✬ ✩ Moduli of Hitchin pairs S. Ramanan Chennai Mathematical Institute Institute of Mathematical Sciences Chennai, India. ✫ ✪ 1
✬ ✩ Vector Bundles over curves We are mainly interested in vector bundles over a projective nonsingular curve over C . One way of constructing holomorphic bundles is via representations of the fundamental group π , namely, bundles which are associated to the universal covering space considered as a principal π -bundle. ✫ ✪ 2
✬ ✩ A.Weil: Two bundles arising from unitary representations are isomorphic if and only if the representations are themselves equivalent. • D.Mumford introduced the notion of stable, semistable and polystable bundles. • Narasimhan, Seshadri: Any polystable vector bundle which is topologically trivial, is isomorphic to a bundle given by a unitary representation of the fundamental group. Conversely the bundle associated to a unitary representation of the fundamental group is polystable. ✫ ✪ 3
✬ ✩ • A Ramanathan constructed the moduli of principal G -bundles for an arbitrary complex reductive algebraic group G and extended the above results to this general case. The original proof of Narasimhan and Seshadri was non-constructive, that is to say, given a polystable bundle, the corresponding representation of the fundamental group was not explicitly constructible. ✫ ✪ 4
✬ ✩ Donaldson gave a different proof of the result of Narasimhan, Seshadri and Ramanathan. This is not entirely constructive either, but at least it reduces it to an optimisation problem. But more than that. It gave rise ultimately to the notion of Higgs Pairs. ✫ ✪ 5
✬ ✩ So we start with Higgs pairs. A Higgs pair consists of a principal G -bundle E and a differential 1 -form Φ with values in the vector bundle Ad ( E ) , namely, the bundle associated to E by the adjoint representation of G in its Lie algebra g . The form Φ is called the Higgs field of the pair. We will often subject Φ to operations which act on elements of the Lie algebra. ✫ ✪ 6
✬ ✩ The deep theorem which culminated due to the efforts of many, like Donaldson, Hitchin, Simpson, Uhlenbeck-Yau, Corlette, ... is: There is a one-one correspondence between the set of homomorphisms of the fundamental group into G whose centraliser is reductive, upto conjugacy in G and the set of isomorphism classes of polystable Higgs pairs. ✫ ✪ 7
✬ ✩ Note that the Higgs field is an element of H 0( C, K ⊗ Ad ( E )) . The tangent space of the moduli space at a stable point E of the moduli space of polystable bundles is canonically H 1( C, Ad ( E )) . Hence by Serre duality, the cotangent space at E is the space H 0( C, K ⊗ Ad ( E )) . In other words, the cotangent bundle over the open set of stable points (without nontrivial automorphisms) of the moduli of G -bundles is contained in the Higgs moduli space. In fact, the above cotangent bundle is an open subset of the Higgs moduli. ✫ ✪ 8
✬ ✩ Now the cotangent bundle of any smooth variety has a canonical symplectic structure, and this structure extends to the whole of the smooth locus of the Higgs moduli space. Given ( E, Φ) we get a natural homomorphism [ . , Φ] Hg : Ad ( E ) Ad ( E ) ⊗ K. ⇒ Treating this as a 2 -term complex of sheaves, the Hypercohomology H 1( C, Hg ) of this complex can be identified as the tangent space to the Higgs Moduli space at a stable point. The Serre dual of this complex is Hg ∗ ⊗ K : Ad ( E ) → Ad ( E ) ⊗ K and the consequent isomorphism of hypercohomology spaces gives the symplectic form required. ✫ ✪ 9
✬ ✩ Unlike the moduli space of vector bundles, the Higgs moduli space is not projective, only quasi-projective. Consider again the adjoint representation of G in g . Then • Weyl: the algebra of invariant polynomials is finitely generated. • Chevalley: the algebra of invariant polynomials is itself a polyomial algebra in l generators where l is the rank of G . If f is a homogeneous adjoint invariant polynomial of degree d , then we get a map H 0( K ⊗ Ad ( E )) → H 0( K d ) by substitution. ✫ ✪ 10
✬ ✩ If the algebra of invariant polynomials is generated by homogeneous invariant polynomials of degrees m 1 , · · · , m l , then we get by the above procedure a morphism of the Higgs moduli space into the affine space i = l � H 0( K m i ) . i =1 This is called the Hitchin map. It was first observed by Nitsure that this map is proper. Hitchin considered this map and showed that the generic fibre is an Abelian variety and is in fact a Lagrangian for the above symplectic structure. ✫ ✪ 11
✬ ✩ In the following one may think of the case of G = SL ( n ) for a quick understanding of what is going on. The Higgs moduli space admits some natural automorphisms. For example, if λ is a nonzero complex number, then the assignment ( E, Φ) �→ ( E, λ Φ) gives rise to an automorphism of the Higgs moduli space. Consider the involution ( E, Φ) �→ ( E, − Φ) in particular. What are its fixed points? Obviously, pairs with the bundle itself polystable and Higgs field 0 , are fixed under the above involution. ✫ ✪ 12
✬ ✩ Noting that the fixed points correspond to unitary representations of π , it is natural to expect and is indeed true that the above involution is given by the automorphism A �→ A ′− 1 on the space of representations of π , under the above bijection. It may indeed be surprising that the induced map is holomorphic on the Higgs side and anti-holomorphic on the representation side! Are there any other fixed points? ✫ ✪ 13
✬ ✩ Actually there are. Let us take G = SL (2) . Let ξ be a line bundle such that ξ 2 is isomorphic to K . Take for E the SL (2) bundle ξ ⊕ ξ − 1 . The Higgs field can then be written as a matrix A B C D where A and D are sections of K , B of K ⊗ ξ 2 and C of K ⊗ ξ − 2 . Since ξ 2 ≃ K by assumption, it makes sense to say that C = Id . If we assume this and that A + D = 0 , then this is a polystable SL (2) Higgs pair. It is easy to see that it is fixed by the above involution. Also any such Higgs is equivalent to one in which A = D = 0 . ✫ ✪ 14
✬ ✩ Notice that the matrix above looks very much like the matrix 0 b , 1 0 which is a companion matrix. It is conjugate to the matrix 0 − b − 1 0 by the matrix diag ( i, − i ) . Also note that the invariant polynomial det resticted to the above matrix is an isomorphism onto C . ✫ ✪ 15
✬ ✩ In much the same way, one can define for arbitrary semi-simple G , a subset of g which is mapped isomorphically onto C l by the set of invariant polynomials. This is called the Kostant section. We will describe this later and in fact lead up to a generalization of this fact. Therefore it is believable and indeed true that there exists a closed subvariey of the Higgs moduli space which is mapped isomorphically by the Hitchin morphism. ✫ ✪ 16
✬ ✩ In fact, Hitchin showed that if we identify the Higgs moduli space with the space of semisimple homomorphisms of π 1 into SL (2 , C ) , then all the Higgs pairs that occur in the Hitchin section are given by representations of π 1 into SL (2 , R ) . Indeed, the subspace of Higgs pairs given by representations into SL (2 , R ) has the Hitchin section as one of its connected components. ✫ ✪ 17
✬ ✩ Note that SL (2 , C ) has, upto isomorphism, two real forms, namely SU (2) and SL (2 , R ) . Representations into these groups, of the fundamental group, give rise precisely to the fixed points under the above involution. In fact, Hitchin determined all the components and described the corresponding Higgs pairs explicitly. These consist of • polystable bundles with Higgs field 0 • bundles E with structure group H = SO (2 , C ) , with Higgs field belonging to H 0( K ⊗ E m ) where E m denotes the bundle associated to the action of H on the space of symmetric matrices of trace 0 . ✫ ✪ 18
✬ ✩ Now the real forms of a semisimple group have been classified by Elie Cartan. The result is the following. Firstly there is a compact real form, unique upto conjugacy. Call the corresponding conjugation of the Lie algebra, or the adjoint group, by τ . After fixing τ , the set of isomorphism classes of real forms, is in canonical bijecton with the conjugacy classes of elements of order 2 in A = Aut ( g ) . We will call this set C 1 . We are equally interested in the set of all involutions in A upto conjugacy by elements of Ad ( G ) . Call this set C 2 . ✫ ✪ 19
✬ ✩ Consider the group Aut ( G ) of automorphisms of G . This group acts on the Higgs moduli space in a natural way. If A is an automorphism of G , it takes a pair ( E, Φ) to ( AE, ( A ⊗ 1)Φ) . It is easy to check that polystable pairs are mapped to polystable pairs and thereby we get an action of Aut ( G ) on the Higgs moduli space. Moreover, it is equally easy to see that Int ( G ) acts trivially on the Higgs moduli and so gives rise to an action of Out ( G ) = Aut ( G ) /Int ( G ) acts on the Higgs moduli space. ✫ ✪ 20
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