [1]. A G-Higgs bundle $(E,\mathrm{\Phi })$ is stable (resp. semistable) if, for every parabolic subgroup $P\subset G$ and every reduction in the structure group $\sigma :X\to E/P$, we have

$$deg\left({\sigma }^{\ast }{T}_{E/P}\right)<0(\mathrm{resp}.\le 0),$$

where $T_{E/P}$ is the relative tangent bundle along the fibers. We denote by $\mathcal{M}\left(G\right)$ the moduli space of semistable G-Higgs bundles over X [3].

The Hitchin fibration is constructed using ad-invariant polynomials on $\mathfrak{g}$. Using the theorem of Chevalley [17], the ring of ad-invariant polynomials on $\mathfrak{g}$ is a polynomial generated by algebraically independent homogeneous polynomials $p_1,\dots ,p_r$ of degrees $d_1,\dots ,d_r$, where $r=\mathrm{rank}\left(G\right)$.

Definition 2

([1]). The Hitchin base is the affine space

$$\mathcal{A}=\underset{i=1}{\overset{r}{⨁}}{H}^0(X,K^{d_i}).$$

The Hitchin map is the morphism

$$h:\mathcal{M}\left(G\right)\to \mathcal{A}$$

defined by $h(E,\mathrm{\Phi })=(p_1\left(\mathrm{\Phi }\right),\dots ,p_r\left(\mathrm{\Phi }\right))$.

The Hitchin map endows $\mathcal{M}\left(G\right)$ with the structure of an algebraically completely integrable system, as elaborated by Liouville [1]. Let ${\mathcal{A}}^{\mathrm{reg}}\subset \mathcal{A}$ denote the open dense subset consisting of points whose associated spectral curves are smooth and irreducible. This subset is labeled as dense according to Bertini’s theorem [6]: the discriminant locus $\Delta$, consisting of points with singular spectral curves, is a proper closed subvariety of a codimension of at least one; hence, its complement ${\mathcal{A}}^{\mathrm{reg}}=\mathcal{A}\setminus \Delta$ is open and dense in the Zariski topology. For $a\in {\mathcal{A}}^{\mathrm{reg}}$, the fiber $h^{-1}\left(a\right)$ is a Lagrangian subvariety of $\mathcal{M}\left(G\right)$ with respect to the natural symplectic structure inherited from the cotangent bundle, and it is biholomorphic to a torsor over an abelian variety (the Jacobian or Prym variety of the associated spectral curve) [1,5].

For classical groups, the spectral curve can be described explicitly. In the case of $G=\mathrm{GL}(n,\mathbb{C})$, the Higgs field $\mathrm{\Phi }\in H^0(X,\mathrm{End}\left(E\right)\otimes K)$ has the characteristic polynomial

$$det(\eta ·\mathrm{id}-\mathrm{\Phi })={\eta }^n+a_1{\eta }^{n-1}+\cdots +a_n,$$

where $a_i\in H^0(X,K^i)$. The spectral curve $S\subset \mathrm{Tot}\left(K\right)$ is defined as the zero locus of this polynomial in the total space of K [1].

Definition 3

([5]). Let $(E,\mathrm{\Phi })$ be a $\mathrm{GL}(n,\mathbb{C})$-Higgs bundle and let $\pi :S\to X$ be its spectral curve. The spectral data consists of the pair $(S,\mathcal{L})$, where $\mathcal{L}$ is a line bundle on S (or, more generally, a rank-one torsion-free sheaf when S is singular) satisfying ${\pi }_{\ast }\mathcal{L}\cong E$.

The Beauville–Narasimhan–Ramanan correspondence [5] establishes a bijection between stable $\mathrm{GL}(n,\mathbb{C})$-Higgs bundles whose spectral curves are smooth and irreducible and line bundles of an appropriate degree on these smooth spectral curves. This correspondence was extended to arbitrary spectral curves (including singular and reducible ones) by Schaub [7], who showed that the fiber of the Hitchin map over any point in $\mathcal{A}$ is a compactification of the Jacobian (or generalized Jacobian) map of the associated spectral curve.

For orthogonal and symplectic groups, the spectral curves admit natural involutions induced by their defining bilinear or symplectic forms. In these cases, the fibers of the Hitchin map over points in ${\mathcal{A}}^{\mathrm{reg}}$ are described by Prym varieties rather than full Jacobians [1].

When the spectral curve S is smooth and irreducible, the fiber of the Hitchin map is an abelian variety. However, when S is singular, the geometry becomes significantly more intricate. Following [8], we recall the following:

Definition 4

([8]). Let S be a (possibly singular) reduced curve, and let $\nu :\tilde{S}\to S$ be its normalization. The non-abelian spectral data on S consists of the following:

(1) 

A line bundle ${\mathcal{L}}_{\mathrm{sm}}$ on the smooth locus $S_{\mathrm{sm}}$;

(2) 

For each singular point $p\in S_{\mathrm{sing}}$ with preimage ${\nu }^{-1}\left(p\right)=\{q_1,\dots ,q_k\}$ in $\tilde{S}$, gluing data specifying isomorphisms between the fibers ${\mathcal{L}}_{\mathrm{sm},q_i}$ for $i=1,\dots ,k$.

When the gluing data at singular points involves non-trivial automorphisms (i.e., not the identity), the spectral data is called genuinely non-abelian.

The study of non-abelian spectral data for $\mathrm{SO}(4,\mathbb{C})\cong (\mathrm{SL}(2,\mathbb{C})\times \mathrm{SL}(2,\mathbb{C}))/{\mathbb{Z}}_2$ in [8] revealed that singular spectral curves with nodal singularities can support non-trivial gluing data, leading the fibers of the Hitchin fibration to display non-abelian structure.

The complex spin group $\mathrm{Spin}(8,\mathbb{C})$ is the simply connected double cover of $\mathrm{SO}(8,\mathbb{C})$, characterized by the short exact sequence

$$1\to \{\pm 1\}\to \mathrm{Spin}(8,\mathbb{C})\to \mathrm{SO}(8,\mathbb{C})\to 1.$$

The Lie algebra $\mathfrak{so}(8,\mathbb{C})$ has type $D_4$ in Cartan classification. Its Dynkin diagram exhibits a unique three-fold symmetry, giving rise to an exceptional outer automorphism of order three [10].

Definition 5

([9,10]). The triality automorphism of $\mathrm{Spin}(8,\mathbb{C})$ is an outer automorphism τ of order 3 that cyclically permutes the following three 8-dimensional irreducible representations: the vector representation V and the two half-spin representations $S^{+}$ and $S^{-}$.

More precisely, the outer automorphism group is $\mathrm{Out}\left(\mathrm{Spin}(8,\mathbb{C})\right)\cong S_3$. The triality automorphism $\tau$ generates a cyclic subgroup of order 3 within this $S_3$. At the Lie algebra level, $\tau$ acts on the root system of $\mathfrak{so}(8,\mathbb{C})$ by permuting the simple roots according to the three-fold symmetry of the $D_4$ Dynkin diagram. A fundamental result of Cartan [9] and Wolf-Gray [18] establishes the following:

Theorem 1

([9,18]). The fixed-point subalgebra of the triality automorphism is

$$\mathfrak{so}{(8,\mathbb{C})}^{\tau }={\mathfrak{g}}_2$$

This is the exceptional Lie algebra of type $G_2$. The corresponding fixed-point subgroup of $\mathrm{Spin}(8,\mathbb{C})$ is the exceptional Lie group $G_2$.

The outer automorphism group $\mathrm{Out}\left(\mathrm{Spin}(8,\mathbb{C})\right)\cong S_3$ acts on the moduli space $\mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$ of Higgs bundles [12]. The triality automorphism induces a forgetful map

$$\iota :\mathcal{M}\left(G_2\right)\to \mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right),$$

whose image consists of fixed points of the triality action [13].

Definition 6

([1,3]). A $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle over X is a pair $(E,\mathrm{\Phi })$ where E is a holomorphic principal $\mathrm{Spin}(8,\mathbb{C})$-bundle over X and $\mathrm{\Phi }\in H^0(X,E\left(\mathfrak{so}(8,\mathbb{C})\right)\otimes K)$ is a Higgs field. The bundle E induces an $\mathrm{SO}(8,\mathbb{C})$-bundle $E_{\mathrm{SO}}=E/\{\pm 1\}$ via the covering map.

Remark 1.

The recent work in [15] provides explicit descriptions of the spectral data associated with fixed points of outer automorphisms of $\mathrm{Spin}(8,\mathbb{C})$, including its triality automorphism. In particular, ref. [15] establishes that fixed points of triality can be characterized through specific constraints on their spectral curves, complementing group-theoretic characterization via $G_2$-reductions.

For $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles, the ring of ad-invariant polynomials on $\mathfrak{so}(8,\mathbb{C})$ is generated by four homogeneous polynomials of degrees $2,4,4,$ and 6 [19]. The Hitchin base is therefore

$${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}=H^0(X,K^2)\oplus H^0(X,K^4)\oplus H^0(X,K^4)\oplus H^0(X,K^6).$$

The spectral curve associated to a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$ is defined as the divisor in $\mathrm{Tot}\left(K\right)$ given by the characteristic polynomial of $\mathrm{\Phi }$ under the adjoint representation. This curve has degree 8 over X [13]. According to Bertini’s theorem [6], there exists a Zariski open dense subset ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}^{\mathrm{reg}}\subset {\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$ such that for all $a\in {\mathcal{A}}_{\mathrm{Spin}\left(8\right)}^{\mathrm{reg}}$, the associated spectral curves are smooth and irreducible. Explicitly, ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}^{\mathrm{reg}}$ is the complement of the discriminant locus $\Delta$, which has codimension one in ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$ as a divisor. Therefore, ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}^{\mathrm{reg}}$ has full dimension $dim{\mathcal{A}}_{\mathrm{Spin}\left(8\right)}=28g-26$ and is dense in the Zariski topology.

Triality automorphism acts on the Hitchin base by permuting the two copies of $H^0(X,K^4)$ while fixing $H^0(X,K^2)$ and $H^0(X,K^6)$ [14]. If we write the Hitchin base in coordinates $(a_2,b_4,c_4,a_6)$, where $b_4,c_4\in H^0(X,K^4)$ corresponds to invariants arising from the two half-spin representations $S^{+}$ and $S^{-}$, then the triality action is

$$\tau :(a_2,b_4,c_4,a_6)\mapsto (a_2,c_4,b_4,a_6).$$

This permutation $(b_4,c_4)\mapsto (c_4,b_4)$ reflects the fact that triality cyclically permutes $V\to S^{+}\to S^{-}\to V$, thereby interchanging the roles of $S^{+}$ and $S^{-}$. This action extends to the fibers of the Hitchin fibration, as established in [13].

3. Main Results

The main results of this research extend the framework of [8] to $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles, incorporating triality automorphism into the analysis of singular spectral curves and non-abelian spectral data.

We begin by characterizing spectral curves that are invariant under triality automorphism.

Theorem 2.

Let $(E,\mathrm{\Phi })\in \mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$ be a Higgs bundle with spectral curve $S\subset \mathrm{Tot}\left(K\right)$. The spectral curve S is invariant under the induced action of the triality automorphism if and only if $(E,\mathrm{\Phi })$ admits a reduction in structure group to $G_2$.

Proof. 

Recall that the Hitchin base for $\mathrm{Spin}(8,\mathbb{C})$ is

$${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}=H^0(X,K^2)\oplus H^0(X,K^4)\oplus H^0(X,K^4)\oplus H^0(X,K^6),$$

where the decomposition corresponds to the degrees of the fundamental invariant polynomials for $\mathrm{Spin}(8,\mathbb{C})$ under the adjoint representation, as established by Chevalley’s theorem [17]. We denote a general element by $(a_2,b_4,c_4,a_6)$, where

$$\begin{array}{cc}\hfill & a_2\in H^0(X,K^2),\hfill \\ \hfill & b_4,c_4\in H^0(X,K^4),\hfill \\ \hfill & a_6\in H^0(X,K^6).\hfill \end{array}$$

The triality automorphism $\tau :\mathrm{Spin}(8,\mathbb{C})\to \mathrm{Spin}(8,\mathbb{C})$ is an outer automorphism of order three that permutes the following three eight-dimensional irreducible representations: the vector representation V and the two half-spin representations $S^{+}$ and $S^{-}$ [10]. This permutation induces an action on the Hitchin base.

According to [14], the action of $\tau$ on ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ is given by

$$\tau (a_2,b_4,c_4,a_6)=(a_2,c_4,b_4,a_6).$$

Note that $\tau$ fixes the coefficients $a_2$ and $a_6$ (corresponding to invariants of degrees 2 and 6) while permuting the two coefficients $b_4$ and $c_4$ in $H^0(X,K^4)$. This permutation reflects the fact that the two half-spin representations $S^{+}$ and $S^{-}$ are interchanged by triality, while the vector representation V contributes to invariants that are fixed. That is, $\tau$ fixes $a_2$ and $a_6$ while permuting $b_4$ and $c_4$. This permutation reflects the exchange of the half-spin representations $S^{+}\leftrightarrow S^{-}$ under triality.

For a Higgs bundle $(E,\mathrm{\Phi })$ with Hitchin invariants $h(E,\mathrm{\Phi })=(a_2,b_4,c_4,a_6)$, the spectral curve $S\subset \mathrm{Tot}\left(K\right)$ is defined by the characteristic equation of the Higgs field. In the total space of the canonical bundle, using the tautological section $\eta$ of the pullback of K, the spectral curve is given by

$$p\left(\eta \right)=det(\eta ·\mathrm{id}-\mathrm{\Phi })={\eta }^8+a_2{\eta }^6+b_4{\eta }^4+c_4{\eta }^2+a_6=0.$$

Note that the structure we use here for the precise form of this equation follows the decomposition of the characteristic polynomial into contributions from the three representations V, $S^{+}$, and $S^{-}$, where $b_4$ and $c_4$ arise from the spinor representations.

When we apply triality automorphism to the Higgs bundle to obtain $\tau (E,\mathrm{\Phi })$, the spectral curve of the transformed bundle is defined by the polynomial

$$p_{\tau }\left(\eta \right)={\eta }^8+a_2{\eta }^6+c_4{\eta }^4+b_4{\eta }^2+a_6=0,$$

where the coefficients $b_4$ and $c_4$ are interchanged.

The spectral curve S is triality-invariant as a divisor in $\mathrm{Tot}\left(K\right)$ if and only if S and $\tau \left(S\right)$ define the same divisor, which occurs if and only if the defining polynomials $p\left(\eta \right)$ and $p_{\tau }\left(\eta \right)$ are equal. Since the polynomials are equal if and only if all corresponding coefficients are equal, we obtain the following:

$$S\mathrm{is}\mathrm{triality}-\mathrm{invariant}\iff b_4=c_4.$$

Assume first that S is triality-invariant. By the analysis above, this means $b_4=c_4$. We must show that $(E,\mathrm{\Phi })$ admits a reduction in structure group to $G_2$.

The exceptional Lie group $G_2$ embeds into $\mathrm{Spin}(8,\mathbb{C})$ as the subgroup of elements fixed by the triality automorphism. This fundamental characterization of $G_2$ goes back to Cartan [9] and is discussed in detail in [10]. Explicitly, $G_2$ can be realized as

$$G_2=\{g\in \mathrm{Spin}(8,\mathbb{C}):\tau \left(g\right)=g\}.$$

The Hitchin base for $G_2$-Higgs bundles is

$${\mathcal{A}}_{G_2}=H^0(X,K^2)\oplus H^0(X,K^6)$$

This reflects the fact that $G_2$ has rank 2 with fundamental invariants of degrees 2 and 6, as established in [11]. The natural embedding $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ induced by the group inclusion $G_2\subset \mathrm{Spin}(8,\mathbb{C})$ is given by

$$j(a_2,a_6)=(a_2,\beta \left(a_6\right),\beta \left(a_6\right),a_6),$$

where $\beta :H^0(X,K^6)\to H^0(X,K^4)$ is a certain linear map determined by the representation theory of $G_2$. The key property is that the image of j lies in the locus $\{b_4=c_4\}$.

According to [13], the forgetful map $\iota :\mathcal{M}\left(G_2\right)\to \mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right)$ is injective, and its image consists precisely of those Higgs bundles whose Hitchin invariants satisfy $b_4=c_4$. Moreover, the Hitchin maps commute with the forgetful map as follows:

$$\begin{array}{ccc}\hfill \mathcal{M}\left(G_2\right)\hfill & \stackrel{\iota }{\to }& \mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right)\\ h_{G_2}\downarrow \hfill & & \hfill \downarrow h_{\mathrm{Spin}(8,\mathbb{C})}\\ \hfill {\mathcal{A}}_{G_2}\hfill & \stackrel{j}{\to }& {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}\end{array}$$

Since $h(E,\mathrm{\Phi })=(a_2,b_4,c_4,a_6)$ with $b_4=c_4$, there exists a point $(a_2^{\prime },a_6^{\prime })\in {\mathcal{A}}_{G_2}$ such that $j(a_2^{\prime },a_6^{\prime })=(a_2,b_4,c_4,a_6)$. By the commutativity of the diagram and the injectivity of $\iota$ (which follows from the injectivity of j, as proven in [13]), there exists a unique $G_2$-Higgs bundle $(E_{G_2},{\mathrm{\Phi }}_{G_2})\in \mathcal{M}\left(G_2\right)$ such that $\iota (E_{G_2},{\mathrm{\Phi }}_{G_2})=(E,\mathrm{\Phi })$.

This establishes that $(E,\mathrm{\Phi })$ admits a reduction in structure group to $G_2$.

Conversely, assume that $(E,\mathrm{\Phi })$ admits a reduction in the structure group to $G_2$. This means there exists a $G_2$-Higgs bundle $(E_{G_2},{\mathrm{\Phi }}_{G_2})$ such that $(E,\mathrm{\Phi })=\iota (E_{G_2},{\mathrm{\Phi }}_{G_2})$ under the forgetful map.

Since the Hitchin maps commute with the forgetful map, we have

$$h(E,\mathrm{\Phi })=h\left(\iota (E_{G_2},{\mathrm{\Phi }}_{G_2})\right)=j\left(h_{G_2}(E_{G_2},{\mathrm{\Phi }}_{G_2})\right).$$

If $h_{G_2}(E_{G_2},{\mathrm{\Phi }}_{G_2})=(a_2,a_6)\in {\mathcal{A}}_{G_2}$, then

$$h(E,\mathrm{\Phi })=j(a_2,a_6)=(a_2,\beta \left(a_6\right),\beta \left(a_6\right),a_6),$$

which by construction satisfies $b_4=c_4=\beta \left(a_6\right)$.

Therefore, the spectral curve S has the defining polynomial

$$p\left(\eta \right)={\eta }^8+a_2{\eta }^6+\beta \left(a_6\right){\eta }^4+\beta \left(a_6\right){\eta }^2+a_6,$$

and the triality-transformed curve has the defining polynomial

$$p_{\tau }\left(\eta \right)={\eta }^8+a_2{\eta }^6+\beta \left(a_6\right){\eta }^4+\beta \left(a_6\right){\eta }^2+a_6.$$

Since $p\left(\eta \right)=p_{\tau }\left(\eta \right)$, the spectral curves coincide, establishing that S is triality-invariant.

This shows that triality invariance of the spectral curve (characterized by $b_4=c_4$ in the Hitchin base) is equivalent to the existence of a $G_2$-reduction, completing the proof of the theorem. □

Remark 2.

Theorem 2 provides an explicit criterion for detecting $G_2$-reductions: A $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$ admits a $G_2$-reduction if and only if its spectral curve S is invariant under the triality action. Since the triality action on spectral curves is explicitly computable from the Hitchin base coordinates via the permutation $b_4\leftrightarrow c_4$, the invariance condition $b_4=c_4$ is immediately verifiable from the coefficients of the characteristic polynomial. This transforms the problem of detecting $G_2$-structures—which requires analyzing the structure group of the principal bundle—into a straightforward calculation on the Hitchin base.

Remark 3.

The result above also clarifies the relationship between the recent work in [15], which characterizes fixed points of triality through explicit spectral data constructions, and the group-theoretic perspective via $G_2$. Theorem 2 establishes that the two possible approaches—the one consisting of spectral invariance, as explained in Remark 2, and the classical $G_2$-reduction—are equivalent.

Remark 4.

From the perspective of the Hitchin integrable system, Theorem 2 identifies the $G_2$-locus as the fixed-point set of the triality action on $\mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$. According to [11], the Hitchin fibration for $G_2$ embeds into the Hitchin fibration for $\mathrm{Spin}(8,\mathbb{C})$, and Theorem 2 shows that this embedding is precisely due to the inclusion of the triality-invariant sublocus.

Theorem 2 provides a stratification of the discriminant locus, established in the following result.

Proposition 1.

The discriminant locus $\Delta \subset {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ decomposes as

$$\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}},$$

where ${\Delta }_{\mathrm{fix}}$ is the image of the discriminant locus of ${\mathcal{A}}_{G_2}$ under the natural embedding $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$, and ${\Delta }_{\mathrm{free}}$ consists of points forming free ${\mathbb{Z}}_3$-orbits under the triality action.

Proof. 

The discriminant locus $\Delta \subset {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ is a subset consisting of points in the Hitchin base for which the corresponding spectral curve is singular. According to the general theory of the Hitchin fibration [1], $\Delta$ is a divisor (a codimension-one subvariety) in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ defined by the vanishing of the discriminant of the universal spectral curve.

More precisely, recall that the Hitchin base is

$${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}=H^0(X,K^2)\oplus H^0(X,K^4)\oplus H^0(X,K^4)\oplus H^0(X,K^6),$$

with coordinates $(a_2,b_4,c_4,a_6)$. The spectral curve corresponding to $(a_2,b_4,c_4,a_6)$ is defined by the equation

$${\eta }^8+a_2{\eta }^6+(b_4+c_4){\eta }^4+a_6=0$$

in the total space of the canonical bundle K, where $\eta$ is the tautological section. The discriminant $\Delta$ is defined by the vanishing of the resultant or discriminant polynomial associated with this equation, which is a polynomial expression with the coefficients $a_2,b_4,c_4,a_6$.

Applying Bertini’s theorem to the Hitchin fibration [6], the locus where the spectral curves are singular forms a proper closed subset of a codimension of at least one in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$. In fact, for the Hitchin system, it is known that $\Delta$ has a pure codimension of one [1].

The Hitchin base for $G_2$-Higgs bundles is

$${\mathcal{A}}_{G_2}=H^0(X,K^2)\oplus H^0(X,K^6),$$

with coordinates $(a_2,a_6)$ corresponding to the degrees of the two basic invariant polynomials for $G_2$, as established in [11]. The natural embedding $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ induced by the inclusion $G_2\subset \mathrm{Spin}(8,\mathbb{C})$ is given by

$$j(a_2,a_6)=(a_2,a_6,a_6,a_6),$$

where we identify the image with the triality-invariant locus $\{b_4=c_4=a_6\}$ in coordinates. This identification follows from the fact that under the embedding $G_2\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$, the characteristic polynomial of a Higgs field for $G_2$ exhibits the symmetry imposed by triality, forcing $b_4=c_4$ in the decomposition of invariants [13].

Define ${\Delta }_{\mathrm{fix}}$ as the image of the discriminant locus in ${\mathcal{A}}_{G_2}$ under the embedding j, as follows:

$${\Delta }_{\mathrm{fix}}=j\left({\Delta }_{G_2}\right),$$

where ${\Delta }_{G_2}\subset {\mathcal{A}}_{G_2}$ is the discriminant locus for $G_2$-Higgs bundles, consisting of points $(a_2,a_6)$ for which the corresponding $G_2$-spectral curve is singular.

We claim that ${\Delta }_{\mathrm{fix}}$ consists precisely of the triality-fixed points in $\Delta$. To see this, let $(a_2,b_4,c_4,a_6)\in \Delta$ be a point with a singular spectral curve. The triality automorphism acts on the Hitchin base by $(a_2,b_4,c_4,a_6)\mapsto (a_2,c_4,b_4,a_6)$, as established in [14]. A point is fixed by this action if and only if $b_4=c_4$.

If $(a_2,b_4,c_4,a_6)\in \Delta$ satisfies $b_4=c_4$, then by Theorem 2, the corresponding spectral curve is triality-invariant, and the Higgs bundle admits a $G_2$-reduction. Therefore, the point lies in the image $j\left({\mathcal{A}}_{G_2}\right)$, and since the spectral curve is singular, it lies in $j\left({\Delta }_{G_2}\right)={\Delta }_{\mathrm{fix}}$.

Conversely, every point in ${\Delta }_{\mathrm{fix}}=j\left({\Delta }_{G_2}\right)$ has the form $(a_2,a_6,a_6,a_6)$ for some $(a_2,a_6)\in {\Delta }_{G_2}$, which is clearly triality-fixed.

Now, define ${\Delta }_{\mathrm{free}}$ as the complement, as follows:

$${\Delta }_{\mathrm{free}}=\Delta \setminus {\Delta }_{\mathrm{fix}}.$$

This consists of points $(a_2,b_4,c_4,a_6)\in \Delta$ with $b_4\ne c_4$. For such points, the triality action produces three distinct points:

$$(a_2,b_4,c_4,a_6),(a_2,c_4,b_4,a_6),(a_2,b_4,c_4,a_6)$$

These points are obtained by applying ${\tau }^0=\mathrm{id}$, $\tau$, and ${\tau }^2$, respectively. Since $b_4\ne c_4$, these three points are distinct pairwise.

We verify that all three points lie in $\Delta$. The discriminant condition (singularity of the spectral curve) is expressed as a polynomial equation with the coefficients $a_2,b_4,c_4,a_6$. Since the spectral curve equation

$${\eta }^8+a_2{\eta }^6+(b_4+c_4){\eta }^4+a_6=0$$

depends symmetrically on $b_4+c_4$, the discriminant polynomial is invariant under the permutation $b_4\leftrightarrow c_4$. Therefore, if $(a_2,b_4,c_4,a_6)\in \Delta$, then $(a_2,c_4,b_4,a_6)\in \Delta$ as well. By the same reasoning, $(a_2,b_4,c_4,a_6)\in \Delta$.

Thus, points in ${\Delta }_{\mathrm{free}}$ form free ${\mathbb{Z}}_3$-orbits under the triality action, meaning each orbit consists of exactly three distinct points.

By construction, ${\Delta }_{\mathrm{fix}}$ consists of points with $b_4=c_4$, while ${\Delta }_{\mathrm{free}}$ consists of points with $b_4\ne c_4$. Therefore, ${\Delta }_{\mathrm{fix}}\cap {\Delta }_{\mathrm{free}}=\varnothing$.

To show that $\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}}$, let $(a_2,b_4,c_4,a_6)\in \Delta$ be arbitrary. Either $b_4=c_4$ or $b_4\ne c_4$. In the first case, $(a_2,b_4,c_4,a_6)\in {\Delta }_{\mathrm{fix}}$, as seen above. In the second case, $(a_2,b_4,c_4,a_6)\in {\Delta }_{\mathrm{free}}$ by definition. Therefore, $\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}}$.

The embedding $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ identifies ${\mathcal{A}}_{G_2}$ with a linear subspace of ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ of codimension

$$\mathrm{codim}\left({\mathcal{A}}_{G_2}\right)=dim\left({\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}\right)-dim\left({\mathcal{A}}_{G_2}\right)=(28g-26)-(14g-12)=14g-14,$$

where we use

$$\begin{array}{cc}\hfill & dim{H}^0(X,K^2)=3g-2,\hfill \\ \hfill & dim{H}^0(X,K^4)=7g-7,\mathrm{and}\hfill \\ \hfill & dim{H}^0(X,K^6)=11g-10.\hfill \end{array}$$

The discriminant locus ${\Delta }_{G_2}\subset {\mathcal{A}}_{G_2}$ is a divisor in ${\mathcal{A}}_{G_2}$ according to [11], and hence, it is of codimension one in ${\mathcal{A}}_{G_2}$. Therefore, ${\Delta }_{\mathrm{fix}}=j\left({\Delta }_{G_2}\right)$ has codimension $1+(14g-14)=14g-13$ in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$.

On the other hand, $\Delta$ itself is codimension one in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$, and hence $dim(\Delta )=28g-27$. Since ${\Delta }_{\mathrm{free}}=\Delta \setminus {\Delta }_{\mathrm{fix}}$ and $\Delta$ is an irreducible divisor (of dimension $28g-27$) while ${\Delta }_{\mathrm{fix}}$ has codimension $14g-13$ in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ (hence dimension $(28g-26)-(14g-13)=14g-13$ if non-empty), we have $dim\left({\Delta }_{\mathrm{fix}}\right)<dim(\Delta )$. Therefore, ${\Delta }_{\mathrm{fix}}$ has a codimension of at least one in $\Delta$, which implies that ${\Delta }_{\mathrm{free}}$ is Zariski open and dense in $\Delta$.

This completes the proof of the decomposition. □

Remark 5.

The decomposition $\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}}$ in Proposition 1 reveals that singular spectral curves split into two disjoint geometric strata with different behavior under triality. The fixed stratum ${\Delta }_{\mathrm{fix}}$ consists of points whose spectral curves are both singular and triality-invariant. Following Theorem 2, triality invariance forces a $G_2$-reduction. Therefore, ${\Delta }_{\mathrm{fix}}$ is precisely the image of ${\Delta }_{G_2}$ under the embedding $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ given by $j(a_2,a_6)=(a_2,a_6,a_6,a_6)$. This shows that ${\Delta }_{\mathrm{fix}}$ parametrizes singular spectral curves that arise from the exceptional group $G_2$ rather than from the generic $\mathrm{Spin}(8,\mathbb{C})$ structure.

The geometry of ${\Delta }_{\mathrm{fix}}$ is governed by singularities occurring in $G_2$-Higgs bundles. According to [11], the discriminant ${\Delta }_{G_2}$ has codimension one in ${\mathcal{A}}_{G_2}=H^0(X,K^2)\oplus H^0(X,K^6)$ and consists of points where the spectral curve acquires singularities of specific types determined by $G_2$ representation theory. The embedding into ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ shows that ${\Delta }_{\mathrm{fix}}$ inherits this codimension-one structure within the triality-invariant sublocus $\{b_4=c_4\}$, but has a higher codimension when measured in the full Hitchin base.

In contrast, the free stratum ${\Delta }_{\mathrm{free}}$ consists of points whose spectral curves are singular but not triality-invariant. Such points form orbits of exactly three elements under the ${\mathbb{Z}}_3$-action. If $(a_2,b_4,c_4,a_6)\in {\Delta }_{\mathrm{free}}$ corresponds to a singular spectral curve S, then $\tau (a_2,b_4,c_4,a_6)=(a_2,c_4,b_4,a_6)$ and ${\tau }^2(a_2,b_4,c_4,a_6)$ correspond to distinct singular curves $\tau \left(S\right)$ and ${\tau }^2\left(S\right)$. The condition $b_4\ne c_4$ ensures these three curves are genuinely different.

Remark 6.

From the perspective of deformation theory, the decomposition $\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}}$ reflects two different mechanisms for creating singularities in spectral curves. Points in ${\Delta }_{\mathrm{fix}}$ arise when the Higgs field Φ, already constrained to a lie in the Lie algebra ${\mathfrak{g}}_2\subset \mathfrak{spin}(8,\mathbb{C})$, takes values that produce singular spectral data within the $G_2$ theory. Points in ${\Delta }_{\mathrm{free}}$ arise when Φ is not constrained by any smaller structure group, and the singularity occurs in the generic $\mathrm{Spin}(8,\mathbb{C})$ context. The triality action permutes such generic singular configurations, but does not fix them.

Remark 7.

The disjointness of the two strata has important implications for understanding the global structure of Δ. While Δ itself is an irreducible hypersurface in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ (as the discriminant of the universal spectral curve), the decomposition shows that it naturally splits into two pieces with different symmetry properties. The closure $\overline{{\Delta }_{\mathrm{free}}}$ may intersect ${\Delta }_{\mathrm{fix}}$, suggesting that there exist limiting processes where free triality orbits of singular curves degenerate to triality-invariant configurations. Understanding these degenerations would require a detailed study of the boundary of ${\Delta }_{\mathrm{free}}$ in Δ, which lies beyond the scope of the present work but represents an interesting direction for future investigation.

Remark 8.

Note that the stratification provided by Proposition 1 is compatible with the stratification by singularity type. Both ${\Delta }_{\mathrm{fix}}$ and ${\Delta }_{\mathrm{free}}$ can be further subdivided according to the type of singularity that occurs in the spectral curve.

We now analyze the structure of triality orbits in the singular locus of the Hitchin fibration.

Theorem 3.

Let $\Delta \subset {\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$ denote the discriminant locus consisting of points whose spectral curves are singular. The triality automorphism τ acts on Δ and induces a decomposition

$$\Delta ={\Delta }_{\mathrm{fix}}\cup {\Delta }_{\mathrm{free}},$$

where

(1) 

${\Delta }_{\mathrm{fix}}=\{a\in \Delta :\tau \left(a\right)=a\}$ consists of fixed points of τ;

(2) 

${\Delta }_{\mathrm{free}}=\{a\in \Delta :|{\mathrm{orb}}_{\tau }\left(a\right)|=3\}$ consists of free ${\mathbb{Z}}_3$-orbits.

Moreover, a point $a\in {\Delta }_{\mathrm{fix}}$ if and only if the fiber $h^{-1}\left(a\right)$ contains a $G_2$-Higgs bundle.

Proof. 

The discriminant locus $\Delta$ is the closed subset of ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$, consisting of points where the associated spectral curve is singular. According to Bertini’s theorem [6], $\Delta$ has a codimension of at least one in ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$. Since the triality action on ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$ is algebraic and preserves the spectral curve construction, it preserves $\Delta$.

For $a=(a_2,b_4,c_4,a_6)\in \Delta$, we have $\tau \left(a\right)=a$ if and only if $\tau (b_4,c_4)=(b_4,c_4)$. Since $\tau$ acts on $H^0(X,K^4)\oplus H^0(X,K^4)$ by cyclic permutation and has order 3, the fixed points are those with $b_4=c_4$. According to Theorem 2 and [13], this occurs if and only if a lies in the image of ${\mathcal{A}}_{G_2}$ under the natural map, which is equivalent to the existence of a $G_2$-Higgs bundle in $h^{-1}\left(a\right)$.

For $a\in \Delta \setminus {\Delta }_{\mathrm{fix}}$, we have $b_4\ne c_4$; hence, $\tau \left(a\right)\ne a$. Since ${\tau }^3=\mathrm{id}$ and $\tau \left(a\right)\ne a$, the orbit $\{{\tau }^j\left(a\right):j=0,1,2\}$ has cardinality 3. These are precisely the free orbits under the ${\mathbb{Z}}_3$-action generated by $\tau$.

The decomposition $\Delta ={\Delta }_{\mathrm{fix}}\cup {\Delta }_{\mathrm{free}}$ is immediate from the fact that every element of $\Delta$ either has a trivial stabilizer (yielding a free orbit) or is fixed by $\tau$. □

We establish the existence of genuinely non-abelian spectral data compatible with triality automorphism.

Theorem 4.

There exist $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles $(E,\mathrm{\Phi })$ with singular spectral curves S having the following properties:

(1) 

S has nodal singularities and is not triality-invariant;

(2) 

The triality automorphism τ permutes the spectral curves $\{S,\tau \left(S\right),{\tau }^2\left(S\right)\}$ as a free ${\mathbb{Z}}_3$-orbit;

(3) 

Each of S, $\tau \left(S\right)$, and ${\tau }^2\left(S\right)$ supports genuinely non-abelian spectral data in the sense of Definition 4.

Proof. 

We construct explicit examples by considering the relationship between $\mathrm{Spin}(4,\mathbb{C})$ and $\mathrm{Spin}(8,\mathbb{C})$ through the isomorphism $\mathrm{Spin}(4,\mathbb{C})\cong \mathrm{SL}(2,\mathbb{C})\times \mathrm{SL}(2,\mathbb{C})$ and standard embedding into $\mathrm{Spin}(8,\mathbb{C})$.

The group $\mathrm{Spin}(4,\mathbb{C})$ embeds naturally into $\mathrm{Spin}(8,\mathbb{C})$ via the inclusion of the corresponding orthogonal groups $\mathrm{SO}(4,\mathbb{C})\hookrightarrow \mathrm{SO}(8,\mathbb{C})$, followed by lifting to the spin groups. Concretely, consider the standard representation $\rho :\mathrm{Spin}(8,\mathbb{C})\to \mathrm{SO}(8,\mathbb{C})$ acting on ${\mathbb{C}}^8$. Fix a decomposition ${\mathbb{C}}^8=V_4\oplus V_4^{\prime }$ where $V_4,V_4^{\prime }$ are four-dimensional subspaces. The subgroup of $\mathrm{SO}(8,\mathbb{C})$ preserving this decomposition orthogonally and acting trivially on $V_4^{\prime }$ is isomorphic to $\mathrm{SO}(4,\mathbb{C})$. Its inverse image under $\rho$ provides an embedding $\iota :\mathrm{Spin}(4,\mathbb{C})\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$.

Under this embedding, the Lie algebra $\mathfrak{spin}(4,\mathbb{C})\cong \mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})$ embeds into $\mathfrak{spin}(8,\mathbb{C})$. Note that this embedding is not preserved by the triality automorphism $\tau$ of $\mathrm{Spin}(8,\mathbb{C})$. Triality permutes the following three eight-dimensional irreducible representations [10]: the vector representation V and the two half-spin representations $S^{+}$ and $S^{-}$. Under this permutation, the decomposition ${\mathbb{C}}^8=V_4\oplus V_4^{\prime }$ is not preserved. Therefore, $\tau \left(\iota \right(\mathrm{Spin}(4,\mathbb{C})\left)\right)$ is conjugate to, but not equal to, $\iota \left(\mathrm{Spin}\right(4,\mathbb{C}\left)\right)$ as a subgroup of $\mathrm{Spin}(8,\mathbb{C})$.

According to [8], there exist $\mathrm{SO}(4,\mathbb{C})$-Higgs bundles $(E_4,{\mathrm{\Phi }}_4)$ over X with nodal spectral curves that support genuinely non-abelian spectral data. Let us fix such a bundle. Since the covering map $\mathrm{Spin}(4,\mathbb{C})\to \mathrm{SO}(4,\mathbb{C})$ is a double cover, any $\mathrm{SO}(4,\mathbb{C})$-bundle admits a lift to a $\mathrm{Spin}(4,\mathbb{C})$-bundle, though the lift may not be unique (the choice of lift is classified by $H^1(X,{\mathbb{Z}}_2)$). We select a lift ${\tilde{E}}_4\to X$ with structure group $\mathrm{Spin}(4,\mathbb{C})$.

The Higgs field ${\mathrm{\Phi }}_4\in H^0(X,E_4\left(\mathfrak{so}(4,\mathbb{C})\right)\otimes K)$ lifts canonically to

$${\tilde{\mathrm{\Phi }}}_4\in H^0(X,{\tilde{E}}_4\left(\mathfrak{spin}(4,\mathbb{C})\right)\otimes K)$$

because the Lie algebra covering map $\mathfrak{spin}(4,\mathbb{C})\to \mathfrak{so}(4,\mathbb{C})$ is an isomorphism. This gives a $\mathrm{Spin}(4,\mathbb{C})$-Higgs bundle $({\tilde{E}}_4,{\tilde{\mathrm{\Phi }}}_4)$.

Using the embedding $\iota :\mathrm{Spin}(4,\mathbb{C})\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$, we extend the structure group via the associated bundle construction. Define $E={\tilde{E}}_4{\times }_{\iota }\mathrm{Spin}(8,\mathbb{C})$, which is a principal $\mathrm{Spin}(8,\mathbb{C})$-bundle over X. The Higgs field extends as follows: the Lie algebra embedding $d\iota :\mathfrak{spin}(4,\mathbb{C})\hookrightarrow \mathfrak{spin}(8,\mathbb{C})$ allows us to view ${\tilde{\mathrm{\Phi }}}_4$ as taking values in $\mathfrak{spin}(8,\mathbb{C})$. Explicitly, we define $\mathrm{\Phi }\in H^0(X,E\left(\mathfrak{spin}(8,\mathbb{C})\right)\otimes K)$ by the composition

$${\tilde{E}}_4\left(\mathfrak{spin}(4,\mathbb{C})\right)\stackrel{d{\iota }_{\ast }}{\to }E\left(\mathfrak{spin}(8,\mathbb{C})\right),$$

where $d{\iota }_{\ast }$ denotes the induced map on associated bundles. Setting $\mathrm{\Phi }=d{\iota }_{\ast }\circ {\tilde{\mathrm{\Phi }}}_4$ yields a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$.

The spectral curve S of $(E,\mathrm{\Phi })$ is defined by the vanishing of the characteristic polynomial $det(\eta ·{\mathrm{id}}_8-\mathrm{\Phi })$ in the total space of the canonical bundle K. Since $\mathrm{\Phi }$ takes values in the image $d\iota \left(\mathfrak{spin}\right(4,\mathbb{C}\left)\right)\subset \mathfrak{spin}(8,\mathbb{C})$, and elements of this subalgebra preserve the decomposition ${\mathbb{C}}^8=V_4\oplus V_4^{\prime }$ when viewed via the vector representation, the characteristic polynomial factors accordingly.

More precisely, under the vector representation $\mathrm{Spin}(8,\mathbb{C})\to \mathrm{GL}(8,\mathbb{C})$, the element $\mathrm{\Phi }$ acts as a block-diagonal matrix with two $4\times 4$ blocks corresponding to the action on $V_4$ and $V_4^{\prime }$, respectively. These two blocks are related to the characteristic polynomial of ${\mathrm{\Phi }}_4$ acting on ${\mathbb{C}}^4$. If the spectral curve $S_4$ of $(E_4,{\mathrm{\Phi }}_4)$ is defined by $p_4\left(\eta \right)=det(\eta ·{\mathrm{id}}_4-{\mathrm{\Phi }}_4)=0$, then the spectral curve S of $(E,\mathrm{\Phi })$ satisfies

$$p_8\left(\eta \right)=det(\eta ·{\mathrm{id}}_8-\mathrm{\Phi })=p_4{\left(\eta \right)}^2=0.$$

Thus, S is the spectral curve $S_4$ taken with a multiplicity of two as a divisor in the total space of K.

More precisely, under the vector representation $\mathrm{Spin}(8,\mathbb{C})\to \mathrm{GL}(8,\mathbb{C})$, the element $\mathrm{\Phi }$ acts as a block-diagonal matrix with two $4\times 4$ blocks, each corresponding to the action on $V_4$ and $V_4^{\prime }$, respectively. Considering this construction, these two blocks are related to the characteristic polynomial of ${\mathrm{\Phi }}_4$ acting on ${\mathbb{C}}^4$. If the spectral curve $S_4$ of $(E_4,{\mathrm{\Phi }}_4)$ is defined by $p_4\left(\eta \right)=det(\eta ·{\mathrm{id}}_4-{\mathrm{\Phi }}_4)=0$, then the spectral curve S of $(E,\mathrm{\Phi })$ satisfies

$$p_8\left(\eta \right)=det(\eta ·{\mathrm{id}}_8-\mathrm{\Phi })=p_4{\left(\eta \right)}^2=0.$$

Thus, S is the spectral curve $S_4$ taken with multiplicity two as a divisor in the total space of K.

Considering the construction in [8], the curve $S_4$ has nodal singularities. The multiplicity-two curve $S=2S_4$ inherits these nodal singularities. At each node $p\in S_4$, the curve S has a singularity locally analytically isomorphic to the node on $S_4$, but with the structure sheaf ${\mathcal{O}}_S$ having a non-reduced structure at p.

Let us establish property (1) of the statement. The curve S has nodal singularities, as seen in the analysis above. To verify that S is not triality-invariant, we use a characterization from the Hitchin base. The Higgs bundle $(E,\mathrm{\Phi })$ maps under the Hitchin map to a point $(a_2,b_4,c_4,a_6)\in {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}=H^0(X,K^2)\oplus H^0(X,K^4)\oplus H^0(X,K^4)\oplus H^0(X,K^6)$.

By construction via $\mathrm{Spin}(4,\mathbb{C})$, the Higgs field $\mathrm{\Phi }$ has only the symmetry of the embedded $\mathrm{Spin}(4,\mathbb{C})$, not the full symmetry of $\mathrm{Spin}(8,\mathbb{C})$. This breaks the symmetry between the two $H^0(X,K^4)$ factors in the Hitchin base. Specifically, the two quartic invariants $b_4$ and $c_4$ arising from the half-spin representations $S^{+}$ and $S^{-}$ are distinguished by the $\mathrm{Spin}(4,\mathbb{C})$-structure, yielding $b_4\ne c_4$.

According to Theorem 2, the triality invariance of S is equivalent to $b_4=c_4$. Since $b_4\ne c_4$, the curve S is not triality-invariant.

For property (2), note that the triality automorphism $\tau$ acts on the Hitchin base by permuting $(a_2,b_4,c_4,a_6)\mapsto (a_2,c_4,b_4,a_6)$, as established in [14]. Since $b_4\ne c_4$, the three points

$$(a_2,b_4,c_4,a_6),(a_2,c_4,b_4,a_6),(a_2,b_4,c_4,a_6)$$

obtained by the actions of ${\tau }^0=\mathrm{id}$, ${\tau }^1$, and ${\tau }^2$ are distinct. These correspond to the three distinct spectral curves S, $\tau \left(S\right)$, and ${\tau }^2\left(S\right)$ in the total space of K.

Since ${\tau }^3=\mathrm{id}$ and the three curves are distinct, they form a free ${\mathbb{Z}}_3$-orbit under the triality action.

Finally, we establish property (3). For the original curve $S_4$, the non-abelian spectral data consists of a rank-one torsion-free sheaf ${\mathcal{L}}_4$ on the nodal curve $S_4$, together with gluing data at each node, as described in [8]. This data is genuinely non-abelian in the sense that it cannot be obtained from a line bundle on any partial normalization of $S_4$; instead, it involves non-trivial Ext-group data at the nodes.

For the $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$, the spectral data on the doubled curve $S=2S_4$ is constructed via pushforward. The spectral sheaf $\mathcal{L}$ on S is a rank-two torsion-free sheaf obtained as the direct image of ${\mathcal{L}}_4$ under the natural doubling morphism $S_4\to S$. At each node of S (corresponding to a node of $S_4$), the sheaf $\mathcal{L}$ inherits gluing data from ${\mathcal{L}}_4$. Since the original gluing data on $S_4$ was non-trivial and non-abelian, the gluing data on S is also non-trivial and non-abelian. This establishes that S supports non-abelian spectral data.

For the transformed curves $\tau \left(S\right)$ and ${\tau }^2\left(S\right)$, the non-abelian property is preserved by the triality action. The triality automorphism $\tau$ induces an isomorphism of spectral curves $S\to \tau \left(S\right)$, and this isomorphism preserves the structure of torsion-free sheaves and gluing data. Since non-abelian gluing data cannot become abelian under isomorphism, the curves $\tau \left(S\right)$ and ${\tau }^2\left(S\right)$ also support non-abelian spectral data. □

Remark 9.

Theorem 4 demonstrates that non-abelian spectral data is compatible with the triality symmetry and can occur in non-trivial ${\mathbb{Z}}_3$-orbits. From this observation, this theorem shows that the non-abelian phenomena discovered by Bradlow, Branco, and Schaposnik [8] for $\mathrm{SO}(4,\mathbb{C})$-Higgs bundles persist in the higher-rank setting of $\mathrm{Spin}(8,\mathbb{C})$. The key difference is that while $\mathrm{SO}(4,\mathbb{C})$ has no outer automorphisms of order three, $\mathrm{Spin}(8,\mathbb{C})$ admits triality automorphism, which acts on the moduli space and partitions it into orbits. The existence of non-abelian data in free ${\mathbb{Z}}_3$-orbits demonstrates that non-abelian spectral structures are not destroyed by the triality symmetry, but rather coexist with it in a non-trivial way.

Remark 10.

The construction made in the proof of Theorem 4 reveals a precise algorithm for producing examples of non-abelian spectral structures: the embedding $\mathrm{Spin}(4,\mathbb{C})\cong \mathrm{SL}(2,\mathbb{C})\times \mathrm{SL}(2,\mathbb{C})\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$ allows us to lift the non-abelian data from rank-4 orthogonal bundles to rank-8 spin bundles. The triality automorphism does not fix $\mathrm{Spin}(4,\mathbb{C})$ as a subgroup, and hence Higgs bundles arising from this construction generally lie in free triality orbits. More precisely, since the condition $b_4=c_4$ defines a codimension-$(7g-7)$ subspace of the Hitchin base (as $dim{H}^0(X,K^4)=7g-7$), and the construction via $\mathrm{Spin}(4,\mathbb{C})$ produces Higgs bundles with $b_4\ne c_4$ for generic choices of the $\mathrm{SO}(4,\mathbb{C})$-data, these bundles lie in the complement of the triality-invariant locus, which is open and dense. This provides an explicit method for constructing $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles with a prescribed orbit structure and spectral behavior.

Remark 11.

According to Theorem 3, the discriminant locus decomposes as $\Delta ={\Delta }_{\mathrm{fix}}\bigsqcup {\Delta }_{\mathrm{free}}$. The existence of non-abelian data in ${\Delta }_{\mathrm{free}}$ shows that for points in ${\Delta }_{\mathrm{free}}$ with nodal spectral curves, the corresponding fiber of the Hitchin map contains not only the expected Prym variety but also additional components parameterizing non-abelian spectral data. Understanding the detailed structure of these components is an important direction for future research.

Remark 12.

From the perspective of moduli theory, Theorem 3 establishes that the locus of Higgs bundles with non-abelian spectral data intersects all three types of triality orbits: fixed points (when the data comes from $G_2$-bundles), orbits of size one (when triality fixes the bundle but not pointwise), and free orbits of size three (as guaranteed by the theorem). This suggests that stratification by spectral data type (abelian vs. non-abelian) is transverse to stratification by triality orbit type, leading to a finer decomposition of the moduli space than either stratification alone would provide.

Remark 13.

Theorem 3 leaves open the question of whether all points in free triality orbits with singular spectral curves support non-abelian data or whether there exist special free orbits where the data remains abelian despite the singularity. This question is related to understanding the precise conditions under which the construction via $\mathrm{Spin}(4,\mathbb{C})$-embeddings can be applied, as well as characterizing which singular spectral curves in ${\Delta }_{\mathrm{free}}$ arise from such embeddings. This would require analyzing the geometry of the triality action on the space of singular spectral curves, building on the techniques of [15] for describing spectral data of triality-fixed points.

Remark 14 (Uniqueness). 

The embedding $\mathrm{Spin}(4,\mathbb{C})\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$ is not unique. Up to conjugacy, there are multiple ways to embed $\mathrm{Spin}(4,\mathbb{C})$ into $\mathrm{Spin}(8,\mathbb{C})$, corresponding to different choices of four-dimensional subspace in ${\mathbb{C}}^8$. All such embeddings have the common property that they are not preserved by triality. This is because triality permutes the three eight-dimensional representations cyclically, and any four-dimensional invariant subspace in one representation does not remain invariant under this permutation. Therefore, for any choice of embedding, the construction in Theorem 4 produces Higgs bundles in free triality orbits.

Remark 15 (Generality). 

Not all non-abelian data in free orbits necessarily arise from $\mathrm{Spin}(4,\mathbb{C})$-embeddings. The construction via $\mathrm{Spin}(4,\mathbb{C})$ provides the existence of such examples, but does not claim to exhaust all possibilities. Other ways to produce non-abelian data in free triality orbits may exist. The question of which singular spectral curves in ${\Delta }_{\mathrm{free}}$ support non-abelian data, and which of these can be realized via embeddings of smaller groups, remains an open problem.

We analyze the effect of triality on Prym varieties arising as fibers of the Hitchin fibration.

Theorem 5.

Let $(E,\mathrm{\Phi })\in \mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$ be a Higgs bundle with a smooth irreducible spectral curve S of genus $g_S$, and let $\sigma :S\to S$ be the involution induced by the covering $\mathrm{Spin}(8,\mathbb{C})\to \mathrm{SO}(8,\mathbb{C})$. Let $P=\mathrm{Prym}(S,\sigma )$ be the corresponding Prym variety:

(1) 

If S is triality-invariant, then τ acts trivially on P;

(2) 

If S is not triality-invariant, then τ acts on P as an automorphism of order dividing 3, and $dim\left(P^{\tau }\right)<dim\left(P\right)$.

Proof. 

According to the Beauville–Narasimhan–Ramanan correspondence [5], the fiber of the Hitchin map $h:\mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right)\to {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ over the point $h(E,\mathrm{\Phi })\in {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ is biholomorphic to the following Prym variety:

$$P=\mathrm{Prym}(S,\sigma )=\{\mathcal{L}\in \mathrm{Jac}\left(S\right):{\sigma }^{\ast }\mathcal{L}\cong {\mathcal{L}}^{-1}\},$$

where $\sigma :S\to S$ is the involution on the spectral curve induced by the central involution in $\mathrm{Spin}(8,\mathbb{C})$. This involution arises from the fact that the center of $\mathrm{Spin}(8,\mathbb{C})$ is ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ and one of its ${\mathbb{Z}}_2$-factors acts as $-\mathrm{id}$ on the adjoint representation, inducing the hyperelliptic-type involution $\sigma$ on the spectral curve.

The Prym variety P is an abelian subvariety of $\mathrm{Jac}\left(S\right)$, defined as the connected component of the identity in the kernel of the norm map ${\mathrm{Nm}}_{\sigma }:\mathrm{Jac}\left(S\right)\to \mathrm{Jac}(S/\sigma )$ given by ${\mathrm{Nm}}_{\sigma }\left(\mathcal{L}\right)=\mathcal{L}\otimes {\sigma }^{\ast }\mathcal{L}$. For a smooth irreducible curve S of genus $g_S=64g-63$ (where g is the genus of X), the Prym variety has dimension

$$dim\left(P\right)=g_S-1-g=(64g-63)-1-g=63g-63.$$

The triality automorphism $\tau :\mathrm{Spin}(8,\mathbb{C})\to \mathrm{Spin}(8,\mathbb{C})$ induces an automorphism of the moduli space $\mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$, according to [12]. A fundamental property established in [13] is that $\tau$ correlates with the Hitchin map:

$$h\circ \tau ={\tau }_{\mathcal{A}}\circ h,$$

where ${\tau }_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}\to {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$ is the induced action on the Hitchin base given by $(a_2,b_4,c_4,a_6)\mapsto (a_2,c_4,b_4,a_6)$, as described in [14].

This commutativity implies that $\tau$ preserves fibers of the Hitchin map. Therefore, for any $(E,\mathrm{\Phi })\in \mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)$, the triality transform $\tau (E,\mathrm{\Phi })$ lies in the same fiber as $(E,\mathrm{\Phi })$ if and only if $h(E,\mathrm{\Phi })$ is fixed by ${\tau }_{\mathcal{A}}$, which occurs precisely when the spectral curve is triality-invariant.

For proving part (1), assume that S is triality-invariant. According to Theorem 2, this is equivalent to $(E,\mathrm{\Phi })$ admitting a reduction in the structure group to $G_2\subset \mathrm{Spin}(8,\mathbb{C})$. Therefore, there exists a $G_2$-Higgs bundle $(E_{G_2},{\mathrm{\Phi }}_{G_2})\in \mathcal{M}\left(G_2\right)$ such that $(E,\mathrm{\Phi })=\iota (E_{G_2},{\mathrm{\Phi }}_{G_2})$ under the forgetful map $\iota :\mathcal{M}\left(G_2\right)\to \mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right)$.

The triality automorphism $\tau$ acts trivially on $G_2$ because $G_2$ is precisely the subgroup of $\mathrm{Spin}(8,\mathbb{C})$ fixed by $\tau$. This follows from the fundamental characterization of $G_2$ as the automorphism group of the octonions, which is preserved by triality [10]. Consequently, $\tau$ acts trivially on the moduli space $\mathcal{M}\left(G_2\right)$, as shown in [13].

The forgetful map $\iota$ is $\tau$-equivariant in the sense that the diagram

$$\begin{array}{ccc}\hfill \mathcal{M}\left(G_2\right)& \stackrel{\iota }{\to }& \mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)\\ \mathrm{id}\downarrow \hfill & & \downarrow \tau \\ \hfill \mathcal{M}\left(G_2\right)& \stackrel{\iota }{\to }& \mathcal{M}\left(\mathrm{Spin}\right(8,\mathbb{C}\left)\right)\end{array}$$

communicates, where the left vertical arrow is the identity because $\tau$ acts trivially on $\mathcal{M}\left(G_2\right)$.

The Hitchin fibrations for $G_2$ and $\mathrm{Spin}(8,\mathbb{C})$ are related by a natural embedding of Hitchin bases $j:{\mathcal{A}}_{G_2}\hookrightarrow {\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$, given in coordinates by $j(a_2,a_6)=(a_2,a_6,a_6,a_6)$, where we identify ${\mathcal{A}}_{G_2}=H^0(X,K^2)\oplus H^0(X,K^6)$, as established in [11]. The image of j is precisely the triality-invariant locus $\{b_4=c_4\}$ in ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}$.

For the Prym varieties, the relationship is as follows. The spectral curve S for $(E,\mathrm{\Phi })$ is the same as the spectral curve for $(E_{G_2},{\mathrm{\Phi }}_{G_2})$ when viewed as a divisor in the total space of K. The involution $\sigma$ on S induced from $\mathrm{Spin}(8,\mathbb{C})$ is restricted to the involution induced from the $G_2$-structure. Therefore, the Prym variety $P=\mathrm{Prym}(S,\sigma )$ can be viewed either as arising from the $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle or from the underlying $G_2$-Higgs bundle.

Since $\tau$ acts trivially on $(E_{G_2},{\mathrm{\Phi }}_{G_2})$ and the construction of the Prym variety from a Higgs bundle is functorial (it depends only on the spectral curve and involution), the induced action of $\tau$ on P is trivial. More precisely, for any $\mathcal{L}\in P$, we have

$${\tau }_{\ast }\left(\mathcal{L}\right)=\mathcal{L}$$

as elements of $\mathrm{Jac}\left(S\right)$, where ${\tau }_{\ast }:P\to P$ denotes the automorphism of the Prym variety induced by $\tau$ via pullback on the spectral data. This establishes part (1).

Now, for part (2) of the statement, assume that S is not triality-invariant. According to Theorem 2, this means $(E,\mathrm{\Phi })$ does not admit a $G_2$-reduction, or equivalently, the Hitchin base element $h(E,\mathrm{\Phi })=(a_2,b_4,c_4,a_6)$ satisfies $b_4\ne c_4$.

The action of $\tau$ on the Hitchin base permutes $b_4\leftrightarrow c_4$ while fixing $a_2$ and $a_6$. Therefore,

$${\tau }_{\mathcal{A}}\left(h(E,\mathrm{\Phi })\right)=(a_2,c_4,b_4,a_6)\ne (a_2,b_4,c_4,a_6)=h(E,\mathrm{\Phi }).$$

This shows that $h\left(\tau \right(E,\mathrm{\Phi }\left)\right)\ne h(E,\mathrm{\Phi })$, so $\tau (E,\mathrm{\Phi })$ and $(E,\mathrm{\Phi })$ lie in different fibers of the Hitchin map.

However, the triality automorphism still acts on each individual fiber of the Hitchin map in the following way. Although $\tau$ moves $(E,\mathrm{\Phi })$ to a different fiber, the spectral curve S is mapped to a different spectral curve $\tau \left(S\right)$ corresponding to the Hitchin base point $(a_2,c_4,b_4,a_6)$. These two curves, S and $\tau \left(S\right)$, are isomorphic as abstract curves (they have the same genus and singularity type), but they are distinct as divisors in the total space of K because they correspond to different elements in the Hitchin base.

The automorphism $\tau$ induces a birational map between fibers:

$${\tau }_{\ast }:P_S=\mathrm{Prym}(S,{\sigma }_S)⤏P_{\tau \left(S\right)}=\mathrm{Prym}(\tau \left(S\right),{\sigma }_{\tau \left(S\right)}).$$

For smooth curves, this birational map is actually an isomorphism of abelian varieties. The key point is that although $\tau$ maps the Prym variety $P_S$ to a different Prym variety $P_{\tau \left(S\right)}$, it also acts on the entire moduli space, and we can track this action.

To make this precise, we use the fact that the triality action preserves the smooth locus $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$, which is an open subset consisting of Higgs bundles with smooth irreducible spectral curves. On this locus, the Hitchin fibration is a smooth fibration, with its fibers being abelian varieties [1].

For a Higgs bundle $(E,\mathrm{\Phi })$ with a non-triality-invariant spectral curve S, the orbit under $\tau$ consists of three elements:

$$(E,\mathrm{\Phi }),\tau (E,\mathrm{\Phi }),{\tau }^2(E,\mathrm{\Phi }),$$

with corresponding spectral curves

$$S,\tau \left(S\right),{\tau }^2\left(S\right).$$

Each of these has an associated Prym variety $P_S$, $P_{\tau \left(S\right)}$, $P_{{\tau }^2\left(S\right)}$. The triality automorphism induces the following isomorphisms:

$${\tau }_{\ast }:P_S\stackrel{\sim }{\to }{P}_{\tau \left(S\right)},{\tau }_{\ast }:P_{\tau \left(S\right)}\stackrel{\sim }{\to }{P}_{{\tau }^2\left(S\right)},{\tau }_{\ast }:P_{{\tau }^2\left(S\right)}\stackrel{\sim }{\to }{P}_S.$$

The composition ${\tau }_{\ast }^3:P_S\to P_S$ is the identity because ${\tau }^3=\mathrm{id}$ on the moduli space. Therefore, ${\tau }_{\ast }$ acts on $P_S$ (after identifying $P_{\tau \left(S\right)}$ and $P_{{\tau }^2\left(S\right)}$ with $P_S$ via the triality isomorphisms) as an automorphism of order dividing 3.

The fixed-point locus $P^{\tau }=\{\mathcal{L}\in P_S:{\tau }_{\ast }\left(\mathcal{L}\right)\cong \mathcal{L}\}$ is a closed abelian subvariety of $P_S$ according to the theory of automorphisms of abelian varieties, as established by Mumford [20].

To show that $dim\left(P^{\tau }\right)<dim\left(P\right)$, we argue by contradiction. Suppose $P^{\tau }=P$, meaning $\tau$ acts trivially on $P_S$. Then, due to the functoriality of the spectral correspondence, this would imply that the spectral data of $(E,\mathrm{\Phi })$ is invariant under $\tau$ up to isomorphism. However, the spectral data includes not just the Prym variety but also the specific line bundle corresponding to $(E,\mathrm{\Phi })$ within that Prym variety.

More precisely, the Beauville–Narasimhan–Ramanan correspondence establishes a bijection between Higgs bundles in the fiber over $h(E,\mathrm{\Phi })$ and points in $P_S$. If $\tau$ acts trivially on $P_S$, then the point corresponding to $(E,\mathrm{\Phi })$ must map to the point corresponding to $\tau (E,\mathrm{\Phi })$. However, since $h\left(\tau \right(E,\mathrm{\Phi }\left)\right)\ne h(E,\mathrm{\Phi })$ (as $b_4\ne c_4$), the bundles $\tau (E,\mathrm{\Phi })$ and $(E,\mathrm{\Phi })$ lie in different fibers and cannot correspond to the same point in any Prym variety under the Beauville–Narasimhan–Ramanan bijection.

This contradiction shows that $\tau$ cannot act trivially on $P_S$, and hence $P^{\tau }\ne P_S$, which implies $dim\left(P^{\tau }\right)<dim\left(P_S\right)$, as claimed. □

Remark 16.

For a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$ with a smooth spectral curve S and associated Prym variety P, using Theorem 5, we may define the triality defect of $(E,\mathrm{\Phi })$ as

$${\delta }_{\tau }(E,\mathrm{\Phi })=dim\left(P\right)-dim\left(P^{\tau }\right).$$

This quantity satisfies the following:

(1) 

${\delta }_{\tau }(E,\mathrm{\Phi })=0$ if and only if $(E,\mathrm{\Phi })$ admits a reduction to $G_2$ (by Theorem 5, part (1));

(2) 

${\delta }_{\tau }(E,\mathrm{\Phi })>0$ if $(E,\mathrm{\Phi })$ does not admit such a reduction (by Theorem 5, part (2));

(3) 

The value ${\delta }_{\tau }(E,\mathrm{\Phi })$ measures how far the Higgs bundle is from being triality-invariant in terms of the dimension of the fixed locus in its associated integrable system fiber.

Since $dim\left(P\right)=g_S-g_X$, where $g_S$ is the genus of the spectral curve and $g_X$ is the genus of X, and the genus formula for the spectral curve of a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle is

$$g_S=1+8^2(g_X-1)=64g_X-63,$$

we have $dim\left(P\right)=64g_X-63-g_X=63g_X-63$. The triality defect, therefore, takes values in the range $0\le {\delta }_{\tau }(E,\mathrm{\Phi })\le 63g_X-63$, providing a stratification of the moduli space based on triality behavior.

4. Examples

In this section, we provide examples illustrating the main results. Throughout, we fix a compact Riemann surface X of genus $g\ge 2$.

We begin with explicit constructions of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles whose spectral curves are triality-invariant.

Example 1.

Let X be a Riemann surface of genus $g=2$. Consider the nilpotent cone in $\mathcal{M}\left(G_2\right)$, consisting of pairs $(E,\mathrm{\Phi })$ where E is a principal $G_2$-bundle and $\mathrm{\Phi }\in H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)$ is nilpotent.

Using the forgetful map $\iota :\mathcal{M}\left(G_2\right)\to \mathcal{M}\left(\mathrm{Spin}(8,\mathbb{C})\right)$ of Definition 6, this $G_2$-Higgs bundle induces a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle. The Hitchin invariants are $(a_2,a_4,a_4,a_6)$, where $a_2\in H^0(X,K^2)$ and $a_6\in H^0(X,K^6)$ are the two generators of the ring of $G_2$-invariant polynomials [19], and $a_4\in H^0(X,K^4)$ appears with a multiplicity of two.

For the nilpotent case, $a_4=a_6=0$ and the only non-zero Hitchin invariant is $a_2$. Since $dim{H}^0(X,K^2)=3g-3=3$ for $g=2$, the nilpotent cone of the $G_2$-Higgs bundles maps to a three-dimensional subvariety of ${\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$.

The spectral curve becomes

$${\eta }^8+a_2{\eta }^6={\eta }^6({\eta }^2+a_2)=0,$$

which is singular, consisting of the zero section of K with multiplicity 6 together with a smooth degree-2 cover ${\eta }^2+a_2=0$ ramified at the zeros of $a_2$. This curve is triality-invariant according to Theorem 2.

Example 2.

Choose generic sections $a_2\in H^0(X,K^2)$ and $a_6\in H^0(X,K^6)$ (meaning sections from a Zariski open dense subset of $H^0(X,K^2)\times H^0(X,K^6)$), and let $a_4\in H^0(X,K^4)$ be determined by the $G_2$-structure via the embedding of Hitchin bases. According to [11], there exists a Zariski open dense subset of ${\mathcal{A}}_{G_2}=H^0(X,K^2)\oplus H^0(X,K^6)$ over which the corresponding spectral curves are smooth and irreducible.

For such $(a_2,a_6)$, the spectral curve

$$S:{\eta }^8+a_2{\eta }^6+a_4{\eta }^4+a_4{\eta }^2+a_6=0$$

in $\mathrm{Tot}\left(K\right)$ is smooth of degree 8 over X. The genus of S is computed using the standard formula for spectral curves [1]: for a degree-n spectral curve over a genus-g base,

$$g_S=n^2(g-1)+1.$$

For $n=8$ and $g=2$, we obtain $g_S=64(2-1)+1=65$.

The spectral curve S admits an involution $\sigma :S\to S$ induced by the covering $\mathrm{Spin}(8,\mathbb{C})\to \mathrm{SO}(8,\mathbb{C})$. The Prym variety $P=\mathrm{Prym}(S,\sigma )$ has dimension

$$dim\left(P\right)=g_S-g=65-2=63.$$

Using Theorem 5, since S is triality-invariant (being associated to a $G_2$-Higgs bundle), the triality automorphism acts trivially on P; hence, the triality defect (Remark 16) is ${\delta }_{\tau }=0$.

Remark 17.

Examples 1 and 2 construct explicit $G_2$-Higgs bundles by imposing triality invariance on the spectral curve equation. This approach is an implication of Theorem 2: to produce a $G_2$-Higgs bundle, one works in the Hitchin base of $\mathrm{Spin}(8,\mathbb{C})$ and imposes the constraint $b_4=c_4$, rather than starting from the $G_2$ framework.

We now analyze an example where the spectral curve acquires singularities.

Example 3.

Let X be an elliptic curve (genus $g=1$) and consider a family of $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles $(E_t,{\mathrm{\Phi }}_t)$ parameterized by $t\in \mathbb{C}$ with ${\mathrm{\Phi }}_t\in H^0(X,{\mathrm{End}}_0\left(E_t\right)\otimes K)$ such that the characteristic polynomial is $det(\eta ·\mathrm{id}-{\mathrm{\Phi }}_t)={\eta }^2-t·q_0$ and $q_0\in H^0(X,K^2)$ is a fixed non-zero section.

For $t\ne 0$, the spectral curve $S_t:{\eta }^2=t·q_0$ is smooth when $t·q_0$ has simple zeros. At $t=0$, the spectral curve degenerates to $S_0:{\eta }^2=0$, which is the zero section of K counted with multiplicity 2.

The corresponding point in the Hitchin base ${\mathcal{A}}_{\mathrm{SL}\left(2\right)}=H^0(X,K^2)$ is the origin, which lies in the discriminant locus.

Example 4.

We now provide an example of a free triality orbit in the discriminant locus, illustrating Theorem 3 and Proposition 1. Let X be a Riemann surface of genus $g=2$ and consider a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$ with Hitchin invariants $(a_2,b_4,c_4,a_6)\in {\mathcal{A}}_{\mathrm{Spin}\left(8\right)}$ where $b_4\ne c_4$ and such that the spectral curve

$$S:{\eta }^8+a_2{\eta }^6+b_4{\eta }^4+c_4{\eta }^2+a_6=0$$

is singular with a nodal singularity at some point $p\in \mathrm{Tot}\left(K\right)$.

Since $b_4\ne c_4$, the point $(a_2,b_4,c_4,a_6)\in \Delta$ belongs to ${\Delta }_{\mathrm{free}}$ according to Proposition 1. The triality automorphism produces three distinct points in the Hitchin base:

$$\begin{array}{cc}\hfill & h_0=(a_2,b_4,c_4,a_6),\hfill \\ \hfill & h_1=\tau \left(h_0\right)=(a_2,c_4,b_4,a_6),\hfill \\ \hfill & h_2={\tau }^2\left(h_0\right)=(a_2,b_4,c_4,a_6).\hfill \end{array}$$

These three points correspond to three distinct singular spectral curves $S_0=S$, $S_1=\tau \left(S\right)$, and $S_2={\tau }^2\left(S\right)$, all with the same singularity type (nodal). The orbit $\{h_0,h_1,h_2\}$ is a free ${\mathbb{Z}}_3$-orbit in Δ under the triality action.

Specifically, we can construct such an example as follows. Let $a_2\in H^0(X,K^2)$ be a section with exactly four simple zeros $\{p_1,p_2,p_3,p_4\}\subset X$ (which exists generically since $dim{H}^0(X,K^2)=3g-2=4$ for $g=2$). Choose $b_4,c_4\in H^0(X,K^4)$ and $a_6\in H^0(X,K^6)$ such that the following is true:

(1) 

$b_4\left(p_1\right)=c_4\left(p_1\right)=a_6\left(p_1\right)=0$ for some point $p_1\in X$ (forcing a singularity in the spectral curve above $p_1$);

(2) 

$b_4\ne c_4$ as sections (ensuring non-triality-invariance).

Such choices exist because $dim{H}^0(X,K^4)=7g-7=7$ for $g=2$, so imposing one vanishing condition $b_4\left(p_1\right)=0$ leaves a six-dimensional space of choices (similarly for $c_4$ and $a_6$, where $dim{H}^0(X,K^6)=11g-10=12$), and the condition $b_4\ne c_4$ is generic (open and dense).

The spectral curve S defined by ${\eta }^8+a_2{\eta }^6+b_4{\eta }^4+c_4{\eta }^2+a_6=0$ is singular because the simultaneous vanishing of all coefficients at the fiber above $p_1$ forces the polynomial to have a repeated root. By construction, S is not triality-invariant since $b_4\ne c_4$.

The three curves in the triality orbit $\{S,\tau \left(S\right),{\tau }^2\left(S\right)\}$ are pairwise and non-isomorphic as divisors in $\mathrm{Tot}\left(K\right)$, but they are isomorphic as abstract curves (same genus, same singularity type). Each fiber $h^{-1}\left(h_i\right)$ of the Hitchin map over $h_i$ contains a Prym variety parameterizing spectral data, and according to Theorem 4, each may support non-abelian spectral data.

We apply Theorem 5 to analyze specific Prym varieties.

Example 5.

Let X be a Riemann surface of genus $g=2$ and consider a smooth $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle $(E,\mathrm{\Phi })$ with Hitchin invariants $(a_2,b_4,c_4,a_6)\in {\mathcal{A}}_{\mathrm{Spin}\left(8\right)}^{\mathrm{reg}}$ where $b_4\ne c_4$. The spectral curve

$$S:{\eta }^8+a_2{\eta }^6+b_4{\eta }^4+c_4{\eta }^2+a_6=0$$

is smooth and irreducible. Its genus is $g_S=64(2-1)+1=65$. The Prym variety $P=\mathrm{Prym}(S,\sigma )$ has dimension $dim\left(P\right)=65-2=63$.

According to Theorem 5 part (2), since $b_4\ne c_4$, we have $dim\left(P^{\tau }\right)<63$. The triality defect satisfies ${\delta }_{\tau }(E,\mathrm{\Phi })=63-dim\left(P^{\tau }\right)>0$. This provides a certain measure of non-triality-invariance.

5. Application to the Stratification by Triality Defect

We introduce a refined stratification of the smooth locus of the Hitchin fibration using the triality defect defined in Remark 16.

Definition 7.

For $0\le k\le 63g-63$, define the triality defect stratum of level k as

$${\mathcal{M}}_k=\{(E,\mathrm{\Phi })\in \mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}:{\delta }_{\tau }(E,\mathrm{\Phi })=k\},$$

where $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$ denotes the open subset consisting of Higgs bundles with smooth irreducible spectral curves.

Proposition 2.

The stratum ${\mathcal{M}}_0$ is non-empty and coincides with the image of $\mathcal{M}{\left(G_2\right)}^{\mathrm{sm}}$ under the forgetful map.

Proof. 

According to Theorem 5 part (1), $(E,\mathrm{\Phi })$ has ${\delta }_{\tau }(E,\mathrm{\Phi })=0$ if and only if the spectral curve is triality-invariant, which according to Theorem 2 occurs precisely when $(E,\mathrm{\Phi })$ admits a $G_2$-reduction. The image of $\mathcal{M}{\left(G_2\right)}^{\mathrm{sm}}$ under the forgetful map is non-empty according to [11]. □

Remark 18.

The stratification ${\left\{{\mathcal{M}}_k\right\}}_{k=0}^{63g-63}$ introduced in Definition 7 provides a decomposition of $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$ by the degree of triality symmetry. Not all integer values k in the range $[0,63g-63]$ necessarily correspond to non-empty strata. Proposition 2 guarantees only that the minimal stratum ${\mathcal{M}}_0$ is non-empty.

Example 6.

For $g=2$, the Prym variety has dimension $dim\left(P\right)=63$. According to Proposition 2, the stratum ${\mathcal{M}}_0\ne \varnothing$ contains all $G_2$-Higgs bundles with smooth spectral curves. For the maximal stratum ${\mathcal{M}}_{63}$, consisting of Higgs bundles with $dim\left(P^{\tau }\right)=0$, and for intermediate strata ${\mathcal{M}}_k$ with $0<k<63$, the question of non-emptiness remains open.

Proposition 3.

The stratum ${\mathcal{M}}_0$ of triality-invariant Higgs bundles is a proper closed subset of $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$. Equivalently, Higgs bundles with positive triality defects form a Zariski open dense subset of the smooth locus.

Proof. 

According to Theorem 2, the stratum ${\mathcal{M}}_0$ consists of Higgs bundles admitting a $G_2$-reduction. These are precisely the Higgs bundles in the image of the forgetful map $\mathcal{M}{\left(G_2\right)}^{\mathrm{sm}}\to \mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$.

In terms of the Hitchin fibration, ${\mathcal{M}}_0$ is the preimage of the triality-invariant locus in the Hitchin base defined by $b_4=c_4$ in coordinates $(a_2,b_4,c_4,a_6)$ on ${\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}=H^0(X,K^2)\oplus H^0(X,K^4)\oplus H^0(X,K^4)\oplus H^0(X,K^6)$, as established in [14].

The equation $b_4=c_4$ defines a linear subspace of $H^0(X,K^4)\oplus H^0(X,K^4)$ of codimension $dim{H}^0(X,K^4)=7g-7$. This is a proper closed subset of the Hitchin base, hence Zariski closed.

Since the Hitchin map is continuous and the triality-invariant locus is closed at the base, its preimage ${\mathcal{M}}_0$ is closed in $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$.

To show that ${\mathcal{M}}_0$ is proper (i.e., not the entire space), observe that the triality-invariant locus has dimension

$$\begin{array}{cc}\hfill dim{H}^0(X,K^2)& +dim{H}^0(X,K^4)+dim{H}^0(X,K^6)\hfill \\ & =(3g-2)+(7g-7)+(11g-10)\hfill \\ & =21g-19,\hfill \end{array}$$

while the full Hitchin base has dimension

$$dim{\mathcal{A}}_{\mathrm{Spin}(8,\mathbb{C})}=(3g-2)+2(7g-7)+(11g-10)=28g-26.$$

Since $21g-19<28g-26$ for all $g\ge 2$, the triality-invariant locus is a proper subset.

Therefore, ${\mathcal{M}}_0$ is a proper closed subset of $\mathcal{M}{\left(\mathrm{Spin}(8,\mathbb{C})\right)}^{\mathrm{sm}}$, and its complement ${\bigcup }_{k=1}^{63g-63}{\mathcal{M}}_k$ is Zariski open and dense. □

Remark 19.

Stratification by triality defect provides a measure of how far a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle is from admitting a $G_2$-reduction. This complements the stratification given by the discriminant locus decomposition in Proposition 1. These two stratifications are related but distinct: the discriminant stratification concerns singularities of spectral curves, while the triality defect stratification concerns the symmetry properties of smooth spectral curves. Proposition 3 shows that Higgs bundles without perfect triality symmetry (i.e., with ${\delta }_{\tau }>0$) form the generic case in the smooth locus.

6. Conclusions

In this work, we studied the moduli space of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface through the interplay between the triality automorphism and singular spectral curves. This extends the Bradlow–Branco–Schaposnik framework from $\mathrm{SO}(4,\mathbb{C})$ to $\mathrm{Spin}(8,\mathbb{C})$ in several directions. While the article of Bradlow–Branco–Schaposnik works with rank-4 orthogonal bundles without exceptional symmetries, we analyze rank-8 spin bundles endowed with the unique triality automorphism of order 3. This triality symmetry has no analogue in orthogonal groups. The triality action induces a novel decomposition of the discriminant locus into fixed points arising from $G_2$-structures and free ${\mathbb{Z}}_3$-orbits, a stratification that is absent in the work of Bradlow–Branco–Schaposnik. We proved that the non-abelian spectral data persists in free triality orbits, demonstrating that triality symmetry and non-abelian structures coexist in a non-trivial way. Furthermore, we established a bridge to the exceptional group $G_2$, providing characterizations of $G_2$-reductions via spectral data.

The central contribution of this paper is the characterization of triality invariance via spectral data. We established that a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle admits a reduction to the exceptional group $G_2$ if and only if its spectral curve is invariant under the induced triality action. The analysis of the discriminant locus under triality also revealed a novel stratification structure. We decomposed the singular locus into two disjoint strata: a fixed stratum arising from $G_2$-Higgs bundles with singular spectral curves and a free stratum consisting of triality orbits of size three. This decomposition shows that singularities in $\mathrm{Spin}(8,\mathbb{C})$-spectral curves arise through two different mechanisms—one governed by the exceptional symmetry of $G_2$, the other exhibiting generic $\mathrm{Spin}(8,\mathbb{C})$ behavior.

We also proved the existence of non-abelian spectral data compatible with triality, specifically that the non-abelian phenomena discovered for $\mathrm{SO}(4,\mathbb{C})$ persist in the presence of triality symmetry, with non-abelian data occurring in free ${\mathbb{Z}}_3$-orbits. The construction via the embedding $\mathrm{Spin}(4,\mathbb{C})\hookrightarrow \mathrm{Spin}(8,\mathbb{C})$ provides an explicit mechanism for producing such examples.

Our introduction of the triality defect invariant provides a measure of symmetry breaking in the moduli space. This invariant, defined as the dimension of the quotient of the Prym variety by its triality-invariant sublocus, takes values in the range $[0,63g-63]$ and captures the degree to which a Higgs bundle deviates from perfect triality symmetry. We established that Higgs bundles with positive triality defects form a Zariski open dense subset of the smooth locus, showing that $G_2$-reductions represent a special, non-generic configuration.

Despite this, several questions remain open and suggest directions for future research. The most immediate is the complete characterization of realized triality defect values. While we have established that zero defects corresponds to $G_2$-reductions and that a positive defect is generic, determining precisely which integer values in the range $[0,63g-63]$ correspond to non-empty strata requires a deeper understanding of how ${\mathbb{Z}}_3$-actions on abelian varieties of dimension $63g-63$ can be realized through spectral curves.

The structure of non-abelian spectral data in free triality orbits deserves further investigation. While we have established existence, the full moduli theory of such data remains to be developed. Specifically, describing when a nodal spectral curve in the free stratum supports non-abelian data, how triality automorphism acts on the space of such data, and whether triality permutes different non-abelian structures on the three curves in the orbit or preserves some finer structure may all be explored in future research.

The relationship between the triality defect and topological invariants from gauge theory presents another line for future research. The moduli space of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles is connected to moduli spaces of connections through the non-abelian Hodge correspondence, and topological invariants such as characteristic classes and Chern–Simons functionals are crucial for understanding the global geometry. It would be valuable to understand whether the triality defect can be expressed in terms of such topological data, or whether it captures purely analytic information about the Hitchin fibration that is not visible at the topological level.

From a broader perspective, the techniques developed here suggest generalizations to other groups with outer automorphisms. While $\mathrm{Spin}(8,\mathbb{C})$ is the only simple complex Lie group admitting an outer automorphism of order three, many other groups possess outer automorphisms of order two, such as the exceptional group $E_6$. It would be interesting to develop a general framework explaining how outer automorphisms interact with singular spectral data and non-abelian structures. This would provide a unified understanding of the symmetries in the Hitchin fibration.

The connection to mathematical physics here, particularly to gauge theory and integrable systems, offers additional directions for further research. Triality automorphism appears in various contexts in theoretical physics, from string theory to supersymmetric gauge theories, and the moduli spaces of Higgs bundles arise as parameter spaces for solutions to certain physical equations. Understanding how the triality defect and the stratification of the discriminant manifest in physical terms could provide new insights into both its geometry and physics.

Finally, the relationship between our results and the geometric Langlands program deserves exploration. The Hitchin fibration plays a central role in geometric Langlands duality, with spectral data providing a bridge between Higgs bundles and the dual group structure. The triality automorphism of $\mathrm{Spin}(8,\mathbb{C})$ permutes the three eight-dimensional representations, suggesting non-trivial implications for the Langlands dual picture. Understanding how triality symmetry and its associated stratifications interact with geometric Langlands duality could provide interesting insights into the Langlands program.