Since Higgs bundles were introduced in 1987 [Hit87], they have found applications in many areas of mathematics and mathematical physics. In particular, Hitchin showed in [Hit87] that their moduli spaces carry a natural hyperkähler structure and furthermore are algebraically completely integrable systems [Hit87a]. More recently Hausel and Thaddeus [HT03] related Higgs bundles to mirror symmetry, and in the work of Kapustin and Witten [KW07] Higgs bundles were used to give a physical derivation of the geometric Langlands correspondence.
Classically a Higgs bundle (E, Φ) on a compact Riemann surface Σ of genus g ≥ 2 is given by a holomorphic vector bundle E on Σ together with a holomorphic section Φ : E → E ⊗ K , for K the canonical bundle of the surface. Moreover, given a complex semisimple Lie group G c one can define G c -Higgs bundles [Hit87a] as pairs (P, Φ) on Σ, where P is a principal G c -bundle and the Higgs field Φ is a holomorphic section of ad(P) ⊗ K , where ad(P) is the adjoint bundle of P. By considering parabolic stability one can construct the moduli space M G c of G c -Higgs bundles (for details see [BiGo08, Section 3] and references therein) which carries a hyperkähler structure.
As a hyperkähler manifold the moduli space M G c carries a natural triple of complex structures I, J, K satisfying the familiar quaternionic relations and a triple of associated Kähler forms ω I , ω J , ω K , following the notation in [Hit87]. It is natural to consider submanifolds of M G c which are Lagrangian or complex with respect to each of these three structures. From the perspective of physics such submanifolds are referred to as A-branes and B-branes respectively. We dedicate this paper to the study of branes of type (A, B, A) arising naturally from anti-holomorphic involutions on the Riemann surface Σ. These (A, B, A)-branes are submanifolds of M G c which are Lagrangian with respect to the two symplectic structures ω I , ω K and complex with respect to J . Such branes are of considerable interest due to the connections believed to exist between Langlands duality and homological mirror symmetry. Additionally the (A, B, A)-branes we construct turn out to be integrable systems making them interesting structures to consider in their own right.
Given an anti-holomorphic involution f on a compact Riemann surface Σ of genus g ≥ 2, one has a natural induced action on the moduli space of G c -Higgs bundles as well as on the moduli space of representations of π 1 (Σ) into G c , which we study in Sects. 2-4. By looking at the fixed point set L G c of this involution, in Sect. 5 we obtain a natural (A, B, A)-brane in the sense of Kapustin and Witten [KW07]: Theorem 14. For each choice of anti-holomorphic involution f on a compact Riemann surface there is a natural (A, B, A)-brane L G c defined in the moduli space of G c -Higgs bundles.
In order to study these branes we consider in Sects. 6 and 7 the spectral data associated to G c -Higgs bundles introduced in [Hit87a], for G c a complex classical Lie group. Looking at L G c as sitting inside the corresponding Hitchin fibration, we obtain new examples of real integrable systems over a real subspace L of the base of the Hitchin fibration:
Theorem 17. If L G c contains smooth points then the restriction of the Hitchin fibration h| L Gc : L G c → L to L G c is a Lagrangian fibration with singularities. The generic fibre is smooth and consists of a finite number of tori.
Through the study of the spectral data associated to such (A, B, A)-branes, we prove in Sect. 7 that L G c always contains smooth points for the classical Lie groups. Hence Theorem 17 applies and we obtain many families of real integrable systems through this construction.
In Sect. 8 we study the connectivity of the fibres of L G c . In the case of G c = G L(n, C), the number of components can be expressed in terms of the number of components of the fixed point set of an induced involution on the spectral curves (Proposition 32). Moreover, in the rank 2 case we are able to reduce this to an exact formula in terms of the associated quadratic differential (Theorem 36), and also obtain exact formulas for G c = SL(2, C) (Theorem 38).
For certain groups G c we find that the Higgs bundles fixed by the induced involution may have a real or quaternionic structure. In particular, we consider this for SL(2, C)-Higgs bundles in Sect. 9. We find that if (E, Φ) is a stable SL(2, C)-Higgs bundle then E carries either a real or quaternionic structure. Moreover, we determine which of the two possibilities occur in terms of the spectral data associated to (E, Φ).
Under Langlands duality, it is known ([KW07, Section 12.4]) that (A, B, A)-branes in M G c map to (A, B, A)-branes in the moduli space ML G c of L G c -Higgs bundles, for L G c the Langlands dual group of G c . We propose that the dual of the (A, B, A)-brane L G c is the corresponding (A, B, A)-brane LL G c defined using the same involution f on the Riemann surface. In Sect. 10 we present some preliminary evidence to support this proposal.
Following the ideas of Kapustin-Witten [KW07] and Gukov [Gu07], the constructions given in this paper can be shown to be closely related to representations of 3-manifolds whose boundary is the Riemann surface Σ. In Sect. 11 we give an outline of this relation, which are developed in the companion paper [BaSch].
Throughout the paper we consider anti-holomorphic involutions f : Σ → Σ on a compact Riemann surface Σ of genus g > 1 always taken to be connected. Such an involution f induces involutions on the moduli space of representations π 1 (Σ) → G c and the moduli space of G c -Higgs bundles on Σ, for a given complex semisimple Lie group G c . In this section we consider the topological properties of such involutions and the induced actions on the fundamental group π 1 (Σ).
Σ → Σ of a compact Riemann surface is a classical result of Klein. The reader should refer to [GH81] for a thorough study of this situation. All such involutions on Σ may be characterised by two integer invariants (n, a) as follows. The fixed point set of f is known to be a disjoint union of copies of the circle embedded in the surface. Let n denote the number of components of the fixed point set which by a classical theorem of Harnack can be at most g + 1. The complement of the fixed point set has one or two components. Set a = 0 if the complement is disconnected and a = 1 otherwise.
Example 1. For Σ of genus 2 and n(Σ) = 1, one may have (Fig. 1):
Remark 2. One should note that an anti-involution f as considered in this paper can also be found in the literature as an anti-conformal map of Σ, whose species Spi( f ) is +k or -k according to whether Σ -Fix( f ) is connected or not (e.g., see [BCGG] and references therein).
). Let g be the genus of Σ. The invariants (n, a) associated to an orientation reversing involution on Σ satisfy the following conditions
Conversely any pair (n, a) satisfying these conditions determines such an orientation reversing involution on Σ, unique up to homeomorphism.
Orientation reversing involutions on a compact Riemann surface Σ may be constructed as follows. Given a compact surface S (not necessarily orientable) with boundary let S be its orientation double cover, which is an oriented surface with boundary. The involution f which interchanges the sheets of the double cover reverses the orientation of S. From each boundary component of S we obtain a corresponding pair of boundary components in S which are exchanged by f . We then define Σ by gluing together the boundary components of S through the restriction of f to the boundary. Moreover such a surface Σ has a natural orientation reversing involution induced by f , and the conditions of Proposition 3 are satisfied by this construction.
In order to see that the above construction produces all such anti-holomorphic involutions on a compact Riemann surface, consider Σ a compact Riemann surface with an anti-holomorphic involution f which has fixed points. In a neighbourhood of a fixed point, one can find a local coordinate z such that the involution f is given by the complex conjugation f : z → z (e.g., see [Se91] or [GH81]). Hence the fixed point set can be seen as a union of copies of the unit circle embedded in the surface. By making cuts in Σ around these circles we obtain a Riemann surface Σ with two boundary components for each cut and a fixed point free anti-holomorphic involution, which permutes each pair of components. Since the induced involution on Σ does not fix any boundary component, it pairs off components and thus Σ is the orientation double cover of a Riemann surface with boundary.
Recall that a spin structure on an oriented Riemannian nmanifold M with SO(n)-frame bundle P may be defined as a class ξ ∈ H 1 (P, Z 2 ) which restricted to any fibre of P → M agrees with the class in H 1 (SO(n), Z 2 ) corresponding to the double cover Spin(n) → SO(n). In the case that M = Σ is a Riemann surface we may identify the frame bundle P with the bundle U Σ of unit tangent vectors, since any unit vector can be uniquely extended to an oriented frame.
Throughout the paper we let K denote the canonical bundle of the Riemann surface Σ. Given a compatible Riemannian metric on Σ, the spin structures correspond to theta characteristics, holomorphic line bundles L for which there is an isomorphism L 2 K . The hermitian structure on K determines a hermitian structure on L such that the bundle of unit vectors in L is a double cover of the unit vectors of K . Then since the unit vectors of K can be naturally identified with the unit tangent bundle U Σ we obtain a double cover of U Σ, which is then a spin structure. The spin structures on Σ for different choices of metric may be canonically identified and the spin structure associated to L does not depend on the choice of metric, for different choices of metrics compatible with the complex structure on Σ.
The induced map f * :
) of an anti-holomorphic involution f on Σ preserves the subset of classes in H 1 (U Σ, Z 2 ) which define spin structures. Therefore the anti-holomorphic involution f has a natural action on the set of spin structures. From [Ati71,GH81] we have the following. Proposition 4. Given Σ a compact oriented surface and f : Σ → Σ an orientation reversing involution, there exists a spin structure preserved by f .
There is also a natural action of f on the theta characteristics. Indeed as f is antiholomorphic there is a natural isomorphism f * (K ) K . It follows that if L is a theta characteristic, then so is f * (L).
Proposition 5. The action of f on the set of spin structures of the Riemann surface Σ interpreted as theta characteristics agrees with the action of f on the set of spin structures considered as a subset of H 1 (U Σ, Z 2 ), where the action of f is induced by the differential f * : U Σ → U Σ.
Proof. Let L be a holomorphic line bundle such that L 2 K . In particular, L inherits a hermitian metric h from the hermitian metric on K . Let Q → U Σ be the corresponding double cover of U Σ and ξ ∈ H 1 (U Σ, Z 2 ) the class defined by Q. The hermitian metric on L determines a metric on f * (L) and it is clear that the induced double cover of
From the above propositions we conclude the existence of theta characteristics
2.3. Action on the fundamental group. Since an anti-holomorphic involution of the Riemann surface f : Σ → Σ is a homeomorphism, it induces an isomorphism f * :
Let . denote the operation of joining paths. Given a path γ joining x 0 to f (x 0 ) we obtain an isomorphism
which sends a loop u based at f (x 0 ) to the loop γ.u.γ -1 based at x 0 . Hence the com-
is an automorphism of π 1 (Σ, x 0 ). Different choices of γ change f by composition with an inner automorphism. Observe that f (γ ) is a path from f (x 0 ) to x 0 so the composition h = γ. f (γ ) is a loop based at x 0 . One sees that f 2 is the inner automorphism u → huh -1 . If the map f has fixed points we can choose x 0 to be a fixed point and γ to be the constant path. Then h is the trivial loop and f is an involutive automorphism of π 1 (Σ). On the contrary if f is fixed point free then Σ is a double cover π : Σ → Σ of a non-orientable surface Σ and thus we get an exact sequence
2.4. Action on principal bundles. Let P → Σ be a principal G c -bundle, for G c a complex connected Lie group and f * (P) be the pullback principal bundle. Since G c is connected the principal bundle P can be trivialized over the 1-skeleton of Σ. The obstruction to extending the trivialization to the 2-skeleton is a cohomology class in
From the above proposition the action of f * can be seen to be the pullback in cohomology
), the induced action f * is multiplication by -1. Therefore a principal G c -bundle P → Σ is fixed by f if and only if its topological class x ∈ π(G c ) is a 2-torsion element.
In this section we shall denote by G c a complex connected Lie group with Lie algebra g, and assume that g admits an invariant symmetric non-degenerate bilinear form B.
3.1. Moduli space of representations. Let Σ be a compact orientable surface of genus g > 1 and let π = π 1 (Σ, x 0 ) be its fundamental group. We denote by Hom(π, G c ) the set of homomorphisms ρ : π → G c with the compact-open topology. Given a representation ρ : π → G c composition with the adjoint representation of G c defines a representation of π on g. We denote by g ρ the Lie algebra g equipped with this representation.
The space Hom(π, G c ) has a natural action of G c by conjugation. To obtain a good quotient one restricts to the subspace Hom + (π, G c ) of reductive representations. Recall that a representation ρ : π → G c is called reductive if g ρ splits into a direct sum of irreducible representations. Restricted to reductive representations the action of G c on Hom + (π, G c ) is proper [Go84], and thus the quotient is a Hausdorff space.
Let Rep + (π, G c ) be the quotient space Hom + (π, G c )/G c , the moduli space of reductive G c -representations of π 1 (Σ). In general Rep + (π, G c ) has singularities, but there is a dense open subset of smooth points over which Rep + (π, G c ) naturally has the structure of a complex manifold. For a representation ρ, let G ρ ⊆ G c be the stabilizer of ρ, and let [Go84]. This condition also ensures that the stabilizer
) is discrete, and hence finite since the action is proper. It follows from general theory (see [MM03]) that around such points the quotient space Rep + (π, G c ) has the structure of an orbifold. Moreover if ρ is a reductive representation with
Definition 7. We say that a reductive representation ρ is simple if G ρ = Z (G c ). In particular if the representation of π on g induced by ρ is irreducible, then ρ is simple.
) is a simple point then the tangent space at ρ is naturally given in terms of group cohomology by H 1 (π, g ρ ). The pairing B : g ⊗ g → C determines a symplectic pairing H 1 (π, g ρ ) ⊗ H 1 (π, g ρ ) → H 2 (π, C) C. This defines a closed complex symplectic form Ω on the smooth part of Rep + (π, G c ) [Go84].
We have seen in Sect. 2.3 that given an orientation reversing involution f : Σ → Σ, there is an induced map f : π → π defined up to an inner automorphism. Accordingly there is an induced action on Hom(π, G c ) sending a representation ρ to ρ • f . This action preserves the subspace of reductive representations and descends through the quotient to an action f :
). We may identify Rep + (π, G c ) with the moduli space of gauge equivalence classes of flat G c -connections on Σ with reductive holonomy. Then f :
) is an involution which preserves the subspace of simple points. On the simple points f is holomorphic and anti-symplectic, that is f * Ω = -Ω.
Proof. Recall that the automorphism f : π → π is such that f 2 is an inner automorphism. It follows that the induced map f :
) is an involution which preserves the space of simple points. It is clear that f acts smoothly on the space of simple points, and its differential is the induced pullback in group cohomology f * : H 1 (π, g ρ ) → H 1 (π, g ρ• f ). Since this is a complex linear map, f :
) is holomorphic on the simple points. Moreover f is orientation reversing from which the anti-symplectic condition f * Ω = -Ω follows.
Lemma 9. Let X be a complex manifold of dimension n and f : X → X an antiholomorphic involution. If the fixed point set of f is non-empty, then it is a real analytic submanifold of real dimension n.
Proof. Let p be a fixed point of f , and let g be a Riemannian metric on X . Replacing g by g + f * (g) we may assume that g is f -invariant. We can also assume that g is analytic near p. Let e : T p X → X be the exponential mapping at p, and C = f * ( p) : T p X → T p X the differential of f at p. Then C is an anti-linear involution on T p X , and since g is f -invariant, for any v ∈ T p X we have that e(C(v)) = f (e(v)). Thus a neighborhood of the origin in T p X defines local analytic coordinates near p such that f corresponds to the anti-linear involution C. Therefore the fixed point set of f near p is a real analytic submanifold of real dimension n.
We shall denote by L G c the fixed point set of the induced involution f :
From Proposition 8 and Lemma 9 we obtain: Proposition 10. If non-empty, the set of simple points of L G c is a smooth complex Lagrangian submanifold of the simple points of Rep + (π, G c ).
Through the theory of spectral curves, we show in Sect. 7 that L G c does indeed contain smooth points for G c = G L(n, C), and for G c a classical complex semi-simple Lie group.
A Higgs bundle on a Riemann surface Σ is a pair (E, Φ), where E is a holomorphic vector bundle on Σ and Φ is a holomorphic section of End(E) ⊗ K . More generally for a complex Lie group G c we define a G c -Higgs bundle to be a pair (P, Φ) where P is a holomorphic principal G c -bundle with adjoint bundle ad(P) and Φ is a holomorphic section of ad(P) ⊗ K . In the following subsections we shall study the geometry of the moduli space of G c -Higgs bundles, which is closely related to the moduli space of surface group representations into G c .
For simplicity assume G c is connected and semisimple. Let G denote the compact real form of G c . Further, we use g c to denote the Lie algebra of G c , and g for the Lie algebra of G. Given a choice of principal G c -bundle P, the Killing form k(x, y) on g c naturally defines a bilinear form on the adjoint bundle ad(P) which we will also denote by k. Suppose P is given a reduction of structure to G. This amounts to equipping the adjoint bundle ad(P) with an antilinear involution ρ : ad(P) → ad(P) such that taking the hermitian adjoint x * of a section x of ad(P) is given by x * = -ρ(x). The associated hermitian form h is given by h
Consider pairs (∂ A , Φ), where ∂ A denotes a ∂-connection on P defining a holomorphic structure, and Φ is a section of Ω 1,0 (Σ, ad(P)). Note that if ∂ A Φ = 0 then (∂ A , Φ) defines a G c -Higgs bundle on P. From [Hit87], the Hitchin equations for a pair (∂ A , Φ) are
where F A is the curvature of the unitary connection
Two solutions to the Hitchin equations on P are considered equivalent if they are related by a G-valued gauge transform. We let M G c (P) denote the moduli space of gauge equivalence classes of solutions to the Hitchin equations on P, and M G c the union of the M G c (P) as P ranges over the set of isomorphism classes of principal G c -bundles. A solution (∂ A , Φ) to the Hitchin equations determines a flat G c -connection ∇ = ∇ A + Φ + Φ * . Using results of Donaldson [D87] and Corlette [Cor88], the assignment (∂ A , Φ) → ∇ gives an isomorphism between the Higgs bundle moduli space M G c of equivalence classes of solutions to the Hitchin equations and the moduli space
The slope of a holomorphic vector bundle E → Σ is defined as the number μ(E) := deg(E)/rank(E). We say that a Higgs bundle (E, Φ) is semi-stable if for each proper, non-zero subbundle F ⊂ E which is Φ-invariant we have μ(F) ≤ μ(E). If this inequality is always strict then (E, Φ) is said to be stable. Finally the Higgs bundle (E, Φ) is poly-stable if it is a sum of stable Higgs bundles of the same slope. It is possible to adapt these definitions to the case of principal G c -Higgs bundles for a complex semisimple Lie group G c (e.g., see [BiGo08, Section 3]).
One may define a moduli space of semi-stable G c -Higgs bundles M Higgs G c , and by a fundamental result of Hitchin [Hit87] and Simpson [S88] there is an isomorphism between M Higgs G c and the moduli space M G c of solutions to the Hitchin equations, when G c is semi-simple. The key result used to establish this is that a G c -Higgs bundle (∂ A , Φ) is gauge equivalent to a solution of the Hitchin equations if and only if it is poly-stable. A similar isomorphism exists when the group is
of Higgs bundles of degree 0.
The moduli space M G c of G c -Higgs bundles, for G c a complex semisimple Lie group is a hyperkähler manifold with singularities, obtained by taking a hyperkähler quotient of the infinite dimensional space of pairs of complex structures and Higgs fields [Hit87,S88].
Recall that a pair (∂ A , Φ) consists of a holomorphic structure ∂ A on the principal bundle P, and a section Φ ∈ Ω 1,0 (Σ, ad(P)). The space of all pairs (∂ A , Φ) on P is an infinite dimensional manifold and an affine space modelled on Ω 0,1 (Σ, ad(P)) ⊕ Ω 1,0 (Σ, ad(P)). We write (Ψ 1 , Φ 1 ), (Ψ 2 , Φ 2 ), . . . for tangent vectors to this space. Furthermore given a pair (∂ A , Φ) we write ∇ A for the unitary connection corresponding to ∂ A , and F A its curvature. Explicitly one has
The metric on this infinite dimensional space is given by
(2)
There are compatible complex structures I, J, K satisfying the quaternionic relations given by
). We shall denote by ω I , ω J , ω K the corresponding Kähler forms defined by
The induced complex symplectic forms Ω I , Ω J , Ω K are given by
Example 11. For example Ω J is given by:
After taking the hyperkähler quotient this agrees up to the factor of -i with the symplectic form Ω defined in Sect. 3.1, that is
To get the moduli space M G c , as seen in [Hit87] one interprets the Hitchin Eq. (1) as the moment map equations for the action of the unitary gauge group and takes the hyperkähler quotient. Given (∂ A , Φ) a solution of the Hitchin Eq. ( 1), the tangent space T (∂ A ,Φ) M G c of the moduli space at a smooth point (∂ A , Φ) can be described as equivalence classes of deformations (Ψ 1 , Φ 1 ) in Ω 0,1 (Σ, ad(P)) ⊕ Ω 1,0 (Σ, ad(P)) satisfying:
Two deformations are equivalent if they are related by an infinitesimal unitary gauge transformation. Thus (Ψ 2 , Φ 2 ) is equivalent to (Ψ 1 , Φ 1 ) if there is a skew-adjoint section ψ ∈ Ω 0 (Σ, ad(P)) of the adjoint bundle such that
Given f : Σ → Σ an anti-holomorphic involution on Σ, we saw in Sect. 3 that f induces an involution on Rep + (π
we obtain an involution on the moduli space of G c -Higgs bundles. From this perspective we can interpret the fixed point set L G c in terms of A-and B-branes following [KW07].
Let P be a principal G c -bundle over Σ and fix a reduction of structure of P to G. For x ∈ M G c a point in the moduli space represented by a G c -Higgs bundle pair (∂ A , Φ) on P, applying a gauge transform we may assume that (∂ A , Φ) satisfies the Hitchin equations. In particular the flat connection ∇ corresponding to (∂ A , Φ) is given by ∇ = ∂ A + ∂ A + Φ + Φ * . The involution f acts on this flat connection by pullback, so we obtain
Higgs bundle on f * (P) which satisfies the Hitchin equations. The flat connection associated to the pair ( f * (∂ A ), f * (Φ * )) is f * ∇, so we have found a Higgs bundle pair 1) is holomorphic with respect to the complex structure J and anti-holomorphic with respect to the complex structures I and K . Moreover f is an isometry with respect to the hyperkähler metric.
Proof. As seen above, the action of f on M G c is given by sending a solution (∂ A , Φ) of the Hitchin equations to the pair
). From this and (3) we see that f is antiholomorphic with respect to I, K and holomorphic with respect to J . Similarly it follows from (2) that f is an isometry.
to the G c -Hitchin equations (1) meets the smooth points in a complex Lagrangian submanifold with respect to J, Ω J .
Proof. Since f is anti-holomorphic in I , we know immediately that its fixed point set
Clearly L G c is a complex submanifold with respect to J , and the symplectic forms ω I , ω K must vanish on L G c . In particular this means the complex symplectic form Ω J = ω K + iω I vanishes on L G c . Thus L G c is a complex Lagrangian submanifold with respect to J .
Following [KW07], we say that L G c is an (A, B, A)-brane with respect to the complex structures I, J and K , and thus have the following: Theorem 14. For each choice of anti-holomorphic involution f on a compact Riemann surface, there is a natural (A, B, A)-brane L G c defined in the moduli space of G c -Higgs bundles.
In subsequent sections we shall describe how the brane L G c lies with respect to the Hitchin fibration for the moduli space M G c (Sect. 6), consider the spectral data for G c -Higgs bundle to show that L G c is non-empty (Sect. 7), and study connectivity of L G c (Sect. 8).
G c be the graded C-algebra of invariant polynomials on g c and Sym * the graded C-algebra with Sym j = H 0 (Σ, K j ). We shall denote by A G c the space Hom(R * , Sym * ) of graded C-algebra homomorphisms from R * to Sym * . Choosing a homogeneous basis of generators p 1 , p 2 , . . . , p l for R * , we have that R * C[ p 1 , . . . , p l ] and hence we obtain a non-canonical isomorphism
where d i is the degree of p i . From [Hit92], we consider the Hitchin fibration h : M G c → A G c , which assigns to a Higgs bundle (P, Φ) the homomorphism h(P, Φ) : R * → Sym * sending an invariant polynomial p to its evaluation p(Φ) on Φ.
The anti-holomorphic involution f on Σ induces a natural anti-holomorphic involution on the spaces H 0 (Σ, K j ) given by sending a holomorphic differential q to f * (q). The induced map f : Sym * → Sym * is an anti-linear graded ring involution. We may also define an anti-linear graded ring involution f : R * → R * as follows. Given p ∈ R * and x ∈ g c , set ( f p)(x) = -p(ρ(x)), for ρ the compact anti-involution defined as in Sect. 4. Combining these actions on R * and Sym * we obtain an anti-linear involution
It is important to note that whilst other anti involutions may be used instead of ρ, the choice ρ is natural, and this will be seen in following sections as well as in [BaSch].
Let (P, Φ) be a Higgs bundle pair which satisfies the Hitchin equations. Then f (P, Φ) has Higgs field f * (Φ * ) =f * (ρΦ). Given any invariant polynomial p, we find that
Therefore the Hitchin map commutes with the involutions:
Remark 15. In the case of G c = G L(n, C), a generating basis of invariant polynomials is given by the traces of powers p j (x) = tr(x j ), and
where we have used ρx = -x * = -x t . Hence for G L(n, C) the above basis of generators for the invariant polynomials are fixed by the involution f : R * → R * .
Let L ⊂ A G c denote the fixed point set of f on A G c . It follows that the Hitchin fibration restricts to a map h| L Gc : L G c → L. For simplicity, we shall drop the subscript of L G c , and refer to the
Proposition 16. Away from the singular fibres of the Hitchin map, the restriction h| L : L → L is a submersion.
Proof. Recall from Proposition 13 that the restriction of L to the smooth points of M G c is a complex submanifold of the smooth part of M G c with respect to J . Thus it is naturally a complex Kähler manifold with complex structure J | L and Kähler form ω J | L . From [Hit92] the non-singular fibres of the Hitchin map are Lagrangian with respect to the complex symplectic form Ω I = ω J + iω K . In particular, ω J vanishes on the fibres of the Hitchin map.
For any smooth point x ∈ L the kernel K x of (h| L ) * is an isotropic subspace of T x L with respect to ω J | L , and thus dim(K x ) ≤ 1 2 dim(L). But on the other hand the image I x of (h| L ) * lies in L, so we also have dim(
Hence we must have dim(I x ) = dim(L) and the restriction h| L is a submersion.
It follows from the above proof that away from the singular fibres of the Hitchin fibration, the fibres of h| L : L → L are Lagrangian submanifolds of L, where the symplectic structure on L is given by ω J | L .
Theorem 17. If L contains smooth points then the restriction of the Hitchin fibration h| L : L → L to L is a Lagrangian fibration with singularities. The generic fibre is smooth and consists of a finite number of tori.
Proof. Assume that L contains smooth points. Then L meets the smooth points of the Higgs bundle moduli space M G c in a smooth real analytic submanifold. At a smooth fixed point p the tangent space to L defines a real structure on T p M G c . On the other hand the smooth points where the Hitchin map is not a submersion form a complex analytic subvariety V . By considering power series one sees that the smooth points of L can not lie entirely in V , so that the Hitchin map is a submersion at a generic smooth point of L. This also shows that the image h(U ) of any non-empty open subset U of L does not lie entirely in the space of critical values of h. Hence the generic fibre of the restricted Hitchin map h| L : L → L is smooth, giving a Lagrangian fibration with singularities. On the generic smooth fibres of h| L we obtain dim(L)/2 commuting vector fields, which are complete since h is a proper map (e.g.: [Hit87] for the rank 2 case). It follows that the generic fibres of h| L consist of finitely many copies of a torus.
⊂ L denote the points of L over which the Hitchin map is not a submersion. Each fibre of L over L\ consists of a finite number of tori. In general L\ may be disconnected and the number of tori in a given fibre varies as one moves between the components of L\ .
The above analysis establishes that the fixed point set L has the structure of a real integrable system with singularities.
As seen in Proposition 12, the map f induces an involution on the moduli space M G c of polystable G c -Higgs bundles
In this section we shall describe how f acts in terms of the spectral data associated to G c -Higgs bundles, for G c a classical complex Lie group and see that the fixed point set L is non-empty (see e.g., [Hit07] for the construction of the spectral data).
makes sense for Higgs bundles of any degree, so that the action of f can be extended to the full moduli space of G L(n, C)-Higgs bundles of arbitrary degree. However we shall see that there can only be fixed points in degree 0. This extended action can be interpreted as the induced action of f on representations of a central extension of π 1 (Σ), following [AB82].
From [Hit87a] the fibre of the Hitchin fibration for G L(n, C)-Higgs bundles is isomorphic to the Jacobian of a curve. To see this let p : K → Σ denote the projection from the total space of K to Σ and let η denote the tautological section of p * K . A G L(n, C)-Higgs bundle (E, Φ) has associated an n-fold cover p : S → Σ in the total space of K , with equation det(η -Φ) = 0, together with a line bundle U on S satisfying p * U = E. Explicitly, S has equation
where a m ∈ H 0 (Σ, K m ). The line bundle U associated to (E, Φ) corresponds to the eigenspaces of Φ and may be defined by the exact sequence [BNR89] 0
Conversely given sections a i ∈ H 0 (Σ, K i ) such that the associated curve S defined by Eq. ( 4) is smooth and a line bundle U on S, one has a corresponding rank n stable Higgs bundle (E, Φ) by defining E = p * (U ) and taking Φ to be the map obtained by pushing forward the tautological section η : U → U ⊗ p * K .
As seen in Sect. 4, we may view a Higgs bundle (∂ A , Φ) as consisting of a holomorphic structure ∂ A and a Higgs field Φ, where Φ ∈ Ω 1,0 (Σ, g c ) is holomorphic with respect to ∂ A . From Proposition 12, the action of the anti-holomorphic involution f sends the pair
The anti-involution f naturally lifts to an anti-holomorphic involution on the total space f :
Definition 20. We denote by f the natural lift of f to all powers K m of K .
Let p K : K → Σ be the projection from the total space of K . Then the map f can be lifted to a natural action f :
and thus the action of f on K restricts to a bijection between the spectral curves for (∂ A , Φ) and f (∂ A , Φ). Let (∂ A , Φ) be a fixed point in the moduli space, so f (∂ A , Φ) is gauge equivalent to (∂ A , Φ) under a complex gauge transformation. In particular, since (∂ A , Φ) and f (∂ A , Φ) are in the same isomorphism class, this requires the spectral curves to coincide.
Proof. Since Φ and Φ * have conjugate eigenvalues we find from (6) that the coefficient of λ n-m in the spectral curve for f (A, Φ) is given by f * (a m ). Hence from (6), the spectral curves for (∂ A , Φ) and f (∂ A , Φ) coincide if and only if a m = f * (a m ) for all m.
Remark 22. When condition (7) holds the map f restricts to an anti-holomorphic involution on the spectral curve S covering the action of f on Σ. Definition 26. Define ι 0 : J ac(S) → J ac(S) as the involution
The involution ι on J ac(S) in Definition 24 can be expressed as
and changes degree according to
Therefore only line bundles of degree n(n -1)(g -1) can be fixed by ι. Let J ac d (S) denote the component of Pic(S) consisting of line bundles of degree d and write J ac(S) for the Jacobian J ac 0 (S). From the above analysis, any fixed point of ι must lie in J ac d (S) for d = n(n -1)(g -1), but we have not yet established the existence of any fixed points. For this let K 1/2 be a theta characteristic corresponding to an f -invariant spin structure on Σ. Such spin structures exist by Proposition 4, and by Proposition 5 this corresponds to the existence of a theta characteristic K 1/2 such that f * (K 1/2 ) K 1/2 . Now since p * K (n-1)/2 has degree d = n(n -1)(g -1), tensoring by p * K (n-1)/2 defines a bijection J ac(S) → J ac d (S). For M a line bundle in J ac(S), the action of ι on M ⊗ p * K (n-1)/2 is
Thus M ⊗ p * K (n-1)/2 is fixed by ι if and only if M is fixed by ι 0 .
Lemma 27. Let A be a complex torus of dimension m and θ : A → A an antiholomorphic group automorphism. Then the fixed point set A θ of A is isomorphic to a product T × (Z 2 ) d of a real torus T of dimension m and a discrete group (Z 2 ) d , for some 0 ≤ d ≤ m. Every component of A θ contains a point of order 2 and there are exactly 2 m+d points of order 2 fixed by θ .
Proof. Since θ is an automorphism the fixed point set A θ is a closed subgroup, hence a compact abelian subgroup. The identity component A θ 0 is therefore a torus which must have real dimension m since θ is an anti-holomorphic involution. Let H = A θ /A θ 0 be the finite abelian group of components of A θ . The exact sequence 0 → A θ 0 → A θ → H → 0 splits because A θ 0 is a divisible group, and every element of H is 2-torsion. Indeed if x ∈ A θ then x + x = x + θ(x) lies in the identity component A θ 0 because it lies in the image of the connected group A under the map A → A θ which sends a point y ∈ A to y + θ(y). It remains only to show that every component of A θ contains a point of order 2. This follows since A θ 0 is a divisible group. The inequality d ≤ m holds since A has 2 2m points of order 2.
) → L is diffeormorphic to a product T × (Z 2 ) d of a torus of real dimension 1 + n 2 (g -1) and a discrete group (Z 2 ) d for some d ≥ 0.
Proof. A generic point in the base L of the restricted Hitchin map L G L(n,C) → L defines a smooth spectral curve S → Σ and the corresponding fibre is diffeomorphic to the fixed point set of the involution ι 0 : J ac(S) → J ac(S), where ι 0 (L) = f * (L * ). Thus ι 0 is an anti-holomorphic group automorphism of J ac(s) and we may apply Lemma 27.
Similar methods can be used to show that L G c is non-empty for classical complex Lie groups G c . For completion, we shall show here how the method applies in the case of G c = SL(n, C).
where E is a rank n vector bundle with trivial determinant and Φ is a trace-free Higgs field. As usual we let ∂ A denote the ∂-operator defining the holomorphic structure on E.
The spectral curve p : S → Σ associated to an SL(n, C)-Higgs pair (∂ A , Φ) has equation
where a m is a holomorphic section in H 0 (Σ, K m ). From Proposition 12 the action of
Thus one can define the spectral curve for f (∂ A , Φ) which has equation det(ηf * (Φ * )) = 0. From Proposition 21, the spectral curves for (∂ A , Φ) and f (∂ A , Φ) coincide if and only if the coefficients a m in (8) satisfy the reality conditions a m = f * (a m ). In such a case we have that f restricts to an anti-holomorphic involution on the spectral curve S covering f .
From [Hit07, Section 2.2], the spectral data in this case is given by an eigen-line bundle U in the Jacobian of S for which U ⊗ p * K (n-1)/2 lies in the Prym variety Prym(S, Σ). Recall that Prym(S, Σ) is defined by the exact sequence
where the norm map N m : Jac(S) → Jac(Σ) is defined by sending a divisor D = i s i on S to the divisor N m(D) = i p(s i ) on Σ. Note that Prym(S, Σ) in this situation is connected, hence a complex torus which is in fact an abelian variety. Through Proposition 23, the spectral data associated to the pair f (∂ A , Φ) is given by the line bundle
To proceed to SL(n, C)-bundles we must understand how the fixed point set of ι 0 meets the Prym variety.
Proposition 29. The action of ι 0 on J ac(S) preserves the Prym variety.
Proof. Recall that the norm map N m may also be defined as the dual map of the pullback p * : J ac(Σ) → J ac(S). Let ι 0 : J ac(Σ) → J ac(Σ) be defined by ι 0 (L) = f * (L * ), then clearly p * ι 0 = ι 0 p * : J ac(Σ) → J ac(S). It follows that ι 0 • N m = N m • ι 0 , and in particular ι 0 preserves the Prym variety, which is given by the kernel of N m.
Applying Lemma 27 we obtain the following:
Proposition 30. The generic fibres of the restriction of the Hitchin fibration L SL(n,C) → L are products T × (Z 2 ) d of a real torus T of dimension (n 2 -1)(g -1) and a discrete group (Z 2 ) d , where the number of points of order 2 in Prym(S, Σ) fixed by ι 0 is 2 (n 2 -1)(g-1)+d .
The above study establishes the existence of fixed points for the action of f on the moduli space of SL(n, C)-Higgs bundles. Moreover, it confirms the picture given in Sect. 6, where the fixed point set is shown to be a real integrable system.
Remark 31. One should note that the moduli space of SU ( p, p)-Higgs bundles can be seen inside the moduli space of SL(2 p, C)-Higgs bundles. In particular from [Sch, Chapter 6], the spectral data for this subspace corresponds to line bundles L of fixed determinant which are preserved by the natural involution σ : η → -η on the corresponding spectral curve. Hence, by looking at the relation between σ and f one can understand the fixed point set of f in the moduli space M SU ( p, p) (equivalently, in M U ( p, p) ).
Applying the same methods for the case of G c = PG L(n, C), Sp(2n, C), SO(2n + 1, C), and SO(2n, C), one can see through the spectral data introduced in [Hit07] that the fixed point set L G c is non-empty and contains smooth points. Moreover the generic fibre of the restricted Hitchin map is the set of real points in an abelian variety, diffeomorphic to 2 d torus components for some d. In the following section we shall study the topology of the (A, B, A)-branes L G c .
In order to study connectivity of the (A, B, A)-brane L G c in the moduli space M G c of G c -Higgs bundles, we consider the fibres of the restricted Hitchin fibration
In particular we shall see that the number of connected components of the fibres depends on the invariants (n, a) associated to the Riemann surface and the spectral curves, as introduced in Sect. 2.1 following [GH81].
Proposition 32. Let S → Σ be a smooth G L(n, C) spectral curve, n S the number of fixed point components of the lifted involution f : S → S, and g S = 1 + n 2 (g -1) the genus of S. Then the number of connected components of the corresponding fibre of
If n S = 0 the number of components is 1 if g S is even, and 2 if g S is odd.
Proof. Recall that ι 0 : J ac(S) → J ac(S) is given by ι 0 (L) = f * (L * ) and that the fibre of the Hitchin map corresponding to the spectral curve S is given by the fixed point set of ι 0 . Thus we must determine the number of components of the fixed point set of ι 0 . From Lemma 27 we see that the number of components of this fixed point set is the same as the number of components of the fixed point set of the involution θ : J ac(S) → J ac(S), where
where it is determined that the number of components of the fixed point set is 2 n S -1 if n S > 0. In the case that n S = 0 the number of components is shown to be 1 or 2 depending on whether the genus of the curve is even or odd.
It remains to determine how n S depends on the pair (Σ, f ) and the coefficients a i ∈ H 0 (Σ, K i ) defining the spectral curve S. As the rank n increases this quickly becomes difficult so we restrict attention to the rank 2 case in Sect. 8.2.
Remark 33. In the case of Accola-Maclachlan and Kulkarni surfaces, the invariants (n, a) are determined in [BBCGG] for any genus g. Thus, the study of the invariants for the ramified covering S in this case gets simplified, and connectivity of L G L(n,C) for these Riemann surfaces can be determined through [GH81]. For hyperelliptic surfaces, a study of the invariants associated to f has been done in [BCGG], and extended to our case in Appendix A.1.
and the corresponding twofold cover p : S → Σ has genus g S = 1 + 4(g -1) and equation η 2 + a 1 η + a 2 = 0. Without loss of generality we assume that a 1 = 0, since we can re-write the previous equation as (η + a 1 /2) 2 + (a 2a 2 1 /4) = 0. The spectral curve S is therefore a double cover given by
for q = -a 2 ∈ H 0 (Σ, K 2 ). Generically q has 4g -4 different zeros, which give the ramification points of the smooth curve S. The reality condition on the defining equation for S is simply that f * (q) = q, and in particular f must act on the zero set of q.
The anti-linear involution f on the Riemann surface Σ induces an involution f : S → S, which from Sect. 6 acts by
The study of the invariant n S depends on how f acts on the zero set of q. First note that since f covers f , any fixed component of f must lie over a fixed component of f . Therefore to determine n S we need only determine how many fixed components of f lie over each fixed component of f . Given a fixed component of f there are two cases to consider depending on whether or not q has zeros along the component.
Proposition 34. If S 1 is a fixed circle component of f in Σ and the differential q does not have any zeros on S 1 , then p -1 (S 1 ) is a disjoint union of two circles. Let x ∈ S 1 be any point of the fixed component. With respect to a local coordinate z centred on x such that f (z) = z, q may be written as q(z) = q 1 (z)(dz) 2 , where q 1 (z) = q 1 (z). Then -the two S 1 factors in S are fixed by f if q 1 (0) > 0 on S 1 , -the two S 1 factors in S are exchanged by f if q 1 (0) < 0 on S 1 . Proof. Let x be a fixed point of f such that q(x) = 0. By choosing an appropriate local holomorphic coordinate z, we can write f (z) = z in a neighbourhood of x, corresponding to z = 0. In local coordinates the differential can be expressed as q(z) = q 1 (z)(dz) 2 for q 1 (0) = 0, and by rescaling z by a real factor if necessary, we may assume that q 1 (0) = ±1. The curve S is thus given by
and so at z = 0 we have η = ±dz for q 1 (0) = 1; ±i dz for q 1 (0) = -1.
In the case of q 1 (0) = 1 the points lying over x are (±dz, 0), and by (9) they are both fixed by f . In the case of q 1 (0) = -1 the points over x are (±i dz, 0), which are interchanged by f .
Consider now S 1 ⊂ Σ a component of the fixed points of f in Σ which does not contain zeros of q. As above, for x ∈ S 1 we can choose a holomorphic local coordinate z such that f (z) = z, and such that the point x corresponds to z = 0. For any other such coordinate w, one can write w = a 0 z + a 1 z 2 + . . . for a 0 = 0, a i ∈ R.
Then dw| x = a 0 dz| x , and this defines a real subbundle K | S 1 (R) of the restriction K | S 1 of K to the fixed circle. Note that since we can find a non-vanishing vector field X ∈ T Σ| S 1 tangent to S 1 , the subbundle K | S 1 (R) is trivial. Therefore since K | S 1 (R) is a trivial real bundle and ±η are non-vanishing along the circle, the double cover p -1 (S 1 ) → S 1 defined by η 2 = q must be a trivial double cover p -1 (S 1 ) = S 1 ∪ S 1 .
We shall now consider the case of a fixed circle S 1 ⊂ Σ which contains one or more zeros of the quadratic differential q. Let x ∈ S 1 be such that q(x) = 0. Choose a local holomorphic coordinate z centred at x satisfying f (z) = z. As in the above proof we can write q(z) = q 1 (z)(dz) 2 , where q 1 (z) = q 1 (z) and now q 1 (0) = 0. Since we are assuming that q has only simple zeros we have dq 1 (z) = 0 at z = 0. Hence we can take q 1 (z) as a local coordinate w = q 1 (z). In terms of this coordinate, the double cover p : S → Σ given by (ηdz, z) → z can be written as
Locally the fixed points of f are the points where w is real and so -for q 1 = w < 0, we have that ±η ∈ iR and f exchanges η and -η, -for q 1 = w > 0, we have that ±η ∈ R and f fixes ±η.
From the above analysis one has the following proposition: Proposition 35. Let S 1 ⊂ Σ be a fixed component of f on which q has at least one zero. Then q has an even number 2k of zeros which divides the circle into 2k segments over which we have q ≥ 0 and q ≤ 0 alternately. For each of the k segments where q ≥ 0, the inverse image of the segment in S is a circle fixed by f . For each of the k segments where q ≤ 0, the inverse image of the segment is a circle such that f acts as a reflection. In particular the inverse image p -1 (S) contains exactly k circles fixed by f . It is possible to strengthen this result to a precise description of S in a neighborhood of the inverse image p -1 (S 1 ). The inverse image p -1 (S 1 ) is the graph obtained by taking 2k circles and joining them in a chain which has Euler characteristic 2k -4k = -2k. We can enlarge each circle of p -1 (S 1 ) to a tubular neighbourhood in S such that their union is a neighbourhood of p -1 (S 1 ) diffeomorphic to a surface of genus k -1 with 4 points removed. Combining Propositions 34 and 35 we obtain: Theorem 36. Let q ∈ H 0 (Σ, K 2 ) be a quadratic differential which has only simple zeros and such that f * (q) = q, let S be the spectral curve given by η 2 = q. Let n + be the number of fixed components of f on which q is non-vanishing and positive, and u the number of zeros of q which are fixed by f . Then the number of components of the induced involution ι 0 : J ac(S) → J ac(S) is 2 d , where d = 2n + + u/2 -1 if 2n + + u/2 > 0, and d = 1 otherwise.
It is natural to ask which pairs (n + , u/2) can occur for a given pair (Σ, f ) with associated invariants (n, a). Note that in particular one has 0 ≤ n + ≤ n and 0 ≤ u/2 ≤ 2g -2. Moreover if u > 0 then n + < n, and if n = 0 then u = 0. Even with these constraints there are a large number of possibilities that can occur. For example in the genus g = 2 case the invariants (n, a, n + , u/2) subject to the above constraints together with the constraints on (n, a) from Proposition 3 give rise to a total of 26 possible cases. To determine which of these possible cases actually occur one can use the explicit description of the space of quadratic differentials on a genus 2 hyperelliptic curve given in Appendix A. Remarkably it turns out that all but one of the 26 cases can actually be realized, the exception being the case (n, a, n + , u/2) = (1, 0, 0, 1).
For SL(2, C)-Higgs bundles we have seen that the fibres of the Hitchin fibration are Prym varieties Prym(S, Σ), for S the spectral curve defined by the point in the Hitchin base. Hence in this case we are interested in the number of connected components of the intersection of Prym(S, Σ) with the real points in the Jacobian of S.
Let q ∈ H 0 (Σ, K 2 ) be a quadratic differential with simple zeros and satisfying f * (q) = q. In particular f preserves the set of 4g -4 branch points of q. For p : S → Σ the branched double cover of degree 2 defined by the equation η 2 = q, its genus is g S = 1 + 4(g -1). The natural action of f on K defines a lift f : S → S of f . We shall denote by σ : S → S the involution which exchanges sheets, which commutes with f .
Since S → Σ is a double cover the Prym variety Prym(S, Σ) may be defined as those line bundles L satisfying σ * L L * and is a complex torus of dimension 3g -3. We are interested in the set of elements L ∈ Prym(S, Σ) which satisfy the reality condition f * (L * ) = L. Since this set is the product of a real torus of dimension 3g -3 and a discrete group (Z 2 ) d , the fixed point set has 2 d components for some d. The strategy we will employ to find d is to look for points of order 2 in Prym(S, Σ) satisfying the reality condition.foot_0 Let P denote the set of such points, which is a subgroup of Prym(S, Σ).
From Lemma 27, the set P is isomorphic to (Z 2 ) 3g-3+d , and thus it will suffice to determine the order of P. By construction, elements of P are line bundles L on S satisfying the following three conditions: L 2 is trivial, σ * (L) = L and f * (L) = L. Since L has order 2, it is equivalent to view L as a flat line bundle with transition functions in Z 2 . Then as flat line bundles L L L * . Thus the set P corresponds to elements of H 1 (S, Z 2 ) which are invariant under the actions of σ and f . Let H 1 (S, Z 2 ) σ denote the subgroup of elements of H 1 (S, Z 2 ) fixed by σ . We will first determine this group and then find the subgroup of elements fixed by f . Let b = 4g -4 be the number of branch points of p, and B = {x 1 , . . . , x b } the set of branch points and xi = p -1 (x i ). For Σ = Σ\B and S = S\ p -1 (B), there is a commutative diagram
Moreover p : S → Σ is a degree 2 covering space, so we obtain an exact sequence
where x ∈ S and x = p( x) ∈ Σ . The map ω : π 1 (Σ , x) → Z 2 is the cohomology class ω ∈ H 1 (Σ , Z 2 ) corresponding to the double cover p : S → Σ .
Proposition 37. There is an exact sequence
where the kernel of p * is generated by ω. Moreover, i * S :
Proof. The Mayer-Vietoris sequence applied to Σ and S gives a commutative diagram with exact rows
Clearly also the image of p * is σ -invariant since pσ = p, so one has the inclusion p * (H 1 (Σ , Z 2 )) ⊆ i * S (H 1 (S, Z 2 ) σ ). From the exact sequence (10) we see that the kernel of p * :
so it remains to show surjectivity of p * . Let x be in a small σ -invariant disc D ⊂ S containing a single branch point x1 , such that x = x1 . Let γ be a path in D\{ x1 } from x to σ ( x). We may also choose D small enough so that the restriction p| D has the form z → z 2 . Define a map σ : π 1 (S, x) → π 1 (S, x) by σ (α) = γ.σ (α).γ -1 . Then σ is an automorphism of π 1 (S, x) whose pullback induces an action on cohomology groups σ * :
Let ρ : π 1 (S) → Z 2 be a homomorphism which is σ * -invariant, so ρ• σ = ρ. We may then define a representation τ : π 1 (Σ ) → Z 2 such that p * (τ ) = i * S (ρ). Indeed consider a loop α in Σ based at x. Then exactly one of the loops α, p(γ ).α lifts to a loop in S based at x. Let α be this lift and set τ (α) = ρ( α). We claim that τ is a homomorphism. This follows from the fact that ρ is σ -invariant, and since ρ(γ.σ (γ )) = 1 because γ.σ (γ ) is null-homotopic in S. By construction it is immediate that p * (τ ) = i * S (ρ).
To find the group of f -invariant elements of H 1 (S, Z 2 ) σ it is equivalent to find the f -invariant elements of H 1 (Σ , Z 2 )/ ω , since we have the commutative diagram
For A = H 1 (Σ , Z 2 )/ ω , the involution f acts on A, and we are after the group A f of f -invariant elements of A. Let A * = Hom(A, Z 2 ). Since A is a Z 2 -vector space, A and A * are isomorphic, and the fixed point subspace A f is isomorphic to the subspace (A * ) f of A * fixed by the dual action of f on A * . The action f * : A * → A * fits into a commutative diagram with exact rows
Thus, to determine (A * ) f it is sufficient to determine the group H 1 (Σ , Z 2 ), the action of the map f * :
Note that the Mayer-Vietoris sequence applied to Σ gives an exact sequence
The above Z b 2 group corresponds to cycles around each of the b branch points, and the boundary ∂ :
Theorem 38. Suppose that at least one branch point of p : S → Σ is fixed by f . Then the fixed point set of the action of f on the Prym variety Prym(S, Σ) has 2 n 0 +u/2-1 connected components, where n 0 is the number of fixed components of f : Σ → Σ which do not contain branch points, and u is the number of branch points which are fixed by f . Proof. Let A * be the kernel of ω : H 1 (Σ , Z 2 ) → Z 2 and (A * ) f the subspace of A * fixed by f * . We must show that (A * ) f has dimension 3g -3 + n 0 + u/2 -1. Consider a cycle D ∈ H 1 (Σ , Z 2 ) around a branch point which is fixed by f . Then f * (D) = D and ω(D) = 1. Therefore it suffices to show that the kernel of θ = f * -1 :
Recall that there are b = 4g -4 branch points and write b = 2t + u, where t is the number of pairs of branch points exchanged by f and u is the number of fixed branch points. For i = 1, . . . , t let C i , C i be cycles around a pair of branch points exchanged by f . Further, for j = 1, . . . , u denote by D j cycles around the fixed branch points. From the Mayer-Vietoris sequence we have a single relation between these
Let a 1 , . . . , a 2g ∈ H 1 (Σ, Z 2 ) be a basis of cycles in Σ. We may assume that the cycles do not touch the branch points so that a 1 , . . . , a 2g are also cycles in Σ . Together with the cycles C i , C i , D j we have a generating set for H 1 (Σ , Z 2 ) satisfying t i=1 C i + C i + u i=1 D j = 0. To proceed we consider three subcases depending on the topology of (Σ, f ). As in previous sections, we let (n, a) be the topological invariants associated to the anti-holomorphic involution f . Case 1: a = 0. Write g = 2s + r , n = r + 1. We have a generating set for H 1 (Σ , Z 2 ) consisting of A i , B i , A i , B i , A j , B j , C k , C k , D l for 1 ≤ i ≤ s, for 1 ≤ j ≤ r , for 0 ≤ k ≤ t, and for 1 ≤ l ≤ u with one relation given by t k=1 C k + C k + u l=1 D l = 0.
The cycles A i , A i are interchanged by f * and similarly for B i , B i . The cycles A i correspond to a choice of r = n -1 of the fixed components of f , perturbed slightly so as to avoid the branch points. Let h∈A j D h denote the sum of cycles D h where D h is a cycle around a branch point which lies in the fixed component of f corresponding to A j . Finally the B j are cycles that cross two of the fixed components of f . We can choose the A j , B j so that θ(A j ) = h∈A j D h and θ(B j ) = 0. It follows that Ker(θ ) has dimension 2s + r + (n 0 -1) + t + u = 3g -3 + n 0 + u/2 as required.
Case 2: a = 1, and gn even. Write g = 2s + r , n = r . We have a generating set for H 1 (Σ , Z 2 ) consisting of A i , B i , A i , B i , A j , B j , C k , C k , D l with one relation as before. As previously, f * exchanges the pairs A i , A i and B i , B i , and the A j correspond to the fixed components of f . We again have θ(A j ) = h∈A j D h , but now θ(B j ) = r j=1 A j + t k=1 C k . As before, we find that Ker(θ ) has dimension 3g -3 + n 0 + u/2. Case 3: a = 1, and gn odd. Write g = 2s + r + 1, n = r . We have a generating set for H 1 (Σ , Z 2 ) consisting of X, Y , and A i , B i , A i , B i , A j , B j , C k , C k , D l , with one relation as before. Moreover, f * (X ) = X and f * (Y ) = Y + r j=1 A j . The pairs A i , A i and B i , B i are exchanged by f * . As before A j correspond to fixed components of the involution f and satisfy θ(A j ) = h∈A j D h . Finally we have θ(B j ) = Y . In this case Ker(θ ) has dimension 3g -3 + n 0 + u/2.
In this section we consider the relation between Higgs bundles fixed by the induced action of an anti-holomorphic involution f : Σ → Σ and bundles with real or quaternionic structure. In particular we focus on the case of SL(2, C)-Higgs bundles.
Let E, F be rank n holomorphic vector bundles on Σ and φ : E → F an anti-linear bundle isomorphism covering f . The map φ can be extended to a map φ : Ω ( p,q) (E) → Ω ( p,q) (F) of form-valued sections as follows. For any point x ∈ Σ choose a local trivialisation of E near x, and a trivialisation of F near f (x). A local section s of E in this trivialisation is a C n -valued function defined near x. The corresponding local section of F near f (x) is of the form φ(s) = g f * (s), for some locally defined G L(n, C)-valued function g. The extension of φ to form-valued sections is given by setting φ(ω ⊗ s) = f * (ω) ⊗ g f * (s), where ω is a form on Σ defined near x. We say that such a map φ : E → F is holomorphic if it sends holomorphic sections to holomorphic sections, or letting ∂ E , ∂ F denote the corresponding ∂-operators on E, F, we have φ
Let (E, Φ) be a rank n Higgs bundle satisfying the Hitchin equations and whose isomorphism class is fixed by f . Since E must have degree 0 we can take it to be the trivial rank n bundle equipped with the constant Hermitian structure. Let ∂ A be the ∂operator defining the holomorphic structure, so ∂ A = ∂+ A for some A ∈ Ω 0,1 (gl(n, C)).
Since (E, Φ) satisfies the Hitchin equations we have that f
where
is isomorphic to (E, Φ) is equivalent to the existence of an anti-linear isomorphism φ : E → E * covering f , holomorphic in the sense described above and such that φ • Φ t = Φ • φ. Note that this gives a purely holomorphic interpretation of the condition for a Higgs bundle (E, Φ) to be fixed by f . Proposition 41. If q has a zero fixed by f , then there exists a lift τ : L → L of τ with τ • τ = 1. In this case E has a real structure.
Proof. Since τ = f σ it follows that τ has fixed points. Let x ∈ S be a fixed point of τ . Since L is a holomorphic line bundle, then there is associated to L a corresponding unitary homomorphism ρ : π 1 (S, x) → U (1). Let S → S be the universal cover viewed as a principal π 1 (S, x)-bundle. Then L may be identified as the quotient of S × C by the relation ( pg, s) ∼ ( p, ρ(g)s), for g ∈ π 1 (S, x). Let τ : π 1 (S, x) → π 1 (S, x) be the automorphism of π 1 (S, x) constructed as in Sect. 2.3. Then since x is a fixed point, τ is an involution. Moreover since τ * (L) = L we have ρ • τ = ρ. We can lift τ to an involution τ : S → S which satisfies τ ( pg) = τ ( p) τ (g). Now define τ : L → L by τ ( p, s) = (τ ( p), s). This is well-defined since ρ • τ = ρ, and is an involution since τ is an involution. Finally note that 1 in Lemma 40 equals 1 since f has fixed points.
Proposition 42. Supppose q has no zeros fixed by f . Let x ∈ S and choose a path l from x to τ (x). Let μ be the loop μ = l.τ (l) and let ρ : π 1 (S, x) → U (1) be the flat unitary structure associated to the holomorphic line bundle L. Then := ρ(μ) = ±1 and there exists a lift τ : L → L of τ with τ • τ = .
Proof. The proof is nearly identical to that of Proposition 41, except now the lift τ : S → S satisfies τ (τ ( p)) = pμ -1 , hence the lift τ : L → L given by τ ( p, s) = (τ ( p), s) squares to ρ(μ) -1 . Note also that since ρ • τ = ρ and τ (μ) = μ, we have that ρ(μ) = ρ(μ) -1 = ±1. (A,B,A
) and restrict attention to the moduli space of degree 0 Higgs bundles M G L(n,C) = M Higgs,0 G L(n,C) . Let p : S → Σ be the spectral curve corresponding to a generic point a ∈ A G L(n,C) . After fixing a choice of spin structure the fibre of M G L(n,C) is given by the Jacobian J ac(S), and the selfduality of M G L(n,C) is reflected in the self-duality of J ac(S). Now, with a slight abuse of notation, let L ⊂ A G L(n,C) be the subspace of A G L(n,C) fixed by the induced action of f , so that if a ∈ L then f lifts to an involution f : S → S. Then L G L(n,C) fibres over L with fibre corresponding to the subspace of J ac(S) fixed by the map ι 0 : J ac(S) → J ac(S) as in Definition 26. Let B ⊂ J ac(S) be the fixed point set of ι 0 .
The self-duality of L G L(n,C) is reflected in the fact that B is a Lagrangian submanifold of J ac(S), where the symplectic structure on J ac(S) is given by the standard principal polarization. This follows since the map H 1 (S, R) → H 1 (S, R) corresponding to ι 0 is given by x → -f * (x), which is anti-symplectic.
To understand the above claim we need a brief digression into Fourier-Mukai duality. Let A, Â be dual abelian varieties. There is a natural dual pairing of H 1 (A, R) and H 1 ( Â, R). Given a subspace V ⊆ H 1 (A, R) there is a corresponding subspace V ⊆ H 1 ( Â, R), namely the annihilator V = V ⊥ of V . Suppose V is the subspace determined by a subtorus B ⊆ A. Then V ⊥ defines a corresponding subtorus B ⊆ Â. Under the cohomological Fourier-Mukai transform [Huy] H * (A, R) → H * ( Â, R), the Poincaré dual cohomology classes η B ∈ H * (A, R) and η B ∈ H * ( Â, R) correspond to one another (up to a sign factor). Based on this we say that closed subtori B ⊂ A and B ⊂ Â are Fourier-Mukai dual if the corresponding subspaces V ⊂ H 1 (A, R) and V ⊂ H 1 ( Â, R) are annihilators of one another. When A is self-dual we can use a principal polarization to identify H 1 (A, R) with H 1 ( Â, R) and we find that a subspace V ⊆ H 1 (A, R) is selfdual if and only if it is a Lagrangian subspace. This explains our claim that L G L(n,C) is a self-dual brane.
Consider now the case G c = SL(n, C), for which L G c = PG L(n, C). Clearly we have an identification of the bases A SL(n,C) A PG L(n,C) . Let a be a generic point of the base with corresponding spectral curve p : S → Σ. To see the duality of the fibres we note that the exact sequence 1 / / Prym(S, Σ) / / Jac(S) N m / / Jac(Σ) / / 1 dualises to 1 / / Jac(Σ) p * / / Jac(S) / / Prym(S, Σ) / / 1.
In particular this shows that the dual of Prym(S, Σ) is Jac(S)/ p * (Jac(Σ)) which is easily seen to describe the spectral data for PG L(n, C)-Higgs bundles.
Let a be a point in the base which is fixed by the action of f , so that f lifts to f : S → S. As before, denote by B ⊂ Jac(S) the fixed point set of ι 0 . Then the fibre of L SL(n,C) over a is B ∩ Prym(S, Σ) and the fibre of L PG L(n,C) is the image of B in Jac(S)/ p * (Jac(Σ)). The components of these fibres are then seen to be dual in the sense described above.
Given an anti-holomorphic involution f : Σ → Σ on a compact Riemann surface, one may construct an associated 3-manifold M with boundary Σ. Moreover we show that there is a close relationship between representations of the fundamental group of M and the (A, B, A)-brane L G c defined by f . To define M consider the product Σ = Σ × [-1, 1]. On Σ there is a natural involution σ : (x, t) → ( f (x), -t), which is orientation preserving and a product action (e.g., [KMcC96, Section 1]). The quotient space M = Σ/σ is a 3-dimensional manifold with boundary ∂ M = Σ. From M we obtain a distinguished subspace of representations of π 1 (Σ), namely those representations which viewed as flat connections on Σ extend to flat connections over M.
Proof. Let i : Σ → Σ be the inclusion i(x) = (x, 0). Then clearly i • f = σ • i. Fix a point x 0 ∈ Σ and choose a path γ in Σ from x 0 to f (x 0 ). Recall from Sect. 2.3 that the automorphism f : π 1 (Σ, x 0 ) → π 1 (Σ, x 0 ) is given by f (u) = γ. f (u).γ -1 . Let m 0 ∈ M be the point of M corresponding to (x 0 , 0) and τ the image of i(γ ) in M, which is a loop at m 0 . Let j : Σ → M be the composition of i with the projection Σ → M. We then have a commutative diagram The study of this correspondence, as well as the further characterisation of the (A, B, A)-branes L G c appears in the companion paper [BaSch].
Acknowledgements. The authors would like to thank S. Gukov and N. Hitchin for inspiring and helpful conversations, and to D. Alessandrini, J. Stix and A. Wienhard for useful comments.
As seen in [GH81, Section 6], in the case of hyperelliptic curves more information can be deduced concerning the topological invariants (n, a). We shall see here some examples.
A.1. Classification. We shall begin by classifying pairs (Σ, f ) where Σ is a hyperelliptic curve of genus g ≥ 2 and f : Σ → Σ an anti-holomorphic involution. Definition 44. Let τ : P 1 → P 1 be the anti-holomorphic involution which in projective coordinates [z 1 , z 2 ] is given by τ ([z 1 , z 2 ]) = [z 1 , z 2 ]. Let α : P 1 → P 1 be given by α([z 1 , z 2 ]) = [z 2 , -z 1 ] so that ατ = τ α is the antipodal map.
Proposition 45. There exists a non-constant meromorphic function z : Σ → P 1 with polar divisor z -1 (∞) of degree 2 such that either z( f (w)) = τ (z(w)) or z( f (w)) = ατ (z(w)). In addition, we may choose z such that ∞ = [0, 1] ∈ P 1 is not a branch point.
Proof. Since Σ is hyperelliptic there exists a meromorphic function z : Σ → P 1 which has two simple poles, or a single pole of order 2. Any other meromorphic function with this property is obtained from z by a Möbius transformation. In particular this applies to τ • z • f , so there exists a matrix Thus z • f = τ α • z as required. The linear transformations in G L(2, C) commuting with the antipodal map (z 1 , z 2 ) → (z 2 , -z 1 ) form the group SU (2). Composing with an element of SU (2) we can ensure that ∞ is not a branch point of z.
We have established that the anti-holomorphic involution f on Σ covers an antiholomorphic involution on P 1 which is either the conjugation map τ or the antipodal map ατ . Let z : Σ → P 1 be the meromorphic function as above. Then z exhibits Σ as a branched double cover of P 1 . We may choose z so that ∞ is not a branch point, and let P 1 , . . . , P 2g+2 ∈ Σ be the branch points with z(P 1 ), . . . , z(P 2g+2 ) ∈ C ⊂ P 1 their images in P 1 . On Σ there is a meromorphic function w satisfying Proof. For simplicity we shall write z in place of τ (z). Since f sends branch points of z to branch points, the set of images z(P 1 ), . . . , z(P 2g+2 ) of branch points is left invariant under the conjugation map τ . Therefore each z(P j ) is either real or occurs in a conjugate pair. The polynomial p(z) = 2g+2 j=1 (zz(P j )) thus satisfies p(z) = p(z). In particular we obtain an anti-holomorphic involution f : Σ → Σ by sending a pair (w, z) such that w 2 = p(z) to the corresponding pair (w, z). Clearly f covers τ . The only other anti-holomorphic involution covering τ is given by sending a pair (w, z) to (-w, z) and this is just σ • f . Thus f is one of f or σ • f . Proposition 47. Let 2k be the number of real roots of p(z). Consider the topological invariants (n Σ , a Σ ) associated to the pair (Σ, f ). If k = 0 then (n Σ , a Σ ) = (1, 0) if g is even, and n Σ = 2 if g is odd. If 0 < k < g + 1 then (n Σ , a Σ ) = (k, 1), and if k = g + 1 then (n Σ , a Σ ) = (g + 1, 0).
Proof. Consider the zeros z(P 1 ), . . . , z(P 2g+2 ) of p(z) in the complex plane. Draw branch cuts between conjugate pairs of roots and joining adjacent pairs of real roots. From this data the invariants (n Σ , a Σ ) can easily be determined.
Similarly, the topological invariants of the involution σ • f can be found.
Proposition 48. Let 2k be the number of real roots of p(z). Consider the topological invariants (n Σ , a Σ ) associated to the pair (Σ, σ • f ). If 0 ≤ k < g +1 then (n Σ , a Σ ) = (k, 1), and if k = g + 1 then (n Σ , a Σ ) = (g + 1, 0).
We shall now consider involutions covering the antipodal map ατ as in Definition 44. In this case the antipodal map must permute the zeros of p(z), so the zeros occur in antipodal pairs. For λ := 2g+2 j=1 z(P j ), it follows that λ is of the form λ = μ (-1/μ) =
One should note that from[Sch11] these points correspond to real points given by SL(2, R)-Higgs bundles.