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Nonabelianization of Higgs bundles

The character varieties of representations of a surface group into the Lie groups SL(m,H), SO(2m,H) and Sp(m,m) have a holomorphic description in terms of the moduli space of Higgs bundles. We show that the fibres of the integrable system in these cases are not abelian varieties, but are instead moduli spaces of rank 2 bundles on a spectral curve, satisfying natural stability conditions.


Introduction
The moduli space of Higgs bundles over a compact Riemann surface Σ of genus g > 0 for a complex group G c has the well-known structure of a completely integrable Hamiltonian system: a proper map to an affine space, whose generic fibre is an abelian variety. For the general linear group such a Higgs bundle consists of a vector bundle V together with a section Φ of End V ⊗ K. The coefficients of the characteristic polynomial of Φ define the map to the base and the corresponding fibre is the Jacobian of an algebraic curve S defined by the equation det(xI − Φ) = 0. This is a covering π : S → Σ on which Φ has a single-valued eigenvalue x, a section of π * K. Conversely, given a line bundle L on S we obtain V as the direct image sheaf of L and Φ as the direct image of x : L → L ⊗ π * K. This abelianization process has been useful in attacking various problems relating to bundles on curves.
It is clear, however, that we could replace L by a rank r bundle E on S and obtain a Higgs bundle on Σ by the same construction. In this case each generic eigenspace of Φ is r-dimensional and det(xI − Φ) = p(x) r for some polynomial p(x). Since the map to the base is surjective such cases certainly occur. What we show here is that they occur, with r = 2, very naturally when considering the Higgs bundles which correspond (by solving the gauge-theoretic Higgs bundle equations) to flat connections on Σ with holonomy in the real Lie groups G r = SL(m, H), SO(2m, H) and Sp(m, m), real forms of G c = SL(2m, C), SO(4m, C) and Sp(4m, C) respectively. The first two groups are often denoted by SU * (2m) and SO * (2m) but the phenomenon we are describing clearly reflects the noncommutativity of the quaternions which justifies the former notation.
The fibres of the integrable system, even those over non-regular values, are always compact. We find here that for SL(m, H) the fibre consists of the moduli space of semi-stable rank 2 bundles with fixed determinant on the spectral curve S. For SO(2m, H) the fibre has many components each of which is a moduli space of semi-stable rank 2 bundles on a quotientS of the spectral curve and for Sp(m, m) it is a Z 2 -quotient of a moduli space of semi-stable rank 2 parabolic bundles onS.
Part of this work is contained in the second author's DPhil thesis [7]. She acknowledges the support of Oxford University, New College and QGM Aarhus. Both authors wish to thank ICMAT, Madrid for support during an activity of the "Nigel Hitchin Laboratory" in March 2013 where this article originated. Work of Ana Peón, to appear in a forthcoming thesis, was also presented there, and gives a characterization using cameral covers of the real forms which exhibit this nonabelianization phenomenon.

Higgs bundles for real forms
Under suitable stability conditions a Higgs bundle defines a solution of equations for a G-connection A, where G is the maximal compact subgroup of G c [5], [8]. For G = U (n) these are F A + [Φ, Φ * ] = 0 and then the connection ∇ A + Φ + Φ * is flat, with holonomy in GL(n, C). To get a flat connection with holonomy in GL(n, R) we take A to be an SO(n)-connection and Φ = Φ T using the transpose defined by the orthogonal structure on V .
For a general real form G r of G c we take a U -connection where U is the maximal compact subgroup of G r and in the decomposition g = u ⊕ m take Φ ∈ H 0 (Σ, m ⊗ K). Much work on the study of connected components of moduli spaces of flat G r -connections using this approach has been carried out by Bradlow, Garcia-Prada et al. (for example in [3], [4]). For the groups in question we have the following descriptions of the corresponding Higgs bundles: • The group SL(m, H) is the subgroup of SL(2m, C) which commutes with an antilinear automorphism J of C 2m such that J 2 = −1. Its maximal compact subgroup is the quaternionic unitary group Sp(m). Since Sp(m) c = Sp(2m, C), the corresponding Higgs bundle consists of a rank 2m symplectic vector bundle (V, ω) and the Higgs field satisfies Φ = Φ T for the symplectic transpose. Using ω to identify V and V * , this means that Φ = ω −1 φ for φ ∈ H 0 (Σ, Λ 2 V ⊗ K).
• The group SO(2m, H) is the subgroup of GL(4m, C) which preserves a complex inner product (u, v) and commutes with an antilinear automorphism J as above for which (u, Jv) is a hermitian form. If (u, Ju) > 0 then (Ju, J 2 u) = −(Ju, u) < 0 and so the form has hermitian signature (2m, 2m). The maximal compact subgroup is U (2m) and the Higgs bundle has the form V = W ⊕ W * for a rank 2m vector bundle W . The inner product is defined by the natural pairing between W and W * . The Higgs field is of the form Φ(w, ξ) = (β(ξ), γ(w)) where β : W * → W ⊗ K and γ : W → W * ⊗ K are skew-symmetric.
• The group Sp(m, m) is the subgroup of GL(4m, C) which preserves a complex symplectic form ω and commutes with an antilinear automorphism J for which ω(u, Jv) is a Hermitian form of signature (2m, 2m). It is the group of quaternionic matrices which are unitary with respect to an indefinite form. The maximal compact subgroup is Sp(m) × Sp(m) and the Higgs bundle is of the form W 1 ⊕ W 2 for symplectic rank 2m vector bundles (W 1 , ω 1 ), (W 2 , ω 2 ). The Higgs field is of the form using the symplectic transpose.
Since SO(2m, H) and Sp(m, m) are subgroups of SL(2m, H) we begin dealing with the first case.
3 Spectral data for SL(m, H) As noted above, in this case we have a rank 2m symplectic vector bundle (V, ω) and a Higgs field which is symmetric with respect to ω. If A is a symmetric endomorphism of a symplectic vector space U of dimension 2m, and α the corresponding element of Λ 2 U * , then xI − A is singular if and only if the exterior product (xω − α) m = 0. The Pfaffian polynomial of xI − A is defined by p(x)ω m = (xω − α) m , and then p(x) 2 = det(xI − A). If p(x) has distinct roots then U decomposes into a sum of m two-dimensional symplectic eigenspaces of A and in this case, and by continuity in all cases, A satisfies the matrix equation p(A) = 0. If A is in the Lie algebra of SL(2m, C) then in addition tr A = 0.
Replacing A by Φ we have a polynomial p(x) = x m +a 2 x m−2 +· · ·+a m where the coefficients a i ∈ H 0 (Σ, K i ). As an SL(2m, C) Higgs bundle, the usual spectral curve is defined by the vanishing of the characteristic polynomial so the coefficients of p(x) 2 , which lie in 2m i=2 H 0 (Σ, K i ), define a point in the base of the fibration. In our case, by a slight abuse of notation, we shall call the curve S defined by p(x) = 0 the spectral curve. Bertini's theorem assures us that for generic a i the curve is nonsingular. It is a ramified m-fold cover of Σ. More precisely, we may interpret the equation p(x) = 0 as the vanishing of a section of π * K m over the total space of the canonical bundle π : K → Σ, where x is the tautological section of π * K. The cotangent bundle of Σ is a symplectic manifold and hence has trivial canonical bundle, so K S ⊗ π * K −m is trivial and K S = π * K m . Taking degrees of both sides this says that the genus of S is given by g S = m 2 (g − 1) + 1.
On the spectral curve S, x is a well-defined eigenvalue of Φ, and the cokernel of xI − Φ is a rank two holomorphic vector bundle E. It then follows, as in [2] (and using p(Φ) = 0 instead of the Cayley-Hamilton theorem), that we can identify V with the direct image π * E and the Higgs field Φ as the direct image of x : E → E ⊗ π * K (recall that the direct image sheaf is defined for each open set U ⊂ Σ by H 0 (U, π * E) = H 0 (π −1 (U ), E)).
If we now start with any rank 2 bundle E on S we can obtain by the same construction a GL(2m, C) Higgs bundle, but we need to determine the conditions on E for this to be the data for the group SL(m, H): be a section of the line bundle π * K m on the cotangent bundle of Σ whose divisor is a smooth curve S, and let E be a rank 2 vector bundle on S. Then the direct image of x : E → E ⊗ π * K defines a semi-stable Higgs bundle on Σ for the group SL(m, H) if and only if Proof: First we define a nondegenerate skew form on π * E. For this, the relative duality theorem gives We should also check that the duality is provided by a skew form, and this requires a concrete expression of relative duality. At a regular value a ∈ Σ of π, This extends for the direct image over branch points. But the isomorphism E * ⊗π * K m−1 ∼ = E is skew-symmetric, showing that V ∼ = V * is also skew. Moreover multiplication by x satisfies xs, s ′ = s, xs ′ which is symmetric and defines Φ as a Higgs field satisfying Φ = Φ T .
The semi-stability condition for Higgs bundles [5] But the characteristic polynomial of Φ is p(x) 2 and S is smooth, and in particular irreducible. So the characteristic polynomial for Φ| W must be p(x). Then (W, Φ) is a rank m Higgs bundle and by [2] is the direct image of a line bundle L ⊂ E.
and so this is equivalent to the semi-stability condition deg L ≤ deg E/2 for the rank 2 bundle E. ✷

Remarks:
1. The degree of E on S is 2m(m − 1)(g − 1) which is even and so the moduli space of semi-stable bundles is singular, the singular locus represented by decomposable bundles E = L ⊕ (L * ⊗ π * K m−1 ). Then relative duality gives π * E = W ⊕ W * and the Higgs field is Φ = (φ, φ T ) for a GL(m, C) Higgs bundle (W, φ).
It follows that we can use the result of the previous section to deduce that det(xI − Φ) = p(x) 2 for some polynomial of degree 2m and, assuming it defines a smooth curve S, Φ has generically two-dimensional eigenspaces. Globally, as before the bundle V can be written π * E for a rank 2 bundle on the spectral curve S, which has genus g S = 4m 2 (g − 1) + 1.
Thus for each two-dimensional generic eigenspace with eigenvalue λ there exists another with eigenvalue −λ. In particular this means that p(x) = x 2m + a 1 x 2m−2 + · · · + a m where a i ∈ H 0 (Σ, K 2i ) and the curve S has an involution σ(x) = −x. We now need to determine the properties of the rank 2 bundle E: Proposition 2 Let p = x 2m + a 2 x 2m−2 + · · · + a m be a section of the line bundle π * K 2m on the cotangent bundle of Σ whose divisor is a smooth curve S and let σ be the involution If E is a rank 2 vector bundle on S, then the direct image of x : E → E ⊗ π * K defines a semi-stable Higgs bundle on Σ for the group SO(2m, H) iff Remark: The bundle π * K 2m−1 is pulled back from Σ and this is why it makes sense to speak of the trivial action of σ, since πσ = π. When E is stable any automorphism is a scalar so the isomorphism gives a lifted action of the involution, well-defined modulo ±1 and in particular the action on Λ 2 E is well-defined. In fact in general all we require is that the action on Λ 2 E at fixed points should be trivial.
Proof: The first part follows from Proposition 1. The bundle E is defined as the cokernel of xI − Φ in V ⊗ K, but using the orthogonal structure on V , this means that E * ⊗ π * K is the kernel of xI + Φ, i.e. generically the two-dimensional eigenspace of Φ with eigenvalue −x. We saw above how (w, ξ) → (w, −ξ) gives an isomorphism between the ±x eigenspaces and so there is an isomorphism σ * E ∼ = E.
Triviality of the action on Λ 2 E = π * K 2m−1 in the statement of the Proposition is a function of the isomorphism, which came to us as in Section 3 from relative duality. This gave us the following formula for the symplectic form over a regular value: where dπ, the derivative, is a section of K S ⊗ π * K −1 and we use the canonical symplectic form of the cotangent bundle of Σ to identify this with π * K 2m−1 . Together with an isomorphism π * K 2m−1 ∼ = Λ 2 E, formula (2) is a well-defined scalar. Now the symplectic form on the cotangent bundle is anti-invariant under σ, scalar multiplication by −1 in the fibres, and the pairing symplectic form on W ⊕ W * is anti-invariant under the map (w, ξ) → (w, −ξ). It follows that the action on Λ 2 E = π * K 2m−1 is trivial. Now letS be the quotient of S by the involution and p : S →S the quotient map. Setting z = x 2 embedsS in the total space of K 2 with equation z m + a 1 z m−1 + · · · + a m = 0. If π : K 2 → Σ is the projection then z is the tautological section ofπ * K 2 and π : S → Σ can be written as π =π • p.
Conversely, suppose we are given S and E as in the statement of the Proposition. Then π * E =π * (E + ⊕E − ) = W 1 ⊕W 2 is a symplectic vector bundle from Proposition 1. Moreover since σ(x) = −x, x : E → E ⊗ π * K maps local invariant sections of E to anti-invariant ones and so the Higgs field has the off-diagonal shape of (1). To obtain V = W ⊕ W * we need to show that W 1 and W 2 are Lagrangian with respect to the symplectic structure on V . Now W 1 is defined by the direct image of an invariant section s. Equation (2) evaluates the symplectic pairing of two sections s, s ′ but if they are both invariant then s ∧ s ′ is invariant. As in the discussion above, if the action on Λ 2 E is trivial, then the denominator in (2) is anti-invariant and so the terms over y and σ(y) cancel and so the symplectic form on V vanishes on W 1 . A similar argument holds for W 2 . They are therefore transverse Lagrangian subbundles and setting W 1 = W , W 2 ∼ = W * .
As a special case of the previous section, Φ = Φ T with respect to the symplectic form but given its shape (1) this means that the terms β and γ are skew-symmetric. ✷ Proposition 2 tells us that the fibre of the integrable system is defined by the fixed points of an involution induced by σ on the moduli space of rank 2 semi-stable bundles on S. There are several components, however. This is clear from the flat connection point of view: the maximal compact subgroup of SO(2m, H) is U (2m) and so any flat SO(2m, H) bundle can be topologically reduced to U (2m) where it has a Chern class. In the Higgs bundle description this is the degree of the vector bundle W . As in the case of U (m, m) dealt with in [7] we can determine this invariant by considering the action at the fixed points of σ on S.
At a fixed point a of σ there is a linear action of σ on the fibre E a . Since the action on Λ 2 E a is trivial this is scalar multiplication ±1 and we can assign to each fixed point this number.
Proof: The fixed point set of σ is the intersection of the zero section of K with S. Setting x = 0 in the equation x 2m + a 1 x 2m−2 + · · · + a m = 0, these points are the images of the 4m(g − 1) zeros of a m ∈ H 0 (Σ, K 2m ) under the zero section. The action is +1 at M of these points.
Choose a line bundle L on Σ of large enough degree such that where the superscript denotes the ± eigenspace under the action of σ. Since V = π * E and H 1 (Σ, V ⊗ L) = 0 the higher cohomology groups vanish and applying the holomorphic Lefschetz formula we obtain and Riemann-Roch gives since V is symplectic and deg V = 0. g)+deg L)+deg W by Riemann-Roch and from these three equations we obtain deg W = 2M − 4m(g − 1). ✷ Remarks: which is the Milnor-Wood inequality for the group SO(2m, H).
2. In the maximal case deg W = 4m(g − 1) all fixed points have action +1 and then the bundle E is pulled back from the curveS. In this case γ : W → W * ⊗K is a homomorphism of bundles of the same degree and so is either everywhere singular or an isomorphism. But S is smooth so a m is not identically zero and hence W ∼ = W * ⊗ K, or setting U = W ⊗ K −1/2 , Ψ = βγ we have a Higgs bundle of the same type as an SL(m, H) bundle but with a K 2twisted Higgs field Ψ. Moreover the spectral data of the previous section holds if one takes the rank 2 bundle E ⊗π * K −1/2 onS. This is a case of the Cayley correspondence of [4].
4. In the stable case the only other action on E is to multiply the given action by −1 which changes M to 4m(g − 1) − M and interchanges the roles of W and W * . Thus there are 2 4m(g−1)−1 components in the fibre.

Spectral data for Sp(m, m)
For this group, the Higgs bundle V = W 1 ⊕ W 2 for symplectic rank 2m vector bundles (W 1 , ω 1 ), (W 2 , ω 2 ). The Higgs field is where β T : W 1 → W 2 ⊗ K is the symplectic adjoint. Since Sp(m, m) ⊂ SL(2m, H) we can apply the results of Section 3 but we need a symplectic structure on V for which Φ = Φ T .
Define ω = (ω 1 , −ω 2 ). Then ω(Φ(u 1 , u 2 ), (w 1 , w 2 )) = ω 1 (βu 2 , w 1 ) + ω 2 (β T u 1 , w 2 ) = ω 1 (u 2 , β T w 1 ) + ω 2 (u 1 , βw 2 ) and this is equal to Here the only difference with the previous case is the action of the involution on E: Proposition 4 Let p = x 2m + a 2 x 2m−2 + · · · + a m be a section of the line bundle π * K 2m on the cotangent bundle of Σ whose divisor is a smooth curve S and let σ be the involution σ(x) = −x. If E is a rank 2 vector bundle on S, then the direct image of x : E → E ⊗ π * K defines a semi-stable Higgs bundle on Σ for the group Sp(m, m) iff Proof: The proof proceeds exactly as in Proposition 2 until the point where we prove that W 1 and W 2 are Lagrangian. With the opposite action on Λ 2 E we deduce instead that W 1 and W 2 are symplectically orthogonal and hence V is the symplectic sum of W 1 and W 2 .
A slightly different and more detailed approach may be found in [7]. ✷

Remarks:
1. Given the action of −1 on Λ 2 E, at a fixed point we have distinct +1 and −1 eigenspaces. Following [1] this defines a rank 2 bundle on the curveS with a parabolic structure at the fixed points defined by the flag given by the −1 eigenspace and the parabolic weight 1/2. As in the previous case the choice of action corresponds to an ordering of W 1 and W 2 so a point in the moduli space for Sp(m, m) determines a point in the quotient of the moduli space of parabolic structures by interchanging the roles of the +1 and −1 eigenspaces.
2. Note that the two groups SO(m, H) and Sp(m, m) correspond to the two equivariant structures on the line bundle Λ 2 E ∼ = π * K 2m−1 and, as in [1], account for all the fixed points in the moduli space of rank 2 bundles over S.
3. We can use the parabolic aspect as a check on the dimension: the parameters for the spectral curve and bundle gives m(8m − 2)(g − 1) as in the previous case but there is a contribution of 1 = dim P 1 for each of the 4m(g − 1) parabolic points giving in total m(8m + 2)(g − 1) = dim Sp(4m, C)(g − 1).

Comments
1. The representation of the moduli space of flat connections as the Higgs bundle moduli space, and in particular the integrable system, depends on the choice of a complex structure on the underlying real surface Σ. Properties of the Higgs bundle can change significantly for the same representation of π 1 (Σ). As an example, the uniformizing representation in P SL(2, R) of a Riemann surface has a nilpotent Higgs field for the natural complex structure but the same representation has a non-singular spectral curve x 2 − a = 0 when we change the complex structure of Σ and hence its Higgs bundle moduli space. It is natural to ask which representations have smooth, or irreducible, spectral curves in some complex structure. The examples given here have reducible spectral curves in any complex structure.
2. The classical abelianization picture of a spectral curve with a line bundle over it is, in the physicists' terminology, a D-brane: a Lagrangian submanifold of the cotangent bundle of Σ together with a flat line bundle over it. However, when two such D-branes coalesce one expects to find a flat higher rank bundle over the resulting curve. Given the stability property of the rank 2 bundle here, it follows from the Narasimhan-Seshadri theorem that this is precisely what we have. A sequence of points in the GL(2m, C) moduli space converging to an SL(m, H) Higgs bundle gives just such a degeneration.