Let X be a compact Riemann surface of genus $g\ge 2$ and G be a semisimple complex Lie group with Lie algebra $\mathfrak{g}$. A G-Higgs bundle over X is defined to be a pair $(E,\phi )$ where E is a holomorphic principal G-bundle over X and $\phi$ is a holomorphic global section of the adjoint bundle of E, $E\left(\mathfrak{g}\right)$, twisted by the canonical bundle, K. The section $\phi$ is called Higgs field of the Higgs bundle. Suitable notions of stability and polystability can be given for G-Higgs bundles that extend that given by Ramanathan [28,29] for principal G-bundles and in [27] for stable principal G-bundles, obtaining that the moduli space of polystable G-Higgs bundles, $\mathcal{M}\left(G\right)$, is a complex algebraic variety of dimension $2dimG(g-1)$.

Higgs bundles were introduced by Hitchin in his groundbreaking 1987 paper [23] and posses a remarkable wealth of geometric structures, so they are of interest in many different areas and have been intensively studied. Indeed, G-Higgs bundles provide the framework for the extension of the theorem of Narasimhan and Seshadri [26], whose analogue states that the moduli space of polystable G-Higgs bundles is isomorphic to the moduli space of reductive representations of the fundamental group ${\pi }_1\left(X\right)$ in G[13,14,31–33]. In other directions, G-Higgs bundles are of interest in different areas of mathematics and physics, including gauge theory, mirror symmetry, Langlands duality, or symplectic, Kähler and hyperkähler geometry [8,20,25].

Hitchin proved the existence of an integrable system in the moduli space of polystable G-Higgs bundles over an algebraic curve for any reductive complex Lie group G[22]. A relevant and classical theorem by Chevalley [12] states that, for any complex reductive Lie group G with adjoint representation $Ad:G\to GL\left(\mathfrak{g}\right)$, the algebra of all Ad-invariant polynomials is finitely generated and the degrees $d_1,\dots ,d_r$, where $r=rkG$, of the elements of a basis of homogeneous polynomials are well-defined. Given any principal G-bundle E over X, an Ad-invariant (or G-invariant, or simply invariant) homogeneous polynomial p of degree d defines a map $p:H^0(X,E\left(\mathfrak{g}\right)\otimes K)\to H^0(X,K^d)$ by the evaluation of p on the corresponding Higgs field $\phi$ (and also a spectral curve S is induced by each G-Higgs bundle $(E,\phi )$ by taking the characteristic polynomial of the Higgs field through the adjoint representation, which is a cover of X whose fibers correspond to the eigenvalues of $\phi$). This can be computed for a basis of invariant polynomials to finally obtain a map This map, called Hitchin map, is ideed proper. The space $B\left(G\right)$ is an affine space called the base of the Hitchin map. For each stable and simple principal G-bundle E (i.e. a stable principal G-bundle whose only automorphisms are those induced by the action of the center of the structure group G, which is a smooth point in the moduli space of principal G-bundles), the space $H^0(X,E\left(\mathfrak{g}\right)\otimes K)$ is isomorphic, by Serre duality, to $H^1{(X,E\left(\mathfrak{g}\right))}^{*}$, the cotangent bundle to the moduli space of stable and simple principal G-bundles at E. This cotangent bundle is naturally embedded in the moduli space $\mathcal{M}\left(G\right)$ of polystable G-Higgs bundles. Hitchin defined a global symplectic structure in $\mathcal{M}\left(G\right)$ that extends the natural symplectic structure of that cotangent space in a way that the Hitchin map defines an integrable system. The Hitchin fibration has proven to be an essential tool for understanding the Geometric Langlands program, as Kapustin and Witten [25] pointed out.

A fruitful way of studying the geometry of the moduli spaces $\mathcal{M}\left(G\right)$ of G-Higgs bundles is by describing the subvarieties and maps between these moduli spaces [2,17,19]. Specifically, given an automorphism of $\mathcal{M}\left(G\right)$, a subvariety is naturally defined by taking the of fixed points of that automorphism. It is then useful to study automorphisms of finite order of $\mathcal{M}\left(G\right)$. The case of involutions of the moduli space of $SL(n,\mathbb{C})$-Higgs bundles was developed by García-Prada in [17], where he related Higgs bundles with representations of the fundamental group of the surface in the real forms of the group. A larger family of finite-order automorphisms is studied in [4], but in the context of orthogonal bundles over a curve. Also, the case of involutions of $\mathcal{G}$ induced by the action of outer automorphisms of order 2 of G have been studied in [19] and, with different techniques for simple classical complex Lie groups in [5].

This paper is interested in Higgs bundles whose structure group is $Spin(8,\mathbb{C})$. This group is the only simple complex Lie group that admits an outer automorphism of order 3, called triality automorphism. This unique fact makes the geometric structures related to this group (including $Spin(8,\mathbb{C})$-Higgs bundles) have both interesting and very specific geometric features, which usually require specific studies [3,24,34]. In particular, in the previous literature it has been proved that the triality automorphism acts on the moduli space $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ and its fixed points can be described as reductions of structure group to the subgroups $G_2$ or $PSL(3,\mathbb{C})$ of $Spin(8,\mathbb{C})$. This leads the existence of two maps of algebraic varieties $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ and $\mathcal{M}(PSL(3,\mathbb{C})\to \mathcal{M}(Spin(8,\mathbb{C}))$. In this work, the study of the first map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ is deepened. The way to carry out this deepening is through the study of the Hitchin integrable system associated to the two moduli spaces involved. Specifically, if B is the basis of the Hitchin map of $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ and $B^{\prime }$ is the basis of the Hitchin map of $\mathcal{M}\left(G_2\right)$, it is proved that the triality automorphism acts on B, that there exists a homomorphism $j:B\to B^{\prime }$ compatible with the map between moduli spaces, and that the image of j is formed by fixed points of the action of the triality automorphism on B (Lemma 4.2). This allows to state that the map between moduli spaces is restricted to maps between Prym varieties $\mathrm{Prym}\left(X_a\right)\to \mathrm{Prym}\left(X_{j\left(a\right)}\right)$, where $a\in B^{\prime }$ and $X_a$ and $X_{j\left(a\right)}$ are the associated spectral curves. Following this, in this paper the geometry of the Prym varieties involved is studied to the extent of providing necessary and sufficient conditions for these Prym varieties to be disjoint (Proposition 5.4). As a consequence of the above results, sufficient conditions are provided for the two Prym varieties involved to be disjoint. Thus, the paper provides innovative results on the geometry of the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ and also provides novel techniques for the mentioned geometryc study (consisting specifically in studying the Hitchin integrable system). In particular, it is intended to provide tools to advance the knowledge about the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ and to be able in the future to prove properties such as, for example, whether it is injective, in the spirit of Serman [30].

In addition, the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ has been considered because the preceding literature provides sufficient results on the Hitchin integrable system of $G_2$[24] and on the relationship between $G_2$-invariant polynomials and $Spin(8,\mathbb{C})$-invariant polynomials [9] to be able to carry out the analysis intended here. Indeed, Hitchin [24] deepened the study of the integralble system of the moduli space of $G_2$-Higgs bundles, because the special characteristics of the group $G_2$ make it of great interest in differential equations [10], or geometry and physics [2,11]. Thus, Hitchin [24] described the spectral curves S associated to the Hitchin fibration for the group $G_2$ and the associated Prym varieties. In particular, he proved the existence of an intermediate curve C such that the covering $S\to X$ factors through C, and an involution $\sigma$ of S so that $\mathrm{Prym}(S,X)$ is given by those $L\in \mathrm{Prym}(S,C)$ satisfying $L^{*}\cong \sigma \left(L\right)$. He also proved that the $G_2$-Higgs bundle can be reconstructed form the associated spectral curve. It has also been proved that every $G_2$-invariant polynomial is also a $Spin(8,\mathbb{C})$-invariant polynomial fixed by the action of the triality automorphism [9].

The article is organized in the following way. In Section 2 some foundations on the geometry of the Lie group $Spin(8,\mathbb{C})$ and the triality automorphism are recalled. The action of the group $Out\left(G\right)$ of outer automorphisms of any semisimple complex Lie group G on the moduli space $\mathcal{M}\left(G\right)$ of G-Higgs bundles over X, introduced in [2], is described and studied in Section 3. It is also explained in a more detailed way the particular case of $G=Spin(8,\mathbb{C})$ and the specific characteristics of the triality automorphism. In Section 4, the action of the triality automorphism on the base and on the fibers of the Hitchin integrable system is constructed. Section 5 is devoted to providing the main geometric features on the Prym varieties coming from the Hitchin integrable system associated to $Spin(8,\mathbb{C})$. Finally, the main conclusions of the paper are drawn.

In this section, some basics on the Lie group $Spin(8,\mathbb{C})$, its subgroups, and the triality automorphism are provided. Suitable references for this topic are [1–3,16]. The group $G=Spin(8,\mathbb{C})$ is the only simple and simply connected complex Lie group of type $D_4$. Its Lie algebra is $\mathfrak{so}(8,\mathbb{C})$, its center is isomorphic to $Z={\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$. It is indeed the double cover of the special orthogonal Lie group $\mathrm{SO}(8,\mathbb{C})$ and then it can be described as an extension of $\mathrm{SO}(8,\mathbb{C})$ by ${\mathbb{Z}}_2$:

Of course, the action of the group $Aut(Spin(8,\mathbb{C}\left)\right)$ of automorphisms of $Spin(8,\mathbb{C})$ leaves the center $Z={\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ invariant, hence there is a homomorphism of $Aut(Spin(8,\mathbb{C}\left)\right)$ into the group $S\left(Z^2\right)$ of permutations of the set $Z^2=Z\setminus \left\{1\right\}$ of central elements of order 2. The subgroup $Inn(Spin(8,\mathbb{C}\left)\right)$ of inner automorphisms clearly acts trivially on Z, thus this induces a homomorphism of the group $Out(Spin(8,\mathbb{C}\left)\right)$ of outer automorphisms of $Spin(8,\mathbb{C})$ into $S\left(Z^2\right)\cong S_3$, which is actually an isomorphism. Recall that $Out(Spin(8,\mathbb{C}\left)\right)$ is the quotient of the group $Aut(Spin(8,\mathbb{C}\left)\right)$ of automorphisms of $Spin(8,\mathbb{C})$ by the normal subgroup $Inn(Spin(8,\mathbb{C}\left)\right)$ of inner automorphisms. The triality automorphism is then a choice $\tau$ of an order 3 outer automorphism (the other outer automorphism of order 3 is ${\tau }^{-1}$).

Since $Spin(8,\mathbb{C})$ is the simply connected complex Lie group with Lie algebra $\mathfrak{so}(8,\mathbb{C})$, there is an isomorphism of short exact sequences of the form

Given any complex Lie algebra $\mathfrak{g}$, if $\alpha ,\beta \in Aut\left(\mathfrak{g}\right)$, it is stated that $\alpha {\sim }_i\beta$ if there exists $\theta \in Inn\left(\mathfrak{g}\right)$ such that $\alpha =\theta \circ \beta \circ {\theta }^{-1}$. Thus defined, ${\sim }_i$ is an equivalence relation such that the obvious map is well-defined [1,3].

Notice that the order of an automorphism of $\mathfrak{g}$ clearly coincides with the order of its class modulo ${\sim }_i$. Then, if ${Aut}_3\left(\mathfrak{so}(8,\mathbb{C})\right)$ denotes the subset of automorphisms of order 3 of $\mathfrak{so}(8,\mathbb{C})$ and an analogous definition is given for ${\left(Aut\left(\mathfrak{so}(8,\mathbb{C})\right)/{\sim }_i\right)}_3$ and for ${Out}_3\left(\mathfrak{so}(8,\mathbb{C})\right)$, it is satisfied that

It is also clear that the automorphisms of order 3 are sent to elements of $Out\left(\mathfrak{so}\right(8,\mathbb{C}\left)\right)$ of order 3 or to the identity through the map defined in (2). This implies that ${Aut}_3\left(\mathfrak{g}\right)/{\sim }_i$ is sent onto ${Out}_3\left(\mathfrak{g}\right)\cup \left\{1\right\}$ through the natural map, that is,

There are exactly two pre-images of the triality automorphism $\tau$ by the map defined in (3) [1,3]. Then there are two possibilities for the subalgebra of fixed points of an automorphism of order 3 of $\mathfrak{so}(8,\mathbb{C})$ representing $\tau$. Wolf and Gray [35] proved that these two different representatives of $\tau$ by the map (3) have ${\mathfrak{g}}_2$ and $\mathfrak{sl}(3,\mathbb{C})$ as subalgebras of fixed points (with simply connected subgroups $G_2$ and $PSL(3,\mathbb{C})$), respectively.

Let X be a compact Riemann surface of genus $g\ge 2$. A principal $\mathrm{SO}(8,\mathbb{C})$-bundle over X is a complex rank-8 and trivial determinant vector bundle equipped with a globally-defined holomorphic non-degenerate symmetric bilinear form. The set of isomorphism classes of principal $\mathrm{SO}(8,\mathbb{C})$-bundles is parametrized by the cohomology set $H^1(X,\mathrm{SO}(8,\mathbb{C}))$. A map ${\omega }_2:H^1(X,\mathrm{SO}(8,\mathbb{C}))\to H^2(X,{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2$ is defined which assigns to each principal $\mathrm{SO}(8,\mathbb{C})$-bundle E its second Stiefel-Whitney class ${\omega }_2\left(E\right)$. The bundle E lifts to a principal $Spin(8,\mathbb{C})$-bundle over X if and only if ${\omega }_2\left(E\right)=0$, two of such lifts differing in a line bundle of order 2. However, every principal $Spin(8,\mathbb{C})$-bundle admits an associated $\mathrm{SO}(8,\mathbb{C})$-bundle through the covering map $Spin(8,\mathbb{C})\to \mathrm{SO}(8,\mathbb{C})$ defined in (1).

In the next definitions, the notions of Higgs bundles are specified for the structure groups $\mathrm{SO}(8,\mathbb{C})$ and $Spin(8,\mathbb{C})$ following the original notion of G-Higgs bundle introduced by Hitchin [23].

For both principal $Spin(8,\mathbb{C})$-Higgs bundles and $\mathrm{SO}(8,\mathbb{C})$-Higgs bundles, the element $\phi$ is called Higgs field and it can be understood on each fiber of the orthogonal bundle E as an 8-dimensional complex matrix of the Lie algebra $\mathfrak{so}(8,\mathbb{C})$.

There are suitable notions of stability and polystability that allows to construct the moduli space of G-Higgs bundles for any complex reductive Lie group G[18], which will be denoted by $\mathcal{M}\left(G\right)$. These notions extend to Higgs pairs the notions given by Ramanathan [28,29] for principal bundles. The moduli space of $Spin(8,\mathbb{C})$-Higgs bundles over X is then the algebraic variety which parameterizes isomorphism classes of polystable $Spin(8,\mathbb{C})$-bundles over X.

Given any $Spin(8,\mathbb{C})$-Higgs bundle $(E,\phi )$, an automorphism of $(E,\phi )$ is an automorphism $f:E\to E$ of the principal $Spin(8,\mathbb{C})$-bundle E such that the following diagram commutes: that is, $(f\otimes 1_K)\circ \phi =\phi \circ f$.

Consider now any semisimple complex Lie group G with Lie algebra $\mathfrak{g}$. In [2] it is proved that the following defines an action of the group $Out\left(G\right)$ on the moduli space $\mathcal{M}\left(G\right)$ of G-Higgs bundles: if $\rho \in Out(Spin(8,\mathbb{C}\left)\right)$ and $(E,\phi )\in \mathcal{M}\left(G\right)$, then $\rho ·(E,\phi )$ is defined to be the G-Higgs bundle where $A\in Aut\left(G\right)$ is an automorphism of G representing $\rho$ and $A\left(E\right)$ is the principal G-bundle whose total space is that of E but it is equipped with the right action of G given by $e\diamond g=e{A}^{-1}\left(g\right)$, for $e\in E$ and $g\in G$. It can be proved that this action preserves the stability and polystability of the G-Higgs bundles and that it does not depend on the choice of the representative A of $\rho$, since inner automorphisms act trivially on G-Higgs bundles [2].

Let $\tau \in Out(Spin(8,\mathbb{C}\left)\right)$ be the triality automorphism and let $(E,\phi )$ be a $Spin(8,\mathbb{C})$-Higgs bundle over X fixed by the action of $\tau$. If $A\in Aut(Spin(8,\mathbb{C}\left)\right)$ is an automorphism of $Spin(8,\mathbb{C})$ representing $\tau$, then $(E,\phi )\cong \left(A\right(E),dA(\phi \left)\right)$. Recall that there are only two possibilities for the group $Fix\left(A\right)$ of fixed points of A depending on the lifting of the triality automorphism by the relation ${\sim }_i$. These two possibilities are $G_2$ or $PSL(3,\mathbb{C})$[35]. In [2] it is proved that the fixed points of the action of the triality automorphism on the moduli space of $Spin(8,\mathbb{C})$-Higgs bundles are those which admit a reduction of structure group to $G_2$ or $PSL(3,\mathbb{C})$. Then there are maps $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ and $\mathcal{M}(PSL(3,\mathbb{C}\left)\right)\to \mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ such that their images complete the subvariety of fixed points of the action of the triality automorphism. In this work, the study of map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ is deepened, taking advantage of the existence of results relating the invariant polynomials of $G_2$ and $Spin(8,\mathbb{C})$, which will be key in the study.

Let G be a semisimple complex Lie group with Lie algebra $\mathfrak{g}$ and $p_1,\dots ,p_r$ be a basis of the ring of invariant homogeneous polynomials of $\mathfrak{g}$, where r is the rank of $\mathfrak{g}$. The action of each polynomial on the Higgs field defines a map where $d_i=deg{p}_i$ for $i=1,\dots ,r$. This map allows to define the so-called Hitchin map, where $\mathcal{M}\left(G\right)$ is the moduli space of G-Higgs bundles. The vector space is called the base of the Hitchin map. In [22] it is proved that $dim\mathcal{M}\left(G\right)=dimB$. If n is the dimension of $\mathcal{M}\left(G\right)$, the Hitchin map induces n complex valued functions $f_1,\dots ,f_n$ defined on $T^{*}{M}_{*}\left(G\right)$, where by $M_{*}\left(G\right)$ we denote the moduli space of stable and simple principal G-bundles, which is a dense open subset of the moduli space $M\left(G\right)$ of principal G-bundles. The tangent space to $M_{*}\left(G\right)$ at an element $E\in M_{*}\left(G\right)$ is isomorphic to $H^1(X,E\left(\mathfrak{g}\right))$, which coincides with $H^0{(X,E\left(\mathfrak{g}\right)\otimes K)}^{*}$ by Serre duality. Hitchin also proved that the n functions $f_i$ Poisson-commute with the canonical symplectic structure of the cotangent bundle [22]. This then defines a completely integrable system on $\mathcal{M}\left(G\right)$ [6].

As will be proved below, the group $Out\left(G\right)$ of outer automorphisms of G acts on the base of the Hitchin map so that this action is compatible qith the action of $Out\left(G\right)$ on $\mathcal{M}\left(G\right)$ when G is simply connected (which is the case of $Spin(8,\mathbb{C})$).

From this, the announced action of $Out\left(G\right)$ on the base B of the Hitchin map can be defined for a semisimple and simply-connected complex Lie group G. Hitchin [22] proved that the Hitchin map $\mathcal{H}:\mathcal{M}\left(G\right)\to B$ is surjective. Then, for each $\alpha \in B$, there exists $(E,\phi )\in \mathcal{M}\left(G\right)$ such that $\alpha =(p_1\left(\phi \right),\dots ,p_r\left(\phi \right))$. If $\rho \in Out\left(G\right)$, it is defined where $f_{\rho }$ is an automorphism of G representing $\rho$.

The above study will now be particularized to the case of $Spin(8,\mathbb{C})$ and $G_2$-Higgs bundles. The algebra of invariant polynomials of $Spin(8,\mathbb{C})$ is generated by four invariant homogeneous polynomials where $deg{p}_i=i$ for each i and $p_4$ is the Pfaffian, so $deg{p}_4^{\prime }=4$ [22]. Indeed, Then the base of the Hitchin map is Given a quadruple $a=(a_2,a_4,a_6,b_4)\in B$, it induces the polynomial in the sense that the characteristic polynomial of the Higgs field of any $(E,\phi )$ in the fiber of the Hitchin map at a is $det(tI-\phi )=b_4^2\left(\phi \right)+a_6\left(\phi \right)t^2+a_4\left(\phi \right)t^4+a_2\left(\phi \right)t^6+t^8$, where each $a_i$ or $b_i$ is a holomorphic global section of $K^i$ over X and $a=(a_2,a_4,a_6,b_4)$ defines the Hitchin map $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)\to B$.

The mentioned characteristic polynomial defines the spectral curve in the total space of the canonical bundle $\pi :K\to X$. Then, given any $a\in B$, it defines the equation of the corresponding spectral curve, which will be called $X_a$ as the divisor of a section of ${\pi }^{*}{K}^8$ defined by the induced polynomial where x is the tautological section of ${\pi }^{*}K$ and it is a single-valued eigenvalue of the Higgs field $\phi$.

Consider now the Lie group $G_2$, seen as a subgroup of $Spin(8,\mathbb{C})$. The algebra of invariant polynomials of $G_2$ is generated by two polynomials, $q_2,q_6$, of degrees 2 and 6, respectively Then the base of the Hitchin map of the moduli space of $G_2$-Higgs bundles is and, given a pair $(a_2,a_6)\in B^{\prime }$, the induced invariant polynomial is $a_6{t}^2+{\displaystyle \frac{a_2^2}{4}}{t}^4+a_2{t}^6+t^8$, so by taking a common factor $t^2$ it follows that the polynomial is also $G_2$-invariant.

Some facts on spectral curves and Prym varieties associated to the moduli spaces $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ and $\mathcal{M}\left(G_2\right)$ will be now recalled. Given a $Spin(8,\mathbb{C})$-Higgs bundle $(E,\phi )$ over X, the characteristic polynomial of the Higgs field $\phi$ defines an element $a\in B$, so a spectral cover $X_{(E,\phi )}=X_a$ and an 8-sheeted covering map ${\pi }_{(E,\phi )}:X_{(E,\phi )}\to X$ are defined, whose fibers are identified with the eigenvalues of $\phi$. A detailed construction of this curve can be found in [7,15,24]. There exists an exact sequence on the spectral curve $X_{(E,\phi )}$ where E denotes both the principal $Spin(8,\mathbb{C})$-bundle and the orthogonal bundle that it induces ([7,24]). By dualizing the sequence (11) and tensoring with ${\pi }^{*}K$ the following is obtained: Here, L is a line bundle which satisfies that $E={\pi }_{*}L$ (as a vector bundle). It is constructed as the coker of ${\pi }^{*}\phi -x$, where x denotes the tautological global section of ${\pi }^{*}K$, and $X_{(E,\phi )}$ can be identified with the support of L. The covering map ${\pi }_{(E,\phi )}:X_{(E,\phi )}\to X$ gives a norm map Nm defined by $\mathrm{Nm}({\sum }_i{a}_i·x_i)={\sum }_i{a}_i{\pi }_{(E,\phi )}\left(x_i\right)$ on divisor classes. The norm defines a map at the level of Jacobian varieties ${\mathrm{Nm}}_J:J\left(X_{(E,\phi )}\right)\to J\left(X\right)$. Then the Prym variety of the spectral curve ${\pi }_{(E,\phi )}:X_{(E,\phi )}\to X$ is defined to be the connected component of the kernel of ${\mathrm{Nm}}_J$. This Prym variety is then an abelian subvariety of the jacobian of the sectral curve $X_{(E,\phi )}$. Observe that, since $\phi$ takes values in $\mathfrak{so}(8,\mathbb{C})$, the opposite of an eigenvalue of $\phi$ is also an eigenvalue of $\phi$, so an involution $\sigma$ of $X_{(E,\phi )}$ is defined by change of sign and the eigenspace V of eigenvalue $\lambda$ of $\phi$ is moved to ${\sigma }^{*}V$ for eigenvalue $-\lambda$. Then $L^{*}\cong L\otimes {\pi }^{*}{K}^{-7}$ by (11) and (12), so $L^2\cong {\pi }^{*}{K}^7$. This means that $M=L\otimes {\pi }^{*}{K}^{-3/2}$ satisfies that $M\otimes {\sigma }^{*}M$ is trivial (the choice of a square root of K is required here). Then the Prym variety is the collection of those elements M of the Jacobian of $X_{(E,\phi )}$ such that $M\otimes {\sigma }^{*}M$ is trivial. This is equivalent to stating that the desired Prym variety coincides with $\mathrm{Prym}(X_{(E,\phi )},X_{(E,\phi )}/\sigma )$, the last Prym variety being defined from the covering map $X_{(E,\phi )}\to X_{(E,\phi )}/\sigma$. Notice that, given any $M\in \mathrm{Prym}(X_{(E,\phi )},X)$, one has that $E\cong {\pi }_{*}L$, where $L={\pi }^{*}{K}^{3/2}\otimes M$ and, since $L^2\cong {\pi }^{*}{K}^7$, then $E\cong E^{*}$ as a consequence of (11) and (12), so the special orthogonal bundle is obtained from the element of the Prym variety. Then the fiber ${\mathcal{H}}^{-1}\left(\mathcal{H}(E,\phi )\right)$ is a $2^{2g}$-sheeted covering of the Prym variety of $X_{(E,\phi )}\to X$ (details on this construction can be found in [24]).

Consider now the Lie group $G_2$. The fundamental complex representation of $G_2$ has rank 7 and defines, in addition, an inclusion $G_2\hookrightarrow \mathrm{SO}(7,\mathbb{C})$. A holomorphic antisymmetric 3-form can be defined in ${\mathbb{C}}^7$ in a way that $G_2$ is the group of elements of $SL(7,\mathbb{C})$ which preserves this 3-form [11]. For any $G_2$-Higgs bundle $(E,\phi )$ over X, Hitchin [24] considered the spectral curve $X_{(E,\phi )}$ assocated to it and constructed an intermediate curve C such that the covering map ${\pi }_{(E,\phi )}$ admits a factorization and proved the existence of an involution $\sigma$ of $X_{(E,\phi )}$ such that $p\circ \sigma =p$ and $\mathrm{Prym}(X_{(E,\phi )},X)$ is the subspace of those $L\in \mathrm{Prym}(X_{(E,\phi )},C)$ for which $L\otimes {\sigma }^{*}L$ is trivial. From this description, Hitchin [24] proved that the globally-defined holomorphic antisymmetric 2-form defined in E by its $G_2$-structure can be reconstructed from the corresponding point of the Prym variety $\mathrm{Prym}(X_{(E,\phi )},X)$. In this way, it was proved that the fiber of the Hitchin map of $\mathcal{M}\left(G_2\right)$ is isomorphic to $\mathrm{Prym}(X_{(E,\phi )},X)$ [24].

Notice that if $(E,\phi )$ is a $G_2$-Higgs bundle, then, seen as a $Spin(8,\mathbb{C})$-Higgs bundle through the contention of groups $G_2\hookrightarrow Spin(8,\mathbb{C})$, it satisfies that the Pfaffian $b_4\left(\phi \right)$ is 0. Then, if a is the $G_2$-invariant polynomial $a_6+{\displaystyle \frac{a_2^2}{4}}{t}^2+a_2{t}^4+t^6$ and $X_a$ is the associated spectral curve, b is the $Spin(8,\mathbb{C})$-invariant polynomial and $X_b$ is the associated spectral curve, the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C})$ restricts to a map

In this section the geometry of the Prym varieties $\mathrm{Prym}\left(X_a\right)$ and $\mathrm{Prym}\left(X_{j\left(a\right)}\right)$ of $\mathcal{M}\left(G_2\right)$ and $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$, respectively, will be studied, where $a\in B^{\prime }$ defined in (9). In particular, necessary and sufficient conditions for the above Prym varieties to be disconnected will be established.