Higgs bundles and exceptional isogenies

We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $SO(4,C)$ and $SO(6,C)$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated to orthogonal Higgs bundles. In the case of $SO(6,C)$ our construction can be interpreted as a new description of Recillas' trigonal construction.

Exceptional isomorphisms between low rank Lie algebras, and the corresponding isogenies between Lie groups, have been a source of fascination since first catalogued by Cartan a century ago ( [6, p. 352-355], see also [12, p. 519]). In this paper we explore the implications for Higgs bundles of the isogenies between Lie groups of rank 2 and 3: as well as their restrictions to the split real forms of these complex semsimple groups. Higgs bundles over a compact Riemann surface Σ of genus g ≥ 2 were introduced in [13] and in a more general setting in [29]. For any Lie group G, a G-Higgs bundle on Σ is a pair (P, Φ) where P is a holomorphic principal bundle and Φ (the Higgs field) is a holomorphic section of an associated bundle twisted by K, the canonical bundle of the surface Σ. If G is a complex group then P is a principal G-bundle, but if G is a real form of a complex group then the structure group of P is the complexification of a maximal compact subgroup of G. In this paper we consider only matrix groups, in particular G = SL(n, C) or SO(2n, C), and real forms of these groups. In these cases (described in §1) the Higgs bundles can be seen as holomorphic vector bundles with extra structure, where the precise nature of the extra structure is determined by the group G, and the Higgs fields are appropriately constrained sections of the endomorphism bundle twisted by K.
The defining data for G-Higgs bundle serves to construct G-local systems on Σ. Indeed this is the crux of Non-Abelian Hodge Theory (NAHT), whereby the moduli space of semistable G-Higgs bundles on Σ is identified with the moduli space of reductive representations of π 1 (Σ) into G. The implications of any group homomorphism h : G 1 → G 2 are clear for surface group representations, since composition with h induces a map from representations into G 1 to representations into G 2 . It follows from NAHT that there must be a corresponding induced map between the Higgs bundle moduli spaces, but the transcendental nature of the NAHT correspondence means that the clarity of the induced map on representations does not transfer so easily to the induced map on Higgs bundles. Our goal is to understand this map in the cases where the group homomorphism is given by the above isogenies.
In [14] Hitchin showed how the defining data of Higgs bundles can be re-encoded into a socalled spectral data set consisting of a ramified covering S of Σ and a bundle on S. For the groups of interest in this paper, the spectral bundles are line bundles, and hence lie in Jacobians of the spectral curve. In fact, they must lie in Prym varieties determined by the S and Σ. These abelian varieties form the generic fibers in a fibration of the Higgs bundle moduli space over a half-dimensional linear space.
In terms of vector bundles, the effect of some low rank isogenies has been studied in [1]. In this paper we expand those result in the case of (2) through the use of Hodge star type operators. For both (1) and (2) we give a novel description of the spectral data correspondences, shedding new light on the maps between Higgs bundles and their moduli spaces. Our key tool for understanding the induced maps on the spectral data for the Higgs bundles is a fibre-product construction, at the heart of which is the following diagram: In our applications, the curves S i and line bundles L i come from spectral data for SL(2, C) or SL(4, C)-Higgs bundles, while the fiber-produc S 1 × Σ S 2 , and the line bundle L yield the spectral data sets for the SO(4, C) or SO(6, C)-Higgs bundles. This construction clearly has wider applicability than the cases inspired by the isogenies (1) and (2). One notable feature of the construction is that it generically yields smooth curves, even though curves defined by spectral data for SO(2n, C)-Higgs bundles are necessarily singular. Indeed, the curves provided by our construction are the normalizations of the singular SO(2n, C)-spectral curves.
As seen in [1], the isogeny I 2 innduces a map on SL(2, C)-Higgs bundles (E i , Φ i ) given by where E 1 ⊗ E 2 has orthogonal structure determined by the symplectic structures on E i . Hence, in the case of the rank 2 isogeny I 2 , on the generic fibers of the Hitchin fibration for the moduli space M SL(2,C)×SL(2,C) , our main result is the following (see Propositions 15,16,34): Let S i be the spectral curve of the SL(2, C)-Higgs bundles (E i , Φ i ), and L i ∈ P rym(S i , Σ) the corresponding spectral line bundle. Then the spectral data for the the image I 2 ((E 1 , Φ 1 ), (E 2 , Φ 2 )) is given by (Ŝ 4 , L) where •Ŝ 4 := S 1 × Σ S 2 is a smooth ramified fourfold cover, and • L := p * 1 (L 1 ) ⊗ p * 2 (L 2 ) where p i : S 1 × Σ S 2 → S i are the projection maps. The map is a 2 2g fold-coverings onto its images. Restricted to SL(2, R) × SL(2, R)-Higgs bundles, the image lies only in components of M SO 0 (2,2) in which the Higgs bundles satisfy a topological constraint.
In the case of the rank 3 isogeny, as seen in [1], the map I 3 induces a morphism on SL(4, C)-Higgs bundles (E, Φ) given by where Λ 2 E has orthogonal structure determined by the isomorphism Λ 2 E * ⊗ det(E) ≃ Λ 2 E. This case differs from the previous one in two notable ways. The first is seen in the restriction of (5) to SL(4, R)-Higgs bundles, where the map reflects the fact that SO(4) is not a simple Lie group. This leads to a decomposition of the bundle Λ 2 E that is analogous to the decomposition of 2-forms on a Riemannian 4-manifold into self-dual and anti-self-dual forms.
The other new wrinkle is in the induced map on spectral data where the fiber product in diagram (3) is now taken on two copies of the same covering, say S. The resulting curve can never be smooth since it has a diagonally embedded copy of S which intersects other components at the ramification points for the covering S → Σ. It turns out though that this component plays no part in the map induced by I 3 . We thus modify our construction by removing the diagonally embedded component in the fiber product of an SL(4, C)-spectral data set with itself (see Section 5, Proposition 18, 26, 40 for details): Theorem 2. Restricted to the Higgs bundles for split real form SL(4, R) ⊂ SL(4, C), the induced map yields where Λ 2 ± E are the ±1-eigenbundles for an involution * : Λ 2 E → Λ 2 E determined by an orthogonal structure on E, with canonically determined orthogonal structures, and with Φ and α related as in Eq. (32). On the generic fibers of the Hitchin fibration for the moduli space M SL(4,C) , given the spectral data (S, L) corresponding to (E, Φ), the spectral data corresponding to the SO(6, C)-Higgs bundle I 3 (E, Φ) is (Ŝ 6 , I 3 (L)), where •Ŝ 6 is the symmetrization of the non-diagonal component in the fibre product S × Σ S; • I 3 (L) is a canonical twist of the line bundle generated by local sections of L = p * 1 (L) ⊗ p * 2 (L) that are anti-invariant with respect to the symmetry of S × Σ S.
The map is a 2 2g fold-coverings onto its images. Restricted to points representing SL(4, R)-Higgs bundles, the image lies only in components of M SO 0 (3,3) in which the Higgs bundles satisfy a topological constraint.
While the two cases covered by (1) and (2) are by no means the only interesting ones, they have some unique features and serve to illustrate phenomena that we expect to apply in greater generality. The group SL(2, R) has a distinguished place in any discussion of surface group representations because of its relation to hyperbolic structures and Teichmuller space, while the group SO 0 (2, 2) is the isomtery group of the anti-de Sitter space AdS 3 . Moreover, both groups are split real forms of complex semisimple groups (viz. SL(2, C) and SO(4, C) respectively) and also groups of Hermitian type. The only other such group is Sp(4, R). For both classes of real Lie group, the representation variety or, equivalently, the moduli space of Higgs bundles has a distinguished set of components. In the case of the split real forms, the distinguished components are called "higher Teichmuller components" because they generalize the copies of Teichmuller space which occur in the case of SL(2, R). For the groups of Hermitian type, the G-Higgs bundles (or the surface group representations) carry a discrete invariant known as a Toledo invariant. This invariant satisfies a so-called Milnor-Wood bound and the distinguished components are those in which the invariant has maximal value.
In the two cases considered in this article, the induced maps on the spectral curves yield a pairing between two different ramified coverings of a common base curve, together with isogenies between the associated Prym varieties. The spectral curves in each pair are, moreover, determined by different representations of the same group. This kind of situation, with pairings between coverings of a given curve and isogenies between the associated Prym varieties, has been considered in the context of integrable systems and also by Donagi (see e.g. [8,9]). As noted in [8, Example 3] the correspondence we see for the pair SL(4, C) and SO(6, C), namely between a four-fold covering and a six-fold covering with a fixed-point-free involution, is essentially the correspondence described by Recillas in his trigonal construction [23] and generalized by Donagi in [9]. The novelty in our version of the correspondence -and resulting relation between Prym varieties -lies in the use of fiber products to get explicit descriptions of the maps.
The first part of this paper covers background material on Higgs bundles (Section 1), spectral curves (Section 2) and the isogenies (Section 3). In Section 4 we describe the maps induced by the isogeny I 2 on both the complex group SL(2, C) × SL(2, C) and its split real form, and in Section 5 we do the same for I 3 . We include a discussion of the relation between our construction on spectral data and the trigonal construction of Recillas, and show how the map we obtain between Prym varieties can be interpreted in terms of a correspondence between curves. We conclude, in Section 6, with a discussion of maps between moduli spaces.

Higgs bundles and the Hitchin fibration
Let Σ be a compact Riemann surface of genus g ≥ 2, and π : K := T * Σ → Σ its canonical bundle. For G c a complex reductive Lie group with Lie algebra g c , from [13] one has the following definition: Definition 3. A G c -Higgs bundle on Σ is given by a pair (P, Φ) for P a principal G c -bundle on Σ, and Φ a holomorphic section of AdP ⊗ K, for AdP = P × Ad g c the adjoint bundle associated to P .
For GL(n, C) one recovers classical Higgs bundles as introduced in [14]. For matrix groups, the definition can be reformulated in terms of vector bundles rather than principal bundles. In particular, • An SL(n, C)-Higgs bundle is a pair (E, Φ) where E is a rank n holomorphic bundle on Σ with fixed trivial determinant, and Φ a traceless holomorphic section of End(E) ⊗ K; • An SO(n, C)-Higgs bundle is a pair (E, Φ) where E is a rank n holomorphic bundle on Σ with an orthogonal structure Q and a compatible trivialization of its determinant bundle 1 , and Φ is a holomorphic section of End(E)⊗K satisfying Q(u, Φv) = −Q(Φu, w). The construction of G-Higgs bundles for a real form G of G c goes back to [13,15] (see also [10,5] Definition 4. Given G a real form of a complex Lie group G c , a principal G-Higgs bundle is a pair (P, Φ) where P is a holomorphic principal H C -bundle on Σ, and Φ is a holomorphic section of P × Ad m C ⊗ K.
Alternatively, as done in [15], we may regard real Higgs bundles as classical Higgs bundles (E, Φ), with extra conditions reflecting the structure of the real group and its isotropy representation. In this paper we shall mainly consider SL(n, R) and SO(n, n)-Higgs bundles, for which we recall their main properties in Sections 2.2 and 2.4 (for further references, see [1,15,26]).
Following [13], a classical Higgs bundle (E, Φ) is said to be (semi)-stable if all subbundles F ⊂ E such that Φ(F ) ⊂ F ⊗ K satisfy deg(F )/rk(F ) < (≤) deg(E)/rk(E). Moreover, the pair is said to be poly-stable if it can be written as a direct sum of stable Higgs bundles (E i , Φ i ) for which deg(E i )/rk(E i ) = deg(E)/rk(E). The notion of stability can be extended to G c -Higgs bundles as in [14], as well as to G-Higgs bundles (e.g. see [4]), and used to construct the corresponding moduli spaces. As explained below in Remark 6 stability considerations will not play a role in this paper. We shall denote by M Gc and M G the moduli space of G c -Higgs bundles and M G -Higgs bundles, respectively.
1.1. The Hitchin fibration and spectral curves. Given a homogenous basis r 1 , . . . , r k of invariant polynomials for the Lie algebra of a complex Lie group G c , let d 1 , . . . , d k be their degrees. Then, the Hitchin fibration (introduced in [14]) is defined as A point in the Hitchin base A Gc determines a section of the line bundle π * K d k defined by η d k + · · · + r k−1 (Φ)η d 1 + r k (Φ), where η the tautological section of π * K. The zero locus of this section is the so-called spectral curve S associated to the Higgs bundle. The curve S lies the total space of K and the projection π : S → Σ is a d k -fold cover of Σ. By considering vector bundles on it (or its desingularization), one can give a geometric description of the fibres of the Hitchin fibration [14]. In the case of classical Higgs bundles the generic fibres are given by the Jacobian varieties of the spectral curves S, on which Φ has a single-valued eigenvalue η. Indeed, given a line bundle L on S one may recover the corresponding classical Higgs bundle (E, Φ) by taking E := π * L and Φ the direct image of η : L → L ⊗ π * K. In Section 2 we describe more fully the spectral data for the specific cases of interest in this paper.
By considering the embedding of M G in M Gc , one may identify the G-Higgs bundles as points in the Hitchin fibration satisfying additional constraints (see [14] for classical complex Lie groups, and [28] and references therein for real Higgs bundles). In particular, in the case of the split real form of G c , the line bundles are the torsion two points in the Jacobian (see [26,Theorem 4.12]).
Remark 5. Spectral curves are defined for all Higgs bundles represented by points in the moduli spaces M G , but they are not necessarily smooth. It follows however from Bertini's theorem that on the generic fiber of the Hitchin fibration the spectral curves are smooth. Throughout the paper we shall consider Higgs bundles over the smooth loci of the Hitchin fibration, i.e. points defining smooth spectral curves, since we will further restrict our attention to those curves for which the most generic type of ramification behaviour occurs.
Remark 6. Note that if the spectral curve for a Higgs bundle is smooth then the characteristic polynomial for the Higgs field is irreducible. The Higgs bundle thus has no invariant subbundles and is therefore automatically stable. Hence, whilst stability is needed to define the moduli spaces, it will play no role in our discussions and we shall omit any further comment on it.

Spectral data for complex and real Higgs bundles
We shall recall here how to study the fibres of the Hitchin fibration through spectral data, which we shall do by reviewing the methods introduced in [14,15] for complex Higgs bundles, and in [26,28] for real Higgs bundles. Since our main focus is on SO(2n, C) and SL(2n, C)-Higgs bundles, we shall restrict attention to those groups and their split real forms.
Here a i ∈ H 0 (Σ, K i ), and η is the tautological section of π * K. The (n − 1)-tuple (a 2 , . . . , a n ) defines a point in the base of the Hitchin fibration (see [14]). Over a dense open set in the base of the Hitchin fibration, the spectral curve S is smooth and the fiber can be identified (biholomorphically) with the Prym variety Prym(S, Σ), the subset of Jac(S) of line bundles whose direct image sheaf has trivial determinant. The relation between L ∈ Prym(S, Σ) and E (once a choice of K 1/2 is made) is then where N m denotes the norm map defined by the projection π : S → Σ, the trivialization of det(E) determined by the trivialization of N m(L). Note that in the case of two-fold covers, a line bundle is in the Prym variety if and only if its dual is isomorphic to the pull back by the involution switching the sheets of the cover.

2.2.
Spectral data for SL(n, R)-Higgs bundles. From Definition 4 , Higgs bundles with structure group SL(n, R) are given by classical Higgs bundles (E, Φ) together with an oriented orthogonal structure on E, with respect to which the Higgs field is traceless and symmetric. Moreover, by [26,Theorem 4.12] one has that the intersection of the moduli space M SL(n,R) with the smooth fibres of the SL(n, C) Hitchin fibration is given by line bundles L ∈ Prym(S, Σ) such that L 2 ∼ = O. Following [3] and [14], a torsion 2 line bundle L induces an SL(n, R)-Higgs bundle (E, Φ) with E = π * (L ⊗ π * K (n−1)/2 ) and Φ the push down of the tautological section η. Moreover, the orthogonal structure on E comes from an O(1) structure on L.

2.3.
Spectral data for SO(2n, C)-Higgs bundles. From the characteristic polynomial of an SO(2n, C)-Higgs field Φ one obtains a 2n-fold cover π : S → Σ whose equation is given by where η is as in (6), the sections b i ∈ H 0 (Σ, K 2i ), and p n ∈ H 0 (Σ, K n ) is the Pfaffian of Φ. This curve has a natural involution σ : η → −η, and is singular at η = p n = 0, i.e., at the fixed points of the σ . The normalization of S, which we denote byπ :Ŝ → Σ, is what we shall refer to as the spectral curve. As in the previous cases,Ŝ is generically smooth, and the involution σ extends to an involutionσ onŜ which does not have fixed points. The generic fibers of the Hitchin fibration can be identified with an abelian variety defined by a connected component of Prym(Ŝ,Ŝ/σ), i.e. the kernel of the norm map N m : P ic(Ŝ) → P ic(Ŝ/σ). Given a line bundle L ∈ Prym(Ŝ,Ŝ/σ) and a choice of K 1/2 , the vector bundle E is recovered as In this case, the orthogonal structure on E comes from the isomorphism σ * L ≃ L −1 (by virtue of which N m(L) = 0). The spectral data associated to an SO(2n, C)-Higgs bundle is defined on the desingularizationŜ of S. From Eq. . This can also be seen as a reflection of the fact that M SO(2n,C) has two components corresponding to the possible values for the second Stieffel-Whitney class of an SO(2n, C)-bundle. We say that the spectral data is given by (Ŝ, L) forŜ the normalized curve and L ∈ Prym(Ŝ,Ŝ/σ).

Remark 7.
For any m-fold ramified cover π : S → Σ, the ramification divisor R in S relates the canonical bundles K S and K of S and Σ, respectively, by denotes the line bundle defined by the divisor R. If S is a spectral curve, i.e. contained in the total space of K, then K S = π * K m , and [R] = π * K (m−1) . The relation between L and E in both (7) and (9) can thus be given 2 as

2.4.
Spectral data for SO 0 (n, n)-Higgs bundles. From Definition 4 one has that SO(n, n)-Higgs bundles can be viewed as SO(2n, C)-Higgs bundles of the form (W 1 ⊕ W 2 , Φ) where W i are vector bundles of rank n with orthogonal structure, say q i , and for α T the orthogonal transpose of α, i.e. α T = q −1 2 · α * · q 1 where α * denotes the dual map. By further requiring that det(W i ) ≃ O, one obtains an SO 0 (n, n) pair. In this case, from [26,Theroem 4.12] and along the lines of Section 2.2 one has that the intersection of the moduli space M SO 0 (n,n) with the smooth fibres of the SO(n, C) Hitchin fibration is given by L ∈ Prym(Ŝ,Ŝ/σ) which satisfy L 2 ∼ = O. As for classical Higgs bundles, given a torsion two line bundle L ∈ Prym(Ŝ,Ŝ/σ) one obtains an SO 0 (n, n)-Higgs bundle (E, Φ) by taking the direct image of the line bundle L ⊗ (KŜ ⊗ π * K * ) 1/2 and the push forward of η.
Given the spectral data (Ŝ, L) of an SO 0 (n, n)-Higgs bundle, since L is of order two and in the Prym variety of a two fold cover p :Ŝ →Ŝ/σ, it is invariant under the involution σ onŜ. Hence its local sections decompose into invariant and anti-invariant local sections , and thus the direct image decomposes as p * L =L 1 ⊕L 2 , where the summands are generated by local invariant and anti-invariant sections (see [27]). Moreover, considering the n-fold coverπ :Ŝ/σ → Σ, the orthogonal bundles W i are recovered by takingπ * (L i ⊗ p * (KŜ ⊗π * K * ) 1/2 ) for i = 1, 2.

Homomorphisms of groups and induced maps
As noted in the Introduction, given a fixed surface Σ and a homomorphism between two Lie groups Ψ : G → G ′ , there is clearly an induced map where Rep(Σ, G) denotes the space of representations modulo conjugation. The correspondence between surface group representations and Higgs bundles thus implies a similar induced map between G-Higgs bundles and G ′ -Higgs bundles.
From Definition 3 one sees that for an isomorphism between complex groups there is in fact an induced map from G-Higgs bundles to G ′ -Higgs bundles given by where dΨ : ad(P G ) → ad(P G ′ ) (12) is the map defined by the derivative at the identity of the map Ψ, i.e. by the map on Lie algebras. Moreover, if the homomorphism Ψ restricts to a map between real forms of G and G ′ which respects the Cartan decompositions of the Lie algebras, then the map Ψ induces a map from G r -Higgs bundles to G ′ r -Higgs bundles. From Definition 4, following the notation of Section 1, the map is given by The maps in (11) and (13) ought to acquire descriptions purely in terms of spectral data , and this shall be investigated in forthcoming sections. The relation between the spectral curves can be deduced from its relation to the eigenvalues of the Higgs field. Thus if the spectral curve S for (P H C , Φ) is defined by where {η 1 , . . . , η n } are the eigenvalues of Φ, then the spectral curve S ′ for Ψ * (P H C , Φ) is defined by an equation of the same form except with the eigenvalues replaced by the eigenvalues of dΨ(Φ), leading to a map S → S ′ . Alternatively one may seek a more intrinsic understanding of the map purely in terms of the information encoded in the geometry of the covering S → Σ and the restrictions on the spectral line bundle L. This is our goal for the special cases of the isogenies in (1) and (2). The isogenies can be described in several ways, including from coincidences of Dynkin diagrams or in terms of representations e.g. in terms of Schur functors as in [21,. For our purposes, the representation theoretic point of view is convenient, as described in the next two sections.
3.1. The isogeny between SL(2, C)× SL(2, C) and SO(4, C). The group SL(2, C)× SL(2, C) The isogeny onto SO(4, C) can be seen as coming from the fact that SL(2, C) ≃ Sp(2, C). If ω is the symplectic form on C 2 preserved by matrices with unit determinant, then Q 4 := ω ⊗ ω, defines a symmetric, non-degenerate bilinear form on C 2 ⊗ C 2 . Hence one has a map where SO(4, C) is the group of (orientation preserving) linear maps C 4 → C 4 preserving the form Q 4 . The derivative at the identity yields an isomorphism of Lie algebras given by , the quadratic form Q 4 has signature (2, 2) and thus the above map (15) between complex Lie groups restricts to where the subscript in SO 0 (2, 2) denotes the connected component of the identity (see [1, Section 5.2] for more details). The map on Lie algebras similarly restricts. Indeed, givenȧ i ∈ sl(2, R) symmetric and trace-free for i = 1, 2, and fixing a basis for R 4 such that the orthogonal structure has the form I 0 0 −I , then the map in (16) is given by where α t denotes the transpose 3 . The precise form of α depends on the orientation chosen for R 4 , i.e. on the identification Λ 4 R 4 ≃ R. Moreover, whilst Pf(dI 2 (ȧ 1 ,ȧ 2 )) 2 = det(α) 2 , the sign of Pf(dI 2 (ȧ 1 ,ȧ 2 )) depends on the choice of orientation.
3.2. The isogeny between SL(4, C) and SO(6, C). The group SL(4, C) of volume-preserving linear maps T : C 4 → C 4 has a 6-dimensional representation on the exterior power Λ 2 C 4 given by g(v ∧ w) := (gv) ∧ (gw). Taking n = 4 and k = 2 in the isomorphism and fixing an identification Λ 4 C 4 ≃ C (i.e. fixing a volume form), one gets a bilinear form Q 6 on Λ 2 C 2 which is symmetric and non-degenerate. Since an element of SL(4, C) preserves the volume form, it preserves a bilinear form, and thus one has a map where SO(6, C) is the group of (orientation preserving) linear maps C 6 → C 6 preserving the nondegenerate symmetric form Q 6 . Moreover, the map gives a double cover SL(4, C) of SO(6, C), thus realising SL(4, C) as Spin (6, C). The derivative at the identity gives the Lie algebra isomorphism where I 4 denotes the identity map on C 4 and the endomorphismȦ ⊗ I 4 + I 4 ⊗Ȧ is understood to be the restriction to Remark 9. IfȦ has eigenvalues {λ a } 4 a=1 , then as a map on Restricted to R 6 , the quadratic form Q 6 has signature (3, 3) and thus I 3 restricts to with a corresponding restriction of (21). In particular, ifȧ ∈ sl(4, R) is symmetric, trace-free, with entries a ij , fixing a basis for R 6 such that the orthogonal structure has the form I 0 0 −I , the map has the form As in the previous case, the precise form of α depends on the orientation chosen for R 6 , i.e. on the identification Λ 6 R 6 ≃ R, and a standard choice yields α =   a 13 + a 24 −a 14 + a 23 a 11 + a 22 −a 12 + a 34 a 11 + a 33 a 14 + a 23 −a 22 − a 33 a 12 + a 34 −a 13 + a 24   .
In all cases, the Pfaffian is given by P f (dI 3 (a)) = ± det(α), where the sign depends on the choice of orientation.
Remark 10. Fixing a maximal compact subgroup SO(4) ⊂ SL(4, R), one obtains extra structure related to the presence of two inequivalent normal SO(3) subgroups, by virtue of which SO(4) fails to be simple. These subgroups have an important influence on the map induced by I 3 on SL(4, R)-Higgs bundles , described in more detail in §5.3, and considered also in §6.2.

The rank 2 isogeny and the Hitchin fibration
In this section we describe the map corresponding to the isogeny I 2 in §3.1 in terms of spectral data. After exploring the maps for the complex groups we examine the extra conditions required to understand the corresponding maps for the split real forms.
Higgs bundle can be viewed as a pair of this type in which the bundles have oriented orthogonal structures and the Higgs fields are traceless and symmetric with respect to the orthogonal structures. Up to isomorphism, we may thus assume in this case that the bundles are of the form E i = N i ⊕ N * i where N i is a line bundle of non-negative degree, the orthogonal structure is defined by the isomorphism and the Higgs field is of the form The characteristic polynomials for the Higgs fields define two spectral curves π i : S i → Σ in the total space of K as in Eq. (6). These are 2-fold covers of the Riemann surface, with equations for a i ∈ H 0 (Σ, K 2 ) and η the tautological section of π * K. Generically the quadratic differentials a i have simple zeros, and thus by Bertini's theorem the curves S i are generically smooth. As seen in Section 2.2, the spectral data associated to these SL(2, C)-Higgs bundles is completed by line bundles L i ∈ Prym(S i , Σ). Remark 11. The SL(2, R)-Higgs bundles (E i , Φ i ) have associated an integer invariant known as the Toledo invariant. This can be defined in several equivalent ways, including as the degree of the line bundle N i or the Euler number of the SO(2)-principal bundle associated to N i ⊕ N * i . It can also be seen in the spectral data where it is detected by the action of the involution on the fibers of the line bundle at fixed points (see [27]). However it is defined, the Toledo invariant is bound by a so-called Milnor-Wood inequality.
4.2. SO(4, C)-Higgs bundles and SO 0 (2, 2)-Higgs bundles. An SO 0 (2, 2)-Higgs bundle can be described as an SO(4, C)-Higgs bundle (E, Φ) where E = W 1 ⊕ W 2 decomposes as the sum of two rank 2 holomorphic oriented orthogonal bundles, and for α T = q −1 2 • α * • q 1 where α T is the dual map and q i is the orthogonal structure of W i . Furthermore, Higgs bundles with structure group SO 0 (2, 2) have isomorphisms δ i : Λ 2 W i ∼ = O. As in the previous section, we may take the rank two orthogonal bundles to be of the form for M i line bundles on Σ with deg(M i ) ≥ 0, and with orthogonal structure as in (25).
As in Eq. (8), the Higgs field Φ in an SO(4, C)-Higgs bundle defines a 4-fold cover π 4 : S 4 → Σ with equation where b 2 is the Pfaffian , b i ∈ H 0 (Σ, K i ), and η the tautological section of π * K. The spectral data for an SO 0 (2, 2)-Higgs bundle is given by the spectral data ( The group SO 0 (2, 2) is both a split real form of SO(4, C) and a group of Hermitian type. As a consequence of being a split real form, the Hitchin fibration admits a section which defines the Hitchin component in M SO 0 (2,2) . By virtue of the properties of groups of Hermitian type the Higgs bundles carry Toledo invariants, i.e. discrete invariants which in the case of SO 0 (2, 2)-Higgs bundles may be taken to be the degrees of the line bundles M 1 and M 2 . The invariants are bounded by a Milnor-Wood type inequality which in this case is (e.g. see [5] or [24, Table  C 4.3. The induced map on the Higgs bundles. From Section 3.1 (see [1] for a detailed study of this), the map on SL(2, C) × SL(2, C)-Higgs bundles induced by I 2 is where the orthogonal structure on E 1 ⊗ E 2 is the tensor product of the symplectic structures ω i on E 1 and E 2 (recall that SL(2, C) ≃ Sp(2, C)).
Following the notation of Section 4.1, the map I 2 can then be put in the form 4.4. The induced map on spectral data. We shall first construct the spectral data (Ŝ 4 , L) associated to the SO(4, C)-Higgs bundle obtained via I 2 ,and then specialize to the spectral data associated to the split real forms SL(2, R) × SL(2, R) and SO 0 (2, 2).
Proposition 13. Let (S i , L i ) be the spectral data corresponding to an SL(2, C)×SL(2, C)-Higgs bundle, where each S i is defined by η 2 + a i = 0 for a i ∈ H 0 (Σ, K 2 ), and L i ∈ Prym(S i , Σ).
Then the pair (Ŝ 4 , L) given by •Ŝ 4 := S 1 × Σ S 2 is the fiber product curve, and is the line bundle as in diagram (3), gives the spectral data associated to an SO(4, C)-Higgs bundle.
Remark 14. The total space of K ⊕K can be identified with the fiber product K × K K ⊂ K ×K. The fiber product S 1 × Σ S 2 may thus be regarded as a subvariety of either K × K K or of the total space of K ⊕ K.
Proof. The fiber productŜ 4 , as a curve in the total space of K ⊕ K, is defined by the conditions where (η 1 , η 2 ) denotes the tautological section on K ⊕ K. It follows that for a generic choice of (a 1 , a 2 ) this curve is smooth. Moreover, if S 4 ⊂ K denotes the image ofŜ 4 under the map given by fiberwise addition, then it is defined by the conditions Hence, the four-fold cover S 4 is defined by the equation which from §2.4 and Eq. (8), is the spectral curve of an SO(4, C)-Higgs bundle. The curve S 4 is generically singular, with singularities over the zeros of a 1 − a 2 , and by construction the the map + :Ŝ 4 → S 4 is an isomorphism on the smooth locus of S 4 . The involution σ : η → −η which preserves S 1 and S 2 induces an involution (σ, σ) on S 4 = S 1 × Σ S 2 . When needed we shall denote the involution on S 1 , S 2 by σ i , for i = 1, 2 and on S 4 byσ 4 . The fixed points of σ i are the zeros of a i , and thus, since the zeros of a 1 and a 2 are generically different, genericallyσ 4 = (σ 1 , σ 2 ) does not have any fixed points. It is clear from (35) thatσ 4 descends to the involution η → −η on the singular curve S 4 , where it has fixed points at the branch locus.
In order to see that L is the spectral line bundle associated to an SO(4, C)-Higgs bundle one has to show that L ∈ Prym(Ŝ 4 ,Ŝ 4 /σ 4 ). Since L i ∈ Prym(S i , Σ) one has that σ * i L i ∼ = L * i and so the line bundle L := p * 1 (L 1 ) ⊗ p * 2 (L 2 ) is sent to its dual by the involutionσ 4 . Hence, the line bundle L onŜ 4 is in Prym(Ŝ 4 ,Ŝ 4 /σ 4 ) as required.
Proposition 15. The spectral data (Ŝ 4 , L) induced by an SL(2, C) × SL(2, C)-Higgs bundle (E 1 , Φ 1 ), (E 2 , Φ 2 ), as in Proposition 13 corresponds to the spectral data of the SO(4, C)-Higgs bundle Proof. As seen in Proposition 13 the spectral curveŜ 4 is the curve associated to an SO(4, C)-Higgs bundle. Furthermore, from Eq.(36) the curve is indeed the one associated to the Higgs bundle in the image of (E 1 , Φ 1 ), (E 2 , Φ 2 ) through I 2 . With the notation of (39) below, in order to corroborate that the line bundle L is indued the spectral line bundle of the image SO(4, C)-Higgs bundle, note that for any line bundles F i on S i one has . Applying this to the line bundles L i and using the relation (7) with n = 2, we get whereas by (9) the vector bundle on Σ defined by L is E = π * (L ⊗ (KŜ 4 ⊗ π * K * ) 1/2 ) . Recall that KŜ 4 ⊗ π * K * corresponds to the ramification divisor R ⊂Ŝ 4 , while the ramification divisors R i ⊂ S i satisfy [R i ] = π * i K. It follows that and hence that E = π * (L) ⊗ K = E 1 ⊗ E 2 as required.

4.5.
The restriction to SL(2, R) × SL(2, R). Let (S i , L i ) be the spectral data of an SL(2, R)-Higgs bundles, for i = 1, 2. Then one has that L 2 ∼ = p 2 Thus as seen in Section 2.4 the line bundle L defines an SO 0 (2, 2)-Higgs bundle. Since σ * i L i ≃ L −1 i ≃ L i , it follows thatσ * 4 L ≃ L −1 ≃ L. This means that under the projection p :Ŝ 4 →Ŝ 4 /σ 4 the direct image sheaf p * L splits as the sum of two line bundles L ± , generated byσ 4 -invariant and anti-invariant local sections. The relation between the different covers of the Riemann surface and the line bundles on them is depicted in the following diagram: In particular, this implies that Moreover,Ŝ 4 is the normalization of the spectral curve S 4 ⊂ π * K and the involutionσ 4 onŜ 4 corresponds to the involution η → −η on S 4 . It follows that multiplication by the tautological section η interchanges L + and L − , and thus that the Higgs field on π * (L) has the form as in (31). We thus get: Proposition 16. Consider ((S 1 , L 1 ), (S 2 , L 2 )) the spectral data for a point in M SL(2,R)×SL(2,R) represented by and let (Ŝ 4 , L) be defined as in Proposition 13. Then (Ŝ 4 , L) is the spectral data for the point in M SO 0 (2,2) represented by where α is as in Eq. (32).

The rank 3 isogeny and the Hitchin fibration
In this section we investigate the induced map I 3 : M SL(4,C) → M SO(6,C) , and its restriction to the split real forms SL(4, R) and SO 0 (3, 3).

SL(4, C)-Higgs bundles and SL(4, R)-Higgs bundles. From Definition 4, an SL(4, R)-
Higgs bundle on Σ is holomorphic SO(4, C)-principal bundle together with a symmetric Higgs field. Equivalently it can be viewed as a pair (E, Φ) where E is an oriented holomorphic rank 4 orthogonal vector bundle, i.e. a vector bundle with a holomorphic symmetric nondegenerate bilinear paring Q, and a compatible isomorphism δ : det(E) ≃ O, and the Higgs field Φ : E → E ⊗ K is traceless and symmetric with respect to Q.
Recall from Section 5.1 that the spectral curve for an SL(4, C)-Higgs bundle (E, Φ) is a ramified 4-fold cover π : S → Σ in the total space of K with equation P 4 (η) := det(ηI − Φ) = η 4 + a 2 η 2 + a 3 η 3 + a 4 = 0, for a i ∈ H 0 (Σ, K i ) and η the tautological section of π * K. For generic choices of {a 2 , a 3 , a 4 } the curve is smooth and has only the most generic ramification, i.e. in fibers over the branch locus there are two unramified points and one order two ramification point. The spectral data is
5.3. The induced map on Higgs bundles. Using the results in Section 3.2 for the induced action of I 3 on vector bundles and Lie algebras, we get the map between complex Higgs bundles The orthogonal structure Q : Λ 2 E → Λ 2 E * is induced by the combination of (19) and the trivialization of det(E). As in the case of the map defined by I 2 (see Remark 12) an isomorphism δ : det(Λ 2 E) → O Σ such that δ 2 agrees with the trivialization of (det(Λ 2 E)) 2 determined by Q, is determined only up to a choice of sign. There are thus two (oppositely oriented) possible conventions for determining the SO(6, C) structure on Λ 2 E. This choice plays a role in the map induced by I 3 on the base of the Hitchin fibrations of the moduli spaces (see Section 74). The vector bundle E of an SL(4, R)-Higgs bundle (E, Φ) has an oriented orthogonal structure, i.e. an associated pair (q, ǫ) where q is a holomorphic orthogonal structure on E, and ǫ is a compatible isomorphism ǫ : det(E) ≃ O trivializing its determinant. The orthogonal structure induces an isomorphism (by abuse of notation, also denoted by q) Using ǫ as the trivialization of det(E) required in the construction of Q, yields an isomorphism * = q −1 · Q : which satisfies 4 q(α, β) = Q(α, * β), where q, Q are regarded as bilinear forms on Λ 2 E, and α, β ∈ Λ 2 E. By using a local oriented orthonormal frame to compute * , it can be seen that * satisfies * 2 = I. Taking the ±1 eigenspaces of * thus gives a decomposition The orthogonal structure on Λ 2 E restricts to orthogonal structures on each summand, so that the structure group reduces to SO(3, C) × SO(3, C).
Remark 17. The structure groups of the bundles Λ 2 E and Λ 2 + E ⊕Λ 2 − E can be reduced to SO(4) and SO(3) × SO(3) respectively. The two copies of SO(3) are precisely the normal subgroups mentioned in Remark 10 by virtue of which SO(4) fails to be simple (see [10]) .
With respect to the reduction in (47), the Higgs field Φ⊗I +I ⊗Φ has the form in (27), where Φ and α are related as in Eq. (24) and −α t is the orthogonal transpose. Denoting orthogonal structures on Λ 2 ± E by q ± , we thus get: Proposition 18. The isogeny I 3 induces the following map between SL(4, R)-Higgs bundles and SO 0 (3, 3)-Higgs bundles: where if E has orthogonal structure q then the bundles Λ 2 ± E have orthogonal structures q ± , and Φ and α are related as in Eq. (23).

5.4.
The induced map on spectral data. Given SL(4, C)-spectral data (S, L), with S defined by (41) and L ∈ Prym(S, Σ), we build SO(6, C)-spectral data using a construction similar to the fiber product construction in Section 4.4, except in this case we take the product of (S, L) with itself, i.e. in diagram (3) we have S 1 = S 2 and L 1 = L 2 . The resulting curve has both singularities and additional symmetries that are absent when S 1 and S 2 are different. Our construction takes both of these features into account in an essential way.
The curve S × Σ S is a 16-fold cover of the Riemann surface Σ. Over a generic point in the Hitchin base, S is smooth and S × Σ S has two smooth components, namely the diagonal S ∆ := {(s, s) ∈ S × Σ S} and another one which we denote by (S × Σ S) 0 . The intersection of these components lies in fibers over the branch locus of the covering π : S → Σ.
Viewing the curve S × Σ S in the total space of K ⊕ K, the involution τ : (x, y) → (y, x) interchanges the copies of S, and thus the fixed point set of τ is S ∆ . The quotient map π τ : (S × Σ S) 0 → (S × Σ S) 0 /τ commutes with the projection onto Σ. It is an unramified double cover on S × Σ S − S ∆ but has ramification points in the fibers over the base locus of π : S → Σ. Using the biholomorphism Sym : we can view the quotient (S × Σ S) 0 /τ as a curve in the total space of K ⊕ K 2 . We definê and denote byπ τ the composition of π τ and Sym. A depiction of the relation between the above curves and projections is given in diagram (51) below. By abuse of notation, we denote by p i : (S × Σ S) 0 → Σ the restrictions to (S × Σ S) 0 of the projection maps to the two factors of the full fiber product: Lemma 20. For generic points in the Hitchin base, the ramification divisors R on S, R 0 on (S × Σ S) 0 andR 6 onŜ 6 for the projections π,π τ andπ 6 respectively, are related as follows: Proof. The ramification divisor for a covering π : X → Y is defined by R = Σ y∈Y R(y) · y where the weights R(y) are such that R(y) + 1 is the multiplicity of π at y. If S is a generic spectral curve and x is a point in the branch locus of π : S → Σ then we can write π −1 (x) = {y 1 , y 2 , y 3 }, with y 1 ∈ R (with weight 1) but π unramified at y 2 , y 3 . Then the fiber of (S × Σ S) 0 /τ over x consists of points  6 , each with weight 1, whileπ −1 τ Sym([y 1 , y 1 ]) = (y 1 , y 1 ) is in R 0 (with weight 1). Notice now that in the fiber over x one has π −1 0 (x) ∩ (p −1 1 (R) + p −1 2 (R)) = 2(y 1 , y 1 ) + (y 1 , y 2 ) + (y 2 , y 1 ) + (y 1 , y 3 ) + (y 3 , y 1 ) = 2(y 1 , y 1 ) +π −1 τ (Sym([y 1 , y 2 ]) + Sym([y 1 , y 3 ])) as required.
Proposition 21. The curveŜ 6 is generically smooth, and the canonical desingularization of its projection to K through the Sym map, which is the spectral curve of an SO(6, C)-Higgs bundle.
Proof. In order to prove the proposition, one needs to show the following hold: (1) the curveŜ 6 is generically smooth; (2) under the projection q 1 : K ⊕ K 2 → K which on each fiber is given by (u, v) → 2u, the image S 6 := q 1 (Ŝ 6 ) is a spectral curve defined by an equation of the form in (42) with (3) the projection q 1 :Ŝ 6 → S 6 is an isomorphism away from the singularities of S 6 at its intersection with the zero section of K. To prove the above items, let S ⊂ K be defined by the zero locus of (41). As a curve in the total space of K ⊕ K, the fiber product S × Σ S is defined by the conditions P 4 (η 1 ) = P 4 (η 2 ) = 0, where η 1 , η 2 denote the tautological sections of the two summands in K ⊕ K. After using the transformation Sym defined in (49) to realize the curve as a subvariety of K ⊕ K 2 , the componentŜ 6 is described locally (i.e. with respect to a trivialization of the bundles) as the zero locus of the map F : C 3 → C 2 given by The projection q 1 onto the total space of K is given locally by (z, u, v) → (z, 2u). Direct computation shows that away from u = 0, the map is a biholomorphism onto the curve define by the equation P 6 (z, η) = η 6 + 2a 2 (z)η 4 + (a 2 (z) 2 − a 4 (z))η 2 + a 2 3 = 0, where η is the local fiber coordinate on K. This curve, which we denote by S 6 , necessarily has singularities at the zeros of a 3 , but is otherwise smooth for generic choice of a 2 , a 3 , a 4 . For such choices, singularities inŜ 6 can occur only at points where u = 0. Direct computation shows that the derivative of F has full rank at such points provided a 3 and a 2 2 − a 4 have no common zeros, proving the proposition.
Proof. On a regular fiber away from the base locus, where we can write π −1 = {y 1 , y 2 , y 3 , y 4 }, the coordinate η on S 6 has values y i + y j for i = j. But y 1 + y 2 + y 3 + y 4 = 0 and hence −(y i + y j ) = (y k + y l ) where {i, j, k, l} = {1, 2, 3, 4}, i.e. η → −η corresponds to the action of σ. Assuming that only the most generic type of ramification occurs, the computation is similar on fibers over the branch locus.
To obtain the correspondence induced by I 3 between spectral line bundles we begin as in Section 4.4, but with L 1 = L 2 , and consider Since L is invariant under τ : (x, y) → (y, x), its direct image onŜ 6 decomposes as a sum of rank one locally free sheaves generated by the invariant and anti-invariant local sections, i.e. Similarly Moreover, if [R 0 ] is the line bundle on (S × Σ S) 0 defined by the divisor R 0 , then (π τ ) * T = [R 0 ] −1 (see for example [18,Lemma 3.1] or [7, p. 49]). We define Remark 23. While we have defined L only on (S × Σ S) 0 ⊂ S × Σ S, this distinction disappears in L − . This is a consequence of the fact that anti-invariant local sections must vanish on the fixed points of τ , so that the sheaf on (S × Σ S)/τ generated by the anti-invariant sections has support only on (S × Σ S) 0 /τ .

Remark 24.
Since L is clearly invariant under pullback by the involution τ and admits a lift of τ which acts as identity on fibers over all fixed points, it follows (see for example [20]) that L =π −1 τ M for some line bundle M onŜ 6 . We thus get with the summands M and M⊗T generated by the τ -invariant and anti-invariant local sections of (π τ ) * (π τ ) * M respectively. With this choice of M we get Proposition 25. If L ∈ Prym(S, Σ) then I 3 (L) ∈ Prym(Ŝ 6 ,Ŝ 6 /σ).
Proof. Let L be defined by the divisor D on S given by For D the divisor p * 1 (D) + p * 2 (D) on (S × Σ S) 0 , one has that D is in the linear system of L. Moreover, when x is not in the branch locus of π 0 , so that π −1 0 (x) has four distinct points {y 1 , y 2 , y 3 , y 4 }, one has that On the other hand, if x is a branch point, then under our genericity assumptions on S, we can assume that π −1 0 (x) has one point (say y 1 ) where π 0 has multiplicity two, and two points (say y 2 , y 3 ) where π 0 is unramified. Then D x =π * τ C x with C x = D(y 1 )·Sym[y 1 , y 1 ]+(D(y 2 )+D(y 3 ))·Sym[y 2 , y 3 ]+[ i=2,3 (D(y 1 )+2D(y i ))·Sym[y 1 , y i ] (61) where the factor 2 in the last term comes from the fact that the projection p 1 is ramified at the points (y 1 , y 2 ) and (y 1 , y 3 ), while p 2 is ramified at the points (y 2 , y 1 ) and (y 3 , y 1 ). Moreover, from Remark 24 one has that L =π * τ (M) and I 3 (L) = M, and hence C is defined by (60) , and (61) is a divisor in the linear system of I 3 (L). Notice that on fibers over branch points of π 0 one has σ[y 1 , y 1 ] = [y 2 , y 3 ]. Then, denoting the branch locus of π 0 by B, one gets where [[y 1 , y 2 ]] denotes the point inŜ 6 /σ and N m σ is the norm map for the coveringŜ 6 →Ŝ 6 /σ. Finally, note that if L ∈ P rym(S, Σ) then we can pick D so that N m π 0 (D x ) = 0 for all x ∈ Σ and hence N m σ (I 3 (L)) = N m σ (C) = 0, as required.
We have shown that the map (S, L) → (Ŝ 6 , I 3 (L)) sends spectral data for an SL(4, C)-Higgs bundle to SO(6, C)-spectral data. We now show that this map is compatible with the map given by (43). Proof. By (43) the Higgs field in I 3 [(E, Φ)] is Φ 6 := Φ ⊗ I + I ⊗ Φ, viewded as a map on Λ 2 E ⊂ E ⊗E. Let η 4 +a 2 η 2 +a 3 η +a 4 and η 6 +b 2 η 4 +b 4 η 2 +b 2 3 be the characteristic polynomials for Φ and Φ 6 respectively. Then a calculation based on Remark 9 shows that the coefficients are related by (53). It follows that the curveŜ 6 is the spectral curve for the SO(6, C)-Higgs bundle I 3 [(E, Φ)]. Moreover, Proposition 21 then shows that Φ 6 can be recovered by pushing down the section (η, η 2 ), where η is tautological section of K.
The direct image π * (I 3 (L) ⊗ [R 6 ] 1/2 ) acquires a special orthogonal structure in the usual way, with the trivialization of its determinant bundle determined by the Prym condition σ * (I 3 (L)) ≃ I 3 (L) * . In order to see that π * (I 3 (L)⊗[R 6 ] 1/2 ) = Λ 2 E, consider the (singular) full fiber product S × Σ S and for any line bundle N on S letÑ = p * 1 (N ) ⊗ p * 2 (N ), where now the projections are from the entire S × Σ S. Arguing as before, sinceÑ is invariant under the involution τ , it follows that its direct image under the projection π τ : S × Σ S → (S × Σ S)/τ decomposes as the sum of rank one coherent sheavesÑ − ⊕Ñ + . Though in principle these summands need not be locally free, their restriction to (S × Σ S) 0 is well-behaved. Indeed, given the projections π : S → Σ andπ : (S × Σ S)/τ → Σ one getŝ But the left-and right-hand sides of this identification are both ±1-eigenspace decompositions for Z 2 -actions that are compatible with the projection map. We can thus identifyπ * (Ñ − ) ≃ Λ 2 (π * (p * 1 (N ) ⊗ p * 2 (N )). Notice, furthermore, that the support ofÑ − is (S × Σ S) 0 , so that N − = N − where dropping the tilde onÑ denotes the restriction to (S × Σ S) 0 . It follows that In particular, taking N = L ⊗ [R] 1/2 where R is the ramification divisor on S, so that π * N = E, it follows that (64) givesπ * N − = Λ 2 E. It thus remains to show that N − = I 3 (L) ⊗ [R 6 ] 1/2 . But by Lemma 20 and (52) , where M is as in Remark 24 and T is as above. It follows that as required.
We shall end this section with a discussion of the relation between our construction and the so-called trigonal construction of Recillas (see [8] or [23]). Given a smooth curve Σ, this construction relates a smooth four-fold cover of Σ to a six-fold cover with a fixed-point-free involution , whose quotient is thus a three-fold smooth cover. Taking S as the four-fold cover, the curve S :=Sym −1 (Ŝ 6 ) = (S × Σ S) 0 /τ is precisely the corresponding six-fold cover with involution. This is most easily seen by considering the fibers of the covering maps onto Σ. At the regular fibers where, if the fiber of S over a point x ∈ Σ consists of points {y 1 (x), y 2 (x), y 3 (x), y 4 (x)} then the fiber of (S × Σ S) 0 /τ over x consists of the points corresponding to the unordered pairs {[y 1 , y 2 ], [y 1 , y 3 ], [y 1 , y 4 ], [y 2 , y 3 ], [y 2 , y 4 ], [y 3 , y 4 ]} (where we have dropped the dependence on x to simplify the notation).
Considering π : S → Σ a smooth cover with only the most generic ramification, if x is in the branch locus and π −1 (x) = {y 1 , y 2 , y 3 } (as in the proof of Proposition 25) , then the fiber of S consists of the unordered pairs {[y 1 , y 1 ], [y 1 , y 2 ], [y 1 , y 3 ], [y 2 , y 3 ]}. This relation between S and S can be regarded as a map from S to the third symmetric product of S, mapping a point in S to the unordered pairs containing that point. This determines a correspondence defined by an effective divisor on S × S, but for the sake of comparison with our construction, we use the map Sym to replace S withŜ 6 .
Notice that ∆ is contained in S × ΣŜ6 and is given explicitly by Here the set B ⊂ Σ, as in (62), is the branch locus of π, and at x ∈ B we label the points in the fiber so that π has degree 2 at y 1 (x).

Remark 28. The map
is injective and its image is precisely the divisor ∆.
Proposition 29. The map C ∆ restricts to a map between Prym varieties, i.e.
This map agrees with the map defined by virtue of Proposition 25, i.e. with L → I 3 (L) given by (58).
Proof. Let L ∈ Jac(S) be defined by divisor D on S. With C as in the proof of Proposition 25, it follows from (60) and (61) that C = N m 2 (π * 1 (D) ∩ ∆), and hence (as in the proof of Proposition 25) that C is in the linear system for I 3 (L). It thus follows from Proposition 25 that C ∆ defines a map between the indicated Prym varieties. Proof. The map L → I 3 (L) defined by (58) is a group homomorphism between Jacobians. Thus, in particular, if L 2 ≃ O S then I 3 (L) 2 = I 3 (L 2 ) = I 3 (O S ) = OŜ 6 . It follows (see [26,Theorem 4.12]) that our construction maps spectral data for SL(4, R)-Higgs bundles to spectral data for SO 0 (3, 3)-Higgs bundles. As in the rank two case discussed in Section 4.5, if I 3 (L) is a point of order two in P rym(Ŝ 6 ,Ŝ 6 /σ) then it is invariant under the involution σ. Under the projection p :Ŝ 6 →Ŝ 6 /σ the direct image sheaf p * I 3 L thus splits as the sum of two line bundles I 3 (L) ± , generated by σ-invariant and anti-invariant local sections. In particular,π We get a diagram similar to (39): Recall from Proposition 26 thatŜ 6 is the normalization of the spectral curve S 6 ⊂ π * K and that the involution σ onŜ 6 corresponds to the involution η → −η on S 6 . It follows that multiplication by the tautological section η interchanges I 3 (L) − and I 3 (L) + , and thus that the Higgs field onπ 6 * I 3 (L) has the form as in (48). Combined with Proposition 18, we thus get: Proposition 31. Let (S, L) be the spectral data for a point in M SL(4,R) represented by a Higgs bundle (E, Φ) with orthogonal structure q on E, and isomorphism δ : det E ≃ O Σ . Let (Ŝ 6 , I 3 (L)) be defined as (50) and (58), i.e.Ŝ 6 := Sym((S × Σ S) 0 /τ ) and I 3 (L) = L − ⊗ T −1 . Then (Ŝ 6 , I 3 (L)) is the spectral data for the point in M SO 0 (3,3) represented by where the bundles have oriented orthogonal structures (q ± , δ ± ) as in §5.3 , and α is as in (48).

Maps between moduli spaces and Hitchin fibrations
Thus far we have examined the maps induced by the isogenies on individual Higgs bundles or their spectral data. In this section we collect together some remarks about the induced maps on the corresponding moduli spaces and on their Hitchin fibrations. We note that whilst the maps on Higgs bundles (given in §4.3 and Proposition 18) are defined for all Higgs bundles , but do not obviously preserve stability properties. On the other hand, the maps on spectral data (see Propositions 13 and 26) automatically preserve stability but apply only to generic points in the moduli spaces -where the stability condition is vacuous. Pending further work we thus limit our remarks to the dense open sets in the moduli spaces which exclude the non-generic fibers of their Hitchin fibrations. We denote these sets byM G ⊂ M G .
Note that for the groups in the isogenies studied in this paper, the special linear groups can be identified as the spin groups for the special orthogonal groups . The induced map M Spin(2n,C) →M SO(2n,C) is a finite map . As seen in [14], for any n, the moduli space M SO(2n,C) has two components corresponding to the two possible values for the second Stieffel-Whitney class of an SO(2n, C)-principal bundle. In contrast, the underlying holomorphic bundles for the Higgs bundles in M SL(2,C)×SL(2,C) and M SL(4,C) have just one topological type, and the moduli spaces are connected. The images of the maps I 2 and I 3 thus see just one of the components in M SO(4,C) or M SO(6,C) . Indeed, this can be understood from the point of view of the surface group representations corresponding to the Higgs bundles via non-Abelian Hodge theory. From this point of view, the component in the image of the map contains precisely the representations in SO(2n, C) which lift to Spin(2n, C). We therefore see only the Higgs bundles in which the underlying holomorphic bundle has w 2 = 0.
The situation is more nuanced for the restriction of the maps to the moduli spaces for the split real forms. In this case, the underlying holomorphic bundles have more complicated topology than in the case of the Higgs bundles for the complex groups. Moreover, fixing the topological type of the bundle does not ensure connectedness of the components. What remains true is that the images of the maps I 2 and I 3 contain only those components of the moduli spaces in which the Higgs bundles correspond to representations which lift to the appropriate spin group.
In terms of Hitchin fibrations, one should note that for each n = 2, 3, the bases of the fibrations of Spin(2n, C) and SO(2n, C)-Higgs bundles are the same . In the case of n = 2 the base is H 0 (Σ, K 2 ) ⊕ H 0 (Σ, K 2 ), and for n = 3 it is H 0 (Σ, K 2 ) ⊕ H 0 (Σ, K 3 ) ⊕ H 0 (Σ, K 4 ). In order to understand the maps induced on these bases it is necessary to understand exactly the relation between coordinates of a point in the base and the coefficients in the defining equation for the spectral curve. This is completely straightforward for SL(n, C), where the two coincide, but less so in the case of SO(2n, C) where the relation is complicated by the role of the Pfaffian. In fact, the maps S →Ŝ 2n (for n = 2, 3) do not unambiguously descend to the base of the fibration. The ambiguity stems from the fact that the induced orthogonal structures on E 1 ⊗ E 2 or Λ 2 E do not have a canonical orientation. The choice of orientation corresponds on the one hand to a choice of trivialization on the determinant bundles, and on the other hand to a choice of sign in the Pfaffian.
Remark 32. One should note that whist not done here, the isogenies could also be understood through the language of Cameral covers.
6.1. The isogeny I 2 on moduli spaces. Once the isomorphism δ : E 1 ⊗ E 2 ≃ O Σ is fixed, the map (30) determines a map on the base of the Hitchin fibrations for M SL(2,C)×SL(2,C) and M SO(4,C) . The explicit form of the map depends on the generators chosen for the rings of invariant polynomials. Taking the coefficients of the characteristic equation det(ϕ − ηI) as generators, it follows from (36) that this map is given on the generic points in the base by (a 1 , a 2 ) → (2(a 1 + a 2 ), ±(a 1 − a 2 )), where the sign in the last component is determined by the isomorphism δ. For example, following the notation of Remark 12, if {e 1 1 ⊗ e 1 2 , e 1 1 ⊗ e 2 2 , e 2 1 ⊗ e 1 2 , e 2 1 ⊗ e 2 2 } is the positively oriented local frame, then Pf(Φ 1 ⊗ I + I ⊗ Φ 2 ) = a 2 − a 1 where a i = det(Φ i ), so the second component in (70) is −(a 1 − a 2 ).
Remark 33. As explained at the beginning of Section 3, the relation between spectral curves and thus the map (70) , can be understood heuristically from the relation between the eigenvalues of the Φ i and those of Φ 1 ⊗ I + I ⊗ Φ 2 . By Remark 8, at least at smooth unramified points, the equations for the spectral curves S i and S 4 are given by from which the relations between (b 1 , b 2 , b 2 3 ) and (a 1 , a 2 ) can be deduced. As described in §2.4, the vector bundle of an SO 0 (2, 2)-Higgs bundles may be expressed as (M 1 ⊕ M * 1 ) ⊕ (M 2 ⊕ M * 2 ), and thus they are labelled by two integer topological invariants, say (c 1 , c 2 ), corresponding to the degrees of the line bundles M 1 and M 2 (or, equivalently by classes in π 1 (SO(2, C)) ∼ = Z). The moduli space M SO 0 (2,2) is thus a disjoint union of (possibly disconnected or empty) components The moduli space M SL(2,R)×SL(2,R) is similarly a disjoint union of (possibly disconnected or empty) components for d 1 , d 2 the degrees of the line bundles defining the SL(2, R) × SL(2, R)-Higgs bundles. As seen in [13], the components are non-empty if and only if |d i | ≤ g − 1, inequalities which are known as the the Milnor-Wood bounds.
Proposition 34. The map I 2 :M SL(2,R)×SL(2,R) →M SO 0 (2,2) is a 2 2g+1 -fold covering onto the components satisfying c 1 = c 2 mod 2 and |c i | ≤ 2g − 2, for i = 1, 2. The map restricts to maps Proposition 35. The isogeny I 2 between moduli spaces of Higgs bundles takes the 2 2g Hitchin components to the one Hitchin component, and the other 2 components to the two components (possibly disconnected) where the two w 2 's are the same.
Remark 37. By realising SO 0 (2, 2)-Higgs bundles in terms of rank 2-Higgs bundles, one can understand the monodromy action studied in [2] for rank 4 Higgs bundles in terms of monodromy of lower rank Higgs bundles. Indeed, taking b 1 and b 2 as in Eq. (36) one recovers the rank 4 monodromy as a product of actions coming from the rank 2-Hitchin systems.
6.2. The isogeny I 3 on moduli spaces. To fully specify the map induced by I 3 (as in (43)) on M SL(4,C) one needs to specify the trivialization δ : det(Λ 2 (E)) ≃ O Σ . Then, using the coefficients of the characteristic equation det(ϕ − ηI) as generators for the rings of invariant polynomials, it follows from (53) that the map on the generic points of the Hitchin base is where the sign in the second component is determined by the isomorphism δ.
Remark 38. For both n=2,3 the variety P rym(Ŝ 2n ,Ŝ 2n / σ )) has two components. In the case n = 3, givenŜ 6 the spectral curve defined by (2a 2 , a 2 2 − 4a 2 , a 2 3 ), the two components occur in the fibers over both points (2a 2 , ±a 3 , a 2 2 − 4a 2 ) in the Hitchin base. However, on each fiber only one of the components is in the image of the map induced by I 3 . A similar phenomenon occurs for n = 2.