LECTURE 5. MODULI SPACES OF VECTOR BUNDLES 323
- as a moduli space of gauge equivalence classes of fl.at connections on E ;
2. - via the Narasimhan-Seshadri theorem [33], M(E, G) (for G = SU(n))
is the moduli space of semistable holomorphic vector bundles of rank n ,degree 0 and fixed determinant on E; M(n, d) is the moduli space of stable
holomorphic vector bundles of rank n, degree d and fixed determinant on E.
These moduli spaces have a symplectic structure which generalizes the symplectic
structure on the Jacobian. For instance, viewing M(E, G) as the moduli space of
gauge equivalence classes of fl.at connections on E , it acquires a symplectic structure
which descends from the quadratic form
w(L:::>jxj, LbkXk) = L 1 < Xj,Xk > aj /\ ak
j k j,k Eon the space A(E, G) = il1 (E) ® Lie( G) of all G connections on E. Here, we
sum over a basis Xj of Lie(G) and the aj and bj are elements of S1^1 (E): we have
introduced an inner product < ., · > on Lie( G) which is invariant under the adjoint
action.
The methods of [23] (summarized in Section 4.4) may be adapted to prove
formulas for intersection numbers in the cohomology of the moduli space M(n, d):
these formulas were discovered originally by Witten [41]. An outline of the proof
was given in the announcement [24], and full details of the proof appear in [25].
5.4. The line bundle over the moduli space of flat connections
We define a line bundle .C over Ap/ Q(E) as a quotient of the trivial line bundle
Ap x C over AF (where AF is the space of fl.at G connections on E and Q(E)
= C^00 (E, G) is the gauge group). We let g: E--+ G act on Ap x C by g: (A, z ) f-+
(A9, G(A, g)z) where the Chern-Simons cocycle G(A, g) is defined (see e.g. [34]) by
(5.4)
Here, N is any three-manifold with 8N = E , A is any connection on N which
extends A, and g is any gauge transformation on N which extends g. One may
check that e is well defined independent of the choices of .A and g, and depends
only on their boundary values.
The line bundle .C is in fact the prequantum line bundle over the moduli space
of fl.at connections on E.