Higgs bundles and exceptional isogenies
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Abstract
We explore relations between Higgs bundles that result from isogenies between lowdimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto \(\mathrm {SO}(4,\mathbb {C})\) and \(\mathrm {SO}(6,\mathbb {C})\) and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated with orthogonal Higgs bundles. In the case of \(\mathrm {SO}(6,\mathbb {C})\), our construction can be interpreted as a new description of Recillas’ trigonal construction.
1 Background
Higgs bundles over a compact Riemann surface \(\varSigma \) of genus \(g\ge 2\) were introduced in [13] and in a more general setting in [28]. For any Lie group G, a GHiggs bundle on \(\varSigma \) is a pair \((P,\varPhi )\) where P is a holomorphic principal bundle and \(\varPhi \) (the Higgs field) is a holomorphic section of an associated bundle twisted by K, the canonical bundle of the surface \(\varSigma \). If G is a complex group, then P is a principal Gbundle, 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 \(G=\mathrm {SL}(n,\mathbb {C})\) or \(\mathrm {SO}(2n,\mathbb {C})\), and real forms of these groups. In these cases (described in Sect. 2), 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 GHiggs bundle serve to construct Glocal systems on \(\varSigma \). Indeed, this is the crux of nonabelian Hodge theory (NAHT), whereby the moduli space of semistable GHiggs bundles on \(\varSigma \) is identified with the moduli space of reductive representations of \(\pi _1(\varSigma )\) into G. The implications of any group homomorphism \(h:G_1\rightarrow G_2\) 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 reencoded into a socalled spectral data set consisting of a ramified covering S of \(\varSigma \) 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 \(\varSigma \). These abelian varieties form the generic fibers in a fibration of the Higgs bundle moduli space over a halfdimensional linear space.
Theorem 1

\(\hat{S}_4:=S_1\times _{\varSigma }S_2\) is a smooth ramified fourfold cover, and

\({\mathcal {L}}:= p_1^*(L_1)\otimes p_2^*(L_2)\) where \(p_i:S_1\times _{\varSigma }S_2\rightarrow S_i\) are the projection maps.
This case differs from the previous one in two notable ways. The first is seen in the restriction of (5) to \(\mathrm {SL}(4,\mathbb {R})\)Higgs bundles, where the map reflects the fact that \(\mathrm {SO}(4)\) is not a simple Lie group. This leads to a decomposition of the bundle \(\varLambda ^2E\) that is analogous to the decomposition of 2forms on a Riemannian 4manifold into selfdual and antiselfdual 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 \(S\rightarrow \varSigma \). It turns out though that this component plays no part in the map induced by \({\mathcal {I}}_3\). We thus modify our construction by removing the diagonally embedded component in the fiber product of an \(\mathrm {SL}(4,\mathbb {C})\)spectral data set with itself (see Sect. 6, Proposition 19, 27, 41 for details):
Theorem 2

\(\hat{S}_6\) is the symmetrization of the nondiagonal component in the fiber product \(S\times _{\varSigma } S\);

\({\mathcal {I}}_3(L)\) is a canonical twist of the line bundle generated by local sections of \({\mathcal {L}}=p_1^*(L)\otimes p_2^*(L)\) that are antiinvariant with respect to the symmetry of \(S\times _{\varSigma } S\).
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 \(\mathrm {SL}(2,\mathbb {R})\) has a distinguished place in any discussion of surface group representations because of its relation to hyperbolic structures and Teichmuller space, while the group \(\mathrm {SO}_0(2,2)\) is the isometry group of the antide Sitter space \(\mathrm{AdS}^3\). Moreover, both groups are split real forms of complex semisimple groups (viz. \(\mathrm {SL}(2,\mathbb {C})\) and \(\mathrm {SO}(4,\mathbb {C})\), respectively) and also groups of Hermitian type. The only other groups that lie in both classes are the symplectic groups \(\mathrm {Sp}(2n,\mathbb {R})\). 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 \(\mathrm {SL}(2,\mathbb {R})\). For the groups of Hermitian type, the GHiggs bundles (or the surface group representations) carry a discrete invariant known as a Toledo invariant. This invariant satisfies a socalled 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 \(\mathrm {SL}(4,\mathbb {C})\) and \(\mathrm {SO}(6,\mathbb {C})\), namely between a fourfold covering and a sixfold covering with a fixedpointfree involution, is essentially the correspondence described by Recillas in his trigonal construction [22] 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 (Sect. 2), spectral curves (Sect. 3) and the isogenies (Sect. 4). In Sect. 5, we describe the maps induced by the isogeny \({\mathcal {I}}_2\) on both the complex group \(\mathrm {SL}(2,\mathbb {C})\times \mathrm {SL}(2,\mathbb {C})\) and its split real form, and in Sect. 6, we do the same for \({\mathcal {I}}_3\). 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 Sect. 7, with a discussion of maps between moduli spaces.
2 Higgs bundles and the Hitchin fibration
Let \(\varSigma \) be a compact Riemann surface of genus \(g\ge 2\), and \(\pi :K:=T^*\varSigma \rightarrow \varSigma \) its canonical bundle. For \(G_c\) a complex reductive Lie group with Lie algebra \(\mathfrak {g}_c\), from [14] one has the following definition:
Definition 3
A \(G_c\)Higgs bundle on \(\varSigma \) is given by a pair \((P,\varPhi )\) for P a principal \(G_c\)bundle on \(\varSigma \), and \(\varPhi \) a holomorphic section of \(\mathrm{Ad}P\otimes K\), for \(\mathrm{Ad}P=P\times _{Ad}\mathfrak {g}_c\) the adjoint bundle associated with P.

An \(\mathrm {SL}(n,\mathbb {C})\)Higgs bundle is a pair \((E,\varPhi )\) where E is a rank n holomorphic bundle on \(\varSigma \) with fixed trivial determinant, and \(\varPhi \) a traceless holomorphic section of \(\mathrm {End}(E)\otimes K\);

An \(\mathrm {SO}(n,\mathbb {C})\)Higgs bundle is a pair \((E,\varPhi )\) where E is a rank n holomorphic bundle on \(\varSigma \) with an orthogonal structure Q and a compatible trivialization of its determinant bundle,^{1} and \(\varPhi \) is a holomorphic section of \(\mathrm {End}(E)\otimes K\) satisfying \(Q(u,\varPhi v)=Q(\varPhi u, w)\).
Definition 4
Given G a real form of a complex Lie group \(G_c\), a principal G Higgs bundle is a pair \((P,\varPhi )\) where P is a holomorphic principal \(H^{\mathbb {C}}\)bundle on \(\varSigma \), and \(\varPhi \) is a holomorphic section of \(P\times _{Ad}\mathfrak {m}^{\mathbb {C}}\otimes K\).
Alternatively, as done in [15], we may regard real Higgs bundles as classical Higgs bundles \((E,\varPhi )\), with extra conditions reflecting the structure of the real group and its isotropy representation. In this paper, we shall mainly consider \(\mathrm {SL}(n,\mathbb {R})\) and \(\mathrm {SO}(n,n)\)Higgs bundles, for which we recall their main properties in Sects. 3.2 and 3.4 (for further references, see [1, 15, 25]).
Following [13], a classical Higgs bundle \((E,\varPhi )\) is said to be (semi)stable if all subbundles \(F\subset E\) such that \(\varPhi (F)\subset F\otimes K\) satisfy \(\deg (F)/\mathrm{rk}(F)< (\le )\deg (E)/\mathrm{rk}(E)\). Moreover, the pair is said to be polystable if it can be written as a direct sum of stable Higgs bundles \((E_i,\varPhi _i)\) for which \(\deg (E_i)/\mathrm{rk}(E_i)=\deg (E)/\mathrm{rk}(E)\). The notion of stability can be extended to \(G_c\)Higgs bundles as in [14], as well as to GHiggs bundles (e.g., see [4]), and used to construct the corresponding moduli spaces. As explained below in Remark 7, stability considerations will not play a role in this paper. We shall denote by \({\mathcal {M}}_{G_c}\) and \({\mathcal {M}}_{G}\) the moduli space of \(G_c\)Higgs bundles and \({\mathcal {M}}_{G}\)Higgs bundles, respectively.
2.1 The Hitchin fibration and spectral curves
By considering the moduli space \({\mathcal {M}}_G\) in \({\mathcal {M}}_{G_c}\), one may identify the GHiggs bundles as points in the Hitchin fibration satisfying additional constraints (see [14] for classical complex Lie groups, and [27] 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 [25, Theorem 4.12]). It should be noted that \({\mathcal {M}}_G\) does not always embed in \({\mathcal {M}}_{G_c}\), and an example of when one does not have an embedding can be seen in [24].
Remark 5
Spectral curves are defined for all Higgs bundles represented by points in the moduli spaces \({\mathcal {M}}_G\), 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 for \(G= \mathrm{GL}(n,\mathbb {C})\), \( \mathrm{SL}(n,\mathbb {C})\), \( \mathrm{SO}(2n+1,\mathbb {C})\), \( \mathrm{Sp}(2n,\mathbb {C})\). Throughout the paper, we shall consider Higgs bundles over the smooth loci of the Hitchin fibration, i.e., points defining smooth spectral curves for the above groups, since we will further restrict our attention to those curves for which the most generic type of ramification behavior occurs.
Remark 6
In the case of \(\mathrm{SO}(2n,\mathbb {C})\)Higgs bundles, the associated spectral curves are always singular, and one needs to work with a canonical normalization, as explained in Sect. 3.3. It should be noted that through the fiber product process described in this paper, one recovers the normalization of the singular curve in a natural way.
Remark 7
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, while 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.
3 Spectral data for complex and real Higgs bundles
We shall recall here how to study the fibers 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 [25, 27] for real Higgs bundles. Since our main focus is on \(\mathrm {SO}(2n,\mathbb {C})\) and \(\mathrm {SL}(2n,\mathbb {C})\)Higgs bundles, we shall restrict attention to those groups and their split real forms.
3.1 Spectral data for \(\mathrm {SL}(n,\mathbb {C})\)Higgs bundles
3.2 Spectral data for \(\mathrm {SL}(n,\mathbb {R})\)Higgs bundles
From Definition 4, Higgs bundles with structure group \(\mathrm {SL}(n,\mathbb {R})\) are given by classical Higgs bundles \((E,\varPhi )\) together with an oriented orthogonal structure on E, with respect to which the Higgs field is traceless and symmetric. Moreover, by [25, Theorem 4.12] one has that the intersection of the moduli space \({\mathcal {M}}_{\mathrm {SL}(n,\mathbb {R})}\) with the smooth fibers of the \(\mathrm {SL}(n,\mathbb {C})\) Hitchin fibration is given by line bundles \(L\in \mathrm{Prym}(S,\varSigma )\) such that \(L^2\cong \mathcal {O}\). Following [3] and [14], a torsion 2 line bundle L induces an \(\mathrm {SL}(n,\mathbb {R})\)Higgs bundle \((E,\varPhi )\) with \(E=\pi _*(L\otimes \pi ^*K^{(n1)/2}) \) and \(\varPhi \) the push down of the tautological section \(\eta \). Moreover, the orthogonal structure on E comes from an O(1) structure on L.
3.3 Spectral data for \(\mathrm {SO}(2n,\mathbb {C})\)Higgs bundles
Remark 8
3.4 Spectral data for \(\mathrm {SO}_0(n,n)\)Higgs bundles
Given the spectral data \((\hat{S},L)\) of an \(\mathrm {SO}_0(n,n)\)Higgs bundle, since L is of order two and in the Prym variety of a twofold cover \(p:\hat{S}\rightarrow \hat{S}/{\sigma }\), it is invariant under the involution \(\sigma \) on \(\hat{S}\). Hence, its local sections decompose into invariant and antiinvariant local sections, and thus, the direct image decomposes as \(p_*L=\hat{L}_1\oplus \hat{L}_2\), where the summands are generated by local invariant and antiinvariant sections (see [26]). Moreover, considering the nfold cover \(\hat{\pi }: \hat{S}/ \hat{\sigma }\rightarrow \varSigma , \) the orthogonal bundles \(W_i\) are recovered by taking \(\hat{\pi }_*(L_i\otimes p_*(K_{\hat{S}}\otimes \hat{\pi }^* K^*)^{1/2})\) for \(i=1,2\).
4 Homomorphisms of groups and induced maps
Alternatively, one may seek a more intrinsic understanding of the map purely in terms of the information encoded in the geometry of the covering \(S\rightarrow \varSigma \) 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 [20, Chapter 9–10]. For our purposes, the representation theoretic point of view is convenient, as described in the next two sections.
4.1 The isogeny between \(\mathrm {SL}(2,{\varvec{\mathbb {C}}}) \times \mathrm {SL}(2,{\varvec{\mathbb {C}}})\) and \( \mathrm {SO}(4,{\varvec{\mathbb {C}}})\)
Remark 9
If \(\dot{A_i}\) has eigenvalues \(\{\lambda ^i_1,\lambda ^i_2\}\), for \(i=1,2\), then the image \( \hbox {d}{{\mathcal {I}}_2} (\dot{A_1},\dot{A_2})\) has eigenvalues \(\{\lambda ^1_a+\lambda ^2_b\ 1\le a,b\le 2\}\). In particular, if \(Tr({\dot{A}}_i)=0\), then \(\lambda ^i_2=\lambda ^i_1\) and the eigenvalues for \( \hbox {d}{{\mathcal {I}}_2} (\dot{A_1},\dot{A_2})\) are \(\{\pm \lambda ^1_1\pm \lambda ^2_1\}\).
4.2 The isogeny between \(\mathrm {SL}(4,{\varvec{\mathbb {C}}})\) and \(\mathrm {SO}(6,{\varvec{\mathbb {C}}})\)
Remark 10
If \({\dot{A}}\) has eigenvalues \(\{\lambda _a\}_{a=1}^{4}\), then as a map on \(\mathbb {C}^6=\varLambda ^2\mathbb {C}^4\) the image \( \hbox {d}{{\mathcal {I}}_3} ({\dot{A}})\) has eigenvalues \(\{\lambda _a+\lambda _b\ 1\le a<b\le 4\}\).
Remark 11
Fixing a maximal compact subgroup \(\mathrm {SO}(4)\subset \mathrm {SL}(4,\mathbb {R})\), one obtains extra structure related to the presence of two inequivalent normal \(\mathrm {SO}(3)\) subgroups, by virtue of which \(\mathrm {SO}(4)\) fails to be simple. These subgroups have an important influence on the map induced by \({\mathcal {I}}_3\) on \(\mathrm {SL}(4,\mathbb {R})\)Higgs bundles, described in more detail in Sect. 6.3, and considered also in Sect. 7.2.
5 The rank 2 isogeny and the Hitchin fibration
In this section, we describe the map corresponding to the isogeny \({\mathcal {I}}_2\) in Sect. 4.1 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.
5.1 \(\mathrm {SL}(2,{\varvec{\mathbb {C}}})\times \mathrm {SL}(2,{\varvec{\mathbb {C}}})\) and \(\mathrm {SL}(2,{\varvec{\mathbb {R}}})\times \mathrm {SL}(2,{\varvec{\mathbb {R}}})\)Higgs bundles
In the case of \(\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})\)Higgs bundles, by [25, Theorem 4.12], the line bundles are required to satisfy \(L_i^2\cong \mathcal {O}_{S_i}\). Note that, since \(L_i\in \mathrm{Prym}(S_i,\varSigma )\) if and only if \(\sigma ^* L\cong L_i^*\), the conditions that \(L_i\in \mathrm{Prym}(S_i,\varSigma )\) and \( L_i^2\cong \mathcal {O}\) are equivalent to the conditions that \(\sigma ^* L_i\cong L_i\) and \(L_i^2\cong \mathcal {O}\).
Remark 12
The \(\mathrm {SL}(2,\mathbb {R})\)Higgs bundles \((E_i,\varPhi _i)\) have associated an integer invariant \(\tau _{E_i}\) 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 \(\mathrm {SO}(2)\)principal bundle associated with \(N_i\oplus N_i^{*}\). 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 [26]). However, it is defined that the Toledo invariant satisfies a socalled Milnor–Wood inequality \(\tau _{E_i}\le 2g2\).
5.2 \(\mathrm {SO}(4,\mathbb {C})\)Higgs bundles and \(\mathrm {SO}_0(2,2)\)Higgs bundles
5.3 The induced map on the Higgs bundles
Remark 13
Notice that the isomorphisms \(\det (E_i)\simeq \mathcal {O}_{\varSigma }\) do not uniquely determine a trivialization \(\delta :\det (E_1\otimes E_2)\simeq \mathcal {O}_{\varSigma }\) compatible with the orthogonal structure on \(E_1\otimes E_2\). Indeed, if \(\{e_i^1,e_i^2\}\) are local oriented frames for \(E_i\) satisfying \(\omega _i(e_i^1,e_i^2)=1\), then both \(\{e_1^1\otimes e^1_2,e_1^1\otimes e^2_2,e_1^2\otimes e^1_2,e_1^2\otimes e^2_2\}\) and \(\{e_1^1\otimes e^1_2,e_1^2\otimes e^1_2,e_1^1\otimes e^2_2, e_1^2\otimes e^2_2\}\) are orthonormal local frames for \(E_1\otimes E_2\) but they have opposite orientations. They determine the two inequivalent choices for \(\delta :\det (E_1\otimes E_2)\simeq \mathcal {O}_{\varSigma }\).
5.4 The induced map on spectral data
We shall first construct the spectral data \((\hat{S}_4, {\mathcal {L}})\) associated with the \(\mathrm {SO}(4,\mathbb {C})\)Higgs bundle obtained via \({\mathcal {I}}_2\)and then specialize to the spectral data associated with the split real forms \(\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})\) and \(\mathrm {SO}_0(2,2)\).
Proposition 14

\(\hat{S}_4:=S_1\times _{\varSigma }S_2\) is the fiber product curve, and

\({\mathcal {L}}= p_1^*(L_1)\otimes p_2^*(L_2)\) is the line bundle
Remark 15
The total space of \(K\oplus K\) can be identified with the fiber product \(K\times _KK\subset K\times K\). The fiber product \(S_1\times _{\varSigma }S_2\) may thus be regarded as a subvariety of either \(K\times _KK\) or of the total space of \(K\oplus K\).
Proof
The involution \(\sigma :\eta \mapsto \eta \) which preserves \(S_1\) and \(S_2\) induces an involution \((\sigma ,\sigma )\) on \(\hat{S}_4=S_1\times _\varSigma S_2\). When needed we shall denote the involution on \(S_1, S_2\) by \(\sigma _i\), for \(i=1,2\) and on \(\hat{S}_4\) by \(\hat{\sigma }_4\). The fixed points of \(\sigma _i\) are the zeros of \(a_i\), and thus, since the zeros of \(a_1\) and \(a_2\) are generically different, generically \(\hat{\sigma }_4=(\sigma _1,\sigma _2)\) does not have any fixed points. It is clear from (35) that \(\hat{\sigma }_4\) descends to the involution \(\eta \mapsto \eta \) on the singular curve \(S_4\), where it has fixed points at the branch locus.
In order to see that \({\mathcal {L}}\) is the spectral line bundle associated with an \(\mathrm {SO}(4,\mathbb {C})\)Higgs bundle, one has to show that \({\mathcal {L}}\in \mathrm {Prym}(\hat{S}_4,\hat{S}_4/\hat{\sigma }_4)\). Since \(L_i\in \mathrm{Prym}(S_i,\varSigma )\), one has that \(\sigma _i^*L_i\cong L_i^*\) and so the line bundle \({\mathcal {L}}:=p_1^*(L_1)\otimes p_2^*(L_2)\) is sent to its dual by the involution \(\hat{\sigma }_4\). Hence, the line bundle \({\mathcal {L}}\) on \(\hat{S}_4\) is in \(\mathrm{Prym}(\hat{S}_4,\hat{S}_4/\hat{\sigma }_4)\) as required.\({\square }\)
Proposition 16
The spectral data \((\hat{S}_4,{\mathcal {L}})\) induced by an \(\mathrm {SL}(2,\mathbb {C})\times \mathrm {SL}(2,\mathbb {C})\)Higgs bundle \((E_1,\varPhi _1),(E_2,\varPhi _2)\), as in Proposition 14, correspond to the spectral data of the \(\mathrm {SO}(4,\mathbb {C})\)Higgs bundle \({\mathcal {I}}_2[(E_1,\varPhi _1),(E_2,\varPhi _2)]\).
Proof
5.5 The restriction to \(\mathrm {SL}(2,{\varvec{\mathbb {R}}})\times \mathrm {SL}(2,{\varvec{\mathbb {R}}})\)
Proposition 17
6 The rank 3 isogeny and the Hitchin fibration
We shall investigate now the induced map \({\mathcal {I}}_3:{\mathcal {M}}_{\mathrm {SL}(4,C)}\rightarrow {\mathcal {M}}_{\mathrm {SO}(6,\mathbb {C})},\) and its restriction to the split real forms \(\mathrm {SL}(4,\mathbb {R})\) and \(\mathrm {SO}_0(3,3)\).
6.1 \(\mathrm {SL}(4,\mathbb {C})\)Higgs bundles and \(\mathrm {SL}(4,\mathbb {R})\)Higgs bundles
From Definition 4, an \(\mathrm {SL}(4,\mathbb {R})\)Higgs bundle on \(\varSigma \) is a holomorphic \(\mathrm {SO}(4,\mathbb {C})\)principal bundle together with a symmetric Higgs field. Equivalently it can be viewed as a pair \((E, \varPhi )\) 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 \(\delta :\det (E)\simeq \mathcal {O}\), and the Higgs field \(\varPhi :E\rightarrow E\,\otimes \,K\) is traceless and symmetric with respect to Q.
6.2 \(\mathrm {SO}(6,\mathbb {C})\)Higgs bundles and \(\mathrm {SO}_0(3,3)\)Higgs bundles
6.3 The induced map on Higgs bundles
Remark 18
The structure groups of the bundles \(\varLambda ^2E\) and \(\varLambda ^2_+ E\oplus \varLambda ^2_E\) can be reduced to \(\mathrm {SO}(4)\) and \(\mathrm {SO}(3)\times \mathrm {SO}(3)\), respectively. The two copies of \(\mathrm {SO}(3)\) are precisely the normal subgroups mentioned in Remark 11 by virtue of which \(\mathrm {SO}(4)\) fails to be simple (see [10]) .
With respect to the reduction in (47), the Higgs field \(\varPhi \otimes I+I\otimes \varPhi \) has the form in (27), where \(\varPhi \) and \(\alpha \) are related as in Eq. (24) and \(\alpha ^t\) is the orthogonal transpose. Denoting orthogonal structures on \(\varLambda ^2_{\pm }E\) by \(q_{\pm }\), we thus get:
Proposition 19
Remark 20
The precise form of \(\alpha \) depends on the choices of the orientations of \(\varLambda ^2E\) and \(\varLambda ^2_{\pm }E\). Different choices will change the sign of \(\det {\alpha }\), i.e., of the Pfaffian of \({\begin{bmatrix}0&\quad \alpha \\ \alpha ^\mathrm{{T}}&\quad 0\end{bmatrix}}\).
6.4 The induced map on spectral data
Given \(\mathrm {SL}(4,\mathbb {C})\)spectral data (S, L), with S defined by (41) and \(L\in \mathrm{Prym}(S,\varSigma )\), we build \(\mathrm {SO}(6,\mathbb {C})\)spectral data using a construction similar to the fiber product construction in Sect. 5.4, 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 \(S\times _{\varSigma }S\) is a 16fold cover of the Riemann surface \(\varSigma \). Over a generic point in the Hitchin base, S is smooth and \(S\times _{\varSigma }S\) has two smooth components, namely the diagonal \(S_{\varDelta }:=\{(s,s)\in S\times _{\varSigma }S\}\) and another one which we denote by \((S\times _{\varSigma }S)_0\). The intersection of these components lies in fibers over the branch locus of the covering \(\pi :S\rightarrow \varSigma \).
Lemma 21
Proof
Proposition 22
The curve \(\hat{S}_6\) is generically smooth and gives the canonical desingularization of its projection to K through the Sym map, which is the spectral curve of an \(\mathrm {SO}(6,\mathbb {C})\)Higgs bundle.
Proof
 1.
the curve \(\hat{S}_6\) is generically smooth;
 2.under the projection \(q_1:K\oplus K^2\rightarrow K\) which on each fiber is given by \((u,v)\mapsto 2u\), the image \(S_6 := q_1(\hat{S}_6)\) is a spectral curve defined by an equation of the form in (42) with$$\begin{aligned} b_1=2a_2, b_2= a_2^24a_4, b^2_3=a^2_3; \end{aligned}$$(53)
 3.
the projection \(q_1:\hat{S}_6\rightarrow S_6\) is an isomorphism away from the singularities of the curve \(S_6\) at its intersection with the zero section of K.
Lemma 23
Proof
From diagram (51), for \(x\in \varSigma \) away from the base locus of \(\pi :S\rightarrow \varSigma \), we can write \(\pi ^{1}(x)=\{y_1,y_2,y_3,y_4\}\), and the coordinate \(\eta \) on \(S_6\) has values \(y_i+y_j\) for \(i\ne j\). 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., \(\eta \mapsto \eta \) corresponds to the action of \(\sigma \). Assuming that only the most generic type of ramification occurs, the computation is similar for \(x\in \varSigma \) in the branch locus of \(\pi \).\(\square \)
Remark 24
While we have defined \({\mathcal {L}}\) only on \((S\times _{\varSigma }S)_0\subset S\times _{\varSigma }S\), this distinction disappears in \({\mathcal {L}}_\). This is a consequence of the fact that antiinvariant local sections must vanish on the fixed points of \(\tau \), so that the sheaf on \((S\times _{\varSigma }S)/{\tau }\) generated by the antiinvariant sections has support only on \((S\times _{\varSigma }S)_0/{\tau }\).
Remark 25
Proposition 26
If \(L \in \mathrm{Prym}(S,\varSigma )\), then \({\mathcal {I}}_3(L)\in \mathrm{Prym}(\hat{S}_6,\hat{S}_6/{\sigma })\).
Proof
We have shown that the map \((S,L)\mapsto (\hat{S}_6,{\mathcal {I}}_3(L))\) sends spectral data for an \(\mathrm {SL}(4,\mathbb {C})\)Higgs bundle to \(\mathrm {SO}(6,\mathbb {C})\)spectral data. We now show that this map is compatible with the map given by (43).
Proposition 27
The spectral data \((\hat{S}_6,{\mathcal {I}}_3(L))\) induced by an \(\mathrm {SL}(4,\mathbb {C})\)Higgs bundle \((E,\varPhi )\) via (50) and (58) correspond to the spectral data of the \(\mathrm {SO}(6,\mathbb {C})\)Higgs bundle \({\mathcal {I}}_3[(E,\varPhi )]\).
Proof
By (43) the Higgs field in \({\mathcal {I}}_3[(E,\varPhi )]\) is \(\varPhi _6:=\varPhi \otimes I + I\otimes \varPhi \), viewed as a map on \(\varLambda ^2E\subset E\otimes E\). Let \(\eta ^4+a_2\eta ^2+a_3\eta +a_4\) and \(\eta ^6+b_2\eta ^4+b_4\eta ^2+b_3^2 \) be the characteristic polynomials for \(\varPhi \) and \(\varPhi _6\), respectively. Then, a calculation based on Remark 10 shows that the coefficients are related by (53). It follows that the curve \(\hat{S}_6\) is the spectral curve for the \(\mathrm {SO}(6,\mathbb {C})\)Higgs bundle \({\mathcal {I}}_3[(E,\varPhi )]\). Moreover, Proposition 22 then shows that \(\varPhi _6\) can be recovered by pushing down \((\eta ,\eta ^2)\), where \(\eta \) is tautological section of K.
We shall end this section with a discussion of the relation between our construction and the socalled trigonal construction of Recillas (see [8] or [22]). Given a smooth curve \(\varSigma \), this construction relates a smooth fourfold cover of \(\varSigma \) to a sixfold cover with a fixedpointfree involution, whose quotient is thus a threefold smooth cover. Taking S as the fourfold cover, the curve \(\mathbb {S}:=Sym^{1}(\hat{S}_6)=(S\times _{\varSigma }S)_0/{\tau }\) is precisely the corresponding sixfold cover with involution. This is most easily seen by considering the fibers of the covering maps onto \(\varSigma \). At a regular fiber of S over a point \(x\in \varSigma \) consisting of points \(\{y_1(x),y_2(x),y_3(x),y_4(x)\}\), the fiber of \((S\times _{\varSigma }S)_0/{\tau }\) 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 \(\pi :S\rightarrow \varSigma \) a smooth cover with only the most generic ramification, if x is in the branch locus and \(\pi ^{1}(x)=\{y_1,y_2,y_3\}\) (as in the proof of Proposition 26), then the fiber of \(\mathbb {S}\) 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 \(\mathbb {S}\) can be regarded as a map from S to the third symmetric product of \(\mathbb {S}\), mapping a point in S to the unordered pairs containing that point. This determines a correspondence defined by an effective divisor on \(S\times \mathbb {S}\), but for the sake of comparison with our construction, we use the map Sym to replace \(\mathbb {S}\) with \(\hat{S}_6\).
Definition 28
Remark 29
Proposition 30
Proof
\(\square \)
6.5 The restriction to the split real form \(\mathrm {SL}(4,\mathbb {R})\)
We now examine the consequences of imposing the additional condition \(L^2=\mathcal {O}_S\) on the spectral data (S, L) described in Sect. 6.4.
Proposition 31
Under the assumptions and notation of Proposition 26, if \(L^2\simeq \mathcal {O}_S\), then \({\mathcal {I}}_3(L)\) is a point of order two in \(Jac(\hat{S}_6)\).
Proof
The map \(L\mapsto {\mathcal {I}}_3(L)\) defined by (58) is a group homomorphism between Jacobians. Thus, in particular, if \(L^2\simeq \mathcal {O}_S\) then \({\mathcal {I}}_3(L)^2={\mathcal {I}}_3(L^2)= {\mathcal {I}}_3(\mathcal {O}_S)=\mathcal {O}_{\hat{S}_6}\). \(\square \)
Proposition 32
Let (S, L) be the spectral data for a point in \({\mathcal {M}}_{\mathrm {SL}(4,\mathbb {R})}\) represented by a Higgs bundle \((E,\varPhi )\) with orthogonal structure q on E, and isomorphism \(\delta :\det E\simeq \mathcal {O}_{\varSigma }\). Let \((\hat{S}_6,{\mathcal {I}}_3(L))\) be defined as (50) and (58), i.e., \( \hat{S}_6:=Sym((S\times _{\varSigma }S)_0/{\tau })\) and \({\mathcal {I}}_3(L)={\mathcal {L}}_\otimes T^{1}.\)
7 Maps between moduli spaces and Hitchin fibrations
Thus far, we have examined the maps induced by the isogenies on individual Higgs bundles and 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 the maps on Higgs bundles (given in Sect. 5.3 and Proposition 19) are defined for all Higgs bundles, but do not obviously preserve stability properties. On the other hand, the maps on spectral data (see Propositions 14 and 27) automatically preserve stability but apply only to generic points in the moduli spaces—where the stability condition is vacuous. Throughout this section, we shall limit our study to the dense open sets in the moduli spaces which exclude the nongeneric fibers of their Hitchin fibrations, for which stability is automatically obtained (see Remark 7). We denote these sets by \({\mathcal {\tilde{M}}}_G\subset {\mathcal {M}}_G\).
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 \({\mathcal {\tilde{M}}}_{\mathrm {Spin}(2n,\mathbb {C})}\rightarrow {\mathcal {\tilde{M}}}_{\mathrm {SO}(2n,\mathbb {C})}\) is a finite map. For any n, the moduli space \({\mathcal {M}}_{\mathrm {SO}(2n,\mathbb {C})}\) has two components corresponding to the two possible values for the second Stiefel–Whitney class of an \(\mathrm {SO}(2n,\mathbb {C})\)principal bundle (see for example [16]). In contrast, the underlying holomorphic bundles for the Higgs bundles in \({\mathcal {M}}_{\mathrm {SL}(2,\mathbb {C})\times \mathrm {SL}(2,\mathbb {C})}\) and \({\mathcal {M}}_{\mathrm {SL}(4,\mathbb {C})}\) have just one topological type, and the moduli spaces are connected. The images of the maps \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\) thus see just one of the components in \({\mathcal {M}}_{\mathrm {SO}(4,\mathbb {C})}\) or \({\mathcal {M}}_{\mathrm {SO}(6,\mathbb {C})}\). Indeed, this can be understood from the point of view of the surface group representations corresponding to the Higgs bundles via nonabelian Hodge theory. From this point of view, the component in the image of the map contains precisely the representations in \(\mathrm {SO}(2n,\mathbb {C})\) which lift to \(\mathrm{Spin}(2n,C)\). 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 \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\) 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 \(\mathrm{Spin}(2n,\mathbb {C})\) and \(\mathrm {SO}(2n,\mathbb {C})\)Higgs bundles are the same. In the case of \(n=2\), the base is \(H^0(\varSigma ,K^2)\oplus H^0(\varSigma ,K^2)\), and for \(n=3\) it is \(H^0(\varSigma ,K^2)\oplus H^0(\varSigma ,K^3)\oplus H^0(\varSigma ,K^4)\). 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 \(\mathrm {SL}(n,\mathbb {C})\), where the two coincide, but less so in the case of \(\mathrm {SO}(2n,\mathbb {C})\) where the relation is complicated by the role of the Pfaffian. In fact, the maps \(S\mapsto \hat{S}_{2n}\) (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 \(E_1\otimes E_2\) or \(\varLambda ^2E\) 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 33
One should note that whist not done here, the isogenies could also be understood through the language of Cameral covers introduced in [8].
7.1 The isogeny \({\mathcal {I}}_2\) on moduli spaces
Remark 34
Proposition 35
The map \({\mathcal {I}}_2: {\mathcal {\tilde{M}}}_{\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})}\rightarrow {\mathcal {\tilde{M}}}_{\mathrm {SO}_0(2,2)}\)
Proof
While the moduli space of \(\mathrm {SL}(4,\mathbb {R})\)Higgs bundles has \(2^{2g}\) Hitchin components, the one for \(\mathrm {SO}_0(3,3)\)Higgs bundles has just one Hitchin component. From the analysis of topological invariants, which are constant on connected components, one has \(4+1=5\) components, 4 coming from the 4 pairs of \(w_2\)’s characterizing \(\mathrm {SO}(3)\times \mathrm {SO}(3)\) bundles.
Proposition 36
The isogeny \({\mathcal {I}}_2\) 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.
As seen before, the map \({\mathcal {I}}_2\) constructed in Sect. 5 is surjective onto some of the components of \({\mathcal {M}}_{\mathrm {SO}_0(2,2)}\). The components in the image correspond to those components in the representation variety \(\mathrm {Rep}(\pi _1(\varSigma ),\mathrm {SO}_0(2,2))\) for which the representations lift to \(\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})\).
We shall denote by \({\mathcal {M}}^0_{\mathrm {SO}_0(2,2)}\) the union of components of \({\mathcal {M}}_{\mathrm {SO}_0(2,2)}\) obtained through (30) and (32) with \(\deg (M_1)=\deg (M_2)\ \mathrm {mod}\ 2.\) Equivalently, let \(\mathrm {Rep}^0(\pi _1(\varSigma ),\mathrm {SO}_0(2,2))\) be the union of components of \(\mathrm {Rep}(\pi _1(\varSigma ),\mathrm {SO}_0(2,2))\) which correspond to the components in \({\mathcal {M}}_0(\mathrm {SO}_0(2,2)\).
Corollary 37
The structure group of an \(\mathrm {SO}_0(2,2)\)Higgs bundle lifts to \(\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})\) if and only if the \(\mathrm {SO}_0(2,2)\)Higgs bundle lies in \({\mathcal {M}}^0_{\mathrm {SO}_0(2,2)}\). Equivalently, a reductive surface group representation into \(\mathrm {SO}_0(2,2)\) lifts to a representation into \(\mathrm {SL}(2,\mathbb {R})\times \mathrm {SL}(2,\mathbb {R})\) if and only if the representation lies \(\mathrm {Rep}^0(\pi _1(\varSigma ),\mathrm {SO}_0(2,2))\).
Remark 38
By realizing \(SO_0(2,2)\)Higgs bundles in terms of rank 2Higgs bundles, one can understand the monodromy action studied in [2] for rank 4 Higgs bundles in terms of monodromy of lowerrank 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 2Hitchin systems.
7.2 The isogeny \({\mathcal {I}}_3\) on moduli spaces
Remark 39
For both n=2,3 the variety \(Prym(\hat{S}_{2n}, \hat{S}_{2n}/_{\sigma }))\) has two components. In the case \(n=3\), given \(\hat{S}_6\) the spectral curve defined by \((2a_2, a_2^24a_2, a_3^2)\), the two components occur in the fibers over both points \((2a_2, \pm a_3, a_2^24a_2)\) in the Hitchin base. However, on each fiber only one of the components is in the image of the map induced by \({\mathcal {I}}_3\). A similar phenomenon occurs for \(n=2\).
Remark 40
The map (73) can be understood heuristically from eigenvalue considerations in the same way as (69), i.e., as explained in Remark 34.
Proposition 41
Proof
It follows from Proposition 19 that the image of \({\mathcal {\tilde{M}}}^a_{\mathrm {SL}(4,\mathbb {R})}\) lies in the component \({\mathcal {\tilde{M}}}^{(b_1,b_2)}_{\mathrm {SO}_0(3,3)}\), where \(b_1=w_2(\varLambda ^2_+(E))\) and \(b_2=w_2(\varLambda ^2_(E))\) for some \(\mathrm {SO}(4,\mathbb {C})\) vector bundle E with \(w_2(E)=a\). Moreover, from [11, Proposition 1.8] it follows that \(w_2(\varLambda ^2(E)_{\pm })=w_2(\varLambda ^2(E))\), so in particular \(b_1=b_2\). However, any pair of \(\mathrm {SO}(3,\mathbb {C})\)bundles with the same second Stiefel–Whitney class arises in this way, i.e., as \(\varLambda ^2(E)_{\pm }\) where E is an \(\mathrm {SO}(4,\mathbb {C})\) bundle. Finally, if \((\varLambda ^2E, \varPhi \otimes I+I\otimes \varPhi )\) represents a point in the image of \({\mathcal {I}}_3\), then the \(2^{2g}\) preimages come from twisting E by any point of order two in \(Jac(\varSigma )\).\({\square }\)
Footnotes
 1.
A trivialization \(\delta :\det (E)\simeq \mathcal {O}_{\varSigma }\) is compatible with Q if \(\delta ^2\) agrees with the trivialization of \((\varLambda ^nE)^2\) given by the discriminant \(Q:\varLambda ^nE\rightarrow \varLambda ^nE^*\) (see Remark 2.6 in [21]).
 2.
 3.
Notice that \(\alpha ^t=(I\cdot \alpha ^t\cdot I)\) so that in this case \(\alpha ^t=\alpha ^T\).
 4.
We have denoted the involution by \(*\) since when E is the cotangent bundle to a 4manifold, the involution is precisely the Hodge star.
Notes
Acknowledgements
We thank David Baraglia, Ron Donagi, Tamas Hausel, Nigel Hitchin, Sheldon Katz, Herbert Lange, Tom Nevins, Tony Pantev, Mihnea Popa, S. Ramanan, Hal Schenck, and Michael Thaddeus, for helpful and enlightening discussions. Both authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties (the GEAR Network).” The second author is partially supported by NSF grant DMS1509693.
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