3.1 Moduli spaces of representations

In Chapter 2 we studied the torus Hl{2, (7(1)) which parametrizes homomorphisms

 

 

We shall now consider the space H\I, G) which parametrizes conjugacy classes of homomorphisms

 

 

where G is any compact simply connected Lie group. For simplicity we shall frequently work with the special case G = SU(n).

 

Now iri(2) has generators Aiy..., Ag,..., B,,..., Bg with the one relation

(3.1.1)

It follows that Hl{2, G) is the quotient by G of the subset of G2g lying over 1 in the map Gg x Gg -> G given by [] [A,, B,]. This shows that H\I, G) is a compact Hausdorff space. More precisely it is a manifold of dimension 2(g-l) dim G at all irreducible points (i.e. where the image of tti(2) generates G). This follows by examining the linearization of (3.1.1). This has been examined in great detail by Narasimhan and Seshadri [22] and also by Newstead [23].

If a: 17"!(.£)-» G is irreducible then the tangent space to H\I,G) at a can be identified with Hl{2, gj, thecohomology of 2 with values in the flat Lie-algebra-valued bundle associated to a.

We now fix, once and for all, a G-invariant metric on the Lie algebra of G. We assume that this is integral in the sense that the corresponding element of H3(G), represented by the invariant 3-form <£ [17, £]), is integral. For simple G there is only a scalar factor to be fixed and we can normalize this by requiring that we get a generator of H3(G, Z). For SU(n) this is given by the standard metric: Trace A2.

Using this metric and the cup product then gives a symplec- tic structure to if'(2, &,). In fact, as we shall see in the next section, this makes the irreducible part of Hl{2, G) a symplec- tic manifold. This generalizes the symplectic structure of the torus H1(2, (7(1)) which we studied in Chapter 2.

Note that g = 0,1 are special cases since all representations are then reducible. Where these low values of g cause prob­lems we will usually assume g sr 2.

Although we have, for simplicity, introduced ir^l), which requires a choice of base point, the space Hl(2, G) is indepen­dent of this choice. This is because we factored out by conju­gation.

It follows that the group DiflT (2) acts on Hl(2, G) and it preserves the symplectic structure.

We shall discuss briefly the generalization of all this for a surface with marked points as previewed in Chapter 2.

Given a marked point P on 2 we associate to it a conjugacy class C of G of order k. Thus, if G = SU(n), C is the class of a matrix with eigenvalues

 

 

Given marked points P,,..., Pr on 2 and associated con­jugacy classes C,,..., Cr we consider homomorphisms

 

 

 

such that the loop around each P, goes into C, . Factoring out by conjugacy gives us a moduli space generalizing H'(2, G). We might denote this by H\l, P, G, C).

 

The dimension of the generalized moduli space is, in general, given by the formula

 

These more general moduli spaces have been studied by Seshadri and others [30] [20]. They share the general proper­ties of the earlier moduli spaces. In particular they are sym­plectic manifolds with singularities. For example if I = S2 then tti(2 - (PiU • • • u Pr)) is free on r- \ generators, and the moduli space is the quotient by G of the fibre over 1 in the multiplication map

The symplectic structure is, as we shall see in § 3.2, partly derived from that of G) and partly derived from the

symplectic structures of the homogeneous spaces Cj.