6.3 Abelianization

The third approach to projective flatness is much more far reaching than the previous two but it has not yetbeen worked out in detail and remains somewhat conjectural at the present stage. Nevertheless it is potentially very impor­tant because it will in principle reduce the whole non-abelian theory to the elementary abelian case which was discussed in Chapter 2. Thus what is being proposed here is an abelianiz- ation procedure. This should be compared with the rep­resentation theory of compact Lie groups. As is well known this theory can, in a sense, be reduced to that of the maximal torus together with the action of the Weyl group. In our case there will also be a discrete group that plays the role of the Weyl group.

The whole programme envisaged here rests on the funda­mental paper [16] of Hitchin, so we begin by reviewing this very briefly. For simplicity we shall discuss only the case G= U(n) or SU(n), but the extension to general G is fairly routine.

Hitchin introduces moduli spaces which generalize those introduced in Chapter 3 and which have both a holomorphic and a representation theory description. The holomorphic description starts from a complex Riemann surface ST and is concerned with 'Higgs bundles' over ST. These are pairs (V, <t>) where V is a holomorphic rank n vector bundle and 0 (the 'Higgs field') is a holomorphic section of (End V) ® K, where K is the canonical line-bundle of 2T.

There is a natural notion of stability for Higgs bundles and a corresponding moduli space M (depending still on r). There is a natural embedding M -* M given by bundles with zero Higgs field. Moreover the cotangent bundle T*MS of the stable points M gives a natural open set in Ji, with the Higgs field being the cotangent vector. In particular this shows that dim Ji= 2 dim M.

The characteristic polynomial of the Higgs field 0

 

 

(6.3.1)

has coefficients a,e H<3(ST, K'). Altogether these define a

x is proper,

the generic fibre is an abelian variety,

dim M. = 2 dim W,

M is an irreducible component of ^"'(0).

The equation det (A - 0) = 0 defines an algebraic curve

 

 

which is an «-fold branched covering of ST depending on a parameter w e W. The fibre of x over a generic point w of W is the Jacobian of £T. Note that the fibre over w = 0 is very degenerate and in particular M is a multiple component.

Thus our moduli space M appears in a degeneration of a family of abelian varieties. Moreover there is a natural line- bundle i? on M whose restriction to M gives our standard line-bundle L over M.

Taking the sections of over the fibres of x we then get a vector bundle over the regular points of W, i.e. over W—D where the discriminant locus D consists of ws W for which the characteristic polynomial (6.3.1) has a double root.

Sections of Lk over M can be pulled back to T*MS and then extended to all of Ji (provided the exceptional set has codimension which holds for g>2). We can therefore identify our Hilbert space H°(M, Lk) as a space of sections of the vector bundle over W— D.

holomorphic map with the following properties:

Now this bundle has the projectively flat connection of the abelian case (given by ©-functions as in Chapter 2). Our aim should be to identify H°(M, Lk), at least projectively, with the covariant constant sections of the bundle over W-D. Equivalently H°(M, Lk) should be the part of

H°{x~l(w),£kw) left fixed by the monodromy group 77 = 77,(W — D, w) based at a generic point we W — D.

If established this would constitute our abelianization, with 77 (or rather its image in the symplectic group associated to the abelian variety) playing the role of the Weyl group.

 

The projective flatness as we now vary re J would follow as a corollary from that of the abelian case. Note that there are two types of variation of abelian variety being used here. First the branched cover £T of 2T varies with the branch points depending on w e WT. Then there is the variation of r itself. The 'Weyl group' should therefore be the group (independent of r) arising from the universal space

Hitchin's moduli space M has many other beautiful proper­ties which are likely to repay further study. In particular M has a description (as a real manifold) by representations, namely as a moduli space of representations i^(.Z)-» GL(n, C). This generalizes the description of the moduli space M as the space of unitary representations of ^,(2).

To deal with the case of surfaces with marked points, the notion of a Higgs bundle has to be generalized by requiring the Higgs field 4> to have simple poles, at the marked points, with nilpotent residues.