1. Introduction
Group
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
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
has complex dimensions 14 and rank 2, and its Lie algebra
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
times longer than the other [
2]. This asymmetry causes the vanishing of the group of outer automorphisms of
.
The geometry of the Lie group
has deep applications in mathematics and theoretical physics, and it has been intensively studied. In particular,
, viewed as the automorphism group of the octonions, has provoked great interest in differential geometry since it gives rise to
-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
-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
, a principal
-bundle over
X is a holomorphic rank-7 vector bundle
E that is equipped with a holomorphic non-degenerate anti-symmetric 3-form
. In addition,
E admits a holomorphic non-degenerate and globally-defined symmetric 2-form
. Ramanathan [
8,
9,
10] gave suitable notions of stability to construct a complex algebraic variety, parametrizing isomorphism classes of polystable principal
-bundles over
X. This variety is called the moduli space of principal
-bundles, is denoted by
, has a complex dimension of
, and admits the subset of stable
-bundles as an open smooth dense subvariety.
There are several ways of studying the geometry of the moduli space of principal
-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
coincides with the group of automorphisms of the base curve in the sense that every automorphism of
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
of principal
-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
of symplectic bundles over a complex projective irreducible curve. They proved that every automorphism of
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
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
-bundle is irreducible, meaning that it does not admit any reduction in the structure group to a proper subgroup of
(Proposition 1).
Group
is also a fixed point of certain order 3 automorphisms of the simple complex Lie group
that lifts the triality automorphism
, viewed as an outer automorphism of
. This gives a well-known inclusion of groups
, which induces an algebraic morphism
between the moduli spaces of principal bundles over
X, called the forgetful map. The image of this map is, indeed, composed of principal
-bundles fixed by the action of
on
[
19]. In general, for a complex semisimple Lie group
G, the fixed points of the action of an outer automorphism of
G in
are reductions in the structure group of the subgroup of fixed points and of some lift of the outer automorphism of
G in
[
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
but also about the geometry of
. In the second main result of the paper, it is proved that the forgetful map
is actually an embedding (Theorem 3). The strategy followed consists of proving that the obstruction for an automorphism between polystable
-bundles (reducing to
-bundles to induce an automorphism of
-bundles) is an element of the non-abelian cohomology set
(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
to be trivial (Proposition 6).
Finally, some consequences of Theorems 2 and 3 are discussed. In particular, it is proved that every automorphism of comes from an automorphism of (Proposition 7) and that this lift is not unique; there is at least one family of such lifts that is parameterized by the group of permutations of 3 elements (Proposition 8). Moreover, subvarieties of are identified, where g is the genus of X, which is pairwise disjointed and isomorphic to (Proposition 9).
The paper is organized as follows. In
Section 2, a description of the group
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
. All this is relevant to present (in
Section 2) an appropriate notion of stability and polystability of principal
-bundles over
X and to state the Torelli theorem for
. Finally, Lemma 1 and Proposition 1, which relate stable, simple, and irreducible
-bundles, are proved.
Section 3 is devoted to describing the Hitchin integrable system associated with
. The action of
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
(Theorem 2) is provided. The second main result of this work (Theorem 3) on the injectivity of the forgetful map
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
and
.
2. Principal -Bundles
In this section, some basics on principal
-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
that will be needed are first presented, mainly following [
21,
22,
23]. For the description of the parabolic subgroups of
, 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
-bundles do not admit any reduction in the structure group to a proper subgroup of
.
2.1. The Simple Complex Lie Group
The complex group
can be viewed as the subgroup of the
of the automorphisms of
that preserve a certain non-degenerate symmetric bilinear form and non-degenerate anti-symmetric trilinear form
[
2,
19,
23]. This embedding
defines the 7-dimensional fundamental irreducible representation of
. Specifically, if
is an orthonormal basis for the bilinear form, and
is the wedge product
of vectors in the dual basis for
; then,
Group
is formally defined as
so
can be viewed through its fundamental 7-dimensional representations as a subgroup
as
Of course, its Lie algebra is a subalgebra of the special orthogonal algebra , given by endomorphisms preserving . Thus defined, group 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
is deeply related to the simple and simply connected complex Lie group
and with the triality automorphism. More precisely, recall that the triality automorphism
is an order 3 outer automorphism of
that admits an order 3 representative in
, the subgroup of which has fixed points and is isomorphic to
. Then,
can also be viewed as the subgroup of
of fixed points for a representative in
of triality automorphism. This defines the inclusion of groups
Since the rank of
is 2, it admits two non-isomorphic maximal parabolic subgroups,
and
, and one non-maximal parabolic subgroup, which is the intersection of
and
. Following the detailed description given in [
4], the action of
induces a 2-graded filtration
on the vector space
, where
is represented through its fundamental 7-dimensional representation and
,
, and
are isotropic for the 3-form
, as well as for
(recall that a subspace
V is isotropic for
if
). Similarly, the action of
gives a filtration of the form
where
,
, and
are, again, isotropic for
. Of course, the action of the nonmaximal parabolic subgroup
induces a filtration
Here, and , and both subspaces are isotropic for .
2.2. Principal -Bundles over a Compact Riemann Surface
Let
X be a compact Riemann surface of genus
. From the definition of
given in (
1) and the matrix representation described in (
2), a principal
-bundle over
X can be viewed as a triple
, where
E is a rank-7 holomorphic vector bundle over
X,
is an anti-symmetric non-degenerate holomorphic trilinear form globally defined on
E, and
is a holomorphic symmetric non-degenerate bilinear form
that is also globally defined on
E such that the pair
is an orthogonal bundle over
X. For simplicity, the principal
-bundles will be called hereafter by referring to the underlying vector bundle
E.
From the above description of the parabolic subgroups of
[
4], the notions of stability and polystability of principal
-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
, 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
-bundle
(as above), a vector sub-bundle
F of
E is isotropic for the trilinear form
if
.
Definition 1. Let be a principal -bundle over the compact Riemann surface X of genus . The principal -bundle is stable if 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, . 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 , 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 of polystable principal -bundles is, then, a complex algebraic variety of dimension , which parametrizes isomorphism classes of polystable principal -bundles over X. It is irreducible since is simply connected, and the subvariety of stable -bundles is an open dense subset of , which is composed by smooth points of .
Since the center of is , a principal -bundle over X is simple if it admits no nontrivial automorphisms. The aim now is to prove that coincides with the subvariety of stable and simple principal -bundles over X (that is, it will be checked that every stable principal -bundle over X is also simple) and that every simple -bundle is irreducible, meaning that it admits no reductions in the structure group to a proper subgroup of .
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
be the Lie algebra of
G and
be the adjoint bundle of
E, the typical fiber of which is
. Since
E is stable,
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
). Then, any global section of
is constant, which corresponds to elements of
that commute with the image of the holonomy homomorphism of
E,
. 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 is trivial, or
H coincides with the center
Z of
G. If
, 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
or
. In any case, the constant sections of
come from elements in
Z. Since the automorphisms of
E are in bijective correspondence with the global sections of
, which are constant sections, these automorphisms must come from
Z. Then,
, so
E is simple. □
Remark 1. Notice that satisfies the conditions of Lemma 1 since is simple and its group of outer automorphisms is trivial. Other simple complex Lie groups like or 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 -bundle over X. Then, E is irreducible, meaning that E admits no reductions in the structure group to a proper subgroup of .
Proof. The principal -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 to a proper subgroup H of . If H is a discrete subgroup, then the fibers of are discrete. If H were infinite, then the automorphisms of would include the translations by elements of H, so it would be infinite. Therefore, H must be finite. In this case, the projection is a finite cover of X, and it induces a representation . If is the universal cover of X, then is a trivial principal H-bundle over . Since is trivial, it admits a nontrivial holomorphic global section s. This section descends to define a nontrivial holomorphic global section of since s is invariant under the action of the finite covering group. Then, provides a nontrivial holomorphic global section of E since 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
with a positive dimension. Then, it may be supposed that
H is a maximal proper subgroup of
. Dynkin [
27] gave a complete description of these maximal proper subgroups, which are
,
, and the proper parabolic subgroups. Moreover, a representation
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,
can be reconstructed by
, where
is the universal cover of
X, and the action of
is obvious, induced by
. Then,
defined by
is a principal
-bundle with
since both
-bundles correspond to the same representation of
on
. However, the group of automorphisms of
is isomorphic to the centralizer of
H in
, which is, of course, strictly greater than the center of
, 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
is as follows.
Theorem 1 ([
30] Theorem 4.1).
Let X and be smooth projective curves of genus g and , respectively, with . Let and be the moduli spaces of principal -bundles over X and , respectively. Syppose that . Then, . The inclusion
described in (
3) gives rise to an algebraic map between moduli spaces
called the forgetful map, which preserves the polystability condition (that is, any polystable
-bundle is moved to a polystable
-bundle). The principal
-bundle
E is moved through this map to the obvious principal
given by the inclusion of the groups above.
3. The Hitchin Integrable System of
Let
X be a compact Riemann surface of genus
,
G be a semisimple complex Lie group, and
be the moduli space of polystable principal
G-bundles over
X. Let
be the moduli space of
G-Higgs bundles over
X and
be the moduli space of stable principal
G-bundles, which is a dense open subset of
. Take
to be a basis of the ring of invariant homogeneous polynomials of the Lie algebra
of
G, where
r is the rank of
. The evaluation of each polynomial on the Higgs field defines a map
where
for
, and
K denotes the canonical line bundle over
X. The map defined above allows for defining the Hitchin map,
The vector space
is called the base of the Hitchin map
. In [
31], it is proved that
. Let
n be the dimension of
. The Hitchin map induces
n complex-valued functions
defined on
. The tangent space to
at an element
is isomorphic to
, which coincides with
by Serre duality. In [
31] (Proposition 4.5), Hitchin proved that the
n functions
Poisson-commute with the canonical symplectic structure of the cotangent bundle. This defines a completely integrable system on
(see [
32] for details).
Consider the group
. The base
of the Hitchin map
is
Here, the element
, where
for
, is associated to the invariant polynomial
in the sense that the characteristic polynomial of the Higgs field of any
in the fiber of the Hitchin map at
a is
.
The fiber of the cotangent bundle
at a stable principal
-bundle
E is identified with the group of Higgs fields
where
denotes the Lie algebra of
.
It is well known that for any complex semisimple Lie group
G, the multiplicative group
acts on
via the product on the Higgs field: if
and
, then
. There is also a natural action of
on the base of the Hitchin map. In the case of
this action can be described as follows: if
and
, then
Proposition 2. Let B be the vector space defined in (6). Let be the Hitchin map. Then, the action of on B defined in (7) is the unique -action on B that makes the Hitchin map -equivariant. Moreover, the subspace of B can be reconstructed from the action of on B. Specifically, for , is the subspace of B, where the ratio of convergence is bigger than when λ goes to 0. Proof Suppose that there exists an action of
on
B that is different from that defined in (
7), which makes
-equivariant. Denote this action by
for
and
. Then, there exist holomorphic functions
such that
. Since
is
-equivariant, it should be
From this, it is clear that
and
are homomorphisms of the multiplicative group
. Then,
and
for some exponents
. According to the
-equivariance of
, the exponents must coincide with the degree of the corresponding invariant so that
and
. Then, the action ⋄ is the action defined in (
7).
Now, the action of
on
B defined in (
7) determines the origin
since it is the only fixed point of the action. The vector space
B is then endowed with an affine structure. Moreover,
can be recovered as the subspace where the ratio of convergence is bigger than
when
goes to 0, as stated. □
Remark 2. Notice that, as a consequence of Proposition 2, defined in (6) is endowed with a structure of an affine space. This follows from the existence of the action of defined in (7). 4. Automorphisms of the Moduli Space of Principal -Bundles
Let
X be a compact Riemann surface of genus
and
be the moduli space of polystable principal
-bundles over
X. Notice that any automorphism
of the base curve
X induces an automorphism of the moduli space
by taking the pull-back:
This section aims to prove that
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
of
induces an automorphism
of the curve
X, which is given by the Torelli theorem (Theorem 1). It will be checked that
, with the notation of (
8), for some
. 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
-bundle is also simple (Lemma 1) and irreducible (Proposition 1). In the situation described, it will be proved that for any stable principal
-bundle
E over
X, the adjoint vector bundles
and
are isomorphic with their structures of Lie algebra bundles, and it must be
as principal
-bundles. The final result (Theorem 2) follows since the open subvariety of stable principal
-bundles is dense in
.
Lemma 2. Let G be a semisimple complex Lie group, be the Lie algebra of G, E be an irreducible principal G-bundle over X, and be the adjoint bundle of E, the typical fiber of which is . Let be any point and be the sheaf associated with the effective divisor defined by x. Then, .
Proof. Suppose that, under the conditions and the notation of the statement, there exists such that . By definition, . Since E is irreducible, 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 with a positive degree, contrary to the stability of . 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 , if E is stable, then it is irreducible by Proposition 1, so the result works for stable -bundles.
Proposition 3. Let B be the base of the Hitchin map of defined in (6), and let be the subspace of B given by Proposition 2. Let E be a stable principal -bundle over X and be the composition of the Hitchin map with the projection of B over . For each , let Proof. Notice that the stable principal
-bundle
E is simple according to Lemma 1 and is also irreducible by Proposition 1. The exact sequence of sheaves
induces an exact sequence of cohomology sets of the form
From Lemma 2 and Serre duality, it follows that
so the map
, which consists of evaluating in
x, is surjective.
Now, take . It is clear that if and only if , where 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 for all t with is (note that if a matrix satisfies that for any with , then for any such B, the whole line through A and , is contained in the zero set of the polynomial ; this necessarily implies ). □
From Proposition 3, a bundle
over
X associated with each stable principal
-bundle
E that has fiber over any point
is defined as follows:
This bundle
is nothing but the kernel of the homomorphism of vector bundles
where
denotes the trivial line bundle over
X.
Proposition 4. Let B be the base of the Hitchin map of defined in (6) and be the corresponding summand in the decomposition of B for . Then, the linear structure and the decomposition are uniquely determined by the isomorphism class of in the sense that an automorphism of induces an automorphism of B, which is diagonal for the decomposition . Proof. Let be an automorphism of . Let E be a stable principal -bundle, and let . From Theorem 1, the automorphism induces an automorphism of the base curve X. By composing with , it may be supposed that this automorphism is the identity.
The isomorphism induces a linear isomorphism that commutes with the Hitchin map.
Let
and
be the bundles over
X defined in (
9) for
E and
, respectively. According to Proposition 3, the automorphism
induces an isomorphism of bundles
. It is easy to prove that there are exact sequences of bundles
and
From the isomorphisms and , an isomorphism is obtained, and also the isomorphism is derived.
Notice that the map
considered above is induced by
and that the Hitchin map
factors as the composition of the map
with the trace map over
B. Then, the isomorphism
descends to a diagonal isomorphism with respect to the natural decompositions of
and
:
Therefore, the automorphism f of B is also diagonal with respect to the decomposition . Then, there exist -equivariant maps for , which induce a weight of f for each . This shows that f is linear for the decomposition of B given above. □
Theorem 2. Let be the moduli space of polystable principal -bundles over the compact Riemann surface X of genus . Let be an automorphism of . Then, there exists an automorphism α of X such that , where is defined in (
8).
Proof. First, notice that the automorphism of X induced by given by Theorem 1 may be supposed to be the identity.
Let
E be a stable principal
-bundle over
X,
be the Hitchin map, and
B be the base of the Hitchin map. For
, let
be the summand
in the decomposition of
B, which is well-defined by Proposition 4. For each
, let
be the composition of the Hitchin map
with the projection map of
B over
. Let
be the automorphism of
B induced by
. Let
. The arguments of Proposition 4 show that
is preserved by
f, so
preserves
. The image of this set under the surjective map
coincides with the nilpotent cone
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 . The discussion above allows us to prove that there is an isomorphism of the nilpotent bundles as bundles over X (since the automorphism of the curve X induced by 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
-bundle according to the reductions in the structure group to a Borel subgroup of
), which will be called
and
. The automorphism
induces a Lie algebra bundle isomorphism
. Notice that giving a Lie algebra bundle structure is equivalent to giving a principal
-bundle together with a reduction in the structure group to
, corresponding to
E. Since
is centerless and admits no outer automorphisms, the Lie algebra automorphism
gives an automorphism of principal
-bundles
.
This holds for any E of the dense open subvariety of stable bundles of , so the result is finally proved. □
5. The Morphism
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
-bundles, the associated
-bundles of which are isomorphic, are also isomorphic, as
-bundles are an element of the non-abelian cohomology set
. Next, sufficient conditions will be given for a complex semisimple Lie group
G and a complex subgroup of it
H to satisfy that
. Finally, it will be checked that the groups
and
satisfy all these conditions.
Proposition 5. Let X be a compact Riemann surface of genus , G be a semisimple complex connected Lie group, and H be a simple maximal complex Lie subgroup of G. Let and be polystable principal H-bundles over X such that the induced polystable G-bundles and are isomorphic. Then, the isomorphism defines an element in the non-abelian cohomology . Moreover, this cohomology class is trivial if and only if the isomorphism lifts to an isomorphism of principal H-bundles.
Proof. Notice that since
H is maximal in
G,
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
and
are polystable as principal
H-bundles, they correspond to certain reductive representations
, where
is the fundamental group of
X [
8]. By composing these representations with the inclusion of groups
, two reductive representations of
in
G are obtained,
and
(since
H is also semisimple, as it is simple), which, of course, correspond to the polystable principal
G-bundles
and
. Since
,
and
are conugate, meaning that there exists
such that
for all
.
Let
M be the homogeneous space
and
be defined by
, where
denotes the class of
in
M. It is easily checked that
s satisfies the cocycle properties since
where · denotes the action on the right of
G on
M. Now, there is a bijective correspondence between cocycles
and rational sections of the
M-bundle associated with
,
(this bijective correspondence is assigned to
, a section
of
, as follows: for each
, if
is a choice of a path from a fixed base point
to
x, then
, where
is in the fiber of
over
; the independence on the choice of
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
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
in the non-abelian cohomology set
.
The cohomology class
is trivial if and only if there exists
such that
for all
. In this case, according to the definition of
s, it is satisfied that
for every
. This leads to the existence of some
such that
It will now be checked that
defines a cocycle
in the sense that
. By taking
in
and applying that
, it is obtained that
Now, by using the expression
twice for
and
and simplifying,
When multiplying by
on the left,
Recall that
, so
When multiplying by
on the left and by
g on the right,
Finally, it is obtained that
since the action of
on
by conjugation is
This finally proves that t defines a cocycle.
Since
, then
Let . It is easily checked that .
Since
for
a,
b, and
c in any group, the following is satisfied:
It has been obtained that , so and are conjugate.
Consider the map
defined by
. This is a constant map since
H is a simple subgroup of
G, and
M is a complex homogeneous variety. Then,
for all
, so for each
, there exists
such that
. By taking
, the existence of a
such that
is obtained. Therefore,
is conjugate in
G to
. However, since both
and
belong to
H, it follows that
where
. This means that
is conjugate in
G to an element of
H for all
. Then,
is conjugate in
H to an element of
H for all
since
is a representation that takes values in
H, and
H is a simple maximal subgroup of
G. From this, there exists an element
such that
. Define
. It is satisfied that
for all
, so
is in the normalizer of
H in
G, which is, indeed,
H since it is a simple maximal subgroup of
G. Then,
, so
and
are conjugate in
H. This proves that
and
are, indeed, isomorphic through an isomorphism that lifts the previous automorphism
. □
Remark 4. Following the proof of Proposition 5, it is possible to give a construction of the isomorphism when the class induced by the isomorphism is trivial. Specifically, given by , 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 and , Here, since is trivial, so h normalizes H. Moreover, if , then obviously , so φ is injective, and for each , there exists such that , so , meaning φ is surjective. All this proves that φ is an isomorphism . The isomorphism induced by φ is given byso it is the original automorphism defined by conjugation by . Proposition 6. Let X be a compact Riemann surface of genus with fundamental group , 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 is effective, meaning that only acts trivially on , and that the action of G on admits no global fixed points, meaning that no element of is fixed for the action of all the elements of G. Then, the non-abelian cohomology set is trivial.
Proof. Under the conditions of the statement,
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,
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
admits no global fixed points. Therefore,
H is contained in a parabolic subgroup
P of
G. Then, there is a natural fibration
with fiber
. It is known that
is a rational variety according to the Bruhat-Tits theorem [
36] and that
is a rational variety since
P is a solvable group. Then,
is a rational variable since rationality is preserved by fibrations, the fiber and base of which are rational variables.
Every element comes from a cocycle E that can be understood as a -bundle. Since is a rational variety, E admits a rational section s, which can be extended to a global section 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
implies that
E is trivial, given that
is a rational variable. Indeed, the map
given by
, where · denotes the action of
G on
E, is an isomorphism. Firstly, if
, then
for some
, and
because
is in the fiber over
x, meaning
is well-defined. Secondly,
is
G-equivariant since
for all
and
. Thirdly, if
and
are such that
, then, of course,
and
, from which
, so
is injective. Fourthly, if
, it is in the fiber of some
, so there exists
such that
, from which
is surjective.
This finally proves that must be trivial. □
It will now be checked that the Lie group and its subgroup satisfy the conditions of Propositions 5 and 6.
Recall that
is semisimple complex and connected, and
is a simple complex subgroup of
. Moreover,
is maximal in
since the Lie algebra
of
appears as one of the maximal proper subalgebras of the Lie algebra
of
in the Dynkin classification of maximal subalgebras of simple complex Lie algebras [
27,
37]. This proves that the Lie group
and its subgroup
satisfy the conditions of Proposition 5.
The group is a complex, connected non-abelian semisimple Lie group and is a closed subgroup of that does not contain any parabolic subgroup of since the dimension of any parabolic subgroup of is greater than (notice that parabolic subgroups of are in bijective correspondence with parabolic subalgebras of ).
Group does not contain any nontrivial normal subgroup of (if admits a normal subgroup of , then it would be a normal subgroup of , which is simple and centerless, so this subgroup is necessarily trivial). Then, the action of on the complex homogeneous variety is effective. Indeed, let . Then, if and only if for all . This implies that for all , so g should be in the normalizer of in for all . However, since contains no nontrivial normal subgroups of , ker must be trivial.
Finally, since is not a normal subgroup of because is simple, the action of on has no global fixed points. Indeed, a global fixed point would satisfy for all . In this situation, for all , there exists such that , that is, . This means that . Since is not normal, this is impossible. All the above proves that and also satisfy the conditions of Proposition 6.
Theorem 3. The forgetful map defined in (4) is an embedding. Proof. It will be proved that the forgetful map is injective. Let and be polystable principal -bundles over X such that the associated polystable principal -bundles, and , are isomorphic. The aim is to prove that and are also isomorphic as -bundles. Let be an isomorphism. According to Proposition 5, a class is induced by , and lifts to an isomorphism if and only if c is trivial. However, c must be trivial since is trivial, according to Proposition 6. Then, and must be isomorphic as H-bundles, according to Proposition 5. □
6. Consequences
In Theorem 3, it is proved that the map
defined in (
4) between moduli spaces over a compact Riemann surface
X of genus
in an embedding. Therefore, the image of this map provides a copy of
included in
. Let
be this copy; that is,
In this section, it is proved that every automorphism of
comes from an automorphism of
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
different form
(indeed, disjointed with it) contained in the moduli space
is proved from Theorem 3, and the consideration of certain well-known automorphisms that
admits are described below.
When considering the moduli space
of principal
-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
is a composition of automorphisms of some of the three following families.
- (A).
The group of automorphisms of the base curve X acts in the moduli space by taking the pull-back of the bundles. If , the induced automorphism of will be denoted by .
- (B).
The group
of outer automorphisms of
acts in
. Then, a family of automorphisms of
parametrized by
is induced, which has been intensively studied [
19,
38,
39]. If
and
, the principal
-bundle
is defined to be the principal
-bundle over
X, the total space of which coincides with that of
E and is equipped with the action of
given by
for
and
, where
R is an automorphism of
representing
. It is easily checked that the isomorphism class of
E does not change by the action of an inner automorphism of
, so the action is well-defined [
19]. Moreover, the order of the automorphism of
thus defined is, of course, the order of the outer automorphism
(the possibilities for this order are 2 or 3 if the outer automorphism is nontrivial since
). The automorphism of
induced by an outer automorphism
of
will also be denoted by
. The action of the triality automorphism
of
, which is a generator of order 3 of
, induces an automorphism
that will be particularly relevant.
- (C).
If
is the center of
, then the group
, which parametrizes isomorphism classes of principal
Z-bundles over
X, acts on
via the tensor product. If
, the automorphism of
induced by the action of
L will be called
and is defined by
As proved in [
19], polystable principal
-bundles that admit a reduction in the structure group to
are fixed points of the action of the triality automorphism
of
on
defined in (
11) since
is one of the possible subgroups of
of the fixed points of an automorphism of
that represents the outer automorphism
. Then, the subvariety
defined in (
10) is composed by fixed points of the action of
on
.
Proposition 7. Let be the moduli space of principal -bundles over X and be the subvariety of defined in (10), which is isomorphic to the moduli space of principal -bundles over X. Then, any automorphism of is the restriction to the of an automorphism of . Proof (Proof). Let
be an automorphism of
. From Theorem 3,
, since there exists an automorphism
such that
by Theorem 2. This automorphism
also induces an automorphism of
, also called
, such that the diagram
commutes. Then, the result holds. □
Remark 5. The subset is described in [19] as the subvariety of fixed points of the action in of the triality automorphism τ of defined in (11). Then, this diagramis commutative. This shows that the automorphism of given by Proposition 7 is not unique since the composition of with the action of τ also restricts it to the original automorphism of . In the next result, it is proved that this happens with the action of every outer automorphism of . The automorphisms of , 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 . Consider the automorphism of induced by ρ, also called ρ. Then, ρ leaves the subvariety defined in (10) invariant. Moreover, the restriction of ρ to is the identity. Proof. Let
be the triality automorphism and
be an outer involution of
such that
and
together generate the group
. It is well-known that
. Let
. Since
is composed by fixed points of the action of
defined in (
11),
, then
so
. Since every outer automorphism of
is generated by
and
, this proves the first part of the statement. For the second, notice that
and, according to Theorem 2,
. For every
, the induced automorphism of
comes from the identity
, so the associated automorphism of
is also the identity. □
Remark 6. As a consequence of Proposition 8, for each automorphism of the subvariety defined in (10), there is, at least, a family of automorphisms of that lift the original automorphism of , which is parametrized by . Let
Z be the center of
, which is isomorphic to
,
, and
be the automorphism defined in (
12). It will now be discussed how the automorphism
moves the subvariety
. Define
Of course, .
Notice that the subvariety
of
defined in (
10) coincides with
, where
denotes the trivial line bundle over the base curve
X. It has, then, been defined as a family
of subvarieties of
parametrized by
, with
Z being the center of
such that every subvariety of the family is isomorphic to
. It is now proved that these subvarieties are pairwise disjointed.
Lemma 3. Let be the subvariety of defined in (10) and , where Z is the center of . Then, if and only if L is trivial, where is the automorphism of defined in (12). Proof. Let be an open covering of X where E and L trivialize with transition functions and , respectively. Then, the transition functions of are . The bundle comes from a -bundle if and only if there exist functions such that for all . Then, induces, for each , an automorphism of via conjugation in and . For to come from a -bundle, the image of must be contained in .
Since admits no nontrivial outer automorphisms and has a trivial center, , is the conjugation according to an element of . However, it is defined as the conjugation by an element of the center Z of , and is trivial. Then, for all , so L is trivial, as stated. □
Proposition 9. The moduli space admits disjoint subvarieties that are isomorphic to .
Proof. Let
, with
being the center of
,
and
be the automorphisms of
defined in (
12), which are induced by
and
, respectively, and
, for
, be the subvarieties of
defined in (
13). Suppose that
. Then,
is not trivial. If
, then there exist
such that
. From this, it is deduced that
and
L is not trivial, which contradicts Lemma 3. Then, such
E does not exist, so
is empty.
On the other hand, via construction, for all , and according to Theorem 3. Then, admits a family of disjointed, closed subvarieties as isomorphic to , which is parametrized by . Since X is a compact Riemann surface of genus , this cohomology group is isomorphic to , so it has elements. This finally proves the result. □