The Moduli Space of Principal G₂-Bundles and Automorphisms

Abstract

Let X be a compact Riemann surface of genus $g\ge 2$ and $M\left(G_2\right)$ be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type $G_2$. In this work, it is proved that the only automorphisms that $M\left(G_2\right)$ admits are those defined as the pull-back action of an automorphism of the base curve X. The strategy followed uses specific techniques that arise from the geometry of the gauge group $G_2$. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of $G_2$-bundles over X have been proved in the paper. The inclusion of groups $G_2\hookrightarrow Spin(8,\mathbb{C})$ where $G_2$ is viewed as the fixed point subgroup of an order of 3 automorphisms of $Spin(8,\mathbb{C})$ that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$. In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of $M\left(G_2\right)$ in relation to $M(Spin(8,\mathbb{C}\left)\right)$.

1. Introduction

Group $G_2$ is one of the five simple complex Lie groups of exceptional type and can be viewed as the automorphism group of a rank-7 vector space preserving a holomorphic non-degenerate anti-symmetric 3-form defined on it. Through this identification, the fundamental, irreducible representation of $G_2$ is 7-dimensional, and it also preserves a holomorphic non-degenerate symmetric 2-form defined on the representational 7-dimensional vector space. Thus defined, the Lie group $G_2$ has complex dimensions 14 and rank 2, and its Lie algebra ${\mathfrak{g}}_2$ is intricately related to the automorphism group of the octonions, the largest of the normed division algebras [1]. Its Dynkin diagram consists of two nodes with a triple bond, induced by a discrepancy in the lengths of the simple roots, as one root is $\sqrt{3}$ times longer than the other [2]. This asymmetry causes the vanishing of the group of outer automorphisms of $G_2$.

The geometry of the Lie group $G_2$ has deep applications in mathematics and theoretical physics, and it has been intensively studied. In particular, $G_2$, viewed as the automorphism group of the octonions, has provoked great interest in differential geometry since it gives rise to $G_2$-manifolds with special holonomy [3]. It is also deeply studied in mathematical physics, especially in dynamical systems [4], and also in string theory and M-theory [3,5,6] since $G_2$-structures play a key role in compactification processes and in the study of exceptional gauge symmetries [7].

Given any compact Riemann surface X of genus $g\ge 2$, a principal $G_2$-bundle over X is a holomorphic rank-7 vector bundle E that is equipped with a holomorphic non-degenerate anti-symmetric 3-form $\mathrm{\Omega }$. In addition, E admits a holomorphic non-degenerate and globally-defined symmetric 2-form $\omega$. Ramanathan [8,9,10] gave suitable notions of stability to construct a complex algebraic variety, parametrizing isomorphism classes of polystable principal $G_2$-bundles over X. This variety is called the moduli space of principal $G_2$-bundles, is denoted by $M\left(G_2\right)$, has a complex dimension of $14(g-2)$, and admits the subset of stable $G_2$-bundles as an open smooth dense subvariety.

There are several ways of studying the geometry of the moduli space of principal $G_2$-bundles over X, such as the exploration of its stratifications [11,12] or the description of its automorphisms [13]. This research fits within the latter approach. In particular, the first main result of this paper proves that the group of automorphisms of $M\left(G_2\right)$ coincides with the group of automorphisms of the base curve in the sense that every automorphism of $M\left(G_2\right)$ is given by the pull-back action of an automorphism of X (Theorem 2). This problem connects with the work of Kouvidakis and Pantev [14], who computed the group of automorphisms of the moduli space of vector bundles with given ranks and degrees. In particular, they described three families of automorphisms of the moduli space $M(SL(n,\mathbb{C}\left)\right)$ of principal $SL(n,\mathbb{C})$-bundles over X that generate the whole group: the automorphism, which consists of taking the dual of the bundle (this is induced by the action of the unique outer involution of the structure group in the moduli space); those defined as the action by a tensor product with an order-n line bundle over X; and the automorphisms, which consist of taking a pull-back via an automorphism of X. The authors use an argument that takes the fibers of the Hitchin map of the moduli space over singular curves with non-generic singularities. Unfortunately, this argument is not easily generalizable, as far as it has been possible to explore.

Hwang and Ramanan [15] used Hecke curves, which are minimal rational curves obtained from Hecke transformations, to form a proof. Other authors have worked on this topic following the path of Hwang and Ramanan [15]. Thus, Biswas, Gómez, and Muñoz [16,17] computed the group of automorphisms of the moduli space $M(Sp(2n,\mathbb{C}\left)\right)$ of symplectic bundles over a complex projective irreducible curve. They proved that every automorphism of $M(Sp(2n,\mathbb{C}\left)\right)$ combines the action of an automorphism of the base curve via pull-back with the action of a line bundle of order 2 via a tensor product. This result is analogous to that of Kouvidakis and Pantev since the symplectic group admits no outer automorphisms. The strategy followed here is an adaptation of that of [16], and the main result (Theorem 2) is covered by a recent, more general result provided by Fringuelli [18], who described the group of automorphisms of $M\left(G\right)$ for any semisimple complex Lie group G. However, some original results are proved here that allow for an adaption of the strategy of Biswas, Gómez, and Muñoz [16] and simplify it; this may be useful in future works. Specifically, sufficient conditions are given on a semisimple complex Lie group G for a stable G-bundle to be simple (Lemma 1), and it is also proved that every stable $G_2$-bundle is irreducible, meaning that it does not admit any reduction in the structure group to a proper subgroup of $G_2$ (Proposition 1).

Group $G_2$ is also a fixed point of certain order 3 automorphisms of the simple complex Lie group $Spin(8,\mathbb{C})$ that lifts the triality automorphism $\tau$, viewed as an outer automorphism of $Spin(8,\mathbb{C})$. This gives a well-known inclusion of groups $G_2\hookrightarrow Spin(8,\mathbb{C})$, which induces an algebraic morphism $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ between the moduli spaces of principal bundles over X, called the forgetful map. The image of this map is, indeed, composed of principal $Spin(8,\mathbb{C})$-bundles fixed by the action of $\tau$ on $M(Spin(8,\mathbb{C}\left)\right)$ [19]. In general, for a complex semisimple Lie group G, the fixed points of the action of an outer automorphism of G in $M\left(G\right)$ are reductions in the structure group of the subgroup of fixed points and of some lift of the outer automorphism of G in $Aut\left(G\right)$ [13]. The description of fixed points of automorphisms of moduli spaces of principal bundles as reductions in the structure group is a fruitful line of research because several automorphisms, apart from the action of an outer automorphism, admit fixed points that are, thus, characterized [20]. The determination of whether the above-mentioned forgetful morphism is injective or not is an interesting and difficult problem that is not only informative about the geometry of $M\left(G_2\right)$ but also about the geometry of $M(Spin(8,\mathbb{C}\left)\right)$. In the second main result of the paper, it is proved that the forgetful map $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ is actually an embedding (Theorem 3). The strategy followed consists of proving that the obstruction for an automorphism between polystable $Spin(8,\mathbb{C})$-bundles (reducing to $G_2$-bundles to induce an automorphism of $G_2$-bundles) is an element of the non-abelian cohomology set $H^1(X,Spin(8,\mathbb{C})/G_2)$ (Proposition 5). Then, sufficient conditions are given on a simple complex Lie group G and a complex subgroup of it H for the non-abelian cohomology $H^1(X,G/H)$ to be trivial (Proposition 6).

Finally, some consequences of Theorems 2 and 3 are discussed. In particular, it is proved that every automorphism of $M\left(G_2\right)$ comes from an automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ (Proposition 7) and that this lift is not unique; there is at least one family of such lifts that is parameterized by the group $S_3$ of permutations of 3 elements (Proposition 8). Moreover, $2^{4g}$ subvarieties of $M(Spin(8,\mathbb{C}\left)\right)$ are identified, where g is the genus of X, which is pairwise disjointed and isomorphic to $M\left(G_2\right)$ (Proposition 9).

The paper is organized as follows. In Section 2, a description of the group $G_2$ is given, where some basics are emphasized, such as the algebra of invariant homogeneous polynomials and the filtrations induced by the maximal parabolic subgroups of $G_2$. All this is relevant to present (in Section 2) an appropriate notion of stability and polystability of principal $G_2$-bundles over X and to state the Torelli theorem for $M\left(G_2\right)$. Finally, Lemma 1 and Proposition 1, which relate stable, simple, and irreducible $G_2$-bundles, are proved. Section 3 is devoted to describing the Hitchin integrable system associated with $M\left(G_2\right)$. The action of ${\mathbb{C}}^{*}$ on this system is introduced, and some auxiliary results are proved, which will be useful in the proof of Theorem 2. In Section 4, a precise description of the group of automorphisms of $M\left(G_2\right)$ (Theorem 2) is provided. The second main result of this work (Theorem 3) on the injectivity of the forgetful map $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ is proved in Section 5. Finally, Section 6 is devoted to drawing some implications of all the above on the geometry of the moduli spaces $M\left(G_2\right)$ and $M(Spin(8,\mathbb{C}\left)\right)$.

2. Principal ${\mathit{G}}_{\mathbf{2}}$-Bundles

In this section, some basics on principal $G_2$-bundles over a curve are discussed to describe specific and simplified notions of stability and polystability for these bundles and define the corresponding moduli space. For that, some relevant results for the complex Lie group $G_2$ that will be needed are first presented, mainly following [21,22,23]. For the description of the parabolic subgroups of $G_2$, the main reference is [4]. As original contributions of this section, sufficient conditions on a semisimple complex Lie group G will be provided for a stable G-bundle to be simple, meaning that it admits no automorphisms other than those induced by the action of the center of G, and simple $G_2$-bundles do not admit any reduction in the structure group to a proper subgroup of $G_2$.

2.1. The Simple Complex Lie Group $G_2$

The complex group $G_2$ can be viewed as the subgroup of the $SL(7,\mathbb{C})$ of the automorphisms of ${\mathbb{C}}^7$ that preserve a certain non-degenerate symmetric bilinear form and non-degenerate anti-symmetric trilinear form $\mathrm{\Omega }$ [2,19,23]. This embedding $G_2\hookrightarrow SL(7,\mathbb{C})$ defines the 7-dimensional fundamental irreducible representation of $G_2$. Specifically, if $\{e_1,\dots ,e_7\}$ is an orthonormal basis for the bilinear form, and $e^{ijk}$ is the wedge product $e_i^{*}\wedge e_j^{*}\wedge e_k^{*}$ of vectors in the dual basis for $i,j,k=1,\dots ,7$; then,

$$\mathrm{\Omega }=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}.$$

Group $G_2$ is formally defined as

$$G_2=\{g\in SL(7,\mathbb{C}):g^{*}\mathrm{\Omega }=\mathrm{\Omega }\},$$

(1)

so $G_2$ can be viewed through its fundamental 7-dimensional representations as a subgroup $SO(7,\mathbb{C})$ as

$$G_2\hookrightarrow SO(7,\mathbb{C}).$$

(2)

Of course, its Lie algebra ${\mathfrak{g}}_2$ is a subalgebra of the special orthogonal algebra $\mathfrak{so}(7,\mathbb{C})$, given by endomorphisms preserving $\mathrm{\Omega }$. Thus defined, group $G_2$ is simply connected and centerless, and it admits no nontrivial outer automorphisms. It has a dimension of 14 and a rank of 2.

As explained in [19], group $G_2$ is deeply related to the simple and simply connected complex Lie group $Spin(8,\mathbb{C})$ and with the triality automorphism. More precisely, recall that the triality automorphism $\tau$ is an order 3 outer automorphism of $Spin(8,\mathbb{C})$ that admits an order 3 representative in $Aut(Spin(8,\mathbb{C}\left)\right)$, the subgroup of which has fixed points and is isomorphic to $G_2$. Then, $G_2$ can also be viewed as the subgroup of $Spin(8,\mathbb{C})$ of fixed points for a representative in $Aut(Spin(8,\mathbb{C}\left)\right)$ of triality automorphism. This defines the inclusion of groups

$$G_2\hookrightarrow Spin(8,\mathbb{C}).$$

(3)

Since the rank of $G_2$ is 2, it admits two non-isomorphic maximal parabolic subgroups, $P_1$ and $P_2$, and one non-maximal parabolic subgroup, which is the intersection of $P_1$ and $P_2$. Following the detailed description given in [4], the action of $P_1$ induces a 2-graded filtration

$$0⊊V_1⊊V_1^{\perp }⊊{\mathbb{C}}^7$$

on the vector space ${\mathbb{C}}^7$, where $G_2$ is represented through its fundamental 7-dimensional representation and $dim{V}_1=1$, $dim{V}_1^{\perp }=6$, and $V_1$ are isotropic for the 3-form $\mathrm{\Omega }$, as well as for $\omega$ (recall that a subspace V is isotropic for $\mathrm{\Omega }$ if $\mathrm{\Omega }(V,V,V)=0$). Similarly, the action of $P_2$ gives a filtration of the form

$$0⊊V_2⊊V_2^{\perp }⊊{\mathbb{C}}^7,$$

where $dim{V}_2=2$, $dim{V}_2^{\perp }=5$, and $V_2$ are, again, isotropic for $\mathrm{\Omega }$. Of course, the action of the nonmaximal parabolic subgroup $P=P_1\cap P_2$ induces a filtration

$$0⊊V_1⊊V_2⊊V_2^{\perp }⊊V_1^{\perp }⊊{\mathbb{C}}^7.$$

Here, $dim{V}_1=1$ and $dim{V}_2=2$, and both subspaces are isotropic for $\mathrm{\Omega }$.

2.2. Principal $G_2$-Bundles over a Compact Riemann Surface

Let X be a compact Riemann surface of genus $g\ge 2$. From the definition of $G_2$ given in (1) and the matrix representation described in (2), a principal $G_2$-bundle over X can be viewed as a triple $(E,\mathrm{\Omega },\omega )$, where E is a rank-7 holomorphic vector bundle over X, $\mathrm{\Omega }$ is an anti-symmetric non-degenerate holomorphic trilinear form globally defined on E, and $\omega$ is a holomorphic symmetric non-degenerate bilinear form $\omega$ that is also globally defined on E such that the pair $(E,\omega )$ is an orthogonal bundle over X. For simplicity, the principal $G_2$-bundles will be called hereafter by referring to the underlying vector bundle E.

From the above description of the parabolic subgroups of $G_2$ [4], the notions of stability and polystability of principal $G_2$-bundles over X can be given in terms of the filtrations of the isotropic sub-bundles of the underlying vector bundles induced by reductions in parabolic subgroups to maximal parabolic subgroups of $G_2$, following [19,24]. These notions of stability, semistability, and polystability are equivalent to those provided by Ramanathan in his seminal works [9,10] and by Subramanian [25]. Recall that, given any principal $G_2$-bundle $(E,\mathrm{\Omega },\omega )$ (as above), a vector sub-bundle F of E is isotropic for the trilinear form $\mathrm{\Omega }$ if $\mathrm{\Omega }(F,F,F)=0$.

Definition 1. 

Let $(E,\mathrm{\Omega },\omega )$ be a principal $G_2$-bundle over the compact Riemann surface X of genus $g\ge 2$. The principal $G_2$-bundle is stable if $degF<0$ for every rank-1 or rank-2 vector sub-bundle F of E, which is isotropic for Ω and ω. It is semistable if, under the same conditions, $degF\le 0$. It is polystable if it is semistable and the underlying vector bundle E can be decomposed as a direct sum of proper vector sub-bundles of the form $E=F\oplus F^{*}\oplus W$, where F is a rank-1 or rank-2 stable vector sub-bundle that is isotropic for both forms, Ω and ω.

The moduli space $M\left(G_2\right)$ of polystable principal $G_2$-bundles is, then, a complex algebraic variety of dimension $14(g-1)$, which parametrizes isomorphism classes of polystable principal $G_2$-bundles over X. It is irreducible since $G_2$ is simply connected, and the subvariety $M_s\left(G_2\right)$ of stable $G_2$-bundles is an open dense subset of $M\left(G_2\right)$, which is composed by smooth points of $M\left(G_2\right)$.

Since the center of $G_2$ is $Z\left(G_2\right)=\left\{1\right\}$, a principal $G_2$-bundle over X is simple if it admits no nontrivial automorphisms. The aim now is to prove that $M_s\left(G_2\right)$ coincides with the subvariety of stable and simple principal $G_2$-bundles over X (that is, it will be checked that every stable principal $G_2$-bundle over X is also simple) and that every simple $G_2$-bundle is irreducible, meaning that it admits no reductions in the structure group to a proper subgroup of $G_2$.

Lemma 1. 

Let G be a simple complex Lie group that admits no nontrivial automorphisms, and let E be a stable principal G-bundle over X. Then, E is simple, meaning that the only automorphisms that E admits are those coming from the center of G.

Proof. 

Let $\mathfrak{g}$ be the Lie algebra of G and $E\left(\mathfrak{g}\right)$ be the adjoint bundle of E, the typical fiber of which is $\mathfrak{g}$. Since E is stable, $E\left(\mathfrak{g}\right)$ is also stable as a vector bundle and has a degree of 0 [26] (this is due to the simplicity of G, which implies that the adjoint representation is irreducible; in general, for semisimple groups, the semistability of E implies the semistability of $E\left(\mathfrak{g}\right)$). Then, any global section of $E\left(\mathfrak{g}\right)$ is constant, which corresponds to elements of $\mathfrak{g}$ that commute with the image of the holonomy homomorphism of E, ${\pi }_1\left(X\right)\to G$. This image is not contained in any proper parabolic subgroup of G due to the stability of E (if that image were contained in a proper subgroup of G, E would admit a reduction in the structure group that E can not admit by stability). Let H be the centralizer in G of the image of the holonomy homomorphism of E. If H were a proper non-central subgroup of G, then the image of the holonomy would be contained in the normalizer of this proper subgroup, which is a parabolic subgroup of G, since G is simple. As the image of the holonomy is not contained in any parabolic subgroup, H cannot be a proper non-central subgroup of G, so there are only three possibilities: $H=G$, H is trivial, or H coincides with the center Z of G. If $H=G$, then the conjugation by elements of the image of the holonomy homomorphism is the identity of G, and this action corresponds to the action of a nontrivial outer automorphism of G. Since, by hypothesis, G admits no nontrivial outer automorphisms, it must be $H=Z$ or $H=1$. In any case, the constant sections of $E\left(\mathfrak{g}\right)$ come from elements in Z. Since the automorphisms of E are in bijective correspondence with the global sections of $E\left(\mathfrak{g}\right)$, which are constant sections, these automorphisms must come from Z. Then, $Aut\left(E\right)=Z$, so E is simple. □

Remark 1. 

Notice that $G=G_2$ satisfies the conditions of Lemma 1 since $G_2$ is simple and its group of outer automorphisms is trivial. Other simple complex Lie groups like $E_7$ or $E_8$ satisfy these conditions; so, according to Lemma 1, stable bundles are also simple for these Lie groups.

Proposition 1. 

Let E be a stable principal $G_2$-bundle over X. Then, E is irreducible, meaning that E admits no reductions in the structure group to a proper subgroup of $G_2$.

Proof. 

The principal $G_2$-bundle E is stable. Then, it is simple to obtain using Lemma 1. Suppose that E is not irreducible. Then, E admits a reduction in the structure group $E_H$ to a proper subgroup H of $G_2$. If H is a discrete subgroup, then the fibers of $E_H$ are discrete. If H were infinite, then the automorphisms of $E_H$ would include the translations by elements of H, so it would be infinite. Therefore, H must be finite. In this case, the projection $E_H\to X$ is a finite cover of X, and it induces a representation $\rho :{\pi }_1\left(X\right)\to H$. If $\pi :\tilde{X}\to X$ is the universal cover of X, then ${\pi }^{*}{E}_H$ is a trivial principal H-bundle over $\tilde{X}$. Since ${\pi }^{*}{E}_H$ is trivial, it admits a nontrivial holomorphic global section s. This section descends to define a nontrivial holomorphic global section ${\pi }_{*}s$ of $E_H\to X$ since s is invariant under the action of the finite covering group. Then, ${\pi }_{*}s$ provides a nontrivial holomorphic global section of E since $E_H$ is a reduction in E to H. Therefore, E is trivial, so it cannot be stable. This is a contradiction, so H must have a positive dimension.

Suppose that H is now a proper subgroup of $G_2$ with a positive dimension. Then, it may be supposed that H is a maximal proper subgroup of $G_2$. Dynkin [27] gave a complete description of these maximal proper subgroups, which are $SL(3,\mathbb{C})$, $SO(4,\mathbb{C})$, and the proper parabolic subgroups. Moreover, a representation $\rho :{\pi }_1\left(X\right)\to H\subset G_2$ is induced by the theorem of Narasimhan and Seshadri for principal bundles [8,28,29]. However, this contradicts the simplicity of E since non-central automorphisms of E are induced. Thus, $E_H$ can be reconstructed by $E_H=(\tilde{X}\times H)/{\pi }_1\left(X\right)$, where $\tilde{X}$ is the universal cover of X, and the action of ${\pi }_1\left(X\right)$ is obvious, induced by $\rho$. Then, $E^{\prime }$ defined by $E^{\prime }=E_H{\times }_H{G}_2$ is a principal $G_4$-bundle with $E^{\prime }\cong E$ since both $G_2$-bundles correspond to the same representation of ${\pi }_1\left(X\right)$ on $G_2$. However, the group of automorphisms of $E^{\prime }$ is isomorphic to the centralizer of H in $G_2$, which is, of course, strictly greater than the center of $G_2$, as can be noted by the explicit description of the maximal proper subgroups above. This finally proves that E is not simple, which contradicts the fact that E is stable. □

In [30] (Theorem 4.1), a Torelli theorem for moduli spaces of principal bundles over Riemann surfaces with a semisimple complex structure group is proved. The particularization of this result to the group $G_2$ is as follows.

Theorem 1 

([30] Theorem 4.1). Let X and $X^{\prime }$ be smooth projective curves of genus g and $g^{\prime }$, respectively, with $g,g^{\prime }\ge 2$. Let $M\left(G_2\right)$ and $M^{\prime }\left(G_2\right)$ be the moduli spaces of principal $G_2$-bundles over X and $X^{\prime }$, respectively. Syppose that $M\left(G_2\right)\cong M^{\prime }\left(G_2\right)$. Then, $X\cong X^{\prime }$.

The inclusion $G_2\hookrightarrow Spin(8,\mathbb{C})$ described in (3) gives rise to an algebraic map between moduli spaces

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

(4)

called the forgetful map, which preserves the polystability condition (that is, any polystable $G_2$-bundle is moved to a polystable $Spin(8,\mathbb{C})$-bundle). The principal $G_2$-bundle E is moved through this map to the obvious principal $Spin(8,\mathbb{C})$ given by the inclusion of the groups above.

3. The Hitchin Integrable System of $\mathit{M}\left({\mathbf{G}}_{\mathbf{2}}\right)$

Let X be a compact Riemann surface of genus $g\ge 2$, G be a semisimple complex Lie group, and $M\left(G\right)$ be the moduli space of polystable principal G-bundles over X. Let $\mathcal{M}\left(G\right)$ be the moduli space of G-Higgs bundles over X and $M_s\left(G\right)$ be the moduli space of stable principal G-bundles, which is a dense open subset of $M\left(G\right)$. Take $p_1,\dots ,p_r$ to be a basis of the ring of invariant homogeneous polynomials of the Lie algebra $\mathfrak{g}$ of G, where r is the rank of $\mathfrak{g}$. The evaluation of each polynomial on the Higgs field defines a map

$$(p_1,\dots ,p_r):H^0(X,E\left(\mathfrak{g}\right)\otimes K)\to \underset{i=1}{\overset{r}{⨁}}{H}^0(X,K^{d_i}),$$

where $d_i=deg{p}_i$ for $i=1,\dots ,r$, and K denotes the canonical line bundle over X. The map defined above allows for defining the Hitchin map,

$$\mathcal{H}:\mathcal{M}\left(G\right)\to \underset{i=1}{\overset{r}{⨁}}{H}^0(X,K^{d_i}).$$

(5)

The vector space

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

is called the base of the Hitchin map $\mathcal{H}$. In [31], it is proved that $dim\mathcal{M}\left(G\right)=dimB$. Let n be the dimension of $\mathcal{M}\left(G\right)$. The Hitchin map induces n complex-valued functions $f_1,\dots ,f_n$ defined on $T^{*}{M}_s\left(G\right)$. The tangent space to $M_s\left(G\right)$ at an element $E\in M_s\left(G\right)$ 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. In [31] (Proposition 4.5), Hitchin proved that the n functions $f_i$ Poisson-commute with the canonical symplectic structure of the cotangent bundle. This defines a completely integrable system on $\mathcal{M}\left(G\right)$ (see [32] for details).

Consider the group $G_2$. The base $B\left(G_2\right)$ of the Hitchin map $\mathcal{H}:T^{*}{M}_s\left(G_2\right)\to B\left(G_2\right)$ is

$$B\left(G_2\right)=H^0(X,K^2)\oplus H^0(X,K^6).$$

(6)

Here, the element $(a_2,a_6)\in B\left(G_2\right)$, where $a_j\in H^0(X,K^j)$ for $j=2,6$, is associated to the invariant polynomial

$$t^6+a_2{t}^4+a_6$$

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 $det(tI-\phi )=t(a_6\left(\phi \right)+a_2\left(\phi \right)t^4+t^6)$.

The fiber of the cotangent bundle $T^{*}{M}_s\left(G_2\right)$ at a stable principal $G_2$-bundle E is identified with the group of Higgs fields

$$T_E^{*}{M}_s\left(G_2\right)\cong H^0(X,E\left({\mathfrak{g}}_{\mathfrak{2}}\right)\otimes K),$$

where ${\mathfrak{g}}_2$ denotes the Lie algebra of $G_2$.

It is well known that for any complex semisimple Lie group G, the multiplicative group ${\mathbb{C}}^{*}$ acts on $T^{*}{M}_s\left(G\right)$ via the product on the Higgs field: if $(E,\phi )\in T^{*}{M}_s\left(G\right)$ and $\lambda \in {\mathbb{C}}^{*}$, then $\lambda ·(E,\phi )=(E,\lambda \phi )$. There is also a natural action of ${\mathbb{C}}^{*}$ on the base of the Hitchin map. In the case of $G_2$ this action can be described as follows: if $(a_2,a_6)\in B\left(G_2\right)$ and $\lambda \in {\mathbb{C}}^{*}$, then

$$\lambda ·(a_2,a_6)=({\lambda }^2{a}_2,{\lambda }^6{a}_6).$$

(7)

Proposition 2. 

Let B be the vector space $B\left(G_2\right)=H^0(X,K^2)\oplus H^0(X,K^6)$ defined in (6). Let $\mathcal{H}:T^{*}{M}_s\left(G_2\right)\to B$ be the Hitchin map. Then, the action of ${\mathbb{C}}^{*}$ on B defined in (7) is the unique ${\mathbb{C}}^{*}$-action on B that makes the Hitchin map ${\mathbb{C}}^{*}$-equivariant. Moreover, the subspace $B_6=H^0(X,K^6)$ of B can be reconstructed from the action of ${\mathbb{C}}^{*}$ on B. Specifically, for $\lambda \in {\mathbb{C}}^{*}$, $B_6$ is the subspace of B, where the ratio of convergence is bigger than ${\lambda }^6$ when λ goes to 0.

Proof 

Suppose that there exists an action of ${\mathbb{C}}^{*}$ on B that is different from that defined in (7), which makes $\mathcal{H}$${\mathbb{C}}^{*}$-equivariant. Denote this action by $\lambda \diamond (a_2,a_6)$ for $\lambda \in {\mathbb{C}}^{*}$ and $(a_2,a_6)\in B$. Then, there exist holomorphic functions $h_2,h_6:{\mathbb{C}}^{*}\to {\mathbb{C}}^{*}$ such that $\lambda \diamond (a_2,a_6)=(h_2\left(\lambda \right)a_2,h_6\left(\lambda \right)a_6)$. Since $\mathcal{H}$ is ${\mathbb{C}}^{*}$-equivariant, it should be

$$\mathcal{H}(\mu \lambda ·(E,\phi \left)\right)=\left(\mu \lambda \right)\diamond \mathcal{H}(E,\phi )=\mu \diamond (\lambda \diamond \mathcal{H}(E,\phi \left)\right).$$

From this, it is clear that $h_2$ and $h_6$ are homomorphisms of the multiplicative group ${\mathbb{C}}^{*}$. Then, $h_2\left(\lambda \right)={\lambda }^{r_2}$ and $h_6\left(\lambda \right)={\lambda }^{r_6}$ for some exponents $r_2,r_6\in \mathbb{Z}$. According to the ${\mathbb{C}}^{*}$-equivariance of $\mathcal{H}$, the exponents must coincide with the degree of the corresponding invariant so that $r_2=2$ and $r_6=6$. Then, the action ⋄ is the action defined in (7).

Now, the action of ${\mathbb{C}}^{*}$ on B defined in (7) determines the origin $0\in B$ since it is the only fixed point of the action. The vector space B is then endowed with an affine structure. Moreover, $B_6$ can be recovered as the subspace where the ratio of convergence is bigger than ${\lambda }^6$ when $\lambda$ goes to 0, as stated. □

Remark 2. 

Notice that, as a consequence of Proposition 2, $B\left(G_2\right)$ defined in (6) is endowed with a structure of an affine space. This follows from the existence of the action of ${\mathbb{C}}^{*}$ defined in (7).

4. Automorphisms of the Moduli Space of Principal ${\mathbf{G}}_{\mathbf{2}}$-Bundles

Let X be a compact Riemann surface of genus $g\ge 2$ and $M\left(G_2\right)$ be the moduli space of polystable principal $G_2$-bundles over X. Notice that any automorphism $\alpha :X\to X$ of the base curve X induces an automorphism of the moduli space $M\left(G_2\right)$ by taking the pull-back:

$${\alpha }^{*}:M\left(G_2\right)\to M\left(G_2\right),E\mapsto {\alpha }^{*}E.$$

(8)

This section aims to prove that $M\left(G_2\right)$ admits no other automorphisms than those coming from the automorphisms of the curve X, as defined in (8). The strategy will be the following: any automorphism $\mathrm{\Phi }$ of $M\left(G_2\right)$ induces an automorphism $\alpha$ of the curve X, which is given by the Torelli theorem (Theorem 1). It will be checked that $\mathrm{\Phi }={\alpha }^{*}$, with the notation of (8), for some $\alpha :X\to X$. This is an adaptation of the analogous result proved in [16] for symplectic groups, which is simplified thanks to the results, proved here; this allows us to conclude that every stable $G_2$-bundle is also simple (Lemma 1) and irreducible (Proposition 1). In the situation described, it will be proved that for any stable principal $G_2$-bundle E over X, the adjoint vector bundles $E\left({\mathfrak{g}}_2\right)$ and $\mathrm{\Phi }\left(E\right)\left({\mathfrak{g}}_2\right)$ are isomorphic with their structures of Lie algebra bundles, and it must be $E\cong \mathrm{\Phi }\left(E\right)$ as principal $G_2$-bundles. The final result (Theorem 2) follows since the open subvariety of stable principal $G_2$-bundles is dense in $M\left(G_2\right)$.

Lemma 2. 

Let G be a semisimple complex Lie group, $\mathfrak{g}$ be the Lie algebra of G, E be an irreducible principal G-bundle over X, and $E\left(\mathfrak{g}\right)$ be the adjoint bundle of E, the typical fiber of which is $\mathfrak{g}$. Let $x\in X$ be any point and $E\left(\mathfrak{g}\right)\left(x\right)$ be the sheaf associated with the effective divisor defined by x. Then, $H^0(X,E\left(\mathfrak{g}\right)\left(x\right))=0$.

Proof. 

Suppose that, under the conditions and the notation of the statement, there exists $s\in H^0(X,E\left(\mathfrak{g}\right)\left(x\right))$ such that $s\ne 0$. By definition, $s\left(x\right)=0$. Since E is irreducible, $End\left(E\right)$ is stable regarding degree 0. The set of zeros of s is closed analytic, so it is discrete, and since X is compact, it must be finite. Then, s defines an effective divisor D, and then a proper sub-bundle of $E\left(\mathfrak{g}\right)$ with a positive degree, contrary to the stability of $E\left(\mathfrak{g}\right)$. Therefore, such a section s may not exist. □

Remark 3. 

Proposition 2 has been proved in the general situation in which E is irreducible. Notice that when G is the group $G_2$, if E is stable, then it is irreducible by Proposition 1, so the result works for stable $G_2$-bundles.

Proposition 3. 

Let B be the base of the Hitchin map of $M\left(G_2\right)$ defined in (6), and let $B_6$ be the subspace of B given by Proposition 2. Let E be a stable principal $G_2$-bundle over X and ${\mathcal{H}}_6:H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to B_6$ be the composition of the Hitchin map $\mathcal{H}$ with the projection of B over $B_6$. For each $x\in X$, let

$$B_{6,x}=H^0\left(K^6(-x)\right)\subseteq B_6.$$

Then,

$$\begin{array}{cc}\hfill H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x))=& \\ & \{s\in H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K):{\mathcal{H}}_6(s+t)\in B_{6,x}\forall t\in {\mathcal{H}}^{-1}\left(B_{6,x}\right)\}.\hfill \end{array}$$

Proof. 

Notice that the stable principal $G_2$-bundle E is simple according to Lemma 1 and is also irreducible by Proposition 1. The exact sequence of sheaves

$$0\to E\left({\mathfrak{g}}_2\right)\otimes K(-x)\to E\left({\mathfrak{g}}_2\right)\otimes K\to {\left.E\left({\mathfrak{g}}_2\right)\otimes K\right|}_x\to 0$$

induces an exact sequence of cohomology sets of the form

$$H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to {\left.E\left({\mathfrak{g}}_2\right)\otimes K\right|}_x\to H^1(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x)).$$

From Lemma 2 and Serre duality, it follows that

$$H^1(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x))=H^0{(X,E\left({\mathfrak{g}}_2\right)\left(x\right))}^{*}=0,$$

so the map $H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to {\left.E\left({\mathfrak{g}}_2\right)\otimes K\right|}_x$, which consists of evaluating in x, is surjective.

Now, take $s\in H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)$. It is clear that ${\mathcal{H}}_6\left(s\right)\in B_6$ if and only if $p_6\left(s\left(x\right)\right)=0$, where $p_6$ is the fundamental homogeneous invariant polynomial with a degree of 6. The result, then, follows from the following easy linear algebra fact: the only possibility for such an element s that satisfies $p_6(s\left(x\right)+t\left(x\right))=0$ for all t with $p_6\left(t\left(x\right)\right)=0$ is $s\left(x\right)=0$ (note that if a matrix $A\in {\mathfrak{g}}_2$ satisfies that $p_6(A+B)=0$ for any $B\in {\mathfrak{g}}_2$ with $p_6\left(B\right)=0$, then for any such B, the whole line through A and $nB$, $n\in \mathbb{N}$ is contained in the zero set of the polynomial $p_6$; this necessarily implies $A=0$). □

From Proposition 3, a bundle $\mathcal{F}$ over X associated with each stable principal $G_2$-bundle E that has fiber over any point $x\in X$ is defined as follows:

$${\mathcal{F}}_x=H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x)).$$

(9)

This bundle $\mathcal{F}$ is nothing but the kernel of the homomorphism of vector bundles

$$H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\otimes \mathcal{O}\to E\left({\mathfrak{g}}_2\right)\otimes K,$$

where $\mathcal{O}\to X$ denotes the trivial line bundle over X.

Proposition 4. 

Let B be the base of the Hitchin map of $M\left(G_2\right)$ defined in (6) and $B_r=H^0(X,K^r)$ be the corresponding summand in the decomposition of B for $r=2,6$. Then, the linear structure and the decomposition $B=B_2\oplus B_6$ are uniquely determined by the isomorphism class of $M\left(G_2\right)$ in the sense that an automorphism of $M\left(G_2\right)$ induces an automorphism of B, which is diagonal for the decomposition $B=B_2\oplus B_6$.

Proof. 

Let $\mathrm{\Phi }$ be an automorphism of $M\left(G_2\right)$. Let E be a stable principal $G_2$-bundle, and let $E^{\prime }=\mathrm{\Phi }\left(E\right)$. From Theorem 1, the automorphism $\mathrm{\Phi }$ induces an automorphism $\alpha$ of the base curve X. By composing with ${\left({\alpha }^{-1}\right)}^{*}$, it may be supposed that this automorphism $\alpha$ is the identity.

The isomorphism $d{\mathrm{\Phi }}^{*}:H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to H^0(X,E^{\prime }\left({\mathfrak{g}}_2\right)\otimes K)$ induces a linear isomorphism $f:B\to B$ that commutes with the Hitchin map.

Let $\mathcal{F}$ and ${\mathcal{F}}^{\prime }$ be the bundles over X defined in (9) for E and $E^{\prime }$, respectively. According to Proposition 3, the automorphism $\mathrm{\Phi }$ induces an isomorphism of bundles $\varphi :\mathcal{F}\to {\mathcal{F}}^{\prime }$. It is easy to prove that there are exact sequences of bundles

$$0\to \mathcal{F}\to H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\otimes \mathcal{O}\to E\left({\mathfrak{g}}_2\right)\otimes K\to 0$$

and

$$0\to {\mathcal{F}}^{\prime }\to H^0(X,E^{\prime }\left({\mathfrak{g}}_2\right)\otimes K)\otimes \mathcal{O}\to E^{\prime }\left({\mathfrak{g}}_2\right)\otimes K\to 0.$$

From the isomorphisms $\varphi$ and $d{\mathrm{\Phi }}^{*}$, an isomorphism $E\left({\mathfrak{g}}_2\right)\otimes K\to E^{\prime }\left({\mathfrak{g}}_2\right)\otimes K$ is obtained, and also the isomorphism $\psi :E\left({\mathfrak{g}}_2\right)\to E^{\prime }\left({\mathfrak{g}}_2\right)$ is derived.

Notice that the map $d{\mathrm{\Phi }}^{*}$ considered above is induced by $\psi$ and that the Hitchin map $\mathcal{H}:H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to B$ factors as the composition of the map

$$H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to \underset{r=2,6,8,12}{⨁}{H}^0\left(X,{\wedge }^{r}E\left({\mathfrak{g}}_2\right)\otimes K^r\right)$$

with the trace map over B. Then, the isomorphism $d{\mathrm{\Phi }}^{*}$ descends to a diagonal isomorphism with respect to the natural decompositions of $H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)$ and $H^0(X,E^{\prime }\left({\mathfrak{g}}_2\right)\otimes K)$:

$$\underset{r=2,6}{⨁}{H}^0(X,{\wedge }^{r}E\left({\mathfrak{g}}_2\right)\otimes K^r)\to \underset{r=2,6}{⨁}{H}^0(X,{\wedge }^r{E}^{\prime }\left({\mathfrak{g}}_2\right)\otimes K^r).$$

Therefore, the automorphism f of B is also diagonal with respect to the decomposition $B={\oplus }_{r=2,6}{B}_r$. Then, there exist ${\mathbb{C}}^{*}$-equivariant maps $f_r:B_r\to B_r$ for $r=2,6$, which induce a weight of f for each $B_r$. This shows that f is linear for the decomposition of B given above. □

Theorem 2. 

Let $M\left(G_2\right)$ be the moduli space of polystable principal $G_2$-bundles over the compact Riemann surface X of genus $g\ge 2$. Let $\mathrm{\Phi }:M\left(G_2\right)\to M\left(G_2\right)$ be an automorphism of $M\left(G_2\right)$. Then, there exists an automorphism α of X such that $\mathrm{\Phi }={\alpha }^{*}$, where ${\alpha }^{*}$ is defined in (8).

Proof. 

First, notice that the automorphism of X induced by $\mathrm{\Phi }$ given by Theorem 1 may be supposed to be the identity.

Let E be a stable principal $G_2$-bundle over X, $\mathcal{H}:H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to B$ be the Hitchin map, and B be the base of the Hitchin map. For $r=2,6$, let $B_r$ be the summand $B_r=H^0(X,K^r)$ in the decomposition of B, which is well-defined by Proposition 4. For each $r=2,6$, let ${\mathcal{H}}_r$ be the composition of the Hitchin map $\mathcal{H}$ with the projection map of B over $B_r$. Let $f:B\to B$ be the automorphism of B induced by $\mathrm{\Phi }$. Let $x\in X$. The arguments of Proposition 4 show that ${\mathcal{H}}_r\left(H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x))\right)$ is preserved by f, so $d{\mathrm{\Phi }}^{*}$ preserves ${\cap }_{r=2,6}{\mathcal{H}}_r^{-1}\left({\mathcal{H}}_r\left(H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K(-x))\right)\right)$. The image of this set under the surjective map $H^0(X,E\left({\mathfrak{g}}_2\right)\otimes K)\to {\left.E\left({\mathfrak{g}}_2\right)\otimes K\right|}_x$ coincides with the nilpotent cone

$$N{\left(E\right)}_x=\{A\in {\left.E\left({\mathfrak{g}}_2\right)\otimes K\right|}_x:A\mathrm{is}\mathrm{nilpotent}\}.$$

Note that the union of all the nilpotent cones when x goes through X gives the nilpotent bundle over X associated with E, which will be called $N\left(E\right)$. The discussion above allows us to prove that there is an isomorphism of the nilpotent bundles $N\left(E\right)\to N\left(\mathrm{\Phi }\right(E\left)\right)$ as bundles over X (since the automorphism of the curve X induced by $\mathrm{\Phi }$ is the identity, as it has been supposed).

The main result of [33] proves that this automorphism between the nilpotent cone bundles induces an automorphism of the corresponding Borel flags varieties (which parametrizes the flags induced in the $G_2$-bundle according to the reductions in the structure group to a Borel subgroup of $G_2$), which will be called $\mathrm{Flag}\left(E\right)$ and $\mathrm{Flag}\left(\mathrm{\Phi }\right(E\left)\right)$. The automorphism $\mathrm{Flag}\left(E\right)\to \mathrm{Flag}\left(\mathrm{\Phi }\right(E\left)\right)$ induces a Lie algebra bundle isomorphism $E\left({\mathfrak{g}}_2\right)\cong \mathrm{\Phi }\left(E\right)\left({\mathfrak{g}}_2\right)$. Notice that giving a Lie algebra bundle structure is equivalent to giving a principal $Aut\left({\mathfrak{g}}_2\right)$-bundle together with a reduction in the structure group to $G_2$, corresponding to E. Since $G_2$ is centerless and admits no outer automorphisms, the Lie algebra automorphism $E\left({\mathfrak{g}}_2\right)\cong \mathrm{\Phi }\left(E\right)\left({\mathfrak{g}}_2\right)$ gives an automorphism of principal $G_2$-bundles $E\cong \mathrm{\Phi }\left(E\right)$.

This holds for any E of the dense open subvariety of stable bundles of $M\left(G_2\right)$, so the result is finally proved. □

5. The Morphism $\mathit{M}\left({\mathit{G}}_{\mathbf{2}}\right)\to \mathit{M}\left(\mathbf{Spin}(\mathbf{8},\mathbb{C})\right)$

The main objective of this section is to show that the forgetful map defined in (4) is injective. The strategy consists of first proving that the obstruction for two polystable $G_2$-bundles, the associated $Spin(8,\mathbb{C})$-bundles of which are isomorphic, are also isomorphic, as $G_2$-bundles are an element of the non-abelian cohomology set $H^1(X,Spin(8,\mathbb{C})/G_2)$. Next, sufficient conditions will be given for a complex semisimple Lie group G and a complex subgroup of it H to satisfy that $H^1(X,G/H)=0$. Finally, it will be checked that the groups $Spin(8,\mathbb{C})$ and $G_2$ satisfy all these conditions.

Proposition 5. 

Let X be a compact Riemann surface of genus $g\ge 2$, G be a semisimple complex connected Lie group, and H be a simple maximal complex Lie subgroup of G. Let $E_1$ and $E_2$ be polystable principal H-bundles over X such that the induced polystable G-bundles $\overline{E_1}$ and $\overline{E_2}$ are isomorphic. Then, the isomorphism $\overline{E_1}\cong \overline{E_2}$ defines an element in the non-abelian cohomology $H^1(X,G/H)$. Moreover, this cohomology class is trivial if and only if the isomorphism $\overline{E_1}\cong \overline{E_2}$ lifts to an isomorphism $E_1\cong E_2$ of principal H-bundles.

Proof. 

Notice that since H is maximal in G, $G/H$ is a homogeneous complex variety and, indeed, a projective algebraic variety. According to the Borel-Remmert theorem [34], it is the product of an abelian variety and a rational variety (so it is a rational variety, given that G is semisimple). Since $E_1$ and $E_2$ are polystable as principal H-bundles, they correspond to certain reductive representations ${\rho }_1,{\rho }_2:{\pi }_1\left(X\right)\to H$, where ${\pi }_1\left(X\right)$ is the fundamental group of X [8]. By composing these representations with the inclusion of groups $i:H\to G$, two reductive representations of ${\pi }_1\left(X\right)$ in G are obtained, ${\sigma }_1=i\circ {\rho }_1$ and ${\sigma }_2=i\circ {\rho }_2$ (since H is also semisimple, as it is simple), which, of course, correspond to the polystable principal G-bundles $\overline{E_1}$ and $\overline{E_2}$. Since $\overline{E_1}\cong \overline{E_2}$, ${\sigma }_1$ and ${\sigma }_2$ are conugate, meaning that there exists $g\in G$ such that ${\sigma }_2\left(\gamma \right)=g{\sigma }_1\left(\gamma \right)g^{-1}$ for all $\gamma \in {\pi }_1\left(X\right)$.

Let M be the homogeneous space $G/H$ and $s:{\pi }_1\left(X\right)\to M$ be defined by $s\left(\gamma \right)=g{\rho }_1\left(\gamma \right)g^{-1}H$, where $g{\rho }_1\left(\gamma \right)g^{-1}H$ denotes the class of $g{\rho }_1\left(\gamma \right)g^{-1}$ in M. It is easily checked that s satisfies the cocycle properties since

$$\begin{array}{cc}\hfill s\left(\gamma \delta \right)& =g{\rho }_1\left(\gamma \delta \right)g^{-1}H=g{\rho }_1\left(\delta \right)g^{-1}H\hfill \\ \hfill & =g{\rho }_1\left(\gamma \right)g^{-1}\left(g{\rho }_1\left(\delta \right)g^{-1}\right))H=s\left(\gamma \right)·({\rho }_1\left(\gamma \right)·s\left(\delta \right)),\hfill \end{array}$$

where · denotes the action on the right of G on M. Now, there is a bijective correspondence between cocycles ${\pi }_1\left(X\right)\to M$ and rational sections of the M-bundle associated with $E_1$, $E_1{\times }_{H}M$ (this bijective correspondence is assigned to $s:{\pi }_1\left(X\right)\to M$, a section $\sigma$ of $E_1{\times }_{H}M$, as follows: for each $x\in X$, if $\gamma$ is a choice of a path from a fixed base point $x_0$ to x, then $\sigma \left(x\right)=[e_0,s\left(\gamma \right)]$, where $e_0\in E_1$ is in the fiber of $E_1$ over $x_0$; the independence on the choice of $\gamma$ follows from the cocycle property). The existence of rational sections is assured because M is a rational variety. Then, s may be understood as a global section $X\to E_1{\times }_{H}M$ through the right action of H on M (as X is a compact Riemann surface, for curves, the existence of a rational section implies the existence of a global section outside a finite set of points, but this can be extended to the whole curve), so it defines a cohomology class $\left[s\right]$ in the non-abelian cohomology set $H^1(X,M)$.

The cohomology class $\left[s\right]$ is trivial if and only if there exists $h\in G$ such that $s\left(\gamma \right)=h^{-1}({\rho }_1\left(\gamma \right)·h)$ for all $\gamma \in {\pi }_1\left(X\right)$. In this case, according to the definition of s, it is satisfied that

$$g{\rho }_1\left(\gamma \right)g^{-1}H=h^{-1}({\rho }_1\left(\gamma \right)·h)$$

for every $\gamma \in {\pi }_1\left(X\right)$. This leads to the existence of some $t\left(\gamma \right)\in H$ such that

$$g{\rho }_1\left(\gamma \right)g^{-1}=h^{-1}{\rho }_1\left(\gamma \right)ht\left(\gamma \right).$$

It will now be checked that $t\left(\gamma \right)$ defines a cocycle $t:{\pi }_1\left(X\right)\to H$ in the sense that $t\left(\alpha \beta \right)=t\left(\alpha \right)({\rho }_1\left(\alpha \right)·t\left(\beta \right))$. By taking $\gamma =\alpha \beta$ in $g{\rho }_1\left(\gamma \right)g^{-1}=h^{-1}{\rho }_1\left(\gamma \right)ht\left(\gamma \right)$ and applying that ${\rho }_1\left(\alpha \beta \right)={\rho }_1\left(\alpha \right){\rho }_1\left(\beta \right)$, it is obtained that

$$g{\rho }_1\left(\alpha \beta \right)g^{-1}=h^{-1}{\rho }_1\left(\alpha \beta \right)ht\left(\alpha \beta \right)=h^{-1}{\rho }_1\left(\alpha \right){\rho }_1\left(\beta \right)ht\left(\alpha \beta \right).$$

Now, by using the expression $g{\rho }_1\left(\gamma \right)g^{-1}=h^{-1}{\rho }_1\left(\gamma \right)ht\left(\gamma \right)$ twice for ${\rho }_1\left(\alpha \right)$ and ${\rho }_1\left(\beta \right)$ and simplifying,

$$\begin{array}{cc}\hfill h^{-1}{\rho }_1\left(\alpha \right){\rho }_1\left(\beta \right)ht\left(\alpha \beta \right)& =\left(h^{-1}{\rho }_1\left(\alpha \right)ht\left(\alpha \right)\right)\left(h^{-1}{\rho }_1\left(\beta \right)ht\left(\beta \right)\right)\hfill \\ & =h^{-1}{\rho }_1\left(\alpha \right)ht\left(\alpha \right)h^{-1}{\rho }_1\left(\beta \right)ht\left(\beta \right).\hfill \end{array}$$

Then,

$${\rho }_1\left(\beta \right)ht\left(\alpha \beta \right)=ht\left(\alpha \right)h^{-1}{\rho }_1\left(\beta \right)ht\left(\beta \right).$$

When multiplying by $h^{-1}$ on the left,

$$h^{-1}{\rho }_1\left(\beta \right)ht\left(\alpha \beta \right)=t\left(\alpha \right)h^{-1}{\rho }_1\left(\beta \right)ht\left(\beta \right).$$

Recall that $h^{-1}{\rho }_1\left(\beta \right)h=g{\rho }_1\left(\beta \right)g^{-1}t{\left(\beta \right)}^{-1}$, so

$$t\left(\alpha \right)\left(g{\rho }_1\left(\beta \right)g^{-1}t{\left(\beta \right)}^{-1}\right)t\left(\beta \right)=g{\rho }_1\left(\beta \right)g^{-1}t\left(\alpha \beta \right).$$

By simplifying,

$$t\left(\alpha \right)g{\rho }_1\left(\beta \right)g^{-1}=g{\rho }_1\left(\beta \right)g^{-1}t\left(\alpha \beta \right).$$

When multiplying by $g^{-1}$ on the left and by g on the right,

$$g^{-1}t\left(\alpha \right)g{\rho }_1\left(\beta \right)={\rho }_1\left(\beta \right)g^{-1}t\left(\alpha \beta \right)g.$$

Finally, it is obtained that

$$t\left(\alpha \beta \right)=g^{-1}t\left(\alpha \right)g{\rho }_1\left(\beta \right)g^{-1}t\left(\beta \right)g=t\left(\alpha \right)({\rho }_1\left(\alpha \right)·t\left(\beta \right)),$$

since the action of ${\rho }_1\left(\alpha \right)$ on $t\left(\beta \right)$ by conjugation is

$$g^{-1}{\rho }_1\left(\alpha \right)gt\left(\beta \right)g^{-1}{\rho }_1{\left(\alpha \right)}^{-1}g=g^{-1}t\left(\alpha \right)g{\rho }_1\left(\beta \right)g^{-1}t\left(\beta \right)g.$$

This finally proves that t defines a cocycle.

Since ${\rho }_2\left(\gamma \right)=g{\rho }_1\left(\gamma \right)g^{-1}$, then

$${\rho }_2\left(\gamma \right)=g{h}^{-1}{\rho }_1\left(\gamma \right)ht\left(\gamma \right)g^{-1}.$$

Let $k=g{h}^{-1}\in G$. It is easily checked that ${\rho }_2\left(\gamma \right)=k{\rho }_1\left(\gamma \right)k^{-1}\left(kt\left(\gamma \right)g^{-1}\right)$.

Since $\left(ab\right)c{\left(ab\right)}^{-1}=a\left(bc{b}^{-1}\right)a^{-1}$ for a, b, and c in any group, the following is satisfied:

$$\begin{array}{cc}\hfill {\rho }_2\left(\gamma \right)& =k{\rho }_1\left(\gamma \right)k^{-1}r=k{\rho }_1\left(\gamma \right)\left(k^{-1}r\right)=k{\rho }_1\left(\gamma \right)\left(k^{-1}\left(kt\left(\gamma \right)g^{-1}\right)\right)\hfill \\ & =k{\rho }_1\left(\gamma \right)\left(t\left(\gamma \right)g^{-1}\right)=k\left({\rho }_1\left(\gamma \right)t\left(\gamma \right)g^{-1}\right)k^{-1}.\hfill \end{array}$$

It has been obtained that ${\rho }_2\left(\gamma \right)=k\left({\rho }_1\left(\gamma \right)t\left(\gamma \right)g^{-1}\right)k^{-1}$, so ${\rho }_2\left(\gamma \right)$ and ${\rho }_1\left(\gamma \right)t\left(\gamma \right)g^{-1}$ are conjugate.

Consider the map $\psi :H\to M$ defined by $\psi \left(x\right)=x{g}^{-1}H$. This is a constant map since H is a simple subgroup of G, and M is a complex homogeneous variety. Then, $x{g}^{-1}H=g^{-1}H$ for all $x\in H$, so for each $x\in H$, there exists $y\in H$ such that $x{g}^{-1}=g^{-1}y$. By taking $x=t\left(\gamma \right)$, the existence of a $t^{\prime }\left(\gamma \right)\in H$ such that $t\left(\gamma \right)g^{-1}=g^{-1}{t}^{\prime }\left(\gamma \right)$ is obtained. Therefore, ${\rho }_2\left(\gamma \right)$ is conjugate in G to ${\rho }_1\left(\gamma \right)g^{-1}{t}^{\prime }\left(\gamma \right)$. However, since both ${\rho }_1\left(\gamma \right)$ and $t^{\prime }\left(\gamma \right)$ belong to H, it follows that

$${\rho }_1\left(\gamma \right)g^{-1}{t}^{\prime }\left(\gamma \right)=g^{-1}\left(g{\rho }_1\left(\gamma \right)g^{-1}\right)t^{\prime }\left(\gamma \right),$$

where $g{\rho }_1\left(\gamma \right)g^{-1}\in H$. This means that ${\rho }_2\left(\gamma \right)$ is conjugate in G to an element of H for all $\gamma \in {\pi }_1\left(X\right)$. Then, ${\rho }_2\left(\gamma \right)$ is conjugate in H to an element of H for all $\gamma$ since ${\rho }_2$ is a representation that takes values in H, and H is a simple maximal subgroup of G. From this, there exists an element $m\in H$ such that ${\rho }_2\left(\gamma \right)=m\left(g{\rho }_1\left(\gamma \right)g^{-1}\right)m^{-1}$. Define $m^{\prime }=mg\in G$. It is satisfied that ${\rho }_2\left(\gamma \right)=m^{\prime }{\rho }_1\left(\gamma \right)m^{\prime -1}$ for all $\gamma$, so $m^{\prime }$ is in the normalizer of H in G, which is, indeed, H since it is a simple maximal subgroup of G. Then, $m^{\prime }\in H$, so ${\rho }_1$ and ${\rho }_2$ are conjugate in H. This proves that $E_1$ and $E_2$ are, indeed, isomorphic through an isomorphism that lifts the previous automorphism $\overline{E_1}\cong \overline{E_2}$. □

Remark 4. 

Following the proof of Proposition 5, it is possible to give a construction of the isomorphism $E_1\cong E_2$ when the class $\left[s\right]$ induced by the isomorphism $\overline{E_1}\cong \overline{E_2}$ is trivial. Specifically, $\phi :E_1\to E_2$ given by $\phi \left(e\right)=e·h^{-1}$, where h is defined in the proof, is the desired isomorphism of H-bundles. Indeed, φ is a morphism of H-bundles since for every $e\in E_1$ and $a\in H$,

$$\phi (e·a)=(e·a)·h^{-1}=e·\left(a{h}^{-1}\right)=e·\left(h^{-1}\left(ha{h}^{-1}\right)\right)=(e·h^{-1})·\left(ha{h}^{-1}\right)=\phi \left(e\right)·ha{h}^{-1}).$$

Here, $ha{h}^{-1}\in H$ since $\left[s\right]$ is trivial, so h normalizes H. Moreover, if $\phi \left(e_1\right)=\phi \left(e_2\right)$, then obviously $e_1=e_2$, so φ is injective, and for each $e^{\prime }\in E_2$, there exists $e\in E_1$ such that $e·h^{-1}=e^{\prime }$, so $\phi \left(e\right)=e^{\prime }$, meaning φ is surjective. All this proves that φ is an isomorphism $E_1\to E_2$. The isomorphism $\mathrm{\Phi }:\overline{E_1}\to \overline{E_2}$ induced by φ is given by

$$\mathrm{\Phi }\left([e,g]\right)=[\phi \left(e\right),g]=[e·h^{-1},g]=[e,h^{-1}g],$$

so it is the original automorphism $\overline{E_1}\to \overline{E_2}$ defined by conjugation by $h^{-1}$.

Proposition 6. 

Let X be a compact Riemann surface of genus $g\ge 2$ with fundamental group ${\pi }_1\left(X\right)$, G be a complex connected non-abelian semisimple Lie group, and H be a closed subgroup of G that does not contain any parabolic subgroup of G. Suppose that, in addition, the obvious action of G on homogeneous variety $G/H$ is effective, meaning that only $1\in G$ acts trivially on $G/H$, and that the action of G on $G/H$ admits no global fixed points, meaning that no element of $G/H$ is fixed for the action of all the elements of G. Then, the non-abelian cohomology set $H^1(X,G/H)$ is trivial.

Proof. 

Under the conditions of the statement, $G/H$ is a complex homogeneous variable; therefore, according to the Borel-Remmert theorem [34], it is the product of an abelian variety and projective variety. Since G is semisimple, $G/H$ must be a projective variety. On the other hand, it will be checked that H is contained in some parabolic subgroup of G. Suppose that this is not true. Then, according to the Borel-Tits theorem [35], H is a reductive subgroup of G, so it contains a maximal torus of G, which contradicts that the action of G on $G/H$ admits no global fixed points. Therefore, H is contained in a parabolic subgroup P of G. Then, there is a natural fibration $G/H\to G/P$ with fiber $P/H$. It is known that $G/P$ is a rational variety according to the Bruhat-Tits theorem [36] and that $P/H$ is a rational variety since P is a solvable group. Then, $G/H$ is a rational variable since rationality is preserved by fibrations, the fiber and base of which are rational variables.

Every element $\left[E\right]\in H^1(X,G/H)$ comes from a cocycle E that can be understood as a $G/H$-bundle. Since $G/H$ is a rational variety, E admits a rational section s, which can be extended to a global section $\overline{s}:X\to E$ because X is a compact Riemann surface. Indeed, the indetermination set of s is discrete, as the complex dimension of X is 1, so it is finite since X is compact. By applying Riemann’s removable singularity theorem, it follows that s extends to a global section.

The existence of such $\overline{s}$ implies that E is trivial, given that $G/H$ is a rational variable. Indeed, the map $\psi :X\times G/H\to E$ given by $\psi (x,gH)=\overline{s}\left(x\right)·g$, where · denotes the action of G on E, is an isomorphism. Firstly, if $g_{1}H=g_{2}H$, then $g_2=g_{1}h$ for some $h\in H$, and $\overline{s}\left(x\right)·g_{1}h=\overline{s}\left(x\right)·g_1$ because $\overline{s}\left(x\right)$ is in the fiber over x, meaning $\psi$ is well-defined. Secondly, $\psi$ is G-equivariant since

$$\psi (x,g_{1}gH)=\overline{s}\left(x\right)·\left(g_{1}g\right)=(\overline{s}\left(x\right)·g_1)·g=\psi (x,g_{1}H)·g$$

for all $g,g_1\in G$ and $x\in X$. Thirdly, if $g_1,g_2\in G$ and $x_1,x_2\in X$ are such that $\psi (x_1,g_{1}H)=\psi (x_2,g_{2}H)$, then, of course, $x_1=x_2$ and $\overline{s}\left(x_1\right)·g_1=\overline{s}\left(x_2\right)·g_2$, from which $g_{1}H=g_{2}H$, so $\psi$ is injective. Fourthly, if $e\in E$, it is in the fiber of some $x\in X$, so there exists $g\in G$ such that $e=s\left(x\right)·g$, from which $\psi$ is surjective.

This finally proves that $\left[E\right]$ must be trivial. □

It will now be checked that the Lie group $G=Spin(8,\mathbb{C})$ and its subgroup $H=G_2$ satisfy the conditions of Propositions 5 and 6.

Recall that $Spin(8,\mathbb{C})$ is semisimple complex and connected, and $G_2$ is a simple complex subgroup of $Spin(8,\mathbb{C})$. Moreover, $G_2$ is maximal in $Spin(8,\mathbb{C})$ since the Lie algebra ${\mathfrak{g}}_2$ of $G_2$ appears as one of the maximal proper subalgebras of the Lie algebra $\mathfrak{so}(8,\mathbb{C})$ of $Spin(8,\mathbb{C})$ in the Dynkin classification of maximal subalgebras of simple complex Lie algebras [27,37]. This proves that the Lie group $G=Spin(8,\mathbb{C})$ and its subgroup $H=G_2$ satisfy the conditions of Proposition 5.

The group $Spin(8,\mathbb{C})$ is a complex, connected non-abelian semisimple Lie group and $G_2$ is a closed subgroup of $Spin(8,\mathbb{C})$ that does not contain any parabolic subgroup of $Spin(8,\mathbb{C})$ since the dimension of any parabolic subgroup of $Spin(8,\mathbb{C})$ is greater than $14=dim{G}_2$ (notice that parabolic subgroups of $Spin(8,\mathbb{C})$ are in bijective correspondence with parabolic subalgebras of $\mathfrak{so}(8,\mathbb{C})$).

Group $G_2$ does not contain any nontrivial normal subgroup of $Spin(8,\mathbb{C})$ (if $G_2$ admits a normal subgroup of $Spin(8,\mathbb{C})$, then it would be a normal subgroup of $G_2$, which is simple and centerless, so this subgroup is necessarily trivial). Then, the action of $Spin(8,\mathbb{C})$ on the complex homogeneous variety $Spin(8,\mathbb{C})/G_2$ is effective. Indeed, let $ker=\{g\in Spin(8,\mathbb{C}):gx{G}_2=x{G}_2\mathrm{for}\mathrm{all}x\in Spin(8,\mathbb{C})\}$. Then, $g\in ker$ if and only if $gx\in x{G}_2$ for all $x\in Spin(8,\mathbb{C})$. This implies that $x^{-1}gx\in G_2$ for all $x\in Spin(8,\mathbb{C})$, so g should be in the normalizer of $G_2$ in $Spin(8,\mathbb{C})$ for all $x\in Spin(8,\mathbb{C})$. However, since $G_2$ contains no nontrivial normal subgroups of $Spin(8,\mathbb{C})$, ker must be trivial.

Finally, since $G_2$ is not a normal subgroup of $Spin(8,\mathbb{C})$ because $Spin(8,\mathbb{C})$ is simple, the action of $Spin(8,\mathbb{C})$ on $Spin(8,\mathbb{C})/G_2$ has no global fixed points. Indeed, a global fixed point $x{G}_2\in Spin(8,\mathbb{C})/G_2$ would satisfy $g\left(x{G}_2\right)=x{G}_2$ for all $g\in Spin(8,\mathbb{C})$. In this situation, for all $g\in Spin(8,\mathbb{C})$, there exists $h\in G_2$ such that $gx=xh$, that is, $x^{-1}gx=h\in G_2$. This means that $x^{-1}Spin(8,\mathbb{C})x\subset G_2$. Since $G_2$ is not normal, this is impossible. All the above proves that $Spin(8,\mathbb{C})$ and $G_2$ also satisfy the conditions of Proposition 6.

Theorem 3. 

The forgetful map $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ defined in (4) is an embedding.

Proof. 

It will be proved that the forgetful map is injective. Let $E_1$ and $E_2$ be polystable principal $G_2$-bundles over X such that the associated polystable principal $Spin(8,\mathbb{C})$-bundles, $\overline{E_1}$ and $\overline{E_2}$, are isomorphic. The aim is to prove that $E_1$ and $E_2$ are also isomorphic as $G_2$-bundles. Let $\mathrm{\Phi }:\overline{E_1}\to \overline{E_2}$ be an isomorphism. According to Proposition 5, a class $c\in H^1(X,Spin(8,\mathbb{C})/G_2)$ is induced by $\mathrm{\Phi }$, and $\mathrm{\Phi }$ lifts to an isomorphism $E_1\to E_2$ if and only if c is trivial. However, c must be trivial since $H^1(X,Spin(8,\mathbb{C})/G_2)$ is trivial, according to Proposition 6. Then, $E_1$ and $E_2$ must be isomorphic as H-bundles, according to Proposition 5. □

6. Consequences

In Theorem 3, it is proved that the map $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ defined in (4) between moduli spaces over a compact Riemann surface X of genus $g\ge 2$ in an embedding. Therefore, the image of this map provides a copy of $M\left(G_2\right)$ included in $M(Spin(8,\mathbb{C}\left)\right)$. Let $M_0$ be this copy; that is,

$$M_0=Im(M\left(G_2\right)\to M(Spin(8,\mathbb{C}))).$$

(10)

In this section, it is proved that every automorphism of $M\left(G_2\right)$ comes from an automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ through the embedding (4). Of course, this is a consequence of the explicit description of the group of automorphisms given in Theorem 2. Moreover, the existence of other copies of $M\left(G_2\right)$ different form $M_0$ (indeed, disjointed with it) contained in the moduli space $M(Spin(8,\mathbb{C}\left)\right)$ is proved from Theorem 3, and the consideration of certain well-known automorphisms that $M(Spin(8,\mathbb{C}\left)\right)$ admits are described below.

When considering the moduli space $M(Spin(8,\mathbb{C}\left)\right)$ of principal $Spin(8,\mathbb{C})$-bundles over X, three main families of automorphisms are found. Indeed, as a consequence of the work of Fringuelli [18], one has that every automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ is a composition of automorphisms of some of the three following families.

(A).

The group $Aut\left(X\right)$ of automorphisms of the base curve X acts in the moduli space $M(Spin(8,\mathbb{C}\left)\right)$ by taking the pull-back of the bundles. If $\alpha \in Aut\left(X\right)$, the induced automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ will be denoted by ${\alpha }^{*}$.

(B).

The group $Out(Spin(8,\mathbb{C}))\cong S_3$ of outer automorphisms of $Spin(8,\mathbb{C})$ acts in $M(Spin(8,\mathbb{C}\left)\right)$. Then, a family of automorphisms of $M(Spin(8,\mathbb{C}\left)\right)$ parametrized by $S_3$ is induced, which has been intensively studied [19,38,39]. If $E\in M(Spin(8,\mathbb{C}\left)\right)$ and $\rho \in Out(Spin(8,\mathbb{C}\left)\right)$, the principal $Spin(8,\mathbb{C})$-bundle $\rho \left(E\right)$ is defined to be the principal $Spin(8,\mathbb{C})$-bundle over X, the total space of which coincides with that of E and is equipped with the action of $Spin(8,\mathbb{C})$ given by $e★g=eR{\left(g\right)}^{-1}$ for $e\in E$ and $g\in Spin(8,\mathbb{C})$, where R is an automorphism of $Spin(8,\mathbb{C})$ representing $\rho$. It is easily checked that the isomorphism class of E does not change by the action of an inner automorphism of $Spin(8,\mathbb{C})$, so the action is well-defined [19]. Moreover, the order of the automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ thus defined is, of course, the order of the outer automorphism $\rho$ (the possibilities for this order are 2 or 3 if the outer automorphism is nontrivial since $\mathrm{Out}(Spin(8,\mathbb{C}))\cong S_3$). The automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ induced by an outer automorphism $\rho$ of $Spin(8,\mathbb{C})$ will also be denoted by $\rho$. The action of the triality automorphism $\tau$ of $Spin(8,\mathbb{C})$, which is a generator of order 3 of $Out(Spin(8,\mathbb{C}\left)\right)$, induces an automorphism

$$\tau :M(Spin(8,\mathbb{C}\left)\right)\to M(Spin(8,\mathbb{C}\left)\right),E\mapsto \tau \left(E\right),$$

(11)

that will be particularly relevant.

(C).

If $Z\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$ is the center of $Spin(8,\mathbb{C})$, then the group $H^1(X,Z)$, which parametrizes isomorphism classes of principal Z-bundles over X, acts on $M(Spin(8,\mathbb{C}\left)\right)$ via the tensor product. If $L\in H^1(X,Z)$, the automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ induced by the action of L will be called $f_L$ and is defined by

$$f_L:M(Spin(8,\mathbb{C}))\to M(Spin(8,\mathbb{C})),f_L\left(E\right)=E\otimes L.$$

(12)

As proved in [19], polystable principal $Spin(8,\mathbb{C})$-bundles that admit a reduction in the structure group to $G_2$ are fixed points of the action of the triality automorphism $\tau$ of $Spin(8,\mathbb{C})$ on $M(Spin(8,\mathbb{C}\left)\right)$ defined in (11) since $G_2$ is one of the possible subgroups of $Spin(8,\mathbb{C})$ of the fixed points of an automorphism of $Spin(8,\mathbb{C})$ that represents the outer automorphism $\tau$. Then, the subvariety $M_0$ defined in (10) is composed by fixed points of the action of $\tau$ on $M(Spin(8,\mathbb{C}\left)\right)$.

Proposition 7. 

Let $M(Spin(8,\mathbb{C}\left)\right)$ be the moduli space of principal $Spin(8,\mathbb{C})$-bundles over X and $M_0$ be the subvariety of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (10), which is isomorphic to the moduli space $M(G_2$ of principal $G_2$-bundles over X. Then, any automorphism of $M_0$ is the restriction to the $M_0$ of an automorphism of $M(Spin(8,\mathbb{C}\left)\right)$.

Proof 

(Proof). Let $\varphi :M_0\to M_0$ be an automorphism of $M_0$. From Theorem 3, $M_0\cong M\left(G_2\right)$, since there exists an automorphism $\alpha :X\to X$ such that $\varphi ={\alpha }^{*}$ by Theorem 2. This automorphism ${\alpha }^{*}$ also induces an automorphism of $M(Spin(8,\mathbb{C}\left)\right)$, also called ${\alpha }^{*}$, such that the diagram

$$\begin{array}{ccc}M(Spin(8,\mathbb{C}\left)\right)& \stackrel{{\alpha }^{*}}{\to }& M(Spin(8,\mathbb{C}\left)\right)\\ \uparrow & & \uparrow & & \\ M_0& \stackrel{{\alpha }^{*}}{\to }& M_0\end{array}$$

commutes. Then, the result holds. □

Remark 5. 

The subset $M_0\cong M\left(G_2\right)$ is described in [19] as the subvariety of fixed points of the action in $M(Spin(8,\mathbb{C}\left)\right)$ of the triality automorphism τ of $Spin(8,\mathbb{C})$ defined in (11). Then, this diagram

$$\begin{array}{ccc}M(Spin(8,\mathbb{C}\left)\right)& \stackrel{\tau }{\to }& M(Spin(8,\mathbb{C}\left)\right)\\ \uparrow & & \uparrow & & \\ M_0& =& M_0\end{array}$$

is commutative. This shows that the automorphism ${\alpha }^{*}$ of $M(Spin(8,\mathbb{C}\left)\right)$ given by Proposition 7 is not unique since the composition of ${\alpha }^{*}$ with the action of τ also restricts it to the original automorphism of $M_0$. In the next result, it is proved that this happens with the action of every outer automorphism of $Spin(8,\mathbb{C})$. The automorphisms of $M(Spin(8,\mathbb{C}\left)\right)$, defined as the composition of the action of an order 3 automorphism of the base curve X, and the action of the triality automorphism are deeply discussed in [38].

Proposition 8. 

Let ρ be any outer automorphism of $Spin(8,\mathbb{C})$. Consider the automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ induced by ρ, also called ρ. Then, ρ leaves the subvariety $M_0$ defined in (10) invariant. Moreover, the restriction of ρ to $M_0$ is the identity.

Proof. 

Let $\tau$ be the triality automorphism and $\sigma$ be an outer involution of $Spin(8,\mathbb{C})$ such that $\tau$ and $\sigma$ together generate the group $Out(Spin(8,\mathbb{C}\left)\right)$. It is well-known that $\tau \sigma =\sigma {\tau }^2$. Let $E\in M_0$. Since $M_0$ is composed by fixed points of the action of $\tau$ defined in (11), $\tau \left(E\right)\cong E$, then

$$\tau \left(\sigma \left(E\right)\right)\cong \sigma \left({\tau }^2\left(E\right)\right)\cong \sigma \left(E\right),$$

so $\sigma \left(E\right)\in M_0$. Since every outer automorphism of $Spin(8,\mathbb{C})$ is generated by $\sigma$ and $\tau$, this proves the first part of the statement. For the second, notice that $M_0\cong M\left(G_2\right)$ and, according to Theorem 2, $Aut\left(M_0\right)\cong Aut\left(X\right)$. For every $\rho \in Out(Spin(8,\mathbb{C}\left)\right)$, the induced automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ comes from the identity $X\to X$, so the associated automorphism of $M_0$ is also the identity. □

Remark 6. 

As a consequence of Proposition 8, for each automorphism of the subvariety $M_0$ defined in (10), there is, at least, a family of automorphisms of $M(Spin(8,\mathbb{C}\left)\right)$ that lift the original automorphism of $M_0$, which is parametrized by $S_3$.

Let Z be the center of $Spin(8,\mathbb{C})$, which is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $L\in H^1(X,Z)$, and $f_L:M(Spin(8,\mathbb{C}))\to M(Spin(8,\mathbb{C}))$ be the automorphism defined in (12). It will now be discussed how the automorphism $f_L$ moves the subvariety $M_0$. Define

$$M_L=f_L\left(M_0\right).$$

(13)

Of course, $M_L\cong M_0\cong M\left(G_2\right)$.

Notice that the subvariety $M_0$ of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (10) coincides with $M_{\mathcal{O}}$, where $\mathcal{O}$ denotes the trivial line bundle over the base curve X. It has, then, been defined as a family $\left\{M_L\right\}$ of subvarieties of $M(Spin(8,\mathbb{C}\left)\right)$ parametrized by $H^1(X,Z)$, with Z being the center of $Spin(8,\mathbb{C})$ such that every subvariety of the family is isomorphic to $M\left(G_2\right)$. It is now proved that these subvarieties are pairwise disjointed.

Lemma 3. 

Let $M_0$ be the subvariety of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (10) and $L\in H^1(X,Z)$, where Z is the center of $Spin(8,\mathbb{C})$. Then, $f_L\left(E\right)\in M_0$ if and only if L is trivial, where $f_L$ is the automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (12).

Proof. 

Let ${\left\{U_i\right\}}_i$ be an open covering of X where E and L trivialize with transition functions $g_{ij}$ and $l_{ij}$, respectively. Then, the transition functions of $f_L\left(E\right)$ are $h_{ij}=g_{ij}{l}_{ij}$. The bundle $f_L\left(E\right)$ comes from a $G_2$-bundle if and only if there exist functions ${\phi }_i:U_i\to Spin(8,\mathbb{C})$ such that ${\varphi }_i{h}_{ij}{\varphi }_j^{-1}\in G_2$ for all $i,j$. Then, $l_{ij}$ induces, for each $x\in X$, an automorphism of $G_2$ via conjugation in $Spin(8,\mathbb{C})$ and ${\phi }_{ij}\left(g\right)=l_{ij}g{l}_{ij}^{-1}$. For $f_L\left(E\right)$ to come from a $G_2$-bundle, the image of ${\phi }_{ij}$ must be contained in $G_2$.

Since $G_2$ admits no nontrivial outer automorphisms and has a trivial center, $Aut\left(G_2\right)\cong G_2$, ${\phi }_{ij}$ is the conjugation according to an element of $G_2$. However, it is defined as the conjugation by an element of the center Z of $Spin(8,\mathbb{C})$, and $Z\cap G_2$ is trivial. Then, $l_{ij}=1$ for all $i,j$, so L is trivial, as stated. □

Proposition 9. 

The moduli space $M(Spin(8,\mathbb{C}\left)\right)$ admits $2^{4g}$ disjoint subvarieties that are isomorphic to $M\left(G_2\right)$.

Proof. 

Let $L_1,L_2\in H^1(X,Z)$, with $Z\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$ being the center of $Spin(8,\mathbb{C})$, $f_{L_1}$ and $f_{L_2}$ be the automorphisms of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (12), which are induced by $L_1$ and $L_2$, respectively, and $M_{L_i}=f_{L_i}\left(M_0\right)$, for $i=1,2$, be the subvarieties of $M(Spin(8,\mathbb{C}\left)\right)$ defined in (13). Suppose that $L_1\ncong L_2$. Then, $L=L_1\otimes L_2^{-1}$ is not trivial. If $E\in M_{L_1}\cap M_{L_2}$, then there exist $E_1,E_2\in M_0$ such that $f_{L_1}\left(E_1\right)=E=f_{L_2}\left(E_2\right)$. From this, it is deduced that $E_2\cong f_L\left(E_1\right)$ and L is not trivial, which contradicts Lemma 3. Then, such E does not exist, so $M_{L_1}\cap M_{L_2}$ is empty.

On the other hand, via construction, $M_P\cong M_0$ for all $P\in H^1(X,Z)$, and $M_0\cong M\left(G_2\right)$ according to Theorem 3. Then, $M(Spin(8,\mathbb{C})$ admits a family of disjointed, closed subvarieties as isomorphic to $M\left(G_2\right)$, which is parametrized by $H^1(X,Z)$. Since X is a compact Riemann surface of genus $g\ge 2$, this cohomology group is isomorphic to $Z^{2g}\cong {\mathbb{Z}}_2^{4g}$, so it has $2^{4g}$ elements. This finally proves the result. □

7. Conclusions

Let X be a compact Riemann surface of genus $g\ge 2$. It has been proved that every stable principal $G_2$-bundle over X is simple and that every simple $G_2$-bundle is irreducible, meaning that it does not admit any reduction in the structure group to a proper subgroup of $G_2$. From this, it is possible to adapt the strategies provided by the previous literature to prove that every automorphism of the moduli space $M\left(G_2\right)$ of polystable $G_2$-bundles over X is the pull-back action of an automorphism of the base curve X. It has also been proved that the forgetful map $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$ induced by the inclusion of groups $G_2\to Spin(8,\mathbb{C})$ is injective. Some consequences have been drawn from the two main results described above. In particular, it is checked that every automorphism of $M\left(G_2\right)$ is the restriction of an automorphism of $M(Spin(8,\mathbb{C}\left)\right)$ and that $M(Spin(8,\mathbb{C}\left)\right)$ contains, at least, $2^{4g}$ pairwise disjointed copies of $M\left(G_2\right)$.