Gauge theory for embedded surfaces, II

68 P. B. Kronheimer and T. S. Mrowka
these classes. By this we mean that, for each point x of a loop γ ⊂ Σ, we form
and J is a stan-
a connection A ′ (x) inB∗ k,l as the sum A#J, whereA∈B∗ k,l−1
dard solution in M α 0,1 (S4 ,S2 ); the resulting connection lives on (X,Σ)#(S4 ,S2 ),
which we identify with (X,Σ). First we check that these loops in B∗ k,l indeed
generate the appropriate part of π1. We can check this after restriction to Σ
where it reduces to the standard fact that the generators of π1(BL) are obtained
by grafting a bundle with c1 = −1 onS2to a bundle with c1 = −l +1 atpoints on the dual loop.
It remains to show that the determinant line bundle is trivial on these loops.
If A ′ (x) ∈Bk,l is the result of the grafting described above, then the determinant
lines of A and A ′ are related by the equation
δA ′ = δA ⊗ det(TxΣ ⊕ R ⊕ R)
where the first factor of R comes from the scale parameter and the second comes
from the tangent space to the gluing variable, a copy of S 1 inside SU(2). This
description shows immediately that the determinant line bundle is trivial if we
drag the standard monopole around a loop in Σ.
(ii) Orienting the moduli space.
Having dealt with the orientability of the determinant line over Bk,l we now
determine the minimal data necessary to orient the determinant line. Recall that
in the untwisted case, to orient the moduli space of anti-self-dual connections
on E, one needs first of all a homology orientation of X, that is an orientation
for H 0 (X) ⊕ H1 (X) ⊕ H + (X). The standard orientation is defined as follows.
First one considers E ′ = C ⊕ c where c =det(E). Now gE = Ē′ ⊕ εR. Butthe
decomposition of E gives Ē′ = εR ⊕ c. The determinant line splits accordingly
and one uses the homology orientation on the trivial real factor and the complex
orientation coming from the line bundle c to orient the determinant line. This
orientation is then propogated by adding instantons to orient E [D3]. For bundles
E,E ′ with
c2(E) − 1
4c21(E)=c2(E ′ )−1 4c21(E ′ )
and
c1(E)=c1(E ′ ) (mod 2)
the corresponding moduli spaces are identified and their orientations compare
via the sign (−1) ((c1(E)−c1(E′ ))/2) 2 ) . In the case of an SU(2) bundle the standard
orientation is the one induced from the decomposition of the trivial bundle E ′ =
C ⊕ C. In the case where X is a Kähler manifold this orientation is the same
as the complex orientation for a particular choice of homology orientation. (See
[D3] for a precise statement.)
The situation in the twisted case is as follows. We again begin with the case of
a connection Aα on a bundle of the form E ′ = C⊕c and require our connections
to respect the splitting on Σ; that is, in the notation of section 2(iv) of [KrM],
we have c = ¯ K. Thus we are considering connections in B α k ′ ,l ′ where k′ = − 1
4 [c]2