1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

322 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

There is an alternative description of t he symplectic structure via de Rham

cohomology: we consider the quadratic form w on A defined by

(5.2) w(a, b) =la Ab

(which yields the cup product in de Rham cohomology). Using Stokes' theorem we

see that this passes to the tangent space H^1 (E,IR) to Hom(H 1 (E),U(l)) to yield

the standard symplectic structure on H^1 (E, IR) given by the cup product.

5.2. The Jacobian as an infinite dimensional symplectic quotient

In t he description (2) above, we introduced the gauge group g = c=(E, U(l)),

which acts on A = il1(E). We now show t hat when the symplectic structure is

defined by (5.2) we have

Proposition 5.1. The moment map for the action of g on A is given by the map

μ:A--+ i1^2 (E) ~ ( n°(E)) * defined by

μ : A f--' -dA,

in other words the map that sends a connection to minus its curvature.

We begin with an easy lemma:

Lemma 5.2. If u E n°(E) = Lie(Q), the vector field u# on A generated by the

action of u is given at A E A by

u~ =A+du.

Proof of Proposition 5.1: We need to check that

w(u#,a) = dμu(a)

for all a E n°(E,JR). We have that

μu(A) = -l uAdA,

so

(dμu)A(a) = -l u Ada= l du A a,

(by Stokes' theorem). By Lemma 5.2 the right hand side agrees with what we find

forw(u#,a). D

5.3. The moduli space of fl.at connections on a Riemann surface

If we instead consider M(E, G) = Hom (1T, G)/G (where G is a compact Lie group,

e.g. G = SU(n)) we find that M(E, G) is no longer smooth. To obtain a smooth

analogue in the case G = SU ( n), we replace 1T by lF 29 (the fundamental group of

a genus g Riemann surface with one boundary component: the fr ee group on 2g

generators x 1 , ... , x2 9 ) and define

g

(5.3) M(n,d) = {p E Hom(lF2 9 ,G): p(IT[x2j-1,X2j]) = c}/G

j=l
where c = e^2 1rid/n1 is a generator of the centre of SU(n) (which is true whenever
n and dare coprime). The spaces M(E, G) (resp. M(n, d)) again have two other
descriptions in addition to the one just given: