1. Introduction
Given a compact Riemann surface
X of genus
and a complex reductive Lie group
G, a principal
G-bundle over
X is a holomorphic complex variety
E together with a projection map
that makes
E locally isomorphic to
such that
E admits a transitive and free right action of
G in it. From Ramanathan’s notion of stability for these objects [
1,
2,
3], the moduli space
that parametrizes
S-equivalence classes of semi-stable principal
G-bundles is a complex algebraic variety admitting the subset of stable bundles as an open dense and smooth subvariety.
The geometry of the space
has been intensively studied, due to its relevance to mathematics and other fields, such as theoretical physics. Indeed, in geometry, principal
G-bundles are the base for constructing
G-Higgs bundles, which naturally arise as solutions of Hitchin’s equations [
4] and whose moduli space construction requires the non-abelian Hodge theory developed by Simpson [
5,
6,
7]. From the perspective of theoretical physics, principal bundles appear in the description of instantons, as solutions to the self-dual Yang–Mills equations [
8], and also in the study of the moduli spaces of monopoles, which are equipped with natural hyperkähler metrics and provide insights into the dynamics and scattering of magnetic monopoles [
9].
The specific case of the structure group
is quite interesting because of its rich geometry. Indeed,
is, of all the complex simple Lie groups of exceptional type, the only one that admits non-trivial outer automorphisms. Specifically, it admits a non-trivial outer involution, coming from the symmetries of its Dynkin diagram, which incorporates geometrically interesting symmetries into the group. The different representatives of the outer involution in
have as a subgroup of fixed points either the simple exceptional complex Lie group
or the projective group
, of type
. From these data, inclusions of groups
and
appear that incorporate
and symplectic geometries in the group
. In addition, the group
is the only simple exceptional complex Lie group that has a non-trivial center, which is isomorphic to
. This also incorporates new symmetries at the level of principal
-bundles [
10]. All of this has led to a great deal of interest in the principal bundles with structure group
, both in geometry and theoretical physics [
11]. Thus, principal
-bundles are employed to describe how the different fundamental forces might emerge from a single fundamental interaction at extremely high energies, so this Lie group arises in the formulation of Grand Unified Theories (GUTs) [
12,
13]. Moreover, in certain formulations of string theory, particularly in the context of heterotic string theory, the group
emerges as a possible gauge symmetry [
14]. Specifically, in the
heterotic string theory, compactification on a six-dimensional Calabi–Yau manifold can lead to an effective four-dimensional theory with a gauge group
, which can further break down to the standard model gauge groups [
14,
15].
The above geometric characteristics have a great impact when studying the fixed points of the automorphisms of the moduli space
of principal
-bundles over
X. For a general reductive complex Lie structure group
G, Fringuelli [
16] proved that the automorphism group of the moduli space
is generated by three families of automorphisms: the action of an outer automorphism of
G, the pull-back action of an automorphism of the base curve
X, and the tensor-product action of an element of
, where
is the center of
Z. This extends previous results concerning vector bundles [
17] and Higgs bundles [
18]. Several works have dealt with automorphisms defined by the action of an inner automorphism of
G, by studying the fixed points of the action of an outer involution [
19,
20] or analyzing the components and the Hitchin integrable system of the fixed point locus in particular interesting cases, such as
[
21]. There are also works that construct fixed points for the action of
, using Prym varieties as the main technique in the case of vector bundles [
22] or specific techniques in the case of the gauge group
[
10]. However, the preceding literature has not addressed the case of the fixed points of the automorphisms defined as the composition of the action of an outer automorphism of the structure group and the pull-back action of an automorphism of the curve, in the case of orthogonal bundles of even rank over a hyperelliptic curve [
23]. This led to the construction of orthogonal
-bundles over
X, where
is an outer involution of the Lie group
and
is an involution of
X.
While the general methodology shares conceptual foundations with previous works such as [
20,
23], this paper extends significantly beyond those studies by applying these techniques to the exceptional Lie group
, which possesses fundamentally different geometric properties from other Lie groups like
. The application to
-bundles is not merely a straightforward extension, as
has a distinctive structure as an exceptional Lie group with a unique outer automorphism pattern and a non-trivial center isomorphic to
. This research also provides novel topological results, including the precise number of connected components, dimensional analysis, characteristic class computations, and a detailed description of the singular locus. These aspects have not been addressed in previous works. Furthermore, this paper establishes an original connection between fixed points and octonionic geometry, demonstrating that fixed points of the automorphism
correspond bijectively to octonionic structures on the quotient curve
. This connection leverages the relationship between
,
, and the geometry of octonions, yielding a geometric interpretation with potential applications in geometry and mathematical physics. Specifically, the octonionic structures are key in string theory and M-theory compactifications due to their relationship with exceptional holonomy manifolds, and this correspondence allows us to understand how octonionic structures emerge from the study of principal bundle automorphisms. The explicit construction of octonionic structures from fixed points (Theorem 2) provides techniques for investigating theoretical models in which
appears as a gauge symmetry, especially in heterotic string theory contexts where compactification on Calabi–Yau manifolds leads to effective theories with
gauge groups.
The main object of study in this research is the moduli space of principal -bundles over a compact Riemann surface X of genus equipped with an involution . Given an automorphism of representing the outer involution that admits, the map defined by , where , is an automorphism of . The main purpose of this paper is to describe and analyze the fixed points of the automorphism . Notice that it results from combining the action of the outer automorphism of with the pull-back action of an automorphism of the curve, which has not been studied before in the preceding literature. In particular, the main result proves that the fixed point subvariety of is isomorphic to the moduli space of H-bundles over the quotient curve , where H is the fixed point subgroup of , either or . It is also proved that the fixed point locus consists of components, where fixes points of X (Theorem 1). For that, the obstruction for the automorphism to admit fixed points is characterized for a general reductive complex group G as an element of the cohomology set , where denotes the subgroup of fixed points of (Proposition 1). Novel results are also provided by computing the dimension of the fixed point subvariety (Proposition 2) and characterizing its singular locus (Proposition 3) and the stability of the fixed points (Proposition 4).
The topology of the fixed point subvariety of is also deeply analyzed in this paper through the study of its characteristic classes. In particular, every fixed point E of induces a principal H-bundle over , where H is the fixed point subgroup of (either of ), and it is proved that the second Chern class of is equal to that of E, if , and half of that of E, if (Proposition 5).
From the above results, a novel geometric application is developed, establishing a concrete correspondence between fixed points of
and octonionic geometric structures on the quotient curve
. Recall that the groups
and
arise naturally from the geometry of octonions. Specifically, they are subgroups of the automorphism group of the complexified algebra of
Hermitian matrices over the octonions (the so-called Albert algebra) [
24,
25]. Octonionic structures on manifolds and bundles have been intensively studied, since they are closely related to manifolds with a holonomy group, like
or
, so they are crucial in string theory and M-theory compactifications [
26]. In this paper, it is proved that the fixed points of
and the octonionic structures on
are in bijective correspondence (Proposition 6), and an explicit construction of octonionic structures is given from the analysis of fixed points made (Theorem 2).
The structure of this paper is as follows. The main result is established and proved in
Section 2, where relevant geometric properties of the fixed point subvariety of
, such as the dimension, stability, and singular locus, are also provided. In
Section 3, the relationship between the Chern classes of a fixed point of
and the Chern class of the induced reduction of structure group to the fixed point subgroup of
is given. The application of the above results of fixed points to the study of octonionic structures, including the connection of fixed points with octonionic structures on
, is provided in
Section 4. In
Section 5, the explicit construction of octonionic structures on bundles, designed from the above study, is developed. Finally, the main conclusions are drawn.
2. Involutions of the Moduli Space of Principal -Bundles
Let
X be a compact Riemann surface of genus
,
G be a complex reductive Lie group, and
be the moduli space of principal
G-bundles over
X. If
is an automorphism of
G, then
acts on
as follows. If
E is a principal
G-bundle over
X, then
has the same total space as
E, but it is equipped with the action of
G given by
, for
and
[
10,
19,
20]. This defines an automorphism of
. Indeed, if
has a finite order, then the induced automorphism of
has the same order as
. Notice also that, if
is an inner automorphism, then
, so it induces the identity on
, but this is not the case if
represents an outer automorphism of
G [
10,
20]. If, in addition,
X is equipped with an automorphism
, then the pull-back action also defines an automorphism of
, with the same order as that of
. The composition of these two actions gives an automorphism
defined by
, where
denotes the pull-back
.
As a first step, in this section, a necessary and sufficient condition for the above automorphism to admit fixed points is provided in the case where both and are involutions. Notice that, in this case, is also an involution. This is performed for a general complex reductive Lie group G. After that, the study is particularized for the case of , providing a characterization of the fixed points of the automorphism when represents the unique outer automorphism that admits (that has order 2) and is an involution of X. In addition, some results are proved concerning the components of the fixed point locus, its dimension, and the stability of the fixed points mentioned above.
Lemma 1. Let X be a compact Riemann surface of genus with an involution , G be a complex semisimple Lie group, be an automorphism of order 2, and be the involution defined by , where . If is a fixed point of Φ, then E admits a reduction of the structure group of E to the fixed point subgroup over the quotient curve .
Proof. Let us establish the correspondence between the fixed points of and the reductions of the structure group. If E is a fixed point of , then , meaning that there exists an isomorphism .
Consider the commutative diagram
where
is the quotient map and
acts on
E by taking the pull-back. Notice that, although the action of
on
E is not free in general, the quotient space
, which forms a bundle over
, can be considered.
For the fixed point E of , the isomorphism can be equivalently expressed as an isomorphism . This isomorphism satisfies the condition that , since is an involution.
Then, this isomorphism
f induces a reduction of the structure group of
E to
over the quotient curve
as follows. Define
as the subset of
E consisting of points that are fixed by the combined action of
and
through
f,
It is easily checked that is a principal -bundle over . Indeed, if and , then , which shows that is preserved by the action of .
The projection map is well defined, since, if , then, the projection of p to X is mapped by to the projection of , which means that the projection of p to X is invariant under and thus descends to .
To fully justify that is indeed a reduction of the structure group to , notice the following. First, we will show that is a principal -bundle over . We have already established that is preserved by the -action. For the principal bundle structure, it will be verified that this action is free and transitive on the fibers.
The action is free because it inherits this property from the free action of G on E. Indeed, if for some and , then (the identity element) since the G-action on E is free.
For transitivity on fibers, let
be points projecting to the same point in
. Their projections to
X must be in the same
-orbit. If they project to the same point in
X, then there exists a unique
such that
(by the principal bundle property of
E). We need to show that
. Since
, we have
and
. This implies that
Using the properties of the bundle isomorphism f and that is an involution, we obtain , which means .
If the projections of p and q to X are different but in the same -orbit, we can use that the bundle E is locally trivial and the isomorphism f preserves fibers to establish the existence of an element in that transforms p into q.
Now, the structure group of is easily extended from to G, obtaining a principal G-bundle over . This extended bundle is , the quotient of by the equivalence relation for . When pulled back to X through the quotient map , this extended bundle is isomorphic to E. The isomorphism maps a point in the pulled-back bundle to in E, and this isomorphism is well defined and respects the G-action.
To complete the proof that represents a genuine reduction of structure group, we will verify that the extension of from a -bundle to a G-bundle, when pulled back to X, is isomorphic to the original bundle E. For this, it will be shown that the map defined by is a well-defined G-bundle isomorphism.
The map is well defined, since, if for , then and . It is also G-equivariant, as for any . Finally, to show it is bijective, we construct an inverse map . For any projecting to , there exists a point projecting to and an element such that . The map is well defined and provides the required inverse.
Conversely, given a principal -bundle F over , extending its structure group to G and pulling back to X yields a principal G-bundle over X that is a fixed point of . The isomorphism f in this case is induced by the action of on F.
Therefore, there is a one-to-one correspondence between the fixed points of and principal -bundles over the quotient curve , or equivalently, reductions of the structure group from G to over , as stated. □
Proposition 1. Let X be a compact Riemann surface of genus with an involution , G be a complex semisimple Lie group, and be an automorphism of order 2. Then, the involution defined by , where , admits a fixed point if and only if the topological obstruction vanishes, where denotes the fixed point subgroup of σ.
Proof. A fixed point of is a principal G-bundle E over X such that , meaning that there exists an isomorphism . By Lemma 1, this isomorphism can be viewed as a reduction of the structure group of E to the fixed point subgroup over the quotient curve .
Let be the quotient map. Choose a covering of with local trivializations of the pull-back bundle . On the overlaps , the transition functions satisfy the cocycle condition . The involution acts on these transition functions via .
For a fixed point of
, there must exist functions
such that
. Define the cohomology class
by the cocycle
on triple intersections
, where
Since has order 2, these cocycles take values in , the component group of the fixed point subgroup. The obstruction vanishes precisely when this cocycle is cohomologically trivial, which completes the proof. □
The simple complex Lie group
is one of the groups of exceptional type classified by Cartan. It can be defined as the group of automorphisms of a certain holomorphic symmetric 3-form defined on a 27-dimensional complex vector space. Specifically,
is the subgroup of
that preserves a non-degenerate, symmetric trilinear form
defined on a 27-dimensional complex vector space
W. Indeed,
W gives the fundamental 27-dimensional representation of
[
11,
27,
28].
The group
is simply connected, and there is only one more simple complex group whose Lie algebra is
, which is
. The rank of
is 6 and admits 72 roots. Since
is simply connected, its group of outer automorphisms,
, coincides with the group of symmetries of its Dynkin diagram. Therefore,
, and the unique non-trivial outer automorphism of
,
, acts on the center
, which is isomorphic to
, by sending each
in
to
[
11,
27].
Wolf and Gray [
29] proved that there are exactly two non-conjugate representatives in
of the outer automorphism
, with fixed point subgroups isomorphic to
and
, respectively. The group
is the rank 4 simple complex Lie group of exceptional type that can be viewed as the group of automorphisms of certain 26-dimensional complex vector spaces that preserves a certain non-degenerate holomorphic symmetric 3-form and a non-degenerate holomorphic symmetric 2-form. On its part,
is the centerless complex Lie group of type
.
More precisely, the first representative of
, denoted by
, corresponds to the diagram automorphism of
that fixes the central node of the Dynkin diagram and interchanges the pairs of nodes at the same distance from it. Explicitly,
interchanges the nodes
and
, and
and
, while fixing
and
, according to
Figure 1.
To identify the fixed point subgroup of
, consider the root system of
as embedded in
with a specific inner product. Precisely, let
be the standard basis of
, and define the subspace
by
Then, the root system of
consists of the vectors in
V defined by
A system of simple roots for
can be chosen as
and the Cartan matrix of
with respect to the simple roots above is
See [
25,
30] for further details and explicit computations. The fixed points of
are precisely those elements in
that commute with the action of
on the root system. Indeed, they form a root system of type
, since the fixed simple roots are
which generate a root system of type
[
25,
30,
31]; thus, the fixed point subgroup of
is isomorphic to the complex Lie group
.
The second representative of the outer involution , denoted , can be constructed by composing with a suitable inner automorphism. Specifically, let be the inner automorphism corresponding to conjugation by the element , where is the coroot associated with a specific root . By choosing appropriately, is another automorphism of order 2 representing the non-trivial outer involution of that is not conjugate to . By analyzing the action of on the root system of , it is easily checked that the fixed roots generate a root system corresponding to the Lie algebra . Since the center of the specific with Lie algebra must be contained in the center of , isomorphic to , it must be isomorphic to .
The combined action of a representative
of the outer involution of
with the pull-back action of an involution
gives an automorphism
of order 2 of the moduli space of principal
-bundles over
X, defined by
where
. In this section, the action of this automorphism on principal
-bundles over the compact Riemann surface is examined. First, it is proved that the obstruction for
to admit fixed points is an element of the cohomology
, where
denotes the subgroup of fixed points of
. Indeed, this obstruction result is proved for a general semisimple group.
In the next result, the fixed points of the combined involution of the moduli space of principal -bundles are studied, specifying the analysis made in Proposition 1 and considering the data concerning the possibilities for the fixed point subgroups of the different representatives of the outer involution that admits.
Theorem 1. Let X be a compact Riemann surface of genus with an involution τ having fixed points, be the outer involution of , and σ be a representative of with fixed point subgroup H (either or ). Then, the fixed point set of the involution of the moduli space defined in (1) is non-empty, it is isomorphic to the moduli space , and it consists of components (in particular, it must be that ). Proof. Let be the quotient curve, which, by the Riemann–Hurwitz formula, has genus . The involution has fixed points, which project to k branch points on Y.
Recall that a principal -bundle E over X is a fixed point of if and only if there exists an isomorphism . Such bundles correspond to reductions of the structure group to the fixed point subgroup H of over the quotient curve Y, by Lemma 1. The obstruction to the existence of a fixed point lies in by Proposition 1 and, since H is connected (notice that both and are connected groups), , so this obstruction vanishes automatically. In particular, the fixed point subvariety is non-empty.
The above reduction of the structure group is unique. To check this, notice that, if
is another fixed point of
, for some
, then,
since
and
acts on the center of
by
. If
is fixed by
, then,
By taking a suitable trivialization of E with open subset U, it follows that , from which . But this is not possible, since S is locally a third root of unity (so different from ), and X is connected, so a locally constant function must be constant. This proves that the reduction of the structure group is unique, so the map is injective and falls into the fixed point subvariety. This gives the isomorphism between the fixed point subvariety and the moduli space announced.
Finally, the moduli space of fixed points then consists of principal
H-bundles over
Y, but with additional twisting coming from the possible liftings of the involution to the bundle. These twistings are classified by
giving a total of
components, each isomorphic to
, for
H equal to
or
, which concludes the result. □
Remark 1. Note that, when a compact Riemann surface admits an involution, the number of fixed points of this involution is always even. This is a result that follows from the Riemann–Hurwitz formula for branched covers. Therefore, the assumption of Theorem 1 that the number of fixed points of τ is does not entail any loss of generality.
Remark 2. In the proof of Theorem 1, it has been used that is a positive integer number, since it is the genus of . Indeed, since τ is a holomorphic map of degree 2 from X to Y with ramification points (the fixed points of τ), each with ramification index , by the Riemann–Hurwitz formula it is obtained thatwhere h denotes the genus of . Simplifying the above equation, it is obtained that Therefore, rearranging to isolate the expression in question, it follows that , so the genus of Y, which must be a positive integer, coincides with the number .
The moduli space of principal -bundles over a compact Riemann surface X of genus has complex dimension , where 78 is the dimension of the Lie algebra . For the components of the fixed point subvariety of the involution given in Theorem 1, the dimension of each of them is computed in the next result.
Proposition 2. Let X be a compact Riemann surface of genus with an involution τ having fixed points and be the outer involution of , represented by the order 2 automorphism σ, with fixed point subgroup H. Then, each component of the fixed point set of Φ on has complex dimension:
- 1.
if .
- 2.
if .
Proof. By Theorem 1, for a fixed point
E of the involution
, there exists a principal
H-bundle
over the quotient curve
such that
E is obtained from
by the extension of the structure group. The dimension of the moduli space of principal
H-bundles over
Y is given by
, where
is the genus of
Y. By the Riemann–Hurwitz formula, the genus
of
satisfies
from which
.
For
, the dimension of the Lie algebra
is 52 [
25,
30]; thus, the dimension of each component is
Similarly, for
, the dimension of the Lie algebra
is 36; thus, the dimension of each component is
concluding the result. □
The fixed point components of the involution given in Theorem 1 may contain singular points, which correspond to principal H-bundles over with non-trivial automorphisms, as proved in the following result.
Proposition 3. Let X be a compact Riemann surface of genus with a fixed non-trivial involution τ, σ be an automorphism of order 2 of , representing its outer involution, with a subgroup of fixed points H, Φ
be the involution of defined in (1), and be a component of the fixed point set of Φ
on the moduli space , corresponding to principal H-bundles over . Then, the singular locus of consists of those -bundles E for which the corresponding principal H-bundle admits non-trivial automorphisms. Moreover, if is a stable principal H-bundle over Y, then it corresponds to a smooth point of . Proof. The moduli space
of principal
H-bundles over
Y is constructed as the quotient of the space of stable principal
H-bundles by the action of the gauge group [
5]. The points in
with non-trivial stabilizers under this action correspond to bundles with non-trivial automorphisms, and these points are precisely the singular points of
. Since each component
is isomorphic to
, the singular locus of
corresponds to those bundles
E for which the associated
H-bundle
has non-trivial automorphisms, as stated.
For a stable principal H-bundle , any automorphism must preserve the stability condition. An automorphism of is a G-equivariant map covering the identity map on Y. Such a map corresponds to a section of the associated bundle , where Ad denotes the adjoint action of H on itself.
Since the action of a non-central element of H would induce reductions of the structure group that violate the stability condition, any such section for a stable bundle must take values in the center of H.
Now, the center of H is trivial for the two possible subgroups of fixed points, or , so stable H-bundles have no non-trivial automorphisms, and the corresponding fixed point components are also smooth. This concludes the result. □
In the following results, the stability conditions of a fixed point of the involution
defined in (
1) as an
-bundle are related to the stability of its corresponding reduction of structure group to
or
, according to Theorem 1.
Proposition 4. Let X be a compact Riemann surface of genus with a fixed non-trivial involution τ, σ be an automorphism of order 2 of , representing its outer involution, and E be a semi-stable principal -bundle over X. If E is a fixed point of the involution defined in (1), then, the corresponding principal H-bundle over given by Theorem 1 is also semi-stable, where H is either or . Proof. Following Ramanathan’s notion of stability for principal bundles [
1,
2,
3], a principal
G-bundle
E over a Riemann surface is semi-stable if and only if for every reduction of the structure group to a parabolic subgroup
with Levi component
L and every anti-dominant character
of
L, the degree of the associated line bundle
is non-positive, where
G is any reductive complex Lie group.
Let E be a semi-stable principal -bundle over X that is a fixed point of the involution , meaning that there exists an isomorphism , and be the corresponding principal H-bundle over , where H is either or . Suppose, for contradiction, that is not semi-stable. Then, there exists a reduction of the structure group to a parabolic subgroup with Levi component and an anti-dominant character of such that . This reduction induces a reduction of the structure group of E to a parabolic subgroup that contains . More precisely, recall that the maximal parabolic subgroups of are in one-to-one correspondence with subsets of the set of simple roots . Under the outer automorphism , the simple roots are permuted as follows: , , , and . For a parabolic subgroup of , the corresponding simple roots in the subsystem of fixed roots are , which map to in the root system of . Thus, a reduction of the structure group of to corresponds to a reduction of the structure group of E to a parabolic subgroup P of that is stable under the action of , and similar reasoning works for .
The character
of
extends to a character of the Levi component
L of
P. Due to the compatibility with the involution
, the degree of the associated line bundle for
E satisfies
since the pull-back map
multiplies degrees by the degree of the covering, which is 2. Now, recalling that
, it follows that
, contradicting the semi-stability of
E. Therefore,
must be semi-stable. □
Corollary 1. Let X be a compact Riemann surface of genus with a fixed non-trivial involution τ and σ be an automorphism of order 2 of , representing its outer involution. Then, the stable fixed point subvariety of the involution Φ
defined in (1) consists of components isomorphic to the moduli space of stable principal H-bundles over , where H is isomorphic to or . Proof. By Proposition 4, if E is a semi-stable principal -bundle that is fixed by , then, the corresponding principal H-bundle over is also semi-stable.
Conversely, given a semi-stable principal
H-bundle
over
Y, the induced principal
-bundle
E over
X obtained by extending the structure group from
H to
is also semi-stable, since the forgetful map
respects semi-stability, where
H is a subgroup of a reductive Lie group
G [
20,
21]. This establishes a bijection between fixed points of
in the moduli space of semi-stable principal
-bundles and the moduli space of semi-stable principal
H-bundles.
For stable bundles, if E is a stable principal -bundle that is a fixed point of , then is also stable. This follows from the fact that stability is a stronger condition than semi-stability, requiring strict inequality in the degree condition for proper parabolic reductions (the same argument of the proof of Proposition 4 works). This concludes the result. □
Remark 3. By Proposition 3 and Corollary 1, it follows that every stable H-bundle over gives a smooth point in the corresponding component of the fixed point subvariety of the automorphism Φ of , where H is the subgroup or of . This smooth point is obtained by extension of the structure group and applying the push-forward , where is the natural projection.
4. Application to Octonionic Structures on the Riemann Surface
In this section, a geometric interpretation of the fixed points of the automorphism
defined in (
1) as octonionic structures on the quotient curve
, where
is an involution of
X, is established. Explicit computations of their characteristic classes are also provided, as well as an explicit construction of the mentioned octonionic structures.
A framework about the octonions that will be useful below is first established. The octonions form an eight-dimensional non-associative division algebra over the real numbers, which are constructed from the quaternions through the Cayley–Dickson process.
Definition 1. The algebra of octonions is the eight-dimensional real vector space with basis , where , and the multiplication of basis elements is given by the following:
for .
for , .
when is one of the triples in the set
While octonions are neither commutative nor associative, they satisfy the weaker property of alternativity, meaning that
for all
.
The automorphism group of the algebra of complex octonions
is isomorphic to the exceptional complex Lie group
[
30]. The simple complex Lie group of exceptional type
is also related to the octonions since
arises as the group of automorphisms of the exceptional Jordan algebra
that preserve a certain symmetric 3-form and certain symmetric 2-form defined on it [
10,
24]. Indeed,
can be constructed as the group of automorphisms of
preserving the above 3-form [
10,
24]. Recall that the exceptional Jordan algebra
, also known as the Albert algebra, is the 27-dimensional real vector space of
Hermitian matrices over the octonions, equipped with the Jordan product defined by
where
denotes the standard matrix multiplication. More precisely, the structure of the exceptional Jordan algebra can be described as
The concept of octonionic vector bundles formalizes the octonionic structures defined in manifolds.
Definition 2. Let M be a holomorphic complex manifold. An octonionic vector bundle over M is a complex vector bundle E of rank 8 together with a holomorphic bundle morphism satisfying the conditions of octonionic multiplication fiberwise.
For the transition to
, notice that
embeds into the complex Lie group
as the fixed point set of a representative
in
of the unique non-trivial outer involution
that
admits [
11,
32].
The isomorphism classes of octonionic structures on a Riemann surface
Y are classified by elements of
, and the isomorphism classes of complexified octonionic structures with a compatible Hermitian form on a Riemann surface
Y are classified by elements of the cohomology
. This classification is connected to the fixed point theory of
-bundles through the commutative diagram [
25,
32]
where
denotes the fixed points of the involution
defined in (
1) in the non-abelian cohomology set
.
Finally, the infinitesimal deformations of a complexified octonionic structure with a compatible Hermitian form on a Riemann surface
Y are parametrized by the cohomology group
, where
is the Lie algebra of
[
25,
32]. In addition, the dimension of this deformation space is, by the Riemann–Roch theorem,
where
denotes the genus of
Y. Notice that, for a generic octonionic structure,
, giving a deformation space of dimension
.
Octonionic structures have geometric significance, since they provide insights into the structure of exceptional Lie groups and manifold geometry. In particular, these structures induce geometric properties that reflect the exceptional nature of octonions as the largest normed division algebra. Thus, the presence of octonionic structures on quotient curves influences the topology and geometry of the underlying space through characteristic classes and curvature properties, thereby providing a geometric interpretation of algebraic invariants. Furthermore, these octonionic structures encode symmetries that are reflected in the deformation theory of the underlying bundles, connecting the discrete symmetries of the surface with the continuous symmetry of the exceptional group .
From the above framework, the relationship between the fixed points of the involution
on
defined in (
1) and the octonionic structures can now be established.
Proposition 6. Let X be a compact Riemann surface of genus with an involution , σ be a representative of the outer involution of with fixed point subgroup , and Φ
be the corresponding involution of defined in (1). Then, there exists a one-to-one correspondence between the fixed points of Φ
and octonionic structures on the quotient curve . Proof. Let
be a fixed point of
. Then, by Theorem 1,
E admits a reduction of the structure group to
over
Y,
. This principal
-bundle defines an octonionic structure on
Y, since complexified octonionic structures on
Y are classified by
[
25,
32]. By Proposition 4, this reduction is semi-stable, so it defines an element in
.
Conversely, given an octonionic structure on Y, the corresponding principal -bundle extends its structure group to give a principal -bundle over X that is fixed by the involution . This construction uses the push-forward operation , where is the quotient map, followed by an extension of the structure group from to , as indicated in Remark 3.
The above correspondence is bijective by the uniqueness of the reduction of the structure group established in Theorem 1, which gives an isomorphism between the fixed point subvariety and the moduli space . □
Remark 4. As a consequence of Proposition 6, Theorem 1, and Proposition 2, the subvariety of fixed points of the automorphism Φ
of defined in (1) is isomorphic to the moduli space of octonionic structures on the quotient curve , each component having complex dimension , where g is the genus of X and k is half the number of fixed points of τ. The above geometric interpretation can be further deepened by discussing the characteristic classes of the octonionic structures, as follows.
Proposition 7. Let X be a compact Riemann surface of genus with an involution , σ be a representative of the outer involution of with fixed point subgroup , Φ
be the corresponding involution of defined in (1), be a fixed point of Φ, and be the corresponding octonionic structure on given by Proposition 6. Then, the second Chern class of E satisfies , where is the quotient map. Proof. By Proposition 5, if E is a fixed point of , then , where is the reduction of the structure group of E to over Y given by Theorem 1. Since the octonionic structure corresponds to the principal -bundle , its second Chern class equals . Therefore, . □
Remark 5. The explicit computation of of Proposition 7 can be made using the Chern character of the octonionic structure. Precisely, for an octonionic structure, the second Chern class can be computed using that is a subgroup of [25]. The embedding of into allows us to relate the second Chern class to the first Pontryagin class through the relation [33]. Since the Pontryagin class is the obstruction to finding a section of the associated bundle of Cayley planes [34], this provides a geometric interpretation of the second Chern class in terms of octonionic geometry. 5. Explicit Construction of Octonionic Structures from Fixed Point Analysis
This section presents an explicit construction of octonionic structures corresponding to fixed points of the automorphism
of
defined in (
1), based on the preceding results. Initially, a hyperelliptic curve
X with hyperelliptic involution
is considered, along with a holomorphic line bundle
L of degree
over
. An octonionic structure on
will be constructed, demonstrating that this bundle is induced by a fixed point of
through Theorem 1, as illustrated in
Figure 2. Subsequently, this construction will be extended to any compact Riemann surface
X of genus
.
Proposition 8. Let X be a hyperelliptic Riemann surface of genus with the hyperelliptic involution τ and L be a line bundle over the quotient curve with . Then, the direct sum of eight copies of L, denoted , can be equipped with an octonionic structure that corresponds to a fixed point of the involution Φ
of induced by the non-trivial outer involution of and the involution τ of X as in (1). Proof. Notice that the direct sum is a rank 8 holomorphic vector bundle over Y. To equip V with an octonionic structure, transition functions valued in will be defined.
Since , it can be covered by two affine charts and with the transition function on the overlap . The line bundle L with has the transition function from to .
To define an octonionic structure on , the transition function for V from to is given by the diagonal matrix . This transition function takes values in (but not necessarily in , in principle). By composing with a carefully chosen map , a new transition function can be defined that takes values in and still defines a vector bundle isomorphic to V.
The existence of such a map is guaranteed since is connected and contains maximal tori that allow for the necessary deformations of transition functions. More precisely, a holomorphic map will be found such that the modified transition function takes values in and defines a vector bundle isomorphic to V.
First, notice that the exceptional Lie group
contains a subgroup isomorphic to
[
25], which in turn contains a maximal torus
. This torus can be embedded in
in such a way that diagonal matrices with entries of the form
can be realized within
[
25]. Consider also the Cartan decomposition of
, which gives the root system defined by
Using this root system, elements in that act diagonally on the octonion algebra , seen as the natural eight-dimensional representation of , can be explicitly constructed.
Now, the transition function
can be rewritten as
where
is the
identity matrix. The scalar factor
can be handled separately. Define a continuous path in the complex plane from
to
, where
is chosen appropriately based on the argument of
z in the overlap region
. Specifically, set
so that
has a positive real value.
Then, construct a map
that sends
z to a diagonal element in the maximal torus with appropriate eigenvalues. Specifically, define
where the phases
are chosen such that
. This is possible because the maximal torus in
has dimension 4, which allows sufficient freedom to satisfy this constraint while ensuring the resulting matrix is in
. The constraint can be satisfied by setting
, ensuring that the product equals
, as required.
Now, define the holomorphic map
as
. The modified transition function then becomes
With the above choice of
, this simplifies to
This matrix represents an element in the maximal torus of and hence is an element of . Moreover, it defines a vector bundle that is isomorphic to V, because the map provides a bundle isomorphism between the bundle defined by the transition function and the bundle defined by the transition function .
The resulting octonionic structure on
V corresponds to a principal
-bundle
over
Y. By Theorem 1 and Remark 3, this principal
-bundle gives a principal
-bundle
E over
X that is fixed by the involution
defined in (
1). □
Remark 6. For computing the second Chern class of the bundle E constructed in Proposition 8, notice that, by Proposition 5, . Since is constructed from the vector bundle , it follows thatwhere it is used that the second Chern class of a direct sum includes the products of first Chern classes. The above construction is now generalized to arbitrary compact Riemann surfaces of genus
(not necessarily hyperelliptic Riemann surfaces). For this generalization, an octonionic structure on a vector bundle over the quotient curve
of the form
, with
, is given, following the construction of Proposition 8, so that this bundle is shown to be induced by a fixed point of the automorphism
defined in (
1), as indicated in
Figure 3.
Theorem 2. Let X be a compact Riemann surface of genus with an involution τ having fixed points. Let be the quotient curve of genus . Then, for any collection of line bundles over Y satisfying , there exists an octonionic structure on the vector bundle that corresponds to a fixed point of the involution Φ
on defined in (1). Proof. Let us construct an explicit octonionic structure on the rank 8 vector bundle over Y and prove that it corresponds to a fixed point of the involution .
First, notice that the genus of Y must be by the Riemann–Hurwitz formula, as in Proposition 2, and that the condition implies that has degree zero. To equip V with an octonionic structure, transition functions valued in must be constructed. This is accomplished by working with a suitable open cover of Y and deforming the natural transition functions of V to take values in , as in Proposition 8.
Choose a standard open cover
of
Y such that each
is trivial over each
. The transition functions of
from
to
are given by holomorphic functions
. The transition functions of
V from
to
are given by diagonal matrices
These transition functions take values in
and can be deformed to take values in
while preserving the isomorphism class of
V. Specifically, for each overlap
, we construct, as in Proposition 8, a holomorphic map
such that the modified transition functions
take values in
and define a vector bundle isomorphic to
V. Then, the modified transition function becomes
Here, is a diagonal matrix that allows us to normalize the transition functions to take values in , the exceptional Lie group that preserves the octonionic structure. The construction ensures that takes values in the subgroup , because preserves the octonionic structure, and the construction of Proposition 8 modifies the original transition functions in a way that maintains this property.
Unlike the situation described in Proposition 8, in which it was not necessary to check a cocycle condition for the constructed transition functions, due to the hyperelliptic nature of the curve
X, the present proof is more complicated due to the need to prove this cocycle condition. In particular, it is required to check that on triple overlaps
it is satisfied that
For that, by expanding the product of the modified transition functions on the triple overlap, it is obtained that
For any point
, define the quantities
These quantities , , and represent the eighth roots of the moduli of the determinants of the original transition matrices.
The cocycle condition for the original transition functions states that
Taking the determinant of both sides, it is obtained that
Since the above quantities are positive real numbers, this means that
Now, recall that
was constructed as a diagonal matrix
where
and
is the argument of
.
For clarity, we define
as the phase adjustment for the
j-th component of the transition function from
to
at point
p. In this specific construction, all these phases are chosen to be equal:
This uniform distribution of phase adjustments across all eight components ensures that the determinant’s argument is compensated while maintaining the octonionic structure.
Using the notation
for the argument of
, it follows that
The cocycle condition for the original transition functions implies that
and
Now, for checking the cocycle condition for the modified transition functions, notice that
Notice that the diagonal matrices
commute with the phase adjustment matrices
. Denote by
the diagonal part of
. Then,
where
is a matrix that accounts for any non-diagonal components of
. In the case under consideration, since the original transition functions
are diagonal matrices (being the direct sum of line bundles), it is satisfied that
and
. Therefore,
By this commutation property, it is satisfied that
Now, by the cocycle condition for the original transition functions,
Expanding the product of the
terms,
Using that
, established before,
Recall that each
is a diagonal matrix with specific eigenvalues
where
The product of these diagonal matrices is
For the specific choice of phases performed, it is satisfied that
Therefore,
where it is used that
. Consequently,
so,
which implies the cocycle condition for the modified transition functions
, confirming that they properly define a holomorphic vector bundle with an octonionic structure corresponding to a principal
-bundle over
Y.
The resulting octonionic structure on V corresponds to a principal -bundle over Y. By the correspondence between principal -bundles over Y and principal -bundles over X fixed by the involution (as established in Theorem 1), this -bundle gives rise to a principal -bundle E over X that is fixed by . □
Remark 7. By Proposition 5 applied to the bundle constructed in Theorem 2, the second Chern class of E is , where is the second Chern class of the -bundle. Since is constructed from the vector bundle , it follows that Given that , the above can be expanded as Solving for the sum over pairs, This gives the second Chern class of the -bundle in terms of the degrees of the line bundles .
The geometric significance of Theorem 2 lies in its establishment of a concrete relationship between the geometry of principal bundles with exceptional structure groups and certain geometric constructions on Riemann surfaces. Specifically, this result shows that octonionic structures corresponding to fixed points of the involution can be realized through direct sums of line bundles satisfying specific degree conditions, providing a geometric characterization of these fixed points in terms of classical geometric objects. The result reveals how the exceptional geometry of manifests in the context of vector bundles over quotient curves, showing that the fixed-point theory of exceptional groups can be understood through more classical geometric structures. Moreover, this construction elucidates the relationship between the branched covering and the induced map on the cohomology with coefficients in exceptional groups, providing geometric insight into the transfer of octonionic structures between different Riemann surfaces. The condition has topological significance, ensuring that the resulting octonionic structure is compatible with the constraints imposed by the geometry.