Hitchin’s Fibrations

4. Compactification And The Geometry Of Hitchin’s Moduli Space

4.3. Hitchin’s Fibrations

Y that we found in complex structure J. Thus all the complex structures Iw, w 6= 0,∞ are equivalent.

In fact, in complex structure I =I_w=0, we can define a symmetry groupU ∼=C^∗ that acts on a Higgs bundle by (E, ϕ)→(E, λϕ),λ ∈C^∗. (The stability condition is invariant under this transformation.) For|λ|= 1, this reduces to the groupU1, described in section 4.1, which is a symmetry group of the hyper-Kahler metric of M_H. For |λ| 6= 1, we do not get a symmetry of the hyper-Kahler metric ofM_H, but as is shown on pp. 107-8 of [20], we get a group of diffeomorphisms of M_H that preserves complex structure I and transforms the other complex structures by

I_w →I_λ⁻¹_w, (4.23)

generalizing eqn. (4.7) to |λ| 6= 1.

The image ofT^∗MinM_H is not all ofM_H because a Hitchin pair (E, ϕ) may be stable even ifE is unstable. However, the stable Hitchin pairs (E, ϕ) for whichE is unstable are a set of very high codimension. Upon throwing away this set, M_H becomes isomorphic to T^∗M, and has a natural map to M by forgetting ϕ. Even though it is only defined away from a set of very high codimension, this map is extremely useful. We will call it Hitchin’s second fibration (the first one being another map that we introduce presently).

The Hitchin Fibration

Another natural operation in complex structure I, apart from mapping ϕ to zero, is to calculate the gauge-invariant polynomials inϕ. ForG=SU(2), this simply means that we consider the quadratic Casimir operator w = Trϕ². Since Dϕ = 0, we have ∂w = 0, so w is a holomorphic quadratic differential, taking values in B = H⁰(C, K_C²) ∼= C^3g−3. The Hitchin fibration, as it is most commonly called, is the map π : M_H → B obtained by mapping the pair (E, ϕ) tow= Trϕ².

For any G, the Hitchin fibration is defined similarly, except that one characterizes ϕ by all of its independent Casimirs (that is, all of the independent G-invariant polynomials in ϕ), not just the quadratic one. For example, for G = SU(N), we define w_n = Trϕⁿ, n = 2, . . . , N, and let B = ⊕^Nn=2H⁰(C, K_Cⁿ). The Hitchin fibration is then the map that takes (E, ϕ) to the point (w2, w3, . . . , wn)∈B. For any G, one considers instead of Trϕⁿ the appropriate independent homogeneous G-invariant polynomials Pi. The number of these polynomials equals the rank r of G, and their degrees d_i obey the identity

Xr i=1

(2di−1) = dim(G). (4.24)

For example, for G = SU(N), the di are 2,3, . . . , N, whence P

i(2di −1) = N² −1 = dim(G). And finally in general we define B =⊕iH⁰(C, K_C^di).

Since dim H⁰(C, K_Cⁿ) = (2n− 1)(g − 1), it follows from (4.24) that the complex dimension of B is (g−1)dim(G), which equals the dimension of M, and one half of the dimension of M_H. The Hitchin fibration π : M_H → B is surjective, as we will discuss momentarily. As the base B of the Hitchin fibration π : M_H → B has one half the dimension of M_H, it follows that the dimension of a typical fiber F of π is also half the dimension of M_H and so equal to the dimension of B:

dim F = dim B= 1

2dim M_H = (g−1) dim G. (4.25)

Let us explain qualitatively why the Hitchin fibration is surjective. For example, take G = SU(2). Pick a stable SU(2) bundle E. Consider the equations Trϕ² = w, where ϕ varies in the (3g−3)-dimensional space H⁰(C, KC ⊗ad(E)) and w is a fixed element of the (3g−3)-dimensional space B = H⁰(C, K_C²). This is a system of 3g−3 quadratic equations for 3g−3 complex variables. The number of solutions is generically 2^3g−3. A similar counting can be made for other G.

Complete Integrability

We now want to explain one of Hitchin’s main results [62]: M_H is a completely integrable Hamiltonian system in the complex structureI. We can find ¹₂dimM_H functions H_a on M_H that are holomorphic in complex structure I, are algebraically independent, and are Poisson-commuting using the Poisson brackets obtained from the holomorphic symplectic form Ω_I.¹⁹

In fact, we can take the H_a to be linear functions on B, since the dimension of B is the same as the desired number of functions. We will explain the construction first for G=SU(2). We begin by picking a basisα_a,a = 1, . . . ,3g−3 of the (3g−3)-dimensional space H¹(C, TC), which is dual to H⁰(C, K_C²)∼=B. (HereTC is the holomorphic tangent bundle to C.) We represent α_a by (0,1)-forms valued in T_C, which we call by the same name, and we define

Ha= Z

C

αa∧Trϕ². (4.26)

We claim that these functions are Poisson-commuting with respect to the holomorphic symplectic form Ω_I.

A natural proof uses the fact that the definition of theHa makes sense on the infinite- dimensional space W, before taking the hyper-Kahler quotient. Using the symplectic structure ΩI on W to define Poisson brackets, the Ha are obviously Poisson-commuting.

For in these Poisson brackets, given the form (4.9) of Ω_I, ϕ_z is conjugate to A_z and commutes with itself. But the H_a are functions ofϕ_z only, not A_z.

19 The reader may be unaccustomed to completely integrable systems in this holomorphic sense.

From such systems, one can extract completely integrable Hamiltonian systems in the ordinary real sense (and moreover, interesting and significant constructions arise in this way; see [63,64]

for some examples). We pick C to admit a real structure – that is, an involution that reverses its complex structure. This induces real structures on M _and M_H. By specializing to a real slice in M_H, one gets then completely integrable Hamiltonian systems in which the phase space coordinates, the symplectic structure, and the commuting Hamiltonians are all real.

The functions Ha can be restricted to the locus with νI = 0, and then, because they are invariant under the GC-valued gauge transformations, they descend to holomorphic functions on M_H. A general property of symplectic reduction (in which one sets to zero a moment map, in this caseνI, and then divides by the corresponding group, in this case the group of GC-valued gauge transformations) is that it maps Poisson-commuting functions to Poisson-commuting functions. So the H_a are Poisson-commuting as functions on M_H. There are enough of them to establish the complete integrability of M_H.

One point to note in this construction is that the H_a, as functions on W, depend on the particular choice of T-valued (0,1)-forms α_a that represent the cohomology classes.

Any choice will do, but we have to make a choice. But after restricting and descending to M_H, the functions we get on M_H depend only on the cohomology classes of the α_a. In fact, once we have Dϕ = 0 and hence ∂Trϕ² = 0, the Ha are clearly invariant under α_a→α_a+∂ǫ_a.

The Poisson-commuting functionsH_agenerate commuting flows onM_H that are holo- morphic in complex structure I. Complex tori admit commuting flows, and one might surmise that the orbits generated by the Ha are complex tori at least generically. This follows from general results about holomorphic symplectic structures and compactness of the fibers of the Hitchin fibration and can also be demonstrated more directly, using the technique of the spectral cover [20,62]. This technique has many applications in further development of this subject, as will be explained elsewhere [26]. In this paper, we will get as far as we can without using the spectral cover construction.

The analog of the above construction for anyGis to replace Trϕ² by a general gauge- invariant polynomial Pi of degree di. The associated commuting Hamiltonians take the form Hα,i = R

CαPi(ϕ), for α ∈ H¹(C, K_C^1−di). A simple dimension counting, using the dimension formula (4.24), shows that these Hamiltonians are precisely sufficient in number to establish the complete integrability of M_H. By using the fact that the fibers of the Hitchin fibration are compact, so that a holomorphic function must be a pullback from the base of this fibration, one can show that the commuting Hamiltonians generate the ring of holomorphic functions on M_H.

One easy and important consequence of complete integrability is that the fibers of the Hitchin fibration are Lagrangian submanifolds in the holomorphic symplectic structure ΩI or equivalently in the real symplectic structures ωJ and ωK. Indeed, a fiber of this fibration is defined by equationsH_k−h_k = 0, where H_k are the commuting Hamiltonians and h_k are complex constants. In general, the zero set of a middle-dimensional collection of Poisson-commuting functions, such as Hk−hk in the present case, is Lagrangian.